101098rspa20020973

A numerical study ofsubmarine-landslide-generated

waves and run-up

By Patrick Lynett a n d Philip L-F L iu

School of Civil and Environmental EngineeringCornell University Ithaca NY 14853 USA

Received 4 October 2001 accepted 18 February 2002 published online 30 September 2002

A mathematical model is derived to describe the generation and propagation ofwater waves by a submarine landslide The model consists of a depth-integratedcontinuity equation and momentum equations in which the ground movement isthe forcing function These equations include full nonlinear but weak frequency-dispersion enotects The model is capable of describing wave propagation from rela-tively deep water to shallow water Simplishy ed models for waves generated by smallsearegoor displacement or creeping ground movement are also presented A numeri-cal algorithm is developed for the general fully nonlinear model Comparisons aremade with a boundary integral equation method model and a deep-water limit forthe depth-integrated model is determined in terms of a characteristic side length ofthe submarine mass The importance of nonlinearity and frequency dispersion in thewave-generation region and on the shoreline movement is discussed

Keywords landslide tsunamis Boussinesq equations wave run-up

1 Introduction

In recent years signishy cant advances have been made in developing mathematicalmodels to describe the entire process of generation propagation and run-up of atsunami event (eg Yeh et al 1996 Geist 1998) These models are based primarilyon the shallow-water wave equations and are adequate for tsunamis generated byseismic searegoor deformation Since the duration of the seismic searegoor deformationis very short the water-surface response is almost instantaneous and the initial water-surface proshy le mimics the shy nal searegoor deformation The typical wavelength of thistype of tsunami ranges from 20 to 100 km Therefore frequency dispersion can beignored in the generation region The nonlinearity is also usually not important in thegeneration region because the initial wave amplitude is relatively small comparedto the wavelength and the water depth However the frequency dispersion becomesimportant when a tsunami propagates for a long distance Nonlinearity could alsodominate as a tsunami enters the run-up phase Consequently a complete model thatcan describe the entire process of tsunami generation evolution and run-up needs toconsider both frequency dispersion and nonlinearity

Tsunamis are also generated by other mechanisms For example submarine land-slides have been documented as one of many possible sources for several destructive

Proc R Soc Lond A (2002) 458 28852910

2885

cdeg 2002 The Royal Society

2886 P Lynett and P L-F Liu

tsunamis (Moore amp Moore 1984 von Huene et al 1989 Jiang amp LeBlond 1992 Tap-pin et al 1999 Keating amp McGuire 2002) On 29 November 1975 a landslide was trig-gered by a 72-magnitude earthquake along the southeast coast of Hawaii A 60 kmstretch of Kilauearsquos south coast subsided 35 m and moved seaward 8 m This land-slide generated a local tsunami with a maximum run-up height of 16 m at Keauhou(Cox amp Morgan 1977) More recently the devastating Papua New Guinea tsunamiin 1998 is thought to have been caused by a submarine landslide (Tappin et al 1999 2001 Keating amp McGuire 2002) In terms of tsunami-generation mechanismstwo signishy cant dinoterences exist between submarine-landslide and coseismic searegoordeformation First the duration of a landslide is much longer and is in the orderof magnitude of several minutes Hence the time-history of the searegoor movementwill anotect the characteristics of the generated wave and needs to be included in themodel Secondly the enotective size of the landslide region is usually much smaller thanthe coseismic searegoor-deformation zone Consequently the typical wavelength of thetsunamis generated by a submarine landslide is also shorter ie ca 110 km There-fore the frequency dispersion could be important in the wave-generation regionThe existing numerical models based on shallow-water wave equations may not besuitable for modelling the entire process of submarine-landslide-generated tsunami(eg Raney amp Butler 1976 Harbitz et al 1993)

In this paper we shall present a new model describing the generation and propaga-tion of tsunamis by a submarine landslide In this general model only the assumptionof weak frequency dispersion is employed ie the ratio of water depth to wavelengthis small or O( middot 2) frac12 1 Until the past decade weakly dispersive models were formu-lated in terms of a depth-averaged velocity (eg Peregrine 1967) Recent work hasclearly demonstrated that modishy cations to the frequency dispersion terms (Madsenamp Sorensen 1992) or expression of the model equations in terms of an arbitrary-level velocity (Nwogu 1993 Liu 1994) can extend the validity of the linear-dispersionproperties into deeper water The general guideline for dispersive properties is thatthe `extendedrsquo versions of the depth-integrated equations are valid for wavelengthsgreater than two water depths whereas the depth-averaged model is valid for lengthsgreater than shy ve water depths (eg Nwogu 1993) Moreover in the model presentedin this paper the full nonlinear enotect is included ie the ratio of wave amplitude towater depth is of order one or = O(1) Therefore this new model is more generalthan that developed by Liu amp Earickson (1983) in which the Boussinesq approx-imation ie O( middot 2) = O() frac12 1 was used In the special case where the searegooris stationary the new model reduces to the model for fully nonlinear and weaklydispersive waves propagating over a varying water depth (eg Liu 1994 Madsen ampSchanoter 1998) The model is applicable for both the impulsive slide movement andcreeping slide movement In the latter case the time duration for the slide is muchlonger than the characteristic wave period

This paper is organized as follows Governing equations and boundary conditionsfor regow motions generated by a ground movement are summarized in the next sectionThe derivation of approximate two-dimensional depth-integrated governing equationsfollows The general model equations are then simplishy ed for special cases A numericalalgorithm is presented to solve the general mathematical model The numerical modelis tested using available experimental data (eg Hammack 1973) for one-dimensionalsituations Employing a boundary integral equation model (BIEM) which solvesfor potential regow in the vertical plane a deep-water limit for waves generated by

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2887

z y

x

h0

l0

z a0

h

Figure 1 Basic formulational set-up

submarine slides is determined for the depth-integrated model The importance ofnonlinearity and frequency dispersion is inferred through numerical simulation of alarge number of dinoterent physical set-ups

2 Governing equations and boundary conditions

As shown in shy gure 1 plusmn 0(x0 y0 t0) denotes the free-surface displacement of a wavetrain propagating in the water depth h0(x0 y0 t0) Introducing the characteristic waterdepth h0 as the vertical length-scale the characteristic length of the submarine slideregion `0 as the horizontal length-scale `0=

pgh0 as the time-scale and the charac-

teristic wave amplitude a0 as the scale of wave motion we can deshy ne the followingdimensionless variables

(x y) =(x0 y0)

`0 z =

z0

h0 t =

pgh0t0

`0

h =h0

h0 plusmn =

plusmn 0

a0 p =

p0

raquo ga0

and

(u v) =(u0 v0)

p

gh0

w =w0

(=middot )p

gh0

(21)

in which (u v) represents the horizontal velocity components w the vertical velocitycomponent and p the pressure Two dimensionless parameters have been introducedin (21) which are

=a0

h0 middot =

h0

`0 (22)

Assuming that the viscous enotects are insignishy cant the wave motion can be describedby the continuity equation and Eulerrsquos equations ie

middot 2r cent u + wz = 0 (23)

ut + u cent ru +

middot 2wuz = iexcl rp (24)

wt + 2u cent rw +2

middot 2wwz = iexcl pz iexcl 1 (25)

Proc R Soc Lond A (2002)

2888 P Lynett and P L-F Liu

where u = (u v) denotes the horizontal velocity vector r = (=x =y) thehorizontal gradient vector and the subscript the partial derivative

On the free surface z = plusmn (x y t) the usual kinematic and dynamic boundaryconditions apply

w = middot 2( plusmn t + u cent r plusmn ) on z = plusmn (26 a)

p = 0 (26 b)

Along the searegoor z = iexcl h the kinematic boundary condition requires

w + middot 2u cent rh +middot 2

ht = 0 on z = iexcl h (27)

For later use we note here that the depth-integrated continuity equation can beobtained by integrating (23) from z = iexcl h to z = plusmn After applying the boundaryconditions (26) the resulting equation reads

r centmiddotZ plusmn

iexclh

u dz

cedil+

1

Ht = 0 (28)

where

H = plusmn + h (29)

We remark here that (28) is exact

3 Approximate two-dimensional governing equations

The three-dimensional boundary-value problem described in the previous sectionwill be approximated and projected onto a two-dimensional horizontal plane In thissection the nonlinearity is assumed to be of O(1) However the frequency dispersionis assumed to be weak ie

O( middot 2) frac12 1 (31)

Using middot 2 as the small parameter a perturbation analysis is performed on theprimitive governing equations The complete derivation is given in Appendix A Theresulting approximate continuity equation is

1

ht + plusmn t + r cent (Hu not )

iexcl middot 2r centfrac12

H

middot( 1

6 (2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent unot )

+ ( 12(plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(32)

Equation (32) is one of three governing equations for plusmn and u not The other twoequations come from the horizontal momentum equation (24) and are given in vector

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2889

form as

unot t + unot cent ru not + r plusmn

+ middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hu not ) +

ht

cedilr

middotr cent (hu not ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hu not )t +

htt

paracedil+ (unot cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[u not cent r(r cent u not )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent unot t iexcl plusmn unot cent r

middotr cent (hunot ) +

ht

cedil+ plusmn

middotr cent (hu not ) +

ht

cedilr cent u not

frac34

+ 3 middot 2rf 12 plusmn 2[(r cent unot )2 iexcl u not cent r(r cent u not )]g = O( middot 4) (33)

Equations (32) and (33) are the coupled governing equations written in terms ofunot and plusmn for fully nonlinear weakly dispersive waves generated by a searegoor move-ment We reiterate here that unot is evaluated at z = z not (x y t) which is a functionof time The choice of z not is made based on the linear frequency-dispersion charac-teristics of the governing equations (eg Nwogu 1993 Chen amp Liu 1995) Assuminga stationary searegoor in order to extend the applicability of the governing equationsto relatively deep water (or a short wave) z not is recommended to be evaluated asz not = iexcl 0531h In the following analysis the same relationship is employed Thesemodel equations will be referred to as FNL-EXT for fully nonlinear `extendedrsquo equa-tions

Up to this point the time-scale of the searegoor movement is assumed to be in thesame order of magnitude as the typical period of generated water wave tw = `0=

pgh0

as given in (21) When the ground movement is creeping in nature the time-scaleof searegoor movement tc could be larger than tw The only scaling parameter thatis directly anotected by the time-scale of the searegoor movement is the characteristicamplitude of the wave motion After introducing the time-scale tc into the timederivatives of h in the continuity equation (32) along with a characteristic changein water depth centh the coemacr cient in front of ht becomes

macr

tw

tc

where macr = centh=h0 To maintain the conservation of mass the above parameter mustbe of order one Thus

= macrtw

tc=

macr l0

tc

pgh0

(34)

The above relationship can be interpreted in the following way During the creepingground movement over the time period t lt tc the generated wave has propagated adistance t

pgh0 The total volume of the searegoor displacement normalized by h0 is

macr l0(t=tc) which should be the same as the volume of water underneath the generated

Proc R Soc Lond A (2002)

2890 P Lynett and P L-F Liu

wave crest ie tp

gh0 Therefore over the ground-movement period t lt tc thewave amplitude can be estimated by (34) Consequently nonlinear enotects becomeimportant only if deshy ned in (34) is O(1) Since by the deshy nition of a creepingslide the value l0=(tc

pgh0) is always less than one fully nonlinear enotects will be

important for only the largest slides The same conclusion was reached by Hammack(1973) using a dinoterent approach The importance of the fully nonlinear enotect whenmodelling creeping ground movements will be tested in x8

4 Limiting cases

In this section the general model is further simplishy ed for dinoterent physical conditions

(a) Weakly nonlinear waves

In many situations the searegoor displacement is relatively small in comparisonwith the local depth and the searegoor movement can be approximated as

h(x y t) = h0(x y) + macr middoth(x y t) (41)

in which macr is considered to be small In other words the maximum searegoor displace-ment is much smaller than the characteristic water depth Since the free-surface dis-placement is directly proportional to the searegoor displacement ie O(plusmn ) = O( macr middoth) ormuch less than the searegoor displacement in the case of creeping ground movementswe can further simplify the governing equations derived in the previous section byallowing

O() = O( macr ) = O( middot 2) frac12 1 (42)

which is the Boussinesq approximation Thus the continuity equation (32) can bereduced to

plusmn t + r cent (Hunot ) +macr

middotht

iexcl middot 2r centfrac12

h0

middot( 1

6h2

0 iexcl 12z2

not )r(r cent u not ) iexcl ( 12h0 + z not )r

micror cent (h0unot ) +

macr

middotht

paracedilfrac34

= O( middot 4 middot 2 macr middot 2)(43)

The momentum equation becomes

u not t + u not cent ru not + r plusmn + middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (h0u not ) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (44)

These model equations will be referred to as WNL-EXT for weakly nonlinear`extendedrsquo equations The linear version of the above will also be used in the followinganalysis and will be referred to as L-EXT for linear `extendedrsquo equations

It is also possible to express the approximate continuity and momentum equationsin terms of a depth-averaged velocity The depth-averaged equations can be derivedusing the same method presented in Appendix A One version of the depth-averaged

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2891

equations will be employed in future sections which is subject to the restraint (42)and is given as

plusmn t + r cent (H middotu) +macr

middotht = 0 (45)

and

middotut + middotu cent r middotu + r plusmn + middot 2

t

frac1212 h2

0r(r cent middotu) iexcl 16 h0r

middotr cent (h0 middotu) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (46)

where the depth-averaged velocity is deshy ned as

middotu(x y t) =1

h + plusmn

Z plusmn

h

u(x y z t) dz (47)

This set of model equations (45) and (46) will be referred to as WNL-DA for weaklynonlinear depth-averaged equations

(b) Nonlinear shallow-water waves

In the case that the water depth is very shallow or the wavelength is very longthe governing equations (32) and (33) can be truncated at O( middot 2) These resultingequations are the well-known nonlinear shallow-water equations in which the searegoormovement is the forcing term for wave generation This set of equations will bereferred to as NL-SW for nonlinear shallow-water equations

5 Numerical model

In this paper a shy nite-dinoterence algorithm is presented for the general model equa-tions FNL-EXT This model has the robustness of enabling slide-generated surfacewaves although initially linear or weakly nonlinear in nature to propagate into shal-low water where fully nonlinear enotects may become important The algorithm isdeveloped for the general two-horizontal-dimension problem however in this paperonly one-horizontal-dimension examples are examined The structure of the presentnumerical model is similar to those of Wei amp Kirby (1995) and Wei et al (1995)Dinoterences between the model presented here and that of Wei et al exist in theadded terms due to a time-dependent water depth and the numerical treatment ofsome nonlinear-dispersive terms which is discussed in more detail in Appendix BA high-order predictor-corrector scheme is used employing a third order in timeexplicit AdamsBashforth predictor step and a fourth order in time AdamsMoultonimplicit corrector step (Press et al 1989) The implicit corrector step must be iter-ated until a convergence criterion is satisshy ed All spatial derivatives are dinoterenced tofourth-order accuracy yielding a model that is numerically accurate to (centx)4 (centy)4

in space and (centt)4 in time The governing equations (32) and (33) are dimen-sionalized for the numerical model and all variables described in this and followingsections will be in the dimensional form Note that the dimensional equations areequivalent to the non-dimensional ones with = middot = 1 and the addition of gravityg to the coemacr cient of the leading-order free-surface derivative in the momentumequation (ie the third term on the left-hand side of (33)) The predictor-corrector

Proc R Soc Lond A (2002)

2892 P Lynett and P L-F Liu

equations are given in Appendix B along with some additional description of thenumerical scheme Run-up and rundown of the waves generated by the submarinedisturbance will also be examined The moving-boundary scheme employed here isthe technique developed by Lynett et al (2002) Founded around the restrictionsof the high-order numerical wave-propagation model the moving-boundary schemeuses linear extrapolation of free surface and velocity through the shoreline into thedry region This approach allows for the shy ve-point shy nite-dinoterence formulae to beapplied at all points even those neighbouring dry points and thus eliminates theneed of conditional statements

In addition to the depth-integrated-model numerical results output from a two-dimensional (vertical-plane) BIEM model will be presented for certain cases ThisBIEM model will be primarily used to determine the deep-water-accuracy limit of thedepth-integrated model The BIEM model solves for inviscid irrotational regows andconverts a boundary-value problem into an integral equation along the boundary ofa physical domain Therefore just as with the depth-integration approach it reducesthe dimension of the problem by one The BIEM model used here solves the Laplaceequation in the vertical plane (x z) and of course is valid in all water depths for allwavelengths Details of this type of BIEM model when used to model water-wavepropagation can be found in Grilli et al (1989) Liu et al (1992) and Grilli (1993)for example The BIEM model used in this work has reproduced the numerical resultspresented for landslide-generated waves in Grilli amp Watts (1999) perfectly

6 Comparisons with experiment and other models

As a shy rst check of the present model a comparison between Hammackrsquos (1973)experimental data for an impulsive bottom movement in a constant water depth ismade The bottom movement consists of a length l0 = 244 water depths whichis pushed vertically upward The change in depth for this experiment macr is 01 sononlinear enotects should play a small role near the source region Figure 2 shows acomparison between the numerical results using FNL-EXT experimental data andthe linear theory presented by Hammack Both the fully nonlinear model and thelinear theory agree well with experiment at the edge of the source region (shy gure 2a)From shy gure 2b a time-series taken at 20 water depths from the edge of the sourceregion the agreement between all data is again quite good but the deviation betweenthe linear theory and experiment is slowly growing The purpose of this comparisonis to show that the present numerical model accurately predicts the free-surfaceresponse to a simple searegoor movement It would seem that if one was interestedin just the wave shy eld very near the source linear theory is adequate However asthe magnitude of the bed upthrust macr becomes large linear theory is not capable ofaccurately predicting the free-surface response even very near the source region Onesuch linear versus nonlinear comparison is shown in shy gure 2 for macr = 06 The motionof the bottom movement is the same as in Hammackrsquos case above Immediately on theoutskirts of the bottom movement there are substantial dinoterences between linearand nonlinear theory as shown in shy gure 2c Additionally as the wave propagatesaway from the source errors in linear theory are more evident

A handful of experimental trials and analytic solutions exist for non-impulsivesearegoor movements However for the previous work that made use of smooth obsta-cles such as a semicircle (eg Forbes amp Schwartz 1982) or a semi-ellipse (eg Lee et

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2886 P Lynett and P L-F Liu

tsunamis (Moore amp Moore 1984 von Huene et al 1989 Jiang amp LeBlond 1992 Tap-pin et al 1999 Keating amp McGuire 2002) On 29 November 1975 a landslide was trig-gered by a 72-magnitude earthquake along the southeast coast of Hawaii A 60 kmstretch of Kilauearsquos south coast subsided 35 m and moved seaward 8 m This land-slide generated a local tsunami with a maximum run-up height of 16 m at Keauhou(Cox amp Morgan 1977) More recently the devastating Papua New Guinea tsunamiin 1998 is thought to have been caused by a submarine landslide (Tappin et al 1999 2001 Keating amp McGuire 2002) In terms of tsunami-generation mechanismstwo signishy cant dinoterences exist between submarine-landslide and coseismic searegoordeformation First the duration of a landslide is much longer and is in the orderof magnitude of several minutes Hence the time-history of the searegoor movementwill anotect the characteristics of the generated wave and needs to be included in themodel Secondly the enotective size of the landslide region is usually much smaller thanthe coseismic searegoor-deformation zone Consequently the typical wavelength of thetsunamis generated by a submarine landslide is also shorter ie ca 110 km There-fore the frequency dispersion could be important in the wave-generation regionThe existing numerical models based on shallow-water wave equations may not besuitable for modelling the entire process of submarine-landslide-generated tsunami(eg Raney amp Butler 1976 Harbitz et al 1993)

In this paper we shall present a new model describing the generation and propaga-tion of tsunamis by a submarine landslide In this general model only the assumptionof weak frequency dispersion is employed ie the ratio of water depth to wavelengthis small or O( middot 2) frac12 1 Until the past decade weakly dispersive models were formu-lated in terms of a depth-averaged velocity (eg Peregrine 1967) Recent work hasclearly demonstrated that modishy cations to the frequency dispersion terms (Madsenamp Sorensen 1992) or expression of the model equations in terms of an arbitrary-level velocity (Nwogu 1993 Liu 1994) can extend the validity of the linear-dispersionproperties into deeper water The general guideline for dispersive properties is thatthe `extendedrsquo versions of the depth-integrated equations are valid for wavelengthsgreater than two water depths whereas the depth-averaged model is valid for lengthsgreater than shy ve water depths (eg Nwogu 1993) Moreover in the model presentedin this paper the full nonlinear enotect is included ie the ratio of wave amplitude towater depth is of order one or = O(1) Therefore this new model is more generalthan that developed by Liu amp Earickson (1983) in which the Boussinesq approx-imation ie O( middot 2) = O() frac12 1 was used In the special case where the searegooris stationary the new model reduces to the model for fully nonlinear and weaklydispersive waves propagating over a varying water depth (eg Liu 1994 Madsen ampSchanoter 1998) The model is applicable for both the impulsive slide movement andcreeping slide movement In the latter case the time duration for the slide is muchlonger than the characteristic wave period

This paper is organized as follows Governing equations and boundary conditionsfor regow motions generated by a ground movement are summarized in the next sectionThe derivation of approximate two-dimensional depth-integrated governing equationsfollows The general model equations are then simplishy ed for special cases A numericalalgorithm is presented to solve the general mathematical model The numerical modelis tested using available experimental data (eg Hammack 1973) for one-dimensionalsituations Employing a boundary integral equation model (BIEM) which solvesfor potential regow in the vertical plane a deep-water limit for waves generated by

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2887

z y

x

h0

l0

z a0

h

Figure 1 Basic formulational set-up

submarine slides is determined for the depth-integrated model The importance ofnonlinearity and frequency dispersion is inferred through numerical simulation of alarge number of dinoterent physical set-ups

2 Governing equations and boundary conditions

As shown in shy gure 1 plusmn 0(x0 y0 t0) denotes the free-surface displacement of a wavetrain propagating in the water depth h0(x0 y0 t0) Introducing the characteristic waterdepth h0 as the vertical length-scale the characteristic length of the submarine slideregion `0 as the horizontal length-scale `0=

pgh0 as the time-scale and the charac-

teristic wave amplitude a0 as the scale of wave motion we can deshy ne the followingdimensionless variables

(x y) =(x0 y0)

`0 z =

z0

h0 t =

pgh0t0

`0

h =h0

h0 plusmn =

plusmn 0

a0 p =

p0

raquo ga0

and

(u v) =(u0 v0)

p

gh0

w =w0

(=middot )p

gh0

(21)

in which (u v) represents the horizontal velocity components w the vertical velocitycomponent and p the pressure Two dimensionless parameters have been introducedin (21) which are

=a0

h0 middot =

h0

`0 (22)

Assuming that the viscous enotects are insignishy cant the wave motion can be describedby the continuity equation and Eulerrsquos equations ie

middot 2r cent u + wz = 0 (23)

ut + u cent ru +

middot 2wuz = iexcl rp (24)

wt + 2u cent rw +2

middot 2wwz = iexcl pz iexcl 1 (25)

Proc R Soc Lond A (2002)

2888 P Lynett and P L-F Liu

where u = (u v) denotes the horizontal velocity vector r = (=x =y) thehorizontal gradient vector and the subscript the partial derivative

On the free surface z = plusmn (x y t) the usual kinematic and dynamic boundaryconditions apply

w = middot 2( plusmn t + u cent r plusmn ) on z = plusmn (26 a)

p = 0 (26 b)

Along the searegoor z = iexcl h the kinematic boundary condition requires

w + middot 2u cent rh +middot 2

ht = 0 on z = iexcl h (27)

For later use we note here that the depth-integrated continuity equation can beobtained by integrating (23) from z = iexcl h to z = plusmn After applying the boundaryconditions (26) the resulting equation reads

r centmiddotZ plusmn

iexclh

u dz

cedil+

1

Ht = 0 (28)

where

H = plusmn + h (29)

We remark here that (28) is exact

3 Approximate two-dimensional governing equations

The three-dimensional boundary-value problem described in the previous sectionwill be approximated and projected onto a two-dimensional horizontal plane In thissection the nonlinearity is assumed to be of O(1) However the frequency dispersionis assumed to be weak ie

O( middot 2) frac12 1 (31)

Using middot 2 as the small parameter a perturbation analysis is performed on theprimitive governing equations The complete derivation is given in Appendix A Theresulting approximate continuity equation is

1

ht + plusmn t + r cent (Hu not )

iexcl middot 2r centfrac12

H

middot( 1

6 (2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent unot )

+ ( 12(plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(32)

Equation (32) is one of three governing equations for plusmn and u not The other twoequations come from the horizontal momentum equation (24) and are given in vector

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2889

form as

unot t + unot cent ru not + r plusmn

+ middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hu not ) +

ht

cedilr

middotr cent (hu not ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hu not )t +

htt

paracedil+ (unot cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[u not cent r(r cent u not )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent unot t iexcl plusmn unot cent r

middotr cent (hunot ) +

ht

cedil+ plusmn

middotr cent (hu not ) +

ht

cedilr cent u not

frac34

+ 3 middot 2rf 12 plusmn 2[(r cent unot )2 iexcl u not cent r(r cent u not )]g = O( middot 4) (33)

Equations (32) and (33) are the coupled governing equations written in terms ofunot and plusmn for fully nonlinear weakly dispersive waves generated by a searegoor move-ment We reiterate here that unot is evaluated at z = z not (x y t) which is a functionof time The choice of z not is made based on the linear frequency-dispersion charac-teristics of the governing equations (eg Nwogu 1993 Chen amp Liu 1995) Assuminga stationary searegoor in order to extend the applicability of the governing equationsto relatively deep water (or a short wave) z not is recommended to be evaluated asz not = iexcl 0531h In the following analysis the same relationship is employed Thesemodel equations will be referred to as FNL-EXT for fully nonlinear `extendedrsquo equa-tions

Up to this point the time-scale of the searegoor movement is assumed to be in thesame order of magnitude as the typical period of generated water wave tw = `0=

pgh0

as given in (21) When the ground movement is creeping in nature the time-scaleof searegoor movement tc could be larger than tw The only scaling parameter thatis directly anotected by the time-scale of the searegoor movement is the characteristicamplitude of the wave motion After introducing the time-scale tc into the timederivatives of h in the continuity equation (32) along with a characteristic changein water depth centh the coemacr cient in front of ht becomes

macr

tw

tc

where macr = centh=h0 To maintain the conservation of mass the above parameter mustbe of order one Thus

= macrtw

tc=

macr l0

tc

pgh0

(34)

The above relationship can be interpreted in the following way During the creepingground movement over the time period t lt tc the generated wave has propagated adistance t

pgh0 The total volume of the searegoor displacement normalized by h0 is

macr l0(t=tc) which should be the same as the volume of water underneath the generated

Proc R Soc Lond A (2002)

2890 P Lynett and P L-F Liu

wave crest ie tp

gh0 Therefore over the ground-movement period t lt tc thewave amplitude can be estimated by (34) Consequently nonlinear enotects becomeimportant only if deshy ned in (34) is O(1) Since by the deshy nition of a creepingslide the value l0=(tc

pgh0) is always less than one fully nonlinear enotects will be

important for only the largest slides The same conclusion was reached by Hammack(1973) using a dinoterent approach The importance of the fully nonlinear enotect whenmodelling creeping ground movements will be tested in x8

4 Limiting cases

In this section the general model is further simplishy ed for dinoterent physical conditions

(a) Weakly nonlinear waves

In many situations the searegoor displacement is relatively small in comparisonwith the local depth and the searegoor movement can be approximated as

h(x y t) = h0(x y) + macr middoth(x y t) (41)

in which macr is considered to be small In other words the maximum searegoor displace-ment is much smaller than the characteristic water depth Since the free-surface dis-placement is directly proportional to the searegoor displacement ie O(plusmn ) = O( macr middoth) ormuch less than the searegoor displacement in the case of creeping ground movementswe can further simplify the governing equations derived in the previous section byallowing

O() = O( macr ) = O( middot 2) frac12 1 (42)

which is the Boussinesq approximation Thus the continuity equation (32) can bereduced to

plusmn t + r cent (Hunot ) +macr

middotht

iexcl middot 2r centfrac12

h0

middot( 1

6h2

0 iexcl 12z2

not )r(r cent u not ) iexcl ( 12h0 + z not )r

micror cent (h0unot ) +

macr

middotht

paracedilfrac34

= O( middot 4 middot 2 macr middot 2)(43)

The momentum equation becomes

u not t + u not cent ru not + r plusmn + middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (h0u not ) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (44)

These model equations will be referred to as WNL-EXT for weakly nonlinear`extendedrsquo equations The linear version of the above will also be used in the followinganalysis and will be referred to as L-EXT for linear `extendedrsquo equations

It is also possible to express the approximate continuity and momentum equationsin terms of a depth-averaged velocity The depth-averaged equations can be derivedusing the same method presented in Appendix A One version of the depth-averaged

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2891

equations will be employed in future sections which is subject to the restraint (42)and is given as

plusmn t + r cent (H middotu) +macr

middotht = 0 (45)

and

middotut + middotu cent r middotu + r plusmn + middot 2

t

frac1212 h2

0r(r cent middotu) iexcl 16 h0r

middotr cent (h0 middotu) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (46)

where the depth-averaged velocity is deshy ned as

middotu(x y t) =1

h + plusmn

Z plusmn

h

u(x y z t) dz (47)

This set of model equations (45) and (46) will be referred to as WNL-DA for weaklynonlinear depth-averaged equations

(b) Nonlinear shallow-water waves

In the case that the water depth is very shallow or the wavelength is very longthe governing equations (32) and (33) can be truncated at O( middot 2) These resultingequations are the well-known nonlinear shallow-water equations in which the searegoormovement is the forcing term for wave generation This set of equations will bereferred to as NL-SW for nonlinear shallow-water equations

5 Numerical model

In this paper a shy nite-dinoterence algorithm is presented for the general model equa-tions FNL-EXT This model has the robustness of enabling slide-generated surfacewaves although initially linear or weakly nonlinear in nature to propagate into shal-low water where fully nonlinear enotects may become important The algorithm isdeveloped for the general two-horizontal-dimension problem however in this paperonly one-horizontal-dimension examples are examined The structure of the presentnumerical model is similar to those of Wei amp Kirby (1995) and Wei et al (1995)Dinoterences between the model presented here and that of Wei et al exist in theadded terms due to a time-dependent water depth and the numerical treatment ofsome nonlinear-dispersive terms which is discussed in more detail in Appendix BA high-order predictor-corrector scheme is used employing a third order in timeexplicit AdamsBashforth predictor step and a fourth order in time AdamsMoultonimplicit corrector step (Press et al 1989) The implicit corrector step must be iter-ated until a convergence criterion is satisshy ed All spatial derivatives are dinoterenced tofourth-order accuracy yielding a model that is numerically accurate to (centx)4 (centy)4

in space and (centt)4 in time The governing equations (32) and (33) are dimen-sionalized for the numerical model and all variables described in this and followingsections will be in the dimensional form Note that the dimensional equations areequivalent to the non-dimensional ones with = middot = 1 and the addition of gravityg to the coemacr cient of the leading-order free-surface derivative in the momentumequation (ie the third term on the left-hand side of (33)) The predictor-corrector

Proc R Soc Lond A (2002)

2892 P Lynett and P L-F Liu

equations are given in Appendix B along with some additional description of thenumerical scheme Run-up and rundown of the waves generated by the submarinedisturbance will also be examined The moving-boundary scheme employed here isthe technique developed by Lynett et al (2002) Founded around the restrictionsof the high-order numerical wave-propagation model the moving-boundary schemeuses linear extrapolation of free surface and velocity through the shoreline into thedry region This approach allows for the shy ve-point shy nite-dinoterence formulae to beapplied at all points even those neighbouring dry points and thus eliminates theneed of conditional statements

In addition to the depth-integrated-model numerical results output from a two-dimensional (vertical-plane) BIEM model will be presented for certain cases ThisBIEM model will be primarily used to determine the deep-water-accuracy limit of thedepth-integrated model The BIEM model solves for inviscid irrotational regows andconverts a boundary-value problem into an integral equation along the boundary ofa physical domain Therefore just as with the depth-integration approach it reducesthe dimension of the problem by one The BIEM model used here solves the Laplaceequation in the vertical plane (x z) and of course is valid in all water depths for allwavelengths Details of this type of BIEM model when used to model water-wavepropagation can be found in Grilli et al (1989) Liu et al (1992) and Grilli (1993)for example The BIEM model used in this work has reproduced the numerical resultspresented for landslide-generated waves in Grilli amp Watts (1999) perfectly

6 Comparisons with experiment and other models

As a shy rst check of the present model a comparison between Hammackrsquos (1973)experimental data for an impulsive bottom movement in a constant water depth ismade The bottom movement consists of a length l0 = 244 water depths whichis pushed vertically upward The change in depth for this experiment macr is 01 sononlinear enotects should play a small role near the source region Figure 2 shows acomparison between the numerical results using FNL-EXT experimental data andthe linear theory presented by Hammack Both the fully nonlinear model and thelinear theory agree well with experiment at the edge of the source region (shy gure 2a)From shy gure 2b a time-series taken at 20 water depths from the edge of the sourceregion the agreement between all data is again quite good but the deviation betweenthe linear theory and experiment is slowly growing The purpose of this comparisonis to show that the present numerical model accurately predicts the free-surfaceresponse to a simple searegoor movement It would seem that if one was interestedin just the wave shy eld very near the source linear theory is adequate However asthe magnitude of the bed upthrust macr becomes large linear theory is not capable ofaccurately predicting the free-surface response even very near the source region Onesuch linear versus nonlinear comparison is shown in shy gure 2 for macr = 06 The motionof the bottom movement is the same as in Hammackrsquos case above Immediately on theoutskirts of the bottom movement there are substantial dinoterences between linearand nonlinear theory as shown in shy gure 2c Additionally as the wave propagatesaway from the source errors in linear theory are more evident

A handful of experimental trials and analytic solutions exist for non-impulsivesearegoor movements However for the previous work that made use of smooth obsta-cles such as a semicircle (eg Forbes amp Schwartz 1982) or a semi-ellipse (eg Lee et

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2887

z y

x

h0

l0

z a0

h

Figure 1 Basic formulational set-up

submarine slides is determined for the depth-integrated model The importance ofnonlinearity and frequency dispersion is inferred through numerical simulation of alarge number of dinoterent physical set-ups

2 Governing equations and boundary conditions

As shown in shy gure 1 plusmn 0(x0 y0 t0) denotes the free-surface displacement of a wavetrain propagating in the water depth h0(x0 y0 t0) Introducing the characteristic waterdepth h0 as the vertical length-scale the characteristic length of the submarine slideregion `0 as the horizontal length-scale `0=

pgh0 as the time-scale and the charac-

teristic wave amplitude a0 as the scale of wave motion we can deshy ne the followingdimensionless variables

(x y) =(x0 y0)

`0 z =

z0

h0 t =

pgh0t0

`0

h =h0

h0 plusmn =

plusmn 0

a0 p =

p0

raquo ga0

and

(u v) =(u0 v0)

p

gh0

w =w0

(=middot )p

gh0

(21)

in which (u v) represents the horizontal velocity components w the vertical velocitycomponent and p the pressure Two dimensionless parameters have been introducedin (21) which are

=a0

h0 middot =

h0

`0 (22)

Assuming that the viscous enotects are insignishy cant the wave motion can be describedby the continuity equation and Eulerrsquos equations ie

middot 2r cent u + wz = 0 (23)

ut + u cent ru +

middot 2wuz = iexcl rp (24)

wt + 2u cent rw +2

middot 2wwz = iexcl pz iexcl 1 (25)

Proc R Soc Lond A (2002)

2888 P Lynett and P L-F Liu

where u = (u v) denotes the horizontal velocity vector r = (=x =y) thehorizontal gradient vector and the subscript the partial derivative

On the free surface z = plusmn (x y t) the usual kinematic and dynamic boundaryconditions apply

w = middot 2( plusmn t + u cent r plusmn ) on z = plusmn (26 a)

p = 0 (26 b)

Along the searegoor z = iexcl h the kinematic boundary condition requires

w + middot 2u cent rh +middot 2

ht = 0 on z = iexcl h (27)

For later use we note here that the depth-integrated continuity equation can beobtained by integrating (23) from z = iexcl h to z = plusmn After applying the boundaryconditions (26) the resulting equation reads

r centmiddotZ plusmn

iexclh

u dz

cedil+

1

Ht = 0 (28)

where

H = plusmn + h (29)

We remark here that (28) is exact

3 Approximate two-dimensional governing equations

The three-dimensional boundary-value problem described in the previous sectionwill be approximated and projected onto a two-dimensional horizontal plane In thissection the nonlinearity is assumed to be of O(1) However the frequency dispersionis assumed to be weak ie

O( middot 2) frac12 1 (31)

Using middot 2 as the small parameter a perturbation analysis is performed on theprimitive governing equations The complete derivation is given in Appendix A Theresulting approximate continuity equation is

1

ht + plusmn t + r cent (Hu not )

iexcl middot 2r centfrac12

H

middot( 1

6 (2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent unot )

+ ( 12(plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(32)

Equation (32) is one of three governing equations for plusmn and u not The other twoequations come from the horizontal momentum equation (24) and are given in vector

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2889

form as

unot t + unot cent ru not + r plusmn

+ middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hu not ) +

ht

cedilr

middotr cent (hu not ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hu not )t +

htt

paracedil+ (unot cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[u not cent r(r cent u not )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent unot t iexcl plusmn unot cent r

middotr cent (hunot ) +

ht

cedil+ plusmn

middotr cent (hu not ) +

ht

cedilr cent u not

frac34

+ 3 middot 2rf 12 plusmn 2[(r cent unot )2 iexcl u not cent r(r cent u not )]g = O( middot 4) (33)

Equations (32) and (33) are the coupled governing equations written in terms ofunot and plusmn for fully nonlinear weakly dispersive waves generated by a searegoor move-ment We reiterate here that unot is evaluated at z = z not (x y t) which is a functionof time The choice of z not is made based on the linear frequency-dispersion charac-teristics of the governing equations (eg Nwogu 1993 Chen amp Liu 1995) Assuminga stationary searegoor in order to extend the applicability of the governing equationsto relatively deep water (or a short wave) z not is recommended to be evaluated asz not = iexcl 0531h In the following analysis the same relationship is employed Thesemodel equations will be referred to as FNL-EXT for fully nonlinear `extendedrsquo equa-tions

Up to this point the time-scale of the searegoor movement is assumed to be in thesame order of magnitude as the typical period of generated water wave tw = `0=

pgh0

as given in (21) When the ground movement is creeping in nature the time-scaleof searegoor movement tc could be larger than tw The only scaling parameter thatis directly anotected by the time-scale of the searegoor movement is the characteristicamplitude of the wave motion After introducing the time-scale tc into the timederivatives of h in the continuity equation (32) along with a characteristic changein water depth centh the coemacr cient in front of ht becomes

macr

tw

tc

where macr = centh=h0 To maintain the conservation of mass the above parameter mustbe of order one Thus

= macrtw

tc=

macr l0

tc

pgh0

(34)

The above relationship can be interpreted in the following way During the creepingground movement over the time period t lt tc the generated wave has propagated adistance t

pgh0 The total volume of the searegoor displacement normalized by h0 is

macr l0(t=tc) which should be the same as the volume of water underneath the generated

Proc R Soc Lond A (2002)

2890 P Lynett and P L-F Liu

wave crest ie tp

gh0 Therefore over the ground-movement period t lt tc thewave amplitude can be estimated by (34) Consequently nonlinear enotects becomeimportant only if deshy ned in (34) is O(1) Since by the deshy nition of a creepingslide the value l0=(tc

pgh0) is always less than one fully nonlinear enotects will be

important for only the largest slides The same conclusion was reached by Hammack(1973) using a dinoterent approach The importance of the fully nonlinear enotect whenmodelling creeping ground movements will be tested in x8

4 Limiting cases

In this section the general model is further simplishy ed for dinoterent physical conditions

(a) Weakly nonlinear waves

In many situations the searegoor displacement is relatively small in comparisonwith the local depth and the searegoor movement can be approximated as

h(x y t) = h0(x y) + macr middoth(x y t) (41)

in which macr is considered to be small In other words the maximum searegoor displace-ment is much smaller than the characteristic water depth Since the free-surface dis-placement is directly proportional to the searegoor displacement ie O(plusmn ) = O( macr middoth) ormuch less than the searegoor displacement in the case of creeping ground movementswe can further simplify the governing equations derived in the previous section byallowing

O() = O( macr ) = O( middot 2) frac12 1 (42)

which is the Boussinesq approximation Thus the continuity equation (32) can bereduced to

plusmn t + r cent (Hunot ) +macr

middotht

iexcl middot 2r centfrac12

h0

middot( 1

6h2

0 iexcl 12z2

not )r(r cent u not ) iexcl ( 12h0 + z not )r

micror cent (h0unot ) +

macr

middotht

paracedilfrac34

= O( middot 4 middot 2 macr middot 2)(43)

The momentum equation becomes

u not t + u not cent ru not + r plusmn + middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (h0u not ) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (44)

These model equations will be referred to as WNL-EXT for weakly nonlinear`extendedrsquo equations The linear version of the above will also be used in the followinganalysis and will be referred to as L-EXT for linear `extendedrsquo equations

It is also possible to express the approximate continuity and momentum equationsin terms of a depth-averaged velocity The depth-averaged equations can be derivedusing the same method presented in Appendix A One version of the depth-averaged

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2891

equations will be employed in future sections which is subject to the restraint (42)and is given as

plusmn t + r cent (H middotu) +macr

middotht = 0 (45)

and

middotut + middotu cent r middotu + r plusmn + middot 2

t

frac1212 h2

0r(r cent middotu) iexcl 16 h0r

middotr cent (h0 middotu) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (46)

where the depth-averaged velocity is deshy ned as

middotu(x y t) =1

h + plusmn

Z plusmn

h

u(x y z t) dz (47)

This set of model equations (45) and (46) will be referred to as WNL-DA for weaklynonlinear depth-averaged equations

(b) Nonlinear shallow-water waves

In the case that the water depth is very shallow or the wavelength is very longthe governing equations (32) and (33) can be truncated at O( middot 2) These resultingequations are the well-known nonlinear shallow-water equations in which the searegoormovement is the forcing term for wave generation This set of equations will bereferred to as NL-SW for nonlinear shallow-water equations

5 Numerical model

In this paper a shy nite-dinoterence algorithm is presented for the general model equa-tions FNL-EXT This model has the robustness of enabling slide-generated surfacewaves although initially linear or weakly nonlinear in nature to propagate into shal-low water where fully nonlinear enotects may become important The algorithm isdeveloped for the general two-horizontal-dimension problem however in this paperonly one-horizontal-dimension examples are examined The structure of the presentnumerical model is similar to those of Wei amp Kirby (1995) and Wei et al (1995)Dinoterences between the model presented here and that of Wei et al exist in theadded terms due to a time-dependent water depth and the numerical treatment ofsome nonlinear-dispersive terms which is discussed in more detail in Appendix BA high-order predictor-corrector scheme is used employing a third order in timeexplicit AdamsBashforth predictor step and a fourth order in time AdamsMoultonimplicit corrector step (Press et al 1989) The implicit corrector step must be iter-ated until a convergence criterion is satisshy ed All spatial derivatives are dinoterenced tofourth-order accuracy yielding a model that is numerically accurate to (centx)4 (centy)4

in space and (centt)4 in time The governing equations (32) and (33) are dimen-sionalized for the numerical model and all variables described in this and followingsections will be in the dimensional form Note that the dimensional equations areequivalent to the non-dimensional ones with = middot = 1 and the addition of gravityg to the coemacr cient of the leading-order free-surface derivative in the momentumequation (ie the third term on the left-hand side of (33)) The predictor-corrector

Proc R Soc Lond A (2002)

2892 P Lynett and P L-F Liu

equations are given in Appendix B along with some additional description of thenumerical scheme Run-up and rundown of the waves generated by the submarinedisturbance will also be examined The moving-boundary scheme employed here isthe technique developed by Lynett et al (2002) Founded around the restrictionsof the high-order numerical wave-propagation model the moving-boundary schemeuses linear extrapolation of free surface and velocity through the shoreline into thedry region This approach allows for the shy ve-point shy nite-dinoterence formulae to beapplied at all points even those neighbouring dry points and thus eliminates theneed of conditional statements

In addition to the depth-integrated-model numerical results output from a two-dimensional (vertical-plane) BIEM model will be presented for certain cases ThisBIEM model will be primarily used to determine the deep-water-accuracy limit of thedepth-integrated model The BIEM model solves for inviscid irrotational regows andconverts a boundary-value problem into an integral equation along the boundary ofa physical domain Therefore just as with the depth-integration approach it reducesthe dimension of the problem by one The BIEM model used here solves the Laplaceequation in the vertical plane (x z) and of course is valid in all water depths for allwavelengths Details of this type of BIEM model when used to model water-wavepropagation can be found in Grilli et al (1989) Liu et al (1992) and Grilli (1993)for example The BIEM model used in this work has reproduced the numerical resultspresented for landslide-generated waves in Grilli amp Watts (1999) perfectly

6 Comparisons with experiment and other models

As a shy rst check of the present model a comparison between Hammackrsquos (1973)experimental data for an impulsive bottom movement in a constant water depth ismade The bottom movement consists of a length l0 = 244 water depths whichis pushed vertically upward The change in depth for this experiment macr is 01 sononlinear enotects should play a small role near the source region Figure 2 shows acomparison between the numerical results using FNL-EXT experimental data andthe linear theory presented by Hammack Both the fully nonlinear model and thelinear theory agree well with experiment at the edge of the source region (shy gure 2a)From shy gure 2b a time-series taken at 20 water depths from the edge of the sourceregion the agreement between all data is again quite good but the deviation betweenthe linear theory and experiment is slowly growing The purpose of this comparisonis to show that the present numerical model accurately predicts the free-surfaceresponse to a simple searegoor movement It would seem that if one was interestedin just the wave shy eld very near the source linear theory is adequate However asthe magnitude of the bed upthrust macr becomes large linear theory is not capable ofaccurately predicting the free-surface response even very near the source region Onesuch linear versus nonlinear comparison is shown in shy gure 2 for macr = 06 The motionof the bottom movement is the same as in Hammackrsquos case above Immediately on theoutskirts of the bottom movement there are substantial dinoterences between linearand nonlinear theory as shown in shy gure 2c Additionally as the wave propagatesaway from the source errors in linear theory are more evident

A handful of experimental trials and analytic solutions exist for non-impulsivesearegoor movements However for the previous work that made use of smooth obsta-cles such as a semicircle (eg Forbes amp Schwartz 1982) or a semi-ellipse (eg Lee et

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2888 P Lynett and P L-F Liu

where u = (u v) denotes the horizontal velocity vector r = (=x =y) thehorizontal gradient vector and the subscript the partial derivative

On the free surface z = plusmn (x y t) the usual kinematic and dynamic boundaryconditions apply

w = middot 2( plusmn t + u cent r plusmn ) on z = plusmn (26 a)

p = 0 (26 b)

Along the searegoor z = iexcl h the kinematic boundary condition requires

w + middot 2u cent rh +middot 2

ht = 0 on z = iexcl h (27)

For later use we note here that the depth-integrated continuity equation can beobtained by integrating (23) from z = iexcl h to z = plusmn After applying the boundaryconditions (26) the resulting equation reads

r centmiddotZ plusmn

iexclh

u dz

cedil+

1

Ht = 0 (28)

where

H = plusmn + h (29)

We remark here that (28) is exact

3 Approximate two-dimensional governing equations

The three-dimensional boundary-value problem described in the previous sectionwill be approximated and projected onto a two-dimensional horizontal plane In thissection the nonlinearity is assumed to be of O(1) However the frequency dispersionis assumed to be weak ie

O( middot 2) frac12 1 (31)

Using middot 2 as the small parameter a perturbation analysis is performed on theprimitive governing equations The complete derivation is given in Appendix A Theresulting approximate continuity equation is

1

ht + plusmn t + r cent (Hu not )

iexcl middot 2r centfrac12

H

middot( 1

6 (2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent unot )

+ ( 12(plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(32)

Equation (32) is one of three governing equations for plusmn and u not The other twoequations come from the horizontal momentum equation (24) and are given in vector

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2889

form as

unot t + unot cent ru not + r plusmn

+ middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hu not ) +

ht

cedilr

middotr cent (hu not ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hu not )t +

htt

paracedil+ (unot cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[u not cent r(r cent u not )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent unot t iexcl plusmn unot cent r

middotr cent (hunot ) +

ht

cedil+ plusmn

middotr cent (hu not ) +

ht

cedilr cent u not

frac34

+ 3 middot 2rf 12 plusmn 2[(r cent unot )2 iexcl u not cent r(r cent u not )]g = O( middot 4) (33)

Equations (32) and (33) are the coupled governing equations written in terms ofunot and plusmn for fully nonlinear weakly dispersive waves generated by a searegoor move-ment We reiterate here that unot is evaluated at z = z not (x y t) which is a functionof time The choice of z not is made based on the linear frequency-dispersion charac-teristics of the governing equations (eg Nwogu 1993 Chen amp Liu 1995) Assuminga stationary searegoor in order to extend the applicability of the governing equationsto relatively deep water (or a short wave) z not is recommended to be evaluated asz not = iexcl 0531h In the following analysis the same relationship is employed Thesemodel equations will be referred to as FNL-EXT for fully nonlinear `extendedrsquo equa-tions

Up to this point the time-scale of the searegoor movement is assumed to be in thesame order of magnitude as the typical period of generated water wave tw = `0=

pgh0

as given in (21) When the ground movement is creeping in nature the time-scaleof searegoor movement tc could be larger than tw The only scaling parameter thatis directly anotected by the time-scale of the searegoor movement is the characteristicamplitude of the wave motion After introducing the time-scale tc into the timederivatives of h in the continuity equation (32) along with a characteristic changein water depth centh the coemacr cient in front of ht becomes

macr

tw

tc

where macr = centh=h0 To maintain the conservation of mass the above parameter mustbe of order one Thus

= macrtw

tc=

macr l0

tc

pgh0

(34)

The above relationship can be interpreted in the following way During the creepingground movement over the time period t lt tc the generated wave has propagated adistance t

pgh0 The total volume of the searegoor displacement normalized by h0 is

macr l0(t=tc) which should be the same as the volume of water underneath the generated

Proc R Soc Lond A (2002)

2890 P Lynett and P L-F Liu

wave crest ie tp

gh0 Therefore over the ground-movement period t lt tc thewave amplitude can be estimated by (34) Consequently nonlinear enotects becomeimportant only if deshy ned in (34) is O(1) Since by the deshy nition of a creepingslide the value l0=(tc

pgh0) is always less than one fully nonlinear enotects will be

important for only the largest slides The same conclusion was reached by Hammack(1973) using a dinoterent approach The importance of the fully nonlinear enotect whenmodelling creeping ground movements will be tested in x8

4 Limiting cases

In this section the general model is further simplishy ed for dinoterent physical conditions

(a) Weakly nonlinear waves

In many situations the searegoor displacement is relatively small in comparisonwith the local depth and the searegoor movement can be approximated as

h(x y t) = h0(x y) + macr middoth(x y t) (41)

in which macr is considered to be small In other words the maximum searegoor displace-ment is much smaller than the characteristic water depth Since the free-surface dis-placement is directly proportional to the searegoor displacement ie O(plusmn ) = O( macr middoth) ormuch less than the searegoor displacement in the case of creeping ground movementswe can further simplify the governing equations derived in the previous section byallowing

O() = O( macr ) = O( middot 2) frac12 1 (42)

which is the Boussinesq approximation Thus the continuity equation (32) can bereduced to

plusmn t + r cent (Hunot ) +macr

middotht

iexcl middot 2r centfrac12

h0

middot( 1

6h2

0 iexcl 12z2

not )r(r cent u not ) iexcl ( 12h0 + z not )r

micror cent (h0unot ) +

macr

middotht

paracedilfrac34

= O( middot 4 middot 2 macr middot 2)(43)

The momentum equation becomes

u not t + u not cent ru not + r plusmn + middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (h0u not ) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (44)

These model equations will be referred to as WNL-EXT for weakly nonlinear`extendedrsquo equations The linear version of the above will also be used in the followinganalysis and will be referred to as L-EXT for linear `extendedrsquo equations

It is also possible to express the approximate continuity and momentum equationsin terms of a depth-averaged velocity The depth-averaged equations can be derivedusing the same method presented in Appendix A One version of the depth-averaged

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2891

equations will be employed in future sections which is subject to the restraint (42)and is given as

plusmn t + r cent (H middotu) +macr

middotht = 0 (45)

and

middotut + middotu cent r middotu + r plusmn + middot 2

t

frac1212 h2

0r(r cent middotu) iexcl 16 h0r

middotr cent (h0 middotu) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (46)

where the depth-averaged velocity is deshy ned as

middotu(x y t) =1

h + plusmn

Z plusmn

h

u(x y z t) dz (47)

This set of model equations (45) and (46) will be referred to as WNL-DA for weaklynonlinear depth-averaged equations

(b) Nonlinear shallow-water waves

In the case that the water depth is very shallow or the wavelength is very longthe governing equations (32) and (33) can be truncated at O( middot 2) These resultingequations are the well-known nonlinear shallow-water equations in which the searegoormovement is the forcing term for wave generation This set of equations will bereferred to as NL-SW for nonlinear shallow-water equations

5 Numerical model

In this paper a shy nite-dinoterence algorithm is presented for the general model equa-tions FNL-EXT This model has the robustness of enabling slide-generated surfacewaves although initially linear or weakly nonlinear in nature to propagate into shal-low water where fully nonlinear enotects may become important The algorithm isdeveloped for the general two-horizontal-dimension problem however in this paperonly one-horizontal-dimension examples are examined The structure of the presentnumerical model is similar to those of Wei amp Kirby (1995) and Wei et al (1995)Dinoterences between the model presented here and that of Wei et al exist in theadded terms due to a time-dependent water depth and the numerical treatment ofsome nonlinear-dispersive terms which is discussed in more detail in Appendix BA high-order predictor-corrector scheme is used employing a third order in timeexplicit AdamsBashforth predictor step and a fourth order in time AdamsMoultonimplicit corrector step (Press et al 1989) The implicit corrector step must be iter-ated until a convergence criterion is satisshy ed All spatial derivatives are dinoterenced tofourth-order accuracy yielding a model that is numerically accurate to (centx)4 (centy)4

in space and (centt)4 in time The governing equations (32) and (33) are dimen-sionalized for the numerical model and all variables described in this and followingsections will be in the dimensional form Note that the dimensional equations areequivalent to the non-dimensional ones with = middot = 1 and the addition of gravityg to the coemacr cient of the leading-order free-surface derivative in the momentumequation (ie the third term on the left-hand side of (33)) The predictor-corrector

Proc R Soc Lond A (2002)

2892 P Lynett and P L-F Liu

equations are given in Appendix B along with some additional description of thenumerical scheme Run-up and rundown of the waves generated by the submarinedisturbance will also be examined The moving-boundary scheme employed here isthe technique developed by Lynett et al (2002) Founded around the restrictionsof the high-order numerical wave-propagation model the moving-boundary schemeuses linear extrapolation of free surface and velocity through the shoreline into thedry region This approach allows for the shy ve-point shy nite-dinoterence formulae to beapplied at all points even those neighbouring dry points and thus eliminates theneed of conditional statements

In addition to the depth-integrated-model numerical results output from a two-dimensional (vertical-plane) BIEM model will be presented for certain cases ThisBIEM model will be primarily used to determine the deep-water-accuracy limit of thedepth-integrated model The BIEM model solves for inviscid irrotational regows andconverts a boundary-value problem into an integral equation along the boundary ofa physical domain Therefore just as with the depth-integration approach it reducesthe dimension of the problem by one The BIEM model used here solves the Laplaceequation in the vertical plane (x z) and of course is valid in all water depths for allwavelengths Details of this type of BIEM model when used to model water-wavepropagation can be found in Grilli et al (1989) Liu et al (1992) and Grilli (1993)for example The BIEM model used in this work has reproduced the numerical resultspresented for landslide-generated waves in Grilli amp Watts (1999) perfectly

6 Comparisons with experiment and other models

As a shy rst check of the present model a comparison between Hammackrsquos (1973)experimental data for an impulsive bottom movement in a constant water depth ismade The bottom movement consists of a length l0 = 244 water depths whichis pushed vertically upward The change in depth for this experiment macr is 01 sononlinear enotects should play a small role near the source region Figure 2 shows acomparison between the numerical results using FNL-EXT experimental data andthe linear theory presented by Hammack Both the fully nonlinear model and thelinear theory agree well with experiment at the edge of the source region (shy gure 2a)From shy gure 2b a time-series taken at 20 water depths from the edge of the sourceregion the agreement between all data is again quite good but the deviation betweenthe linear theory and experiment is slowly growing The purpose of this comparisonis to show that the present numerical model accurately predicts the free-surfaceresponse to a simple searegoor movement It would seem that if one was interestedin just the wave shy eld very near the source linear theory is adequate However asthe magnitude of the bed upthrust macr becomes large linear theory is not capable ofaccurately predicting the free-surface response even very near the source region Onesuch linear versus nonlinear comparison is shown in shy gure 2 for macr = 06 The motionof the bottom movement is the same as in Hammackrsquos case above Immediately on theoutskirts of the bottom movement there are substantial dinoterences between linearand nonlinear theory as shown in shy gure 2c Additionally as the wave propagatesaway from the source errors in linear theory are more evident

A handful of experimental trials and analytic solutions exist for non-impulsivesearegoor movements However for the previous work that made use of smooth obsta-cles such as a semicircle (eg Forbes amp Schwartz 1982) or a semi-ellipse (eg Lee et

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2889

form as

unot t + unot cent ru not + r plusmn

+ middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hu not ) +

ht

cedilr

middotr cent (hu not ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hu not )t +

htt

paracedil+ (unot cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[u not cent r(r cent u not )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent unot t iexcl plusmn unot cent r

middotr cent (hunot ) +

ht

cedil+ plusmn

middotr cent (hu not ) +

ht

cedilr cent u not

frac34

+ 3 middot 2rf 12 plusmn 2[(r cent unot )2 iexcl u not cent r(r cent u not )]g = O( middot 4) (33)

Equations (32) and (33) are the coupled governing equations written in terms ofunot and plusmn for fully nonlinear weakly dispersive waves generated by a searegoor move-ment We reiterate here that unot is evaluated at z = z not (x y t) which is a functionof time The choice of z not is made based on the linear frequency-dispersion charac-teristics of the governing equations (eg Nwogu 1993 Chen amp Liu 1995) Assuminga stationary searegoor in order to extend the applicability of the governing equationsto relatively deep water (or a short wave) z not is recommended to be evaluated asz not = iexcl 0531h In the following analysis the same relationship is employed Thesemodel equations will be referred to as FNL-EXT for fully nonlinear `extendedrsquo equa-tions

Up to this point the time-scale of the searegoor movement is assumed to be in thesame order of magnitude as the typical period of generated water wave tw = `0=

pgh0

as given in (21) When the ground movement is creeping in nature the time-scaleof searegoor movement tc could be larger than tw The only scaling parameter thatis directly anotected by the time-scale of the searegoor movement is the characteristicamplitude of the wave motion After introducing the time-scale tc into the timederivatives of h in the continuity equation (32) along with a characteristic changein water depth centh the coemacr cient in front of ht becomes

macr

tw

tc

where macr = centh=h0 To maintain the conservation of mass the above parameter mustbe of order one Thus

= macrtw

tc=

macr l0

tc

pgh0

(34)

The above relationship can be interpreted in the following way During the creepingground movement over the time period t lt tc the generated wave has propagated adistance t

pgh0 The total volume of the searegoor displacement normalized by h0 is

macr l0(t=tc) which should be the same as the volume of water underneath the generated

Proc R Soc Lond A (2002)

2890 P Lynett and P L-F Liu

wave crest ie tp

gh0 Therefore over the ground-movement period t lt tc thewave amplitude can be estimated by (34) Consequently nonlinear enotects becomeimportant only if deshy ned in (34) is O(1) Since by the deshy nition of a creepingslide the value l0=(tc

pgh0) is always less than one fully nonlinear enotects will be

important for only the largest slides The same conclusion was reached by Hammack(1973) using a dinoterent approach The importance of the fully nonlinear enotect whenmodelling creeping ground movements will be tested in x8

4 Limiting cases

In this section the general model is further simplishy ed for dinoterent physical conditions

(a) Weakly nonlinear waves

In many situations the searegoor displacement is relatively small in comparisonwith the local depth and the searegoor movement can be approximated as

h(x y t) = h0(x y) + macr middoth(x y t) (41)

in which macr is considered to be small In other words the maximum searegoor displace-ment is much smaller than the characteristic water depth Since the free-surface dis-placement is directly proportional to the searegoor displacement ie O(plusmn ) = O( macr middoth) ormuch less than the searegoor displacement in the case of creeping ground movementswe can further simplify the governing equations derived in the previous section byallowing

O() = O( macr ) = O( middot 2) frac12 1 (42)

which is the Boussinesq approximation Thus the continuity equation (32) can bereduced to

plusmn t + r cent (Hunot ) +macr

middotht

iexcl middot 2r centfrac12

h0

middot( 1

6h2

0 iexcl 12z2

not )r(r cent u not ) iexcl ( 12h0 + z not )r

micror cent (h0unot ) +

macr

middotht

paracedilfrac34

= O( middot 4 middot 2 macr middot 2)(43)

The momentum equation becomes

u not t + u not cent ru not + r plusmn + middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (h0u not ) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (44)

These model equations will be referred to as WNL-EXT for weakly nonlinear`extendedrsquo equations The linear version of the above will also be used in the followinganalysis and will be referred to as L-EXT for linear `extendedrsquo equations

It is also possible to express the approximate continuity and momentum equationsin terms of a depth-averaged velocity The depth-averaged equations can be derivedusing the same method presented in Appendix A One version of the depth-averaged

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2891

equations will be employed in future sections which is subject to the restraint (42)and is given as

plusmn t + r cent (H middotu) +macr

middotht = 0 (45)

and

middotut + middotu cent r middotu + r plusmn + middot 2

t

frac1212 h2

0r(r cent middotu) iexcl 16 h0r

middotr cent (h0 middotu) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (46)

where the depth-averaged velocity is deshy ned as

middotu(x y t) =1

h + plusmn

Z plusmn

h

u(x y z t) dz (47)

This set of model equations (45) and (46) will be referred to as WNL-DA for weaklynonlinear depth-averaged equations

(b) Nonlinear shallow-water waves

In the case that the water depth is very shallow or the wavelength is very longthe governing equations (32) and (33) can be truncated at O( middot 2) These resultingequations are the well-known nonlinear shallow-water equations in which the searegoormovement is the forcing term for wave generation This set of equations will bereferred to as NL-SW for nonlinear shallow-water equations

5 Numerical model

In this paper a shy nite-dinoterence algorithm is presented for the general model equa-tions FNL-EXT This model has the robustness of enabling slide-generated surfacewaves although initially linear or weakly nonlinear in nature to propagate into shal-low water where fully nonlinear enotects may become important The algorithm isdeveloped for the general two-horizontal-dimension problem however in this paperonly one-horizontal-dimension examples are examined The structure of the presentnumerical model is similar to those of Wei amp Kirby (1995) and Wei et al (1995)Dinoterences between the model presented here and that of Wei et al exist in theadded terms due to a time-dependent water depth and the numerical treatment ofsome nonlinear-dispersive terms which is discussed in more detail in Appendix BA high-order predictor-corrector scheme is used employing a third order in timeexplicit AdamsBashforth predictor step and a fourth order in time AdamsMoultonimplicit corrector step (Press et al 1989) The implicit corrector step must be iter-ated until a convergence criterion is satisshy ed All spatial derivatives are dinoterenced tofourth-order accuracy yielding a model that is numerically accurate to (centx)4 (centy)4

in space and (centt)4 in time The governing equations (32) and (33) are dimen-sionalized for the numerical model and all variables described in this and followingsections will be in the dimensional form Note that the dimensional equations areequivalent to the non-dimensional ones with = middot = 1 and the addition of gravityg to the coemacr cient of the leading-order free-surface derivative in the momentumequation (ie the third term on the left-hand side of (33)) The predictor-corrector

Proc R Soc Lond A (2002)

2892 P Lynett and P L-F Liu

equations are given in Appendix B along with some additional description of thenumerical scheme Run-up and rundown of the waves generated by the submarinedisturbance will also be examined The moving-boundary scheme employed here isthe technique developed by Lynett et al (2002) Founded around the restrictionsof the high-order numerical wave-propagation model the moving-boundary schemeuses linear extrapolation of free surface and velocity through the shoreline into thedry region This approach allows for the shy ve-point shy nite-dinoterence formulae to beapplied at all points even those neighbouring dry points and thus eliminates theneed of conditional statements

In addition to the depth-integrated-model numerical results output from a two-dimensional (vertical-plane) BIEM model will be presented for certain cases ThisBIEM model will be primarily used to determine the deep-water-accuracy limit of thedepth-integrated model The BIEM model solves for inviscid irrotational regows andconverts a boundary-value problem into an integral equation along the boundary ofa physical domain Therefore just as with the depth-integration approach it reducesthe dimension of the problem by one The BIEM model used here solves the Laplaceequation in the vertical plane (x z) and of course is valid in all water depths for allwavelengths Details of this type of BIEM model when used to model water-wavepropagation can be found in Grilli et al (1989) Liu et al (1992) and Grilli (1993)for example The BIEM model used in this work has reproduced the numerical resultspresented for landslide-generated waves in Grilli amp Watts (1999) perfectly

6 Comparisons with experiment and other models

As a shy rst check of the present model a comparison between Hammackrsquos (1973)experimental data for an impulsive bottom movement in a constant water depth ismade The bottom movement consists of a length l0 = 244 water depths whichis pushed vertically upward The change in depth for this experiment macr is 01 sononlinear enotects should play a small role near the source region Figure 2 shows acomparison between the numerical results using FNL-EXT experimental data andthe linear theory presented by Hammack Both the fully nonlinear model and thelinear theory agree well with experiment at the edge of the source region (shy gure 2a)From shy gure 2b a time-series taken at 20 water depths from the edge of the sourceregion the agreement between all data is again quite good but the deviation betweenthe linear theory and experiment is slowly growing The purpose of this comparisonis to show that the present numerical model accurately predicts the free-surfaceresponse to a simple searegoor movement It would seem that if one was interestedin just the wave shy eld very near the source linear theory is adequate However asthe magnitude of the bed upthrust macr becomes large linear theory is not capable ofaccurately predicting the free-surface response even very near the source region Onesuch linear versus nonlinear comparison is shown in shy gure 2 for macr = 06 The motionof the bottom movement is the same as in Hammackrsquos case above Immediately on theoutskirts of the bottom movement there are substantial dinoterences between linearand nonlinear theory as shown in shy gure 2c Additionally as the wave propagatesaway from the source errors in linear theory are more evident

A handful of experimental trials and analytic solutions exist for non-impulsivesearegoor movements However for the previous work that made use of smooth obsta-cles such as a semicircle (eg Forbes amp Schwartz 1982) or a semi-ellipse (eg Lee et

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2890 P Lynett and P L-F Liu

wave crest ie tp

gh0 Therefore over the ground-movement period t lt tc thewave amplitude can be estimated by (34) Consequently nonlinear enotects becomeimportant only if deshy ned in (34) is O(1) Since by the deshy nition of a creepingslide the value l0=(tc

pgh0) is always less than one fully nonlinear enotects will be

important for only the largest slides The same conclusion was reached by Hammack(1973) using a dinoterent approach The importance of the fully nonlinear enotect whenmodelling creeping ground movements will be tested in x8

4 Limiting cases

In this section the general model is further simplishy ed for dinoterent physical conditions

(a) Weakly nonlinear waves

In many situations the searegoor displacement is relatively small in comparisonwith the local depth and the searegoor movement can be approximated as

h(x y t) = h0(x y) + macr middoth(x y t) (41)

in which macr is considered to be small In other words the maximum searegoor displace-ment is much smaller than the characteristic water depth Since the free-surface dis-placement is directly proportional to the searegoor displacement ie O(plusmn ) = O( macr middoth) ormuch less than the searegoor displacement in the case of creeping ground movementswe can further simplify the governing equations derived in the previous section byallowing

O() = O( macr ) = O( middot 2) frac12 1 (42)

which is the Boussinesq approximation Thus the continuity equation (32) can bereduced to

plusmn t + r cent (Hunot ) +macr

middotht

iexcl middot 2r centfrac12

h0

middot( 1

6h2

0 iexcl 12z2

not )r(r cent u not ) iexcl ( 12h0 + z not )r

micror cent (h0unot ) +

macr

middotht

paracedilfrac34

= O( middot 4 middot 2 macr middot 2)(43)

The momentum equation becomes

u not t + u not cent ru not + r plusmn + middot 2

t

frac1212z2

not r(r cent u not ) + z not rmiddotr cent (h0u not ) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (44)

These model equations will be referred to as WNL-EXT for weakly nonlinear`extendedrsquo equations The linear version of the above will also be used in the followinganalysis and will be referred to as L-EXT for linear `extendedrsquo equations

It is also possible to express the approximate continuity and momentum equationsin terms of a depth-averaged velocity The depth-averaged equations can be derivedusing the same method presented in Appendix A One version of the depth-averaged

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2891

equations will be employed in future sections which is subject to the restraint (42)and is given as

plusmn t + r cent (H middotu) +macr

middotht = 0 (45)

and

middotut + middotu cent r middotu + r plusmn + middot 2

t

frac1212 h2

0r(r cent middotu) iexcl 16 h0r

middotr cent (h0 middotu) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (46)

where the depth-averaged velocity is deshy ned as

middotu(x y t) =1

h + plusmn

Z plusmn

h

u(x y z t) dz (47)

This set of model equations (45) and (46) will be referred to as WNL-DA for weaklynonlinear depth-averaged equations

(b) Nonlinear shallow-water waves

In the case that the water depth is very shallow or the wavelength is very longthe governing equations (32) and (33) can be truncated at O( middot 2) These resultingequations are the well-known nonlinear shallow-water equations in which the searegoormovement is the forcing term for wave generation This set of equations will bereferred to as NL-SW for nonlinear shallow-water equations

5 Numerical model

In this paper a shy nite-dinoterence algorithm is presented for the general model equa-tions FNL-EXT This model has the robustness of enabling slide-generated surfacewaves although initially linear or weakly nonlinear in nature to propagate into shal-low water where fully nonlinear enotects may become important The algorithm isdeveloped for the general two-horizontal-dimension problem however in this paperonly one-horizontal-dimension examples are examined The structure of the presentnumerical model is similar to those of Wei amp Kirby (1995) and Wei et al (1995)Dinoterences between the model presented here and that of Wei et al exist in theadded terms due to a time-dependent water depth and the numerical treatment ofsome nonlinear-dispersive terms which is discussed in more detail in Appendix BA high-order predictor-corrector scheme is used employing a third order in timeexplicit AdamsBashforth predictor step and a fourth order in time AdamsMoultonimplicit corrector step (Press et al 1989) The implicit corrector step must be iter-ated until a convergence criterion is satisshy ed All spatial derivatives are dinoterenced tofourth-order accuracy yielding a model that is numerically accurate to (centx)4 (centy)4

in space and (centt)4 in time The governing equations (32) and (33) are dimen-sionalized for the numerical model and all variables described in this and followingsections will be in the dimensional form Note that the dimensional equations areequivalent to the non-dimensional ones with = middot = 1 and the addition of gravityg to the coemacr cient of the leading-order free-surface derivative in the momentumequation (ie the third term on the left-hand side of (33)) The predictor-corrector

Proc R Soc Lond A (2002)

2892 P Lynett and P L-F Liu

equations are given in Appendix B along with some additional description of thenumerical scheme Run-up and rundown of the waves generated by the submarinedisturbance will also be examined The moving-boundary scheme employed here isthe technique developed by Lynett et al (2002) Founded around the restrictionsof the high-order numerical wave-propagation model the moving-boundary schemeuses linear extrapolation of free surface and velocity through the shoreline into thedry region This approach allows for the shy ve-point shy nite-dinoterence formulae to beapplied at all points even those neighbouring dry points and thus eliminates theneed of conditional statements

In addition to the depth-integrated-model numerical results output from a two-dimensional (vertical-plane) BIEM model will be presented for certain cases ThisBIEM model will be primarily used to determine the deep-water-accuracy limit of thedepth-integrated model The BIEM model solves for inviscid irrotational regows andconverts a boundary-value problem into an integral equation along the boundary ofa physical domain Therefore just as with the depth-integration approach it reducesthe dimension of the problem by one The BIEM model used here solves the Laplaceequation in the vertical plane (x z) and of course is valid in all water depths for allwavelengths Details of this type of BIEM model when used to model water-wavepropagation can be found in Grilli et al (1989) Liu et al (1992) and Grilli (1993)for example The BIEM model used in this work has reproduced the numerical resultspresented for landslide-generated waves in Grilli amp Watts (1999) perfectly

6 Comparisons with experiment and other models

As a shy rst check of the present model a comparison between Hammackrsquos (1973)experimental data for an impulsive bottom movement in a constant water depth ismade The bottom movement consists of a length l0 = 244 water depths whichis pushed vertically upward The change in depth for this experiment macr is 01 sononlinear enotects should play a small role near the source region Figure 2 shows acomparison between the numerical results using FNL-EXT experimental data andthe linear theory presented by Hammack Both the fully nonlinear model and thelinear theory agree well with experiment at the edge of the source region (shy gure 2a)From shy gure 2b a time-series taken at 20 water depths from the edge of the sourceregion the agreement between all data is again quite good but the deviation betweenthe linear theory and experiment is slowly growing The purpose of this comparisonis to show that the present numerical model accurately predicts the free-surfaceresponse to a simple searegoor movement It would seem that if one was interestedin just the wave shy eld very near the source linear theory is adequate However asthe magnitude of the bed upthrust macr becomes large linear theory is not capable ofaccurately predicting the free-surface response even very near the source region Onesuch linear versus nonlinear comparison is shown in shy gure 2 for macr = 06 The motionof the bottom movement is the same as in Hammackrsquos case above Immediately on theoutskirts of the bottom movement there are substantial dinoterences between linearand nonlinear theory as shown in shy gure 2c Additionally as the wave propagatesaway from the source errors in linear theory are more evident

A handful of experimental trials and analytic solutions exist for non-impulsivesearegoor movements However for the previous work that made use of smooth obsta-cles such as a semicircle (eg Forbes amp Schwartz 1982) or a semi-ellipse (eg Lee et

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2891

equations will be employed in future sections which is subject to the restraint (42)and is given as

plusmn t + r cent (H middotu) +macr

middotht = 0 (45)

and

middotut + middotu cent r middotu + r plusmn + middot 2

t

frac1212 h2

0r(r cent middotu) iexcl 16 h0r

middotr cent (h0 middotu) +

macr

middotht

cedilfrac34

= O( middot 4 middot 2 macr middot 2) (46)

where the depth-averaged velocity is deshy ned as

middotu(x y t) =1

h + plusmn

Z plusmn

h

u(x y z t) dz (47)

This set of model equations (45) and (46) will be referred to as WNL-DA for weaklynonlinear depth-averaged equations

(b) Nonlinear shallow-water waves

In the case that the water depth is very shallow or the wavelength is very longthe governing equations (32) and (33) can be truncated at O( middot 2) These resultingequations are the well-known nonlinear shallow-water equations in which the searegoormovement is the forcing term for wave generation This set of equations will bereferred to as NL-SW for nonlinear shallow-water equations

5 Numerical model

In this paper a shy nite-dinoterence algorithm is presented for the general model equa-tions FNL-EXT This model has the robustness of enabling slide-generated surfacewaves although initially linear or weakly nonlinear in nature to propagate into shal-low water where fully nonlinear enotects may become important The algorithm isdeveloped for the general two-horizontal-dimension problem however in this paperonly one-horizontal-dimension examples are examined The structure of the presentnumerical model is similar to those of Wei amp Kirby (1995) and Wei et al (1995)Dinoterences between the model presented here and that of Wei et al exist in theadded terms due to a time-dependent water depth and the numerical treatment ofsome nonlinear-dispersive terms which is discussed in more detail in Appendix BA high-order predictor-corrector scheme is used employing a third order in timeexplicit AdamsBashforth predictor step and a fourth order in time AdamsMoultonimplicit corrector step (Press et al 1989) The implicit corrector step must be iter-ated until a convergence criterion is satisshy ed All spatial derivatives are dinoterenced tofourth-order accuracy yielding a model that is numerically accurate to (centx)4 (centy)4

in space and (centt)4 in time The governing equations (32) and (33) are dimen-sionalized for the numerical model and all variables described in this and followingsections will be in the dimensional form Note that the dimensional equations areequivalent to the non-dimensional ones with = middot = 1 and the addition of gravityg to the coemacr cient of the leading-order free-surface derivative in the momentumequation (ie the third term on the left-hand side of (33)) The predictor-corrector

Proc R Soc Lond A (2002)

2892 P Lynett and P L-F Liu

equations are given in Appendix B along with some additional description of thenumerical scheme Run-up and rundown of the waves generated by the submarinedisturbance will also be examined The moving-boundary scheme employed here isthe technique developed by Lynett et al (2002) Founded around the restrictionsof the high-order numerical wave-propagation model the moving-boundary schemeuses linear extrapolation of free surface and velocity through the shoreline into thedry region This approach allows for the shy ve-point shy nite-dinoterence formulae to beapplied at all points even those neighbouring dry points and thus eliminates theneed of conditional statements

In addition to the depth-integrated-model numerical results output from a two-dimensional (vertical-plane) BIEM model will be presented for certain cases ThisBIEM model will be primarily used to determine the deep-water-accuracy limit of thedepth-integrated model The BIEM model solves for inviscid irrotational regows andconverts a boundary-value problem into an integral equation along the boundary ofa physical domain Therefore just as with the depth-integration approach it reducesthe dimension of the problem by one The BIEM model used here solves the Laplaceequation in the vertical plane (x z) and of course is valid in all water depths for allwavelengths Details of this type of BIEM model when used to model water-wavepropagation can be found in Grilli et al (1989) Liu et al (1992) and Grilli (1993)for example The BIEM model used in this work has reproduced the numerical resultspresented for landslide-generated waves in Grilli amp Watts (1999) perfectly

6 Comparisons with experiment and other models

As a shy rst check of the present model a comparison between Hammackrsquos (1973)experimental data for an impulsive bottom movement in a constant water depth ismade The bottom movement consists of a length l0 = 244 water depths whichis pushed vertically upward The change in depth for this experiment macr is 01 sononlinear enotects should play a small role near the source region Figure 2 shows acomparison between the numerical results using FNL-EXT experimental data andthe linear theory presented by Hammack Both the fully nonlinear model and thelinear theory agree well with experiment at the edge of the source region (shy gure 2a)From shy gure 2b a time-series taken at 20 water depths from the edge of the sourceregion the agreement between all data is again quite good but the deviation betweenthe linear theory and experiment is slowly growing The purpose of this comparisonis to show that the present numerical model accurately predicts the free-surfaceresponse to a simple searegoor movement It would seem that if one was interestedin just the wave shy eld very near the source linear theory is adequate However asthe magnitude of the bed upthrust macr becomes large linear theory is not capable ofaccurately predicting the free-surface response even very near the source region Onesuch linear versus nonlinear comparison is shown in shy gure 2 for macr = 06 The motionof the bottom movement is the same as in Hammackrsquos case above Immediately on theoutskirts of the bottom movement there are substantial dinoterences between linearand nonlinear theory as shown in shy gure 2c Additionally as the wave propagatesaway from the source errors in linear theory are more evident

A handful of experimental trials and analytic solutions exist for non-impulsivesearegoor movements However for the previous work that made use of smooth obsta-cles such as a semicircle (eg Forbes amp Schwartz 1982) or a semi-ellipse (eg Lee et

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2892 P Lynett and P L-F Liu

equations are given in Appendix B along with some additional description of thenumerical scheme Run-up and rundown of the waves generated by the submarinedisturbance will also be examined The moving-boundary scheme employed here isthe technique developed by Lynett et al (2002) Founded around the restrictionsof the high-order numerical wave-propagation model the moving-boundary schemeuses linear extrapolation of free surface and velocity through the shoreline into thedry region This approach allows for the shy ve-point shy nite-dinoterence formulae to beapplied at all points even those neighbouring dry points and thus eliminates theneed of conditional statements

In addition to the depth-integrated-model numerical results output from a two-dimensional (vertical-plane) BIEM model will be presented for certain cases ThisBIEM model will be primarily used to determine the deep-water-accuracy limit of thedepth-integrated model The BIEM model solves for inviscid irrotational regows andconverts a boundary-value problem into an integral equation along the boundary ofa physical domain Therefore just as with the depth-integration approach it reducesthe dimension of the problem by one The BIEM model used here solves the Laplaceequation in the vertical plane (x z) and of course is valid in all water depths for allwavelengths Details of this type of BIEM model when used to model water-wavepropagation can be found in Grilli et al (1989) Liu et al (1992) and Grilli (1993)for example The BIEM model used in this work has reproduced the numerical resultspresented for landslide-generated waves in Grilli amp Watts (1999) perfectly

6 Comparisons with experiment and other models

As a shy rst check of the present model a comparison between Hammackrsquos (1973)experimental data for an impulsive bottom movement in a constant water depth ismade The bottom movement consists of a length l0 = 244 water depths whichis pushed vertically upward The change in depth for this experiment macr is 01 sononlinear enotects should play a small role near the source region Figure 2 shows acomparison between the numerical results using FNL-EXT experimental data andthe linear theory presented by Hammack Both the fully nonlinear model and thelinear theory agree well with experiment at the edge of the source region (shy gure 2a)From shy gure 2b a time-series taken at 20 water depths from the edge of the sourceregion the agreement between all data is again quite good but the deviation betweenthe linear theory and experiment is slowly growing The purpose of this comparisonis to show that the present numerical model accurately predicts the free-surfaceresponse to a simple searegoor movement It would seem that if one was interestedin just the wave shy eld very near the source linear theory is adequate However asthe magnitude of the bed upthrust macr becomes large linear theory is not capable ofaccurately predicting the free-surface response even very near the source region Onesuch linear versus nonlinear comparison is shown in shy gure 2 for macr = 06 The motionof the bottom movement is the same as in Hammackrsquos case above Immediately on theoutskirts of the bottom movement there are substantial dinoterences between linearand nonlinear theory as shown in shy gure 2c Additionally as the wave propagatesaway from the source errors in linear theory are more evident

A handful of experimental trials and analytic solutions exist for non-impulsivesearegoor movements However for the previous work that made use of smooth obsta-cles such as a semicircle (eg Forbes amp Schwartz 1982) or a semi-ellipse (eg Lee et

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2893

- 002

0

002

004

006

008z

h

0 20 40 - 02

0

02

04

06

zh

t (gh)05 - x h

0 20 40

t (gh)05 - x h

(a) (b)

(c) (d)

Figure 2 (a) (b) Comparison between Hammackrsquo s (1973) experimental data (dots) for an impul-sive seadeg oor upthrust of macr = 01 FNL-EXT numerical simulation (solid line) and linear theory(dashed line) (a) Time-series at x=h = 0 (b) time-series at x=h = 20 where x is the distancefrom the edge of the impulsive movement (c) (d) FNL-EXT (solid line) and L-EXT (dashedline) numerical results for Hammackrsquo s set-up except with macr = 06

al 1989) the length of the obstacle is always less than 125 water depths or middot gt 08Unfortunately these objects will create waves too short to be modelled accuratelyby a depth-integrated model

Watts (1997) performed a set of experiments where he let a triangular block free falldown a planar slope In all the experiments the front (deep-water) face of the blockwas steep and in some cases vertical Physically as the block travels down a slopewater is pushed out horizontally from the vertical front Numerically however usingthe depth-integrated model the dominant direction of water motion near the verticalface is vertical This can be explained as follows Examining the depth-integrated-model equations starting from the leading-order shallow-water-wave equations theonly forcing term due to the changing water depth appears in the continuity equationThere is no forcing term in the horizontal momentum equation Therefore in the non-dispersive system any searegoor bottom cannot directly create a horizontal velocityThis concept can be further illuminated by the equation describing the vertical proshy leof horizontal velocity

u(x y z t) = unot (x y t) + O( middot 2) (61)

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

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096

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102

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0

5 10 15 2010

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max

run

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undo

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WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2894 P Lynett and P L-F Liu

- 10

- 08

- 06

- 04

- 02

0

02

z

x

Ls = L2 = L1

L2L1

0 02 04 06 08 10x

0 02 04 06 08 10

- d hd t - d hd t

Ls = L2 lt L1

L2L1

(a) (b)

Figure 3 Graphical demacrnition of the characteristic side length of a slide mass The slide mass attime t0 is shown by the solid line while the promacrle at some time t gt t0 is shown by the dashedline The negative of the change in water depth (or the approximate free-surface response in thenon-dispersive equation model) during the increment t iexcl t0 is shown by the thick line plottedon z = 01

Again the changing searegoor bottom cannot directly create a horizontal velocitycomponent for the non-dispersive system All of the searegoor movement whether it isa vertical or translational motion is interpreted as strictly a vertical motion whichcan lead to a very dinoterent generated wave pattern

When adding the weakly dispersive terms the vertical proshy le of the horizontalvelocity becomes

u(x y z t) = unot (x y t) iexcl middot 2

frac1212 z2 iexcl z2

not r(rcentunot )+(z iexcl z not )rmiddotrcent(hu not )+

ht

cedilfrac34+O( middot 4)

(62)Now with the higher-order dispersive formulation there is the forcing term rhtwhich accounts for the enotects of a horizontally moving body Keep in mind how-ever that this forcing term is a second-order correction and therefore should rep-resent only a small correction to the horizontal velocity proshy le Thus with rapidtranslational motion andor steep side slopes of a submarine slide the regow motionis strongly horizontal locally and the depth-integrated models are not adequateIn slightly dinoterent terms let the slide mass have a characteristic side length L s A side length is deshy ned as the horizontal distance between two points at whichh=t = 0 This deshy nition of a side length is described graphically in shy gure 3Figure 3a shows a slide mass that is symmetric around its midpoint in the hori-zontal direction where the back (shallow-water) and front (deep-water) side lengthsare equal Figure 3b shows a slide mass whose front side is much shorter that theback Note that for the slide shown in shy gure 3b the side lengths measured in the

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2895

z

x x1 xc xr

h0 (x)

hc (t)

b

D h h (x t)

d (t)

Figure 4 Set-up for submarine landslide comparisons

direction parallel to the slope are equal whereas for the slide in shy gure 3a theslide lengths are equal when measured in the horizontal direction An irregular slidemass will have at least two dinoterent side lengths In these cases the characteris-tic side length Ls is the shortest of all sides When L s is small compared to acharacteristic water depth h0 that side is considered steep or in deep water andthe shallow-water-based depth-integrated model will not be accurate For the ver-tical face of Wattsrsquos experiments L s = 0 and therefore Ls =h0 = 0 and the sit-uation resembles that of an inshy nitely deep ocean The next section will attemptto determine a limiting value of L s =h0 where the depth-integrated model begins tofail

7 Limitations of the depth-integrated model

Before using the model for practical applications the limits of accuracy of the depth-integrated model must be determined As illustrated above just as there is a short-wave accuracy limit (wave should be at least two water depths long when applyingthe `extendedrsquo model) it is expected that there is also a slide length-scale limitationBy comparing the outputs of this model to those of the BIEM model a limiting valueof Ls =h0 can be inferred The high degree of BIEM model accuracy in simulatingwave propagation is well documented (eg Grilli 1993 Grilli et al 1995)

The comparison cases will use a slide mass travelling down a constant slope Theslide mass moves as a solid body with velocity described following Watts (1997)This motion is characterized by a decreasing acceleration until a terminal velocity isreached All of the solid-body motion coemacr cients used in this paper are identical tothose employed by Grilli amp Watts (1999) Note that all of the submarine landslidesimulations presented in this paper are non-breaking

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2896 P Lynett and P L-F Liu

0 1 2 3 4 5 6 7x b

t = 0

z d 0

002

- 002

- 006

D h

d 02

01

0

(a)

(b)

(c)

(d)

(a) (b) (c) (d)

Ls hc = 61

Ls hc = 45

Ls hc = 31

Ls hc = 25

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

z d 0

002

- 002

- 006

(e)

Figure 5 Free-surface snapshots for BIEM (solid line) and depth-integrated (dashed line) resultsat t(g=d0)1=2 values of (a) 106 (b) 21 (c) 316 and (d) 41 (e) The location of the slide mass ineach of the four snapshots above

The set-up of the slide mass on the slope is shown in shy gure 4 The time-history ofthe searegoor is described by

h(x t) = h0(x) iexcl 12centh

middot1 + tanh

microx iexcl xl(t)

S

paracedilmiddot1 iexcl tanh

microx iexcl xr(t)

S

paracedil (71)

where centh is the maximum vertical height of the slide xl is the location of the tanhinregection point of the left side of the slide xr is the location of the inregection pointon the right side and S is a shape factor controlling the steepness of the slide sidesThe left and right boundaries and steepness factor are given by

xl(t) = xc(t) iexcl 12b cos( sup3 ) xr(t) = xc(t) + 1

2b cos( sup3 ) S =

05

cos( sup3 )

where xc is the horizontal location of the centre point of the slide and is determinedusing the equations governing the solid body motion of the slide The angle of theslope is given by sup3 The thickness of the `slidelessrsquo water column or the baseline waterdepth at the centre point of the slide is deshy ned by hc(t) = h0(xc(t)) = centh + d(t)With a specishy ed depth above the initial centre point of the slide mass d0 = d(t = 0)the initial horizontal location of the slide centre xc(t = 0) can be found The lengthalong the slope between xl and xr is deshy ned as b and all lengths are scaled by b

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2897

- 003

- 002

- 001

0

0 5 10

- 004

- 002

0

vert

sho

relin

e m

ovem

ent

d0

- 002

- 001

0

0 5 10 - 003

- 002

- 001

0

- 0012

- 0008

- 0004

0

0 5 10

- 0012

- 0008

- 0004

0

Ls hc0 = 34

z d 0

Ls hc0 = 24 Ls hc0

= 15

Ls hc0 = 34 Ls hc0

= 24 Ls hc0 = 15

(a)

(b)

(c)

(d)

(e)

( f )

t (gd0)12 t (gd0)12 t (gd0)12

Figure 6 Time-series above the initial centre point of the slide ((a) (c) (e)) and vertical move-ment of the shoreline ((b) (d) (f )) for a 20macr slope and a slide mass with a maximum heightcent h = 01 BIEM results are shown by the solid line depth-integrated results by the dashed line(a) (b) d0 =b = 04 (c) (d) d0=b = 06 and (e) (f ) d0 =b = 10

For the shy rst comparison a slide with the parameter set sup3 = 6macr d0=b = 02 andcenth=b = 005 is modelled with FNL-EXT and BIEM With these parameters thecharacteristic horizontal side length of the slide mass L s =b is 17 Ls is deshy ned as inshy gure 3 or specishy cally the horizontal distance between two points at which h=tis less than 1 of the maximum h=t value Note that a 6macr slope is roughly 1

10 Figure 5 shows four snapshots of the free-surface elevation from both models Thelowest panel in the shy gure shows the initial location of the slide mass along withthe locations corresponding to the four free-surface snapshots Initially as shownin shy gure 5a b where Ls =hc = 61 and 45 respectively the two models agree andthus are still in the range of acceptable accuracy of the depth-integrated model Inshy gure 5c as the slide moves into deeper water where L s =hc = 31 the two modelsbegin to diverge over the source region and by shy gure 5d the free-surface responsesof the two models are quite dinoterent These results indicate that in the vicinity ofx=b = 5 the depth-integrated model becomes inaccurate At this location hc=b = 05and Ls =hc = 34

Numerous additional comparison tests were performed and all indicated that thedepth-integrated model becomes inaccurate when L s =hc lt 335 One more of thecomparisons is shown here Examining a 20macr slope and a slide mass with a maxi-mum height centh=b = 01 the initial depth of submergence d0=b will be successivelyincreased from 04 to 06 to 10 The corresponding initial L s =hc values are 34 24

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2898 P Lynett and P L-F Liu

and 15 respectively Time-series above the initial centre point of the slide massesand vertical shoreline movements are shown in shy gure 6 The expectation is thatthe shy rst case (L s =hc = 34 initially) should show good agreement the middle case(L s =hc = 24 initially) marginal agreement and the last case (L s =hc = 15 initially)bad agreement The time-series above the centre shy gure 6a c e clearly agree with thestated expectation Various dinoterent z not levels were tested in an attempt to better theagreement with the BIEM-model results for the deeper water cases but z not = iexcl 0531hprovided the most accurate output Rundown as shown in shy gure 6b d f shows goodagreement for all the trials The explanation is that the wave that creates the run-down is generated from the back face of the slide mass This wave sees a characteristicwater depth that is less than hc and thus this back face wave remains in the regionof accuracy of the depth-integrated model whereas the wave motion nearer to thefront face of the slide is inaccurate This feature is also clearly shown in shy gure 5Thus if one was solely interested in the leading wave approaching the shoreline thecharacteristic water depth should be interpreted as the average depth along the backface of the slide instead of hc The inaccurate elevation waves created by the frontface of the moving mass could be absorbed numerically such as with a sponge layerso that they do not enotect the simulation

A guideline that the depth-integrated `extendedrsquo model will yield accurate resultsfor Ls =hc gt 35 is accepted This restriction would seem to be more stringent than the`extendedrsquo model frequency-dispersion limitation which requires that the free-surfacewave be at least two water depths long In fact the slide length-scale limitation ismore in line with the dispersion limitations of the depth-averaged (conventional)model The limitations of the various model formulations ie `extendedrsquo and depthaveraged are discussed in the next section

8 Importance of nonlinearity and frequency dispersion

Another useful guideline would be to know when nonlinear enotects begin to playan important role This can be determined by running numerous numerical trialsemploying the FNL-EXT WNL-EXT and L-EXT equation models These threeequation sets share identical linear-dispersion properties but have varying levels ofnonlinearity The linear-dispersion limit of these `extendedrsquo equations for the rigidbottom case is near kh = 3 where k is the wavenumber Nonlinearity howeveris only faithfully captured to near kh = 10 for the FNL-EXT model and to aneven lesser value for WNL-EXT (Gobbi et al 2000) The source-generation accuracylimitation of the model is such that the side length of the landslide over the depthmust be greater than 35 If the slide is symmetric in the horizontal direction which isthe only type of slide examined in this section then the wavelength of the generatedwave will be 2pound35poundh or roughly kh = 1 Thus up to the accuracy limit found in theprevious section nonlinearity is expected to be well captured The FNL-EXT modelwill be considered correct and any dinoterence in output compared to the other modelswith lesser nonlinearity would indicate that full nonlinear enotects are important

The importance of nonlinearity will be tested through examination of variouscenth=d0 combinations using the slide mass described in the previous section The valueof centh=d0 can be thought of as an impulsive nonlinearity as this value represents themagnitude of the free-surface response if the slide motion was entirely vertical andinstantaneous The procedure will be to hold the value hc0

= hc(t = 0) = centh + d0

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2899

0 1 2

- 03

- 02

- 01

0

0 1 2

- 06

- 04

- 02

0

- 02

- 01

0

- 02

- 01

0

- 015

- 010

- 005

0

- 02

- 01

0

0 02 04 06 08 10

- 006

- 004

- 002

0

- 015

- 010

- 005

0

D hd0 D hd0

D hd0

0 02 04 06 08 10D hd0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

max

imum

dep

ress

ion

d0

max

imum

run

-dow

n d

0

(a) (b)

(c) (d)

(e) ( f )

(g) (h)

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 30ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 15ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 055

Ls hc0 = 35

slope = 5ordm hc0 b = 015

Ls hc0 = 13

slope = 5ordm hc0 b = 015

Ls hc0 = 13

Figure 7 Maximum depression above the initial centre point of the slide mass and maximumrundown for four direg erent trial sets FNL-EXT results indicated by the solid line WNL-EXTby the dashed line and L-EXT by the dotted line

constant for a given slope angle while altering centh and d0 Two output values willbe compared between all the simulations maximum depression above the initialcentre point of the slide and maximum rundown For all simulations presented inthis section centx=b = 0003 and centt

pghc0

=b = 00003

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2900 P Lynett and P L-F Liu

Table 1 Characteristics of the simulations performed for the nonlinearity test

slopeset no (deg) hc0 =b L s =hc0

1 30 055 35

2 15 055 35

3 5 055 35

4 5 015 13

Figure 7 shows the output from four sets of comparisons whose characteris-tics are given in table 1 Figure 7a b show the depression above the centre pointand the rundown for set 1 shy gure 7c d for set 2 shy gure 7e f for set 3 and shy g-ure 7g h for set 4 Examining the maximum depression plots for sets 13 it is clearthat the trends between the three sets are very similar with FNL-EXT predictingthe largest depression and L-EXT predicting the smallest The dinoterence betweenFNL-EXT and WNL-EXT is solely due to nonlinear-dispersive terms which areof O(middot 2) while the dinoterence between WNL-EXT and L-EXT is caused by thenonlinear-divergence term in the continuity equation and the convection term inthe momentum equation which are of O() The relative dinoterences in the maxi-mum depression predicted between FNL-EXT and WNL-EXT are roughly the sameas the dinoterences between WNL-EXT and L-EXT for sets 1 2 and 3 Thereforein the source region for L s =hc0

values near the accuracy limit of the `extendedrsquomodel (near 35) the nonlinear-dispersive terms are as necessary to include inthe model as the leading order nonlinear terms As the L s =hc value is increasedthe slide produces an increasingly longer (shallow-water) wave Frequency disper-sion plays a lesser role and thus the nonlinear-dispersive terms become expect-edly less important This can be seen in the maximum depression plot for set 4For this set Ls =hc0

= 13 and the FNL-EXT and WNL-EXT results are nearlyindistinguishable

Inspecting the maximum rundown plots for sets 1 2 and 3 it seems that thetrends between the three dinoterent models have changed Now WNL-EXT predictsthe largest rundown while L-EXT predicts the smallest It is hypothesized that thedocumented over-shoaling of WNL-EXT (Wei et al 1995) cancels out the lesser waveheight generated in the source region compared to FNL-EXT leading to rundownheights that agree well between the two models As the slope is decreased the errorin the L-EXT rundown prediction increases This is attributed to a longer distanceof shoaling before the wave reaches the shoreline As the slope is decreased whilehc0

is kept constant the horizontal distance from the shoreline to the initial centrepoint of the slide increases The slide length is roughly the same for the three setstherefore the generated wavelength is roughly the same Thus with a lesser slope thegenerated wave shoals for a greater number of wave periods During this relativelylarger distance of shoaling nonlinear enotects and in particular the leading-ordernonlinear enotects accumulate and yield large errors in the linear (L-EXT) simulationsThis trend is also evident in the rundown plot for set 4 Also note that in set 4where the nonlinear-dispersive terms are very small the FNL-EXT and WNL-EXTrundowns are identical

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2901

- 02

- 01

0

092

094

096

098

100

102

- 02

- 01

0

5 10 15 2010

11

12

13

max

run

-dow

nm

ax r

undo

wn

WN

L-E

XT

- 8

- 6

- 4

- 2

0

zd 0

acute 10

-3

max

imum

run

dow

n d

0

slope = 15ordmD h b = 005Ls b = 185

Ls hc0

max

dep

m

ax d

ep W

NL

-EX

Tm

axim

um d

epre

ssio

n d

0

slope = 15ordmD h b = 005Ls b = 185

0 10 20 30 40 50

- 002

- 001

0

shor

elin

e m

ovem

ent

d0

Ls hc0 = 35

time (g d0)12

Ls hc0 = 35

(a)

(b)

(c)

(d)( f )

(e)

Figure 8 Maximum depression above the initial centre point of the slide mass (a) and max-imum rundown (d) for a set of numerical simulations on a 15macr slope (b) (e) The maximumdepression and maximum rundown scaled by the corresponding values from the WNL-EXTmodel Time-series comparisons for L s =hc0 = 35 showing the free-surface elevation above thecentre point (c) and vertical shoreline movement (f ) are given on the right WNL-EXT resultsindicated by the solid line WNL-DA by the dashed line and NL-SW by the dotted line

A deep-water limit has been determined for the `extendedrsquo model (L s =hc gt 35)but it would also be interesting to know the limits of applicability of the depth-averaged (WNL-DA) and shallow-water (NL-SW) models The only dinoterences

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2902 P Lynett and P L-F Liu

between these three models (the weakly nonlinear `extendedrsquo weakly nonlinear depthaveraged and nonlinear shallow water) are found in the frequency-dispersion terms|the nonlinear terms are the same The testing method to determine the deep-waterlimits of the various model types will be to shy x both a slope of 15macr and a slide masswith centh=b = 005 and L s =b = 185 while incrementally increasing the initial waterdepth above the centre point of the slide d Figure 8 shows a summary of the com-parisons of the three models Figure 8a d show the maximum free-surface depressionmeasured above the initial centre point of the slide and the maximum rundown forvarious L s =hc0

combinations WNL-EXT solutions are indicated by solid lines WNL-DA by dashed lines and NL-SW by the dotted lines Also shown in shy gure 8b e are themaximum depression and rundown results from WNL-DA and NL-SW relative to theresults from WNL-EXT thereby more clearly depicting the dinoterences between themodels These shy gures show WNL-EXT and WNL-DA agreeing nearly exactly whilethe errors in NL-SW decrease with increasing L s =hc0

The NL-SW results do notconverge with the WNL-EXT results until Ls =hc0

amp 15 Figure 8c f are time-seriesof the free-surface elevation above the initial centre point of the slide and the verti-cal movement of the shoreline for the case of L s =hc0

= 35 respectively Dinoterencesbetween NL-SW and WNL-EXT are clear with NL-SW under-predicting the freesurface above the slide but over-predicting the rundown due to over shoaling in thenon-dispersive model The only signishy cant dinoterence between the WNL-EXT andWNL-DA results come after the maximum depression in shy gure 8c where WNL-DApredicts an oscillatory train following the depression These results indicate that tothe deep-water limit that WNL-EXT was shown to be accurate WNL-DA is accurateas well As mentioned previously altering the level on which z not is evaluated in the`extendedrsquo model does not increase the deep-water accuracy limit for slide-generatedwaves

In summary the nonlinear-dispersive terms are important for slides near thedeep-water limit (L s =hc = 35) whose heights or centh=d0 values are large (greaterthan 04) For shallow-water slides (L s =hc gt 10) the nonlinear-dispersive terms arenot important near the source even for the largest slides The `extendedrsquo formula-tion of the depth-integrated equations does not appear to onoter any beneshy ts over thedepth-averaged formulation in regards to modelling the generation of waves in deeperwater The `extendedrsquo model would be useful if one was interested in modelling thepropagation of shallow-water slide-generated waves into deeper water which is notthe focus of this paper The shallow-water-wave equations are only valid for slides invery shallow water where L s =hc0

amp 15

9 Conclusions

A model for the creation of fully nonlinear long waves by searegoor movement and theirpropagation away from the source region is presented The general fully nonlinearmodel can be truncated so as to only include weakly nonlinear enotects or model anon-dispersive wave system Rarely will fully nonlinear enotects be important abovethe landslide region but the model has the advantage of allowing the slide-generatedwaves to become fully nonlinear in nature without requiring a transition amonggoverning equations

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2903

A high-order shy nite-dinoterence model is developed to numerically simulate wavegeneration by searegoor movement The numerical generation of waves by both impul-sive and creeping movements agrees with experimental data and other numericalmodels A deep-water accuracy limit of the model Ls =hc gt 35 is adopted Withinthis limitation the `extendedrsquo formulation of the depth-integrated equations showsno beneshy t over the `conventionalrsquo depth-averaged approach near the source regionLeading-order nonlinear enotects were shown to be important for prediction of shorelinemovement and the fully nonlinear terms are important for only the thickest slideswith relatively short length-scales Although only one-horizontal-dimension problemsare examined in this paper slides in two horizontal dimensions have been analysedby the authors but due to paper length limitations will be presented in a futurepublication As a shy nal remark it is noted that prediction of landslide tsunamis inreal cases is subject to the large uncertainty inherent in knowing the time-evolutionof a landslide Extensive shy eld research of high-risk sites is paramount to reducingthis uncertainty

The research reported here is partly supported by grants from the National Science Foundation(CMS-9528013 CTS-9808542 and CMS 9908392) and a subcontract from the University ofPuerto Rico The authors thank Ms Yin-yu Chen for providing the numerical results based onher BIEM model

Appendix A Derivation of approximatetwo-dimensional governing equations

In deriving the two-dimensional depth-integrated governing equations the frequencydispersion is assumed to be weak ie

O( middot 2) frac12 1 (A 1)

We can expand the dimensionless physical variables as power series of middot 2

f =

1X

n = 0

middot 2nfn (f = plusmn p u) (A 2)

w =

1X

n = 1

middot 2nwn (A 3)

Furthermore we will assume the regow is irrotational Zero horizontal vorticity yieldsthe following conditions

zu0 = 0 (A 4)

zu1 = rw1 (A 5)

Consequently from (A 4) the leading-order horizontal velocity components are inde-pendent of the vertical coordinate ie

u0 = u0(x y t) (A 6)

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2904 P Lynett and P L-F Liu

Substituting (A 2) and (A 3) into the continuity equation (23) and the boundarycondition (27) we collect the leading-order terms as

r cent u0 + w1z = 0 iexcl h lt z lt plusmn (A 7)

w1 + u0 cent rh +ht

= 0 on z = iexcl h (A 8)

Integrating (A 7) with respect to z and using (A 8) to determine the integrationconstant we obtain the vertical proshy le of the vertical velocity components

w1 = iexcl zr cent u0 iexcl r cent (hu0) iexcl ht

(A 9)

Similarly integrating (A 5) with respect to z with information from (A 8) we canshy nd the corresponding vertical proshy les of the horizontal velocity components

u1 = iexcl 12z2r(r cent u0) iexcl zr

middotr cent (hu0) +

ht

cedil+ C1(x y t) (A 10)

in which C1 is a unknown function to be determined Up to O(middot 2) the horizontalvelocity components can be expressed as

u = u0(x y t)

+ middot 2

frac12iexcl 1

2z2r(r cent u0) iexcl zrmiddotr cent (hu0) +

ht

cedil+ C1(x y t)

frac34+ O( middot 4)

iexcl h lt z lt plusmn (A 11)

Now we can deshy ne the horizontal velocity vector u not (x y z not (x y t) t) evaluatedat z = z not (x y t) as

u not = u0 + middot 2

frac12iexcl 1

2z2not r(rcentu0) iexcl z not r

middotrcent (hu0)+

ht

cedil+C1(x y t)

frac34+O( middot 4) (A 12)

Subtracting (A 12) from (A 11) we can express u in terms of unot as

u = u not iexcl middot 2

frac1212z2 iexcl z2

not r(r cent u not ) + (z iexcl z not )rmiddotr cent (hunot ) +

ht

cedilfrac34+ O( middot 4) (A 13)

Note that u not = u0 + O( middot 2) has been used in (A 13)The exact continuity equation (28) can be rewritten approximately in terms of plusmn

and u not Substituting (A 13) into (28) we obtain

1

Ht + r cent (Hu not ) iexcl middot 2r cent

frac12H

middot( 1

6(2 plusmn 2 iexcl plusmn h + h2) iexcl 12z2

not )r(r cent u not )

+ ( 12 (plusmn iexcl h) iexcl z not )r

micror cent (hu not ) +

ht

paracedilfrac34= O( middot 4)

(A 14)

in which H = h + plusmn Equation (A 14) is one of three governing equations for plusmn and u not The other two

equations come from the horizontal momentum equation (24) However we must

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2905

shy nd the pressure shy eld shy rst This can be accomplished by approximating the verticalmomentum equation (25) as

pz = iexcl 1 iexcl middot 2(w1t + 2u0 cent rw1 + 2w1w1z) + O( middot 4) iexcl h lt z lt plusmn (A 15)

We can integrate the above equation with respect to z to shy nd the pressure shy eld as

p =

microplusmn iexcl z

para

+ middot 2

frac1212 (z2 iexcl 2 plusmn 2)r cent u0t + (z iexcl plusmn )

middotr cent (hu)0t +

htt

cedil

+ 12(z2 iexcl 2 plusmn 2)u0 cent r(r cent u0) + (z iexcl plusmn )u0 cent r

middotr cent (hu0) +

ht

cedil

+ 12(2 plusmn 2 iexcl z2)(r cent u0)2 + (plusmn iexcl z)

middotr cent (hu0) +

ht

cedilr cent u0

frac34

+ O( middot 4)(A 16)

for iexcl h lt z lt plusmn We remark here that (A 11) has been used in deriving (A 16) Toobtain the governing equations for unot we shy rst substitute (A 13) and (A 16) into (24)and obtain the following equation up to O( middot 2)

u not t + u not cent ru not + r plusmn

+ middot 2

frac1212 z2

not r(r cent unot t) + z not rmiddotr cent (hunot )t +

htt

cedilfrac34

+ middot 2z not t

frac12z not r(r cent unot ) + r

middotr cent (hu not ) +

ht

cedilfrac34

+ middot 2

frac12middotr cent (hunot ) +

ht

cedilr

middotr cent (hunot ) +

ht

cedil

iexcl rmiddotplusmn

micror cent (hunot )t +

htt

paracedil+ (u not cent rz not )r

middotr cent (hu not ) +

ht

cedil

+ z not rmiddotunot cent r

micror cent (hunot ) +

ht

paracedil+ z not (u not cent rz not )r(r cent u not )

+ 12z2

not r[unot cent r(r cent unot )]

frac34

+ 2 middot 2rfrac12

iexcl 12 plusmn 2r cent u not t iexcl plusmn u not cent r

middotr cent (hu not ) +

ht

cedil

+ plusmn

middotr cent (hunot ) +

ht

cedilr cent unot

frac34

+ 3 middot 2rfrac12

12 plusmn 2[(r cent u not )2 iexcl u not cent r(r cent u not )]

frac34= O( middot 4) (A 17)

Equations (A 14) and (A 17) are the coupled governing equations written in termsof u not and plusmn for fully nonlinear weakly dispersive waves generated by a submarinelandslide

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2906 P Lynett and P L-F Liu

Appendix B Numerical scheme

To simplify the predictor-corrector equations the velocity time derivatives in themomentum equations are grouped into the dimensional form

U = u + 12(z2

not iexcl plusmn 2)uxx + (z not iexcl plusmn )(hu)xx iexcl plusmn x[ plusmn ux + (hu)x] (B 1)

V = v + 12(z2

not iexcl plusmn 2)vyy + (z not iexcl plusmn )(hv)yy iexcl plusmn y [ plusmn vy + (hv)y ] (B 2)

where subscripts denote partial derivatives Note that this grouping is dinoterent fromthat given in Wei et al (1995) The grouping given above in (B 1) and (B 2) incor-porates nonlinear terms which is not done in Wei et al These nonlinear timederivatives arise from the nonlinear-dispersion terms r[ plusmn (r cent (hu not )t + htt=)] andr( 1

2 plusmn 2r cent u not t) which can be reformulated using the relation

rmiddotplusmn

micror cent (hunot )t +

htt

paracedil

= rmiddotplusmn

micror cent (hu not ) +

ht

paracedil

t

iexcl rmiddotplusmn t

micror cent (hu not ) +

ht

paracedilr( 1

2 plusmn 2r cent u not t)

= r( 12 plusmn 2r cent unot )t iexcl r( plusmn plusmn tr cent u not )

The authors have found that this form is more stable and requires less iterations toconverge for highly nonlinear problems as compared to the Wei et al formulationThe predictor equations are

sup2 n+ 1ij = sup2 n

ij + 112centt(23En

ij iexcl 16Eniexcl1ij + 5Eniexcl2

ij ) (B 3)

U n+ 1ij = Un

ij + 112centt(23F n

ij iexcl 16F niexcl1ij + 5F niexcl2

ij ) + 2(F1)nij iexcl 3(F1)niexcl1

ij + (F1)niexcl2ij

(B 4)

V n+ 1ij = V n

ij + 112centt(23Gn

ij iexcl 16Gniexcl1ij + 5Gniexcl2

ij ) + 2(G1)nij iexcl 3(G1)niexcl1

ij + (G1)niexcl2ij

(B 5)

where

E = iexcl ht iexcl [( plusmn + h)u]x iexcl [( plusmn + h)v]y

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2not )Sx + ( 1

2( plusmn iexcl h) iexcl z not )Tx]gx

+ f(h + plusmn )[( 16( plusmn 2 iexcl plusmn h + h2) iexcl 1

2z2

not )Sy + ( 12( plusmn iexcl h) iexcl z not )Ty]gy (B 6)

F = iexcl 12 [(u2)x + (v2)x] iexcl gplusmn x iexcl z not hxtt iexcl z not thxt + ( plusmn htt)x iexcl [E( plusmn S + T )]x

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]x iexcl [(z not iexcl plusmn )(uTx + vTy)]x iexcl 12 [(T + plusmn S)2]x

(B 7)

F1 = 12( plusmn 2 iexcl z2

not )vxy iexcl (z not iexcl plusmn )(hv)xy + plusmn x[ plusmn vy + (hv)y ] (B 8)

G = iexcl 12[(u2)y + (v2)y ] iexcl gplusmn y iexcl z not hytt iexcl z not thyt + ( plusmn htt)y iexcl [E( plusmn S + T )]y

iexcl [12(z2

not iexcl plusmn 2)(uSx + vSy)]y iexcl [(z not iexcl plusmn )(uTx + vTy)]y iexcl 12[(T + plusmn S)2]y

(B 9)

G1 = 12 ( plusmn 2 iexcl z2

not )uxy iexcl (z not iexcl plusmn )(hu)xy + plusmn y [ plusmn ux + (hu)x] (B 10)

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2907

and

S = ux + vy T = (hu)x + (hv)y + ht (B 11)

All terms are evaluated at the local grid point (i j) and n represents the currenttime-step when values of plusmn u and v are known The above expressions (B 6)(B 11)are for the fully nonlinear problem if a weakly nonlinear or non-dispersive systemis to be examined the equations should be truncated accordingly The fourth-orderimplicit corrector expressions for the free-surface elevation and horizontal velocitiesare

sup2 n + 1ij = sup2 n

ij + 124centt(9En+ 1

ij + 19Enij iexcl 5Eniexcl1

ij + Eniexcl2ij ) (B 12)

Un + 1ij = Un

ij + 124 centt(9F n + 1

ij + 19F nij iexcl 5F niexcl1

ij + F niexcl2ij ) + (F1)n+ 1

ij iexcl (F1)nij (B 13)

V n + 1ij = V n

ij + 124centt(9Gn + 1

ij + 19Gnij iexcl 5Gniexcl1

ij + Gniexcl2ij ) + (G1)n + 1

ij iexcl (G1)nij

(B 14)

The system is solved by shy rst evaluating the predictor equations then u and v aresolved via (B 1) and (B 2) respectively Both (B 1) and (B 2) yield a diagonal matrixafter shy nite dinoterencing The matrices are diagonal with a bandwidth of shy ve (due toshy ve-point shy nite dinoterencing) and an emacr cient LU decomposition can be used At thispoint in the numerical system we have predictors for plusmn u and v Next the correctorexpressions are evaluated and again u and v are determined from (B 1) and (B 2)The relative errors in each of the physical variables is found in order to determineif the implicit correctors need to be reiterated This relative error is given as

wn + 1 iexcl wn+ 1curren

wn + 1 (B 15)

where w represents plusmn u and v and w curren is the previous iterations value The correctorsare recalculated until all errors are less than 10iexcl4 Note that inevitably there willbe locations in the numerical domain where values of the physical variables areclose to zero and applying the above error calculation to these points may lead tounnecessary iterations in the corrector loop Thus it is required that

macrmacr plusmn

a

macrmacr

macrmacr u v

degp

gh

macrmacr gt 10iexcl4

for the corresponding error calculation to proceed where a is determined from equa-tion (34) for a creeping slide For the model equations linear stability analysis givesthat centt lt centx=2c where c is the wave celerity in the deepest water Note that whenmodelling highly nonlinear waves a smaller centt is usually required for stability Inthis analysis centt = centx=4c produced stable and convergent results for all trails

For the numerical exterior boundaries two types of conditions are applied reregec-tive and radiation The reregective or no-regux boundary condition for the Boussinesqequations has been examined by previous researchers (Wei amp Kirby 1995) and theirmethodology is followed here For the radiation or open boundary condition asponge layer is used The sponge layer is applied in the manner recommended byKirby et al (1998) Run-up and rundown are modelled with the `extrapolationrsquomoving-boundary algorithm described in Lynett et al (2002)

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2908 P Lynett and P L-F Liu

Nomenclature

a wave amplitude

b length along the slope between xl and xr for the tanh slide

c wave celerity

d depth of water above the centre point of the slide function of time

d0 initial depth of water above the centre point of the slide ie at t = 0

g gravity

h0 characteristic water depth or baseline water depth function of space

h water depth proshy le function of space and timemiddoth the changing part of the water depth proshy le ((h iexcl h0)=macr )

hc baseline water depth at the centre point of the slide (centh + d)

hc0initial baseline water depth at the centre point of the slide (centh + d0)

H total water depth (h + plusmn )

l0 characteristic horizontal length-scale of the submarine slide

Ls characteristic horizontal side length of the submarine slide

p depth-dependent pressure

S shape factor for tanh slide

t time

tc time-scale of searegoor motion

tw typical period of wave generated by a specishy ed searegoor motion

u v w depth-dependent components of velocity in x y z

u not v not magnitude of horizontal velocity components u v evaluated on z not

middotu middotv depth-averaged horizontal velocity components

u horizontal velocity vector (u v)

xc yc horizontal coordinates of the midpoint of the searegoor movement

xl xr locations of the left and right inregection points for the tanh slide proshy le

z not arbitrary level on which the `extendedrsquo equations are derived

macr scaled characteristic change in water depth

due to searegoor motion (centh=h0)

centh characteristic or maximum change in water depth

due to searegoor motion

centt time-step in numerical model

centx centy space steps in numerical model

nonlinearity parameter (a=h0)

r horizontal gradient vector

raquo density of water

sup3 slope angle

middot frequency-dispersion parameter (h0=l0)

plusmn free-surface displacement

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

Submarine-landslide-generated waves and run-up 2909

References

Chen Y amp Liu P L-F 1995 Modimacred Boussinesq equations and associated parabolic modelfor water wave propagation J Fluid Mech 228 351381

Cox D C amp Morgan J 1977 Local tsunamis and possible local tsunamis in Hawaii Reportno HIG 77-14 Hawaii Institute of Geophysics University of Hawaii

Forbes L K amp Schwartz L W 1982 Free-surface deg ow over a semicircular obstruction J FluidMech 114 299314

Geist E L 1998 Local tsunami and earthquake source parameters Adv Geophys 39 117209

Gobbi M Kirby J T amp Wei G 2000 A fully nonlinear Boussinesq model for surface wavesPart 2 Extension to O(kh4 ) J Fluid Mech 405 181210

Grilli S T 1993 Modeling of nonlinear wave motion in shallow water In Computational methodsfor free and moving boundary problems in heat and deg uid deg ows (ed I C Wrobel amp C ABrebbia) ch 3 pp 3765 Elsevier

Grilli S T amp Watts P 1999 Modeling of waves generated by a moving submerged bodyApplications to underwater landslides Engng Analysis Bound Elem 23 645656

Grilli S T Skourup J amp Svendsen I A 1989 An eplusmn cient boundary element method fornonlinear waves Engng Analysis Bound Elem 6 97107

Grilli S T Subramanya R Svendsen I A amp Veeramony J 1995 Shoaling of solitary waveson plane beaches J Wtrwy Port Coastal Ocean Engng 120 7492

Hammack H L 1973 A note on tsunamis their generation and propagation in an ocean ofuniform depth J Fluid Mech 60 769799

Harbitz C B Pedersen G amp Gjevik B 1993 Numerical simulation of large water waves dueto landslides J Hydraul Engng 119 13251342

Jiang L amp LeBlond P H 1992 The coupling of a submarine slide and the surface waves whichit generates J Geophys Res 97 12 73112 744

Keating B H amp McGuire W J 2002 Island edimacrce failures and associated hazards In Pureand applied geophysics special issue landslide and tsunamis (In the press)

Kirby J T Wei G Chen Q Kennedy A B amp Dalrymple R A 1998 FUNWAVE 10 fullynonlinear Boussinesq wave model documentation and userrsquos manual Newark DE Universityof Delaware

Lee S-J Yates G T amp Wu T Y 1989 Experiments and analyses of upstream-advancingsolitary waves generated by moving disturbances J Fluid Mech 199 569593

Liu P L-F 1994 Model equations for wave propagations from deep to shallow water InAdvances in coastal and ocean engineering (ed P L-F Liu) vol 1 pp 125158 WorldScientimacrc

Liu P L-F amp Earickson J 1983 A numerical model for tsunami generation and propagation InTsunamis their science and engineering (ed J Iida amp T Iwasaki) pp 227240 HarpendenTerra Science

Liu P L-F Hsu H-W amp Lean M H 1992 Applications of boundary integral equationmethods for two-dimensional non-linear water wave problems Int J Numer Meth Fluids15 11191141

Lynett P Wu T-W amp Liu P L-F 2002 Modeling wave runup with depth-integrated equa-tions Coastal Engng 46 89107

Madsen P A amp Schareg er H A 1998 Higher-order Boussinesq-type equations for surface gravitywaves derivation and analysis Phil Trans R Soc Lond A 356 21233184

Madsen P A amp Sorensen O R 1992 A new form of the Boussinesq equations with improvedlinear dispersion characteristics Part II A slowly varying bathymetry Coastal Engng 18183204

Moore G amp Moore J G 1984 Deposit from giant wave on the island of Lanai Science 22213121315

Proc R Soc Lond A (2002)

2910 P Lynett and P L-F Liu

Nwogu O 1993 Alternative form of Boussinesq equations for nearshore wave propagation JWtrwy Port Coastal Ocean Engng 119 618638

Peregrine D H 1967 Long waves on a beach J Fluid Mech 27 815827

Press W H Flannery B P amp Teukolsky S A 1989 Numerical recipes the art of scientimacrccomputing pp 569572 Cambridge University Press

Raney D C amp Butler H L 1976 Landslide generated water wave model J Hydraul Div 10212691282

Tappin D R (and 18 others) 1999 Sediment slump likely caused 1998 Papua New Guineatsunami Eos 80 329

Tappin D R Watts P McMurtry G M Lafoy Y amp Matsumoto T 2001 The SissanoPapau New Guinea tsunami of July 1998|oreg shore evidence on the source mechanism MarGeol 175 123

von Huene R Bourgois J Miller J amp Pautot G 1989 A large tsunamigetic landslide anddebris deg ow along the Peru trench J Geophys Res 94 17031714

Watts P 1997 Water waves generated by underwater landslides PhD thesis California Instituteof Technology

Wei G amp Kirby J T 1995 A time-dependent numerical code for extended Boussinesq equationsJ Wtrwy Port Coastal Ocean Engng 120 251261

Wei G Kirby J T Grilli S T amp Subramanya R 1995 A fully nonlinear Boussinesq modelfor surface waves Part 1 Highly nonlinear unsteady waves J Fluid Mech 294 7192

Yeh H Liu P L-F amp Synolakis C (eds) 1996 Long-wave runup models In Proc 2nd IntWorkshop on Long-wave Runup Models World Scientimacrc

As this paper exceeds the maximum length normally permittedthe authors have agreed to contribute to production costs

Proc R Soc Lond A (2002)