tolrol¿Õ i

TIDAL PROPAGATION

IN THE

GULF OF CARPENTARIA

by

Michele Marie RieneckerB.Sc. (ttons. ), University of Queensland

Thesis submitted for the degree ofDoctor of PhilosoPhy

in the

University of AdelaideDepartment of Applied Mathematics

0,q

J

December L978

[^t.ro{r.i tli" 11ì1-qT0

TABLE OF CONTENTS

SUMMARY

SIGNED STATEMENT

ACKNOIIILEDGEMENTS

CHAPTER 1 INTRODUCTION

CHAPTER 2

CHAPTER 3

CHAPTER 4

CHAPTER 5

(i)( ii)(íii)

I

THE PROBLEM: A RECTÆ{GULAR RESONATOR ON A SEMI-INFINITECHANNEL

2.L The Tidal Equations 5

2.2 Frequency Response Analysis 6

2.3 Reformulation of the Equations for an Analytic Model 8

THE ANA],YTIC SOLUTION

3.1 The Method of Solution and the Boundary Conditions

3.2 Solution for the Channel Region

3.3 Solution for the ResonaÈor Region

3.4 Solution folthe Junction Region

3.5 The Remaining Matchíng Conditions and theGalerkin Technique

AN EXTENSION TO THE MODEL: AN ADJOINING CHANNEL

4.1 The Equatíons

4,2 The Solutions

4.3 Determination of ô

TI4IO NUMERICAL MODELS

5.1 A Linear Fínite-oifference Numeríca1 Model

5,2 The EVP Method

5.3 Stability, Consistency and Convergence

5.4 The Friction Parameter

5.5 A Non-Linear Model

5.6 Consistency, Convergence and Stability

10

11

13

15

20

35

37

46

47

4B

52

55

59

6L

64

CHAPTER 6 APPLICATION TO TIIE GI]LF OF CARPENTARIA

6.1 The Gulf of CarPentaría

6.2 The AnalYtic Model of ChaPter 3

6.3 The AnalYtic Model of ChaPter 4

6.4 The Linear Numerical Model

6.5 The Non-linear Numerical Model

6.6 The Programs

6.7 The Response of the Gulf to Tidal Forcing

CHAPTER 7 CONCLUSION

APPENDIX 1 The Representation of Bottom Friction

APPENDIX 2 \te Galerkin and Collocation Methods

APPENDIX 3 Evaluation of the Integral Form for 6z(x,y)

68

72

89

95

105

110

110

116

1t_8

L27

130

138

APPENDIX 4

BIBLIOGRAPHY

The Classes of Elements for the Non-linear Model and

Their Associated Finite-Difference Equations 135

(i)

SUMMARY

This thesis considers tidal propagation in a rectangular

resonator-channel system, with specific reference to the Gulf of

Carpentaria, situated to Èhe North of Australia.

The linearízed form of the trdo-dimensional depth-averaged

equations of continuity and momenEum conservation is used. An

analytic solution is found by dividing the area into regions of

constant deprh. In this manner, a solution is found for Èhe case

of a síngle connecting channel and then for the exEended case of two

connecting channels, associated with either neglecÈing or including

the effect of tidal flux from Torres Strait into the Gulf.

Results from the analytic model are used to provide tidal

inputs for two numerical models, both of which use explicir finite-

difference approximations. The first numerical model is linear and

is developed to account for realistic boundary and bathynetry variations,

with the emphasis on obtaining a model with small- compuÈer time and

memory requirements. Since, in shallower coastal areas, the non-

linear Lerms in Èhe continuity and momentum equations become more

important, the second numerical model developed includes these terms

to determíne their effect on the resonator as a whole. The two models

show favourable agreement, thus verifying the usefulness of the linear

mode 1 .

( ii)

SIGNED STATEMENT

I hereby declare that this thesis contains no material which has

been accepted for the award of any other degree or diploma in any

University and, to the best of my knowledge, it contains no material

previously published by any other person, except where due reference ís

made in the text of the thesie.

M.M. RIENECKER.

(iii)

ACKNOI,üLEDGEMENTS

I would like to thank my st¡pervisor, Dr. B.J. Noye, for his

advice and guidance in completíng this thesis.

Many thanks are also due Èo Dr. Michael Teubner for his help

and encouragement throughout, Èo Mrs. Angela McKay for her excellent,

accurate typing, to Mr, G. de Vries for preparing the diagrams and

to Mr. Phil Leppard for his advice on computing.

The work associated wíth this thesis was carried Òut from

February 1975 to November 1978, during which time I was financed by

a Commonwealth Postgtaduate Research Award'

M.M. Rietrecker -

1

CHAPTER 1

INTRODUCTION

The Gulf of Carpentaria, located in tl're North-Eastern part of

Australia, is an area of difficult access by either land or water and

hence remains relatively uncharted and unstudied. However, over the

past few years, there has been an increase in the nuirber of research

programs in the area as its potential for the lucrative rnining and

fishing industries has been recognized. Lirnited sectors, such as at

lrleipa, PorÈ McArthur, Groote Eylandt and Gove Peninsular, have been

surveyed as port facilities l{ere required by mining companies. The

C.S.I.R.O. has reported on the hydrology of the region (Rochford (1966),

Newell (1973)) and has made a preliminary study of the circulation in the

Gulf (Cresswell (1971)). Further work by this organízatíoa is currently

under way. The Gulf r^ras also chosen by Teleki et al (1973), as the test

site for assessing the usefulness of satellite irnagery to the rnapping of

hydrological parameters in areas of difficult access.

Tides and wind are the only two mechanisms which generate Èhe

currenÈs in the atea. For several months of the year it is the influence

of the strong, sÈeady trade winds which drive the circulation. However,

overall, the circulation and mixing of waters ín the Gulf are governed

by the clockwise motíon of tides and by density gradients resulting from

the stratification of the water masses. Stratification is partly induced

by differential evaporation rates betvreen the Northern and Southern halves

of the bay (Teleki et al (1973)). According to Newell (1973), "the annual

evaporatiorr/precipitation budget of Èhe Gulf forms a very sma1l part of

?-

the total water exchange but is of great importance in influencing water

movement.tt

One examPle of the effect, in

evaporation is the striking feature

the Gulf, of winds and precipitation/

of the annual cycle of about .7m rn

the tides, the lowesttídes being r:ecorded at the end of the drl' season

and the highest during the wet season' It is during the wet season, which

occurs in the summer months, that the activity of cyclones sometimes has

a disastrous effect, causing large surges. some description of this is

given by Easton (1970) who outlines the general tidal features of the

Gulf.

Realistic and accurate modelling of the waÈer movement in the Gulf

of Carpentaria is obviously an intricate affair, it being necessary to

incorporate the effects of tide, wind, stratification, precipitation/

evaporation and Pressure surges, not to mention ríver run-off and sediment

transport. However, whereas the effects of the other mechanisms diminish

at certain times of the year, Eidal forcing is always Present; and it is

only the response of the Gulf to tidal forcing which is investigated in

this thesis. Once the Èi-dal response is understood, it is easier to

sËudy the coupling of the tide with other effsu[s'

The tides are caused by the movement of the sun and the moon and

their changing gravitational pull on the $Iater of the earth; however, in

coastal areas, such as gulfs and estuaries, a\¡Iay from the deep oceans,

astronomical tidal forcing can usually be neglected compared to the direct

forcing from the motion of adjacent r¡raters. From the results of Hamblin

(1976) for different size basins, the maximum amplitude of the resPonse of

the Gulf of Carpentaría to direct astronomical forcing could be expected

to be only about 3 cm, a very sma1l contribution to the response as a whole

In this thesis, Eidal propagation in Gulf systems is investigated

by finding the resPonse of Ëhe system Èo tidal forcing on an open boundary'

3

One of Èhe main problems in ascertaining the accuracy or limitations of

any model of an area is that reliable input data across the open boundary

is rarely available, especially if the boundary is wide. Data is usually

available from coastal areas, but this may be disturbed by local effects

and is not always representative for the open sea. Values may be inter-

polated between coastal areas, but this does not normally take into account

the disÈribution of depth (see Hansen (1962)). A more accurate input may

be obtained from the results of an analytic model which produces the input

from the solution to the model equations. This still does not account for

the true distribution of depth, but avoids the need to interpolate over

wide areas.

Analytic models have their limiEations in that they provide solutions

only for simplified situations; however, they can be very useful in providing

an insight into the important feaÈures of a model area. As well as providing

the tidal forcing daÈa along an open boundary for numerical models, they

may also act as a guideline to the accuracy or validity of these more

complicated models.

lüith this in mind, two analytic models of the tidal propagation in

the Gulf of Carpentaria are developed. They a, e essentially extensions

of Taylorrs (I92Oa) problem of the reflection of a Kelvin wave by an end

barrier in a semi-infinite channel.

tr{illiams (L972) modelled the Gulf as a rectangular resonator on a

serni-infinite channel. His first model neglects the effect of the Coriolis

force; his second considers the frequency of rotation of the earEh to be

small compared to the frequency of the tidal motion.

The models in this thesis are extensions of his work, incorporating,

amongst other things, the dissipation of energy by bottom friction. Although

Èhe Coriolis parameter is taken to be a constant, no assumption is made as

4

to its value relative to the forcing frequency and, hence, this model

is applicable to more general situations.

The solutions are found for the linearized form of the two-

dimensional depth-integrated equations of continuity and momentum

conservation. I^Ihen it is impossible to find a solution which satisfies

a boundary condiÈion exactly, a Galerkin technique is used to find an

approxímate solution.

Ttre first model, in keeping with lüilliams (1972) , considers no

flux through Torres Strait, while the second allows for the presence of

tidal forci.ng through this Straít by incorporating a second semi-infinite

channel in the model. Torres Strait is a shallow region with an intricate

array of íslands, reefs and atolls. .tt is not considered that the tides

in this area have been modelled accurately; only the effect of motion

through the Strait on the tides in the Gulf is of interest.

The results from the second rnodel are used as input for a linear

numerical model which accounts for more complicated boundaries and bottom

topography. This numerical model is a frequency-response scheme, based

on the EVP method described by Roache (1972), rather than a time-stepping

mode1, and has the ariset of requiring very little computer time and storage.

Its results also compare very well with a more complicaÈed non-linear

numerical scheme which is developed to determine the effects of the non-

linear terms on the tidal motion in the Gulf and to assess Lhe usefulness

of the more simple linear model. Both numerical models use a finite-

difference approximation to the two-dimensional depth-integrated form of

the equations of continuity and momentum conservation which govern fluid

motion.

5

CHAPTER 2

THE PROBLEM: A RECTANGULAR RESONATOR ON A

SEMI-INFINI.TE CHANNEL

2.L The Tidal Equations

The general two-dimensional depth-averaged equations governing

fluid motion have been derived by such authors as Dronkers (1964) and

Nihoul (1975). These equations, as given by Nihoul, may be written

in vector form as

=0

ò

= (U,V) is the depth-averaged horízonLal velocity,

is the depÈh at mean-sea-leve1,

is the surface elevation above mean-sea-leve1,

is the time coordinate,

is the Coriolis Parameter '

is the unit vector in the vertical direction,

is the acceleration due to gravitY,

is a bottom friction Parameter,

$tt*zl + V.{(rr+z¡01 2.L,IA

2.I.lbg+ q.Vq + I b^_e = - v(ez) - ;L S S + [ðt

where g

h

Z

t

f

5

e

Y

V= tãI,¡T'xry are orthogonaL Cartesian coordinates, positively increasing

to the East and North resPecÈivelY,

E represents the contribution from other factors such as external

forces, atmospheric pressure gradients, wind stress effects and

t.urbulence and shear effects.

and

6

In this study of the response of some systems to tidal forcing

on an open boundary, the effects of the t.erm { are neglected. The

equations 2.1.L can be further simplified to yield linear equations

r¿hich have the advantage of superposition of solutions. The assumptions

(and a discussion of their validity) implicit in such an approximation

may be found, for example, inHendershottand Munk (1970) and Noye and

Tronson (1978).

The linearízed shallow r¡rater r¡rave equations may then be written

3cnul*3(r,vl =-Yâx "'" ðy ".' ' ðt

AU

ðt*;u - f v = - c #

#.;u+fu= -tK

2.I.2a

2.1.2b

2 .I.2c

in which the Coriolis parameter is considered to be constant and the

friction parametèr, r, is some linear approximaEion to yllqll A discussion

of this línearízation of the quadratic friction term is given in Appendi-x 1

along with different forms which may be used to model Y

2.2 Frequency Res Ana 1ys is

Equations 2,L2 are used to model the tidal propagation in a channel-

resonator system, as shown in Figure 2.1, where the motion is produced in

response to an input ne{60(xry).t* }, of period T = 2rfu, travelling

along the channel in the positive x-direction. The equations are solved

subject to the input eoeÅ-t and the real Parts of the soluÈions for

Z, U and V will give the elevation and velocity fields at any insËant

in time. (i = ,/-l¡.

7

b CHANNEL(Region 3)

JUNCTION(Region 2) v

d

x

RESONATOR(Region 1)

a

A rectangular resonator on a semi-infinite channel.

The area is divided into three regions with a

depËh-step at the common boundaries of each region.

Figure 2. 1

8

Since the equations are linear, it follows that

Z(x,y,t) = 6(x,y)"-lt''tt

U(x,y,t) = u(x,y)e-iu,t

V(x,y, t) = v(x,y).-i<'rt

so that the explicit time dependence in 2.1.2 may be removed. The

equations governing the spatial variation of the fluid motion are

therefore

(-it¡.¡+r/h)u - f ., = -, t

(-ir¡+r/h)v+f,r=-*Foây

3rn'¡ * 3(n*,)dx dyit¡6

2.2.1a

2.2.rb

2.2.Ic

2.2.2a

2.2.2b

2.2 .2c

2.2.2d,

2 .2.2e

subject to the boundary condiÈions

-d<y<b

-d<y<o

-a{x<0

x<0

x(-a

u(O,y) = o,

u(-a,y) = o,

v(x,-d) = 0,

v(x,b) = 0,

v(x,O) = 0,

and a radiation condition that the input wave does not excite other $/aves

travelling in Èhe same direction.

2.3 Reformulation of the EquaËions for an Analytic Model

The equations 2.2.1 can be solved analytically if it is assumed

that the depth is a constant; the equations being forrnulated in terms of

6(xry) only and the solution found by a separation of variables technique

Manipulation of equations 2.2.|a and b yields the relations

9

2.3.lb

and substitution of these expressions into 2.2.\c yields the differential

equation governing Ç :-

u(x,y) = - g{(-io+r/h)2 +f'}-t{(-ir¡+r/h)ff.f fft

v(x,y) = - g{(-io+r/h) 2 * f'}-t {- f * + (-ir¡+r/h)#}

v'Ç*#f'{ffiffi.l6=o

2.3.La

2.3.2

2.3.3a

2.3.3b

2.3.3c

2.3 .4a

2 "3.4b

2.3.5

Defining

q = r/htrt

0=flw

these equations rnay be rewritten as

u(x,y) = - fit(r+i6¡'-o'\-'{i(r+iq>Nf, - t #}

v(x,y) = - fit(l+io)'-a'j-'{u t + i(r+i4)ff}

Y'e*x'e=o

and solved subject to the condiÈions 2.2.2.

10

CHAPTER 3

THE ANALYTIC SOLUTION

3.1 The MeÈhod of SoluËion and the Boundary Conditions

To find Èhe solution to equations 2:3.4 and 2.3.5 in the area

depicted in Figure 2.1, the system is divided inÈo three constant depth

regions as shown - the channel, the junction and the resonator. Ihe

equations are solved (as far as possible) in each separate region and

then the elevations and volume transports are matched at the colnmon

boundaries of adjacent regions.

Thus, the solutions are required to Ehe equations

subject Ëo the

(i)

( ii)( iii)( iv)

(v)

(vi)

(vii)

(vl11)

( ix)

and (x)

(rn the

and j=3 to

found from the

lvz+{)e, =0, j=r,2,3

conditions

va(x,O) = vs(x,b) = 0, x < -a

v2(xrb)=0, -a(x<0

u2(0,y)=0, 0<y<b

v1(x,-d)=0, -a<x<0

ur(O,y) = ur(-a,y) = 0, -d < y < 0

6r(x,o) = Ez(x,O)' -a < x < 0

h¡v1(x,0) = h2v2(x,0), -a < x < 0

ÇzGa,y) = ÇtGa,Y), 0 < y < b

h2u2(-ary) = hgug(-a,y), 0 { y < b

a radiation condition in the channel region.

above, j= 1 refers Ëo theresonator, j = 2 to the junction

the channel, and each of the u. (xry), v, (x,y) may be

appropriate forms of 2,3.4.)

11

The condition at the corner (-a,0) is given as vs(-a,0)

= ,rr(-a,0) = 0, while the restrictions v2(-a,0) = hlv1(-a,O) /hz and

uz(-a,0) = hsus(-a,0)/hz do not necessarily drive these latter velocities

to zero and hence could provide a shear effect. lrlhereas it is recognised

that this may be erroneous, the exact condition on the velocity at such

corners is not known and the use of the above condition is not considered

to be too detrimental to the solution as a whole.

The inclusion of the Coriolis force (0 I 0) prohibits the finding

of an exact solution to this system. Even for Ehe simpler problem of a

constant depth channel with a barrier aE one end (Taylor (I92Oa), Defant

(1961)) one has to resort to an approximate method to find a solution which

satisfies Ehe zero normal velocity condition at the closed end of the

channel. Here, for those boundary or matching conditions which cannot be

satisfied exactly, a Galerkin method, which is an approximaÈion technique,

ís used. This technique is discussed in Appendix 2.

3.2 Solution for the Channel Resion

In the channel region, the equation which governs the surface

elevation is

YzÇt + xlÇ, = o, 3.2.I

where

^.2 _ or2 {(r+iqù2-e2}^s

- cr\ --( l.Io'f- 3.2.2

subject to

vg(x,O) = vs(x,b) = 0, for x d -a

and the radiation condition.

3.2.3

L2

The general solution, found by separation of variables, is

( o)

4r(x,y)=60(x,y)+Aoe -ik x+Ks y3

æ

I -| co

(o)k (r+iqr¡

1.rv6--L-b

( e) bs .0n

1ß3

k '-{

'{*

(u2

ttn '

3 .2.4

3.2.5

3 .2.6a

3 .2.7

+ & e kI

where Eo(x,y) is the input described in Section 2.2 and

Yz

K3=

Y2

3ì.I

Lez+3 .2.6b

with n.{t![) ] > O so Ehar rhe radiation condition is satisfied. It

inrnediately follows that f*{tlf) } > O so that the waves with coefficient

fo, whích travel- back up the channel, have a finite amplitude as x + -æ'

The input ldave must also satisfy 3.2"1and 3'2'3, so thaÈ

( o)

6o (x,y) = doêik x; K¡ Y

. (r)K3 {r:- , ur/Ð'f

h@

Ie

k : o'

,.

{"."&'ry - "+)

( [)k3

b.-s1y-Tt+ e

Each component in 3.2.7 decays exPonentially as x increases. If the

rate of decay of the zetoth mode is less than that of the higher modes

and the channel is sufficienEly long, the contribution to 4o(x,y) from

the higher modes (1, > 0) can be neglected relative Ëo the contríbution

from the ltave correspondíng to [ = 0. Hence, the inpuÈ wave is aPProx-

ímated as

Ço(x,y) = "tnlo)*'K'" 3.2.8

13

and Lhe surface elevation in the channel region may be written

ik(o) ( o)

6r(x,y) = e-ik x+K5y

3x- Kry*

-ik& e

Ao e

æ ( [)

{co +.Q,nv

êJ"b.([) b .Q,nvìK3 I; "t"-E.-i+ I

l= t¡X 3 .2.9

3. 2. 10

3.3.1

3.3.2

3.3.3

3.3.4

The coefficient 0,s is omitted in 3.2.8 and 3.2.9 since the system is

linear and os serves only as a scaling factor. Relative amplitudes

and phases at different locations may be obtained by using 3.2.8 as

input, and the actual amplitude or phase at any location may be found

if the results are scaled according to some reference point.

Using the relations 2.3.4, the velocities are found to be

ua(x,y) = ,=19, - [- n!o) "'*Ío)x'K¡Y* aot(ro)

"-t ulo)*+K'Y

t¡(1+i0g) [ *'

-ik& e

noe-ik( [)

*[ Kz

Yzer*xl6r=0,

where

62 {(r+i4 t)2-02}Bhr ( r+iOt ¡

must be solved, subject to the condiÈions

1o'.1n,0, "o"ff .e # É "i .Q.nvno¿)l

(tJ']"'"\'

æF+L

2=t

vg(x,y)x 3.2.tL+

3.3 Solution for the Resonator Reeion

To determine the surface elevation in the resonator, the equation

2

ur (-a ry)

vr (xr-d)

=u¡(Qry)=0, -d<y<0

= 0, -a < x < 0.

T4

The separation of variables technique is used once again to obtain

a solution which satisfies 3.3.3, resulting in the expression

E, (x,y) = Bo {cosh Krx cos po(y+d) + i sinh Ktx sin Po(y+d)

. oÏ,

uo{"."T cos p'(y+a) . 1,.fu h o,. l¡rxs1n- sln

a Pq (Y+d)

3.3.5

3.3.6

3.3 .7 a

3.3.7b

3.3.8

)

¡Co cosh K1x sin ps(y+d) - i sinh K1x cos po(y+d){

.ico[= r

I L¡rx i0 a

fcos- sin po(y+¿) - I1.TõJ G pl tp (y+d). [nxs l-n- cos

a

. lnxs1n-a

where K ^( u2 | )v"= ut--lehr (1+i0r)J

fficl+io,))

{'t -(Ð'}', Le z+

Po

Pp

The condition 3.3.4, which cannot be satisfied exactly, is treated, using

the Galerkin process, in Section 3.5.

The velocities may be found from 2.3.4 and are

a(-trnl

oo

i *î . (Ð'){* cos pr(y+d)Q=t

* Ct sin pO (y+d)

vr(x,y) =ffil - nopo{sinh K1x cos p'(y+d) + i cosh K1x sin no(r+a)}

*; * "i.r [r* cos pt (v+d) + ' Pt "o"'l'rr sin Po (v.d)]

%{iQ=t

+ Copo{- sint K¡x sin po(y+d) + i cosh K¡x cos noly*a)}

- oi,

.r{Ë fi sin [r* "i' eo(r+d) - ' pQ "o"'tr* cos pr(y+d)

3. 3.9

15

3.4 Solution for the Junction Region

The surface elevation in the junction must satisfy the equation

= 0,

where

v"Ç" * x|e, 3.4.1

3.4.2

3 .4.3

3.4.4

3 .4.7

3.4.8

,,2 = 62 {(r+iOz)2-o2iL2 Et'z ( 1+i0z )

and the conditions

From 3.4.5 ít is found thaË

-æ

= 0, -a < x < 0

= o, o < Y < b

vz(x,b)

uz(0,y)

The solution to 3.4.1, which also satisfies 3.4.3¡ mâY be written as

r3.4.5

1r(x,y) ß(À¡"{)t*,À0 sinh s(y-b) + s(1+i0z)cosh s(y-b)}aÀ

where À2 - s, = X| 3.4.6

and ß(À) is an unknown complex function of À

Only a discrete seÈ of À-values satisfies the matching condition

(vii) given in Section 3.1,

hrvr(x,0) = hzvz(xrO), -a < x < 0.

o frt",0) + i(l+iQrlfff",ol

r-I ß(À).{À* {x2ez - "r(1+iQ2)2} "int sb dÀ

so that 3.4.7 may be wriÈten as

T6

.4

=o

in which

Ilence

J-u, ^, "

I

-iÀx II

l.Ixzoz - "2{t+íqr¡' sinh sb dÀ

u2atrrt G cos Pl d sin T * t no sin pou "'" T)

1

[- urnr{"inh K1x cos ped + i cosh r1x sin psd}

oo

ie

Br 0{1

+ Copo - sinh Krx sin Pod + i cosh K1x cos P¡dII

.r{t *fi sin pqd st" T - 1 Pg cos Pp- .Q,nxdcos-

a3 .4.9

3 .4. 10

3,4"11

L

{u'u' - "'(1+iQ2)'z}"i't, sb du dx+ I" .'iÀx r

-a -oo

rCI iÀxe

2tr

ß(u)e-lttx

gs cosh K¡x + f s sinh K1x

+e Il'lTX) e cos-+Þ"fte I

N'ITXt sln -

æ

n= I

oo

n= I )u"n a

where

go i lo [- no sin pod + Co cos Pod]

fo = - po[Bo cos pod + Co sin p

gn = i Pn [- B' sin P,rd + Cn cos

a tiLn nn gh1 [B' cos pnd + cf

{3t0

d1o-+

P.d], ¡ e Z'

,, sin pnd] , ,, € z*0

To evaluate the left-hand-side of the equaliÈy sign in 3.4.10,

the definition of vz(x,O) is extended to Èhe infinite domain by

, X(-â2(xr0), -â { x < 0

, x)0vz(x,0)

t7

The definition for x ( -a is essentially an analytic continuaÈion

into region 3. The extension for x > 0 is quite arbitrary, but, as

long as the results are not used outside the original domain of definition

of vz(x,y), this device causes no loss ín generality'

Thus, equation 3.4.10 reduces to

ß(À) í{x202 - "2(1+iQ2)2}"intr "b

* Fret- iÀ * .-iÀn {iÀ cosh K¡a + K1 sinh rra}l

#[xr - "-^' {iÀ sinh K1a + Kl cosh K1a}]+

2 ¡ÍLT(d

æ

i grr [1 - (-l)""-i1\a ,íÀ

n

e

n= I

æ

I (-1)"

À

iÀx À +iK fK

+ 3.4.12

The expression for ß(À) which is obtained from 3.4.I2 may be substituted

into 3 .4.5 to yield the new expression for Çz(x,y) which now satisfies

3.4.7, the condition which ensures the continuity of volume flux from

region 1 to regio¡ 2:-

i t nn/?. , tl _ (_l)n"-iÀ" l,,=", " À'-(a!!)-

a

LI

Çz(x"y) I À0( À) dÀ

I

r

I

+ .-iÀ(x+a) {go []. cosh Kra - it<, sinh Kla]

- f o [À sính K1a - iKr cosh K1a] ]ffiulI

[- --,^* {le'+iTr"}

o(À) dÀJ " )r, 1*rl

\v\^,' u/\

-æA

J-+

n= I

-tÀ( x+a)e

a

3.4.L3

À0 sinh s( -b

18

+s ( 1+i )cosh s( -b)

where

r¿ith

where

0(À) = 1+i0,

b

sinh sb

+ Eg

l=tæ

+ I Gg

l=t

ebIt-Io,I ln

3.4.14

3 .4.L6a

cos

iCI so(1+iôz)

a

The detaí1s of the contour integration used to evaluate these integrals

are given in Appendix 3, the result being

Çr(x,y) = Eo "inlo)*'K'(v-b) * Go "{nlo)**K'(v'b)

oo ( r)w_ À r.!e) "i" rylJl,r' bl

ie

k ,XI cos

( [) tlry.-tk ,X &!rb b

. ( Q)K2 Sr-n+e

+ oofcosh K,x cosh so(l-b) . ;#øsinhK,x sinh sr{v-u)]

î ^ t [r*. "o"h

so(y-b) - i0 .^'tr -..- + sinh so(y-b)]* ,=1,

Do [cos

s cosh so (v-b) ;{ "1"

+ ro[sinh K,x cosh so(v-b) .;ffi; cosh Krx sinh sr{v-u)]

. oi,

ro["ir, [r* "o"¡ so(v-b) . G# in

"o" [r*

"i"¡ "o(r-u)] '

3 .4. 15

(oo,ro) (Bo,fo)

(o0,tr'o) = -

tozr!+sfi ( 1+iQr) 2l sính sob

ifi (1+iQ2)sO(go,fo), Le z+ , 3.4.16b

rr$l' - "'o(1+i4r)2lsintr sot

T9

lu'IXzso

-*;\,

- (Ð'r,

2

+

. ( o)kz ffirr+io'z))

sg

. ( r)K2

r/-

Lez,

9"e2,= {*i+

K2

and the fn and gn are as defined in 3'4'LL'

The velocities, which are obtained using the relations 2.3.4,

,t2 ( x, y) ,db {- uour,o', *!o) x- K,( v- b)

e

l.'Í

0 r¡2 l.tx I+

- - cos

- sinh sn 1y-b) I

rlgltzaLl

3.4.17

are

3.4. 18

EI

+ cs k{zo) "- nlo) *+K'( v'b)

.,*Ír).[- u!0, "o, {r * onng * "t" T]æ

¡æ

I

e

I

+

+ Gr "-' *Í o'.

[u(e)2 cos +.'#É"'"ry]

sinh K1x cosh so(y-b) . * É cosh K1x sính s't'-O)]+Do iKr

oo

+

l=t

a Fo

æ

I Dr. 9'tr l,'t¡xs].n

-AAcosh sO (y-b)

iKl cosh K¡x cosh se(v-b) . * # sinh K1x sinh s'f'-o)]

,[, T "o" @ cosh so(y-b) . ** "ir,$ sinh sort-o)l]

+ FI9.= t

vz(x,fi = rffr6)

I (Ð']"'" Urb

ikEre .* [^;

1o',. oI.e,nL

2

< rl

20

2

+

e. I

Gg

@\"+L

Q=t

-; . (+)']"'"

[*î . (nt" )']"t"n r<¡x sinh se(v-b)

-ik ¡rve b

rl* lo "o I

*' . (uf," )

2

(

.oï,"i[(-tcosh K1x sinh so(Y-b)

2

) (T)']"'" T sinh so(v-b)o

* ,o*

q)

Ie

1

) (Ð']"t" T sinh socr-o))( o)k 2+ Fg

"Q 3.4.19

llilliams (Lg72) has omitte<l the wave forms associated with the

coefficients Fr since their Presence is not due to the inclusion

of friction, nor to Èhe absence of any condition on 0 (I^lilliams assumes

that0maybeconsideredsmallenoughtoneglecÈ'whenconvenient'

terms of order 02), their omission apPears to be inconsistent with his

earlier analYsis using 0 = 0'

tchins Conditions and the Galerkin Te chnioue

1

3 5 The Remaini ne Ma

Not all the conditions

by the exPressions found for

unknown comPlex coefficients '

ate

(1)

(2)

continuit.y of elevation at x = -a' 0 d y < b

that is, Çz(-a,Y) = 6g(-a,Y), o ( Y < b

continuity of volume flux at x = -a' 0 ( y < b

that ís, in2u2(-arY) = h3ua(-ary), 0 { y < b

continuity of elevation at y = 0, -ê ( x < 0

that is, 6r(x,o) = 4z(x,0), -a < x < 0

listed in SecÈion 3.1 have been satisfied

Çt,Çz and Çs which still involve, as yet '

The conditions which remain to be satisfied

3.5.1

3.5.2

(3)3. 5.3

(4)

(5)

2l

zero normal velocity at the boundary

that is, u2(0,y) = 0' 0 < y < b

zero normal velocity at the boundary

that is, Vl (xr-d) = 0, -a < x < 0'

x=0

y=-d

3.5.4

3.5.5

Since the expressions for l-he elevaËions (3.2.9,3.3.5,3.4.15)

and velocities (3.2.10, 3.2.IL, 3.3.8, 3.3.9, 3.4.18, 3.4.19) involve

infinite sums, no explicit expression or value can be found for each

coefficient, and it is obvious that no finite combination of terms will

satisfy the above conditions exactly. Hence, some approximation technique

must be used. Techniques widely used in such circumstances belong to the

Method of !üeighted Residuals (Finlayson, L972), from r¿hich class, the

most cormnonly used are probably Collocation and the Galerkin method'

These, with particular emphasis on the latter ' are discussed in Appendíx 2'

The Galerkin technique is used here'

Each of the series in the expressions for elevations or velocitíes

is truncated after l, = N; and, for each of the five conditions above'

1¡+1) weighting functions are used" The resulting equations, together

with the appropriate 2(N+1) equations from 3.4.16, yield a system of

71¡+1) simulÈaneous linear equations in the 7(t't+t) unknown coefficients

s, Bl , Cg., Dl , EQ, F[, Gl , [ = 0'1'""N'

The five conditions are no\'/ treated in turn'

22

(1) Continuity of elevation at x = -ê, 0<y<b

Substirution of 3.2.9 and 3.4.15 into equation 3.5.1 yields

( [)æ

Eo ea-Kr(v-b) + [ Er e

Q= r

+Goe Gr

Arb

. lnvls r-n b4l

.tk ( o)

Ar

ß1

+ ße sln tTtf_bcos

ioK r(t+i4r¡"0

Fi0

r ( 1+i0z

( o)ik a+K3 y

b+ p1 sln

ge0b= Gîot l,rr

9g0b

ÎT+ïõ--J [1T

trG",y)dy =

-ik ,ã cos

k( 1)

,*:o)a+Kr(v-b) te &yiI+ "à

+Do cosh K¡a cosh so(y-b)

- sinh K1a cosh ss(y-b)

IDr (-1) cosh so (y-b)

sinh Kla sinh so(y-b)

sinh sO (y-b)

b

2t

+ Fo

k( r)

i0K,. n#O}.; cosh K1a sinh so(Y-b)

I[=r

+.i[=t

[n (-r)""I

9

+ Aoe

æ

¡I+

ie ao Ut W 3.5.6

3.5.7 a

3.5.7b

3.5.9

cos b

where

( r) +k Lez

( r) +!"ezk3

The Galerkin equations are produced using

ro

u) (-a,y)dy, R = 0,1,...,N, 3. 5.8Ëm ?

where the u) are chosen to bem

t^)o

K_y=e,

mfiy + p sinm

mlTy ,m)0W=cosm bb

and the ËrGary), tr(-ary) are the truncated-series forms of the

expressions for Çr(-a,y), 6r(-a,l) given in 3"5'6 (this notational

convention is carríed Èhroughout without further explanation) '

23

1 tk

The resulting equations are

( 0)a*t

o -"*'o

1+Go

Ks+Kz

Ka+ atlLn

b

K_b -K^b-e

k t8( 0)

r

aa e

( -1)

E o K3-K2e

- -tk*)Ete 1

x2*(4)'3Þ

- sinh K1a t, ]

'-1]

(Note thaÈ, if hg = hz, the term with coefficient Es is Esb e

= Kr("K'b - cosh sob) - so sinh sob ,

""'o -r]a

2

([)

( r)- ik+)Gte

(o)=be -ik a

where

and

[*, - ß, f][t-'r',r""'o -r]

'.-Frf. o,

** ["o"r, K," r,o

30

. t, ;ft [ - "l"n Kra rroiOK '. Tffi cosh Kta t,]

+ I Dr(-r)r _r*, T,.n * I Fq dJ #( -1)r*1-"i

Trn

*Ao2Kt

.t nl o) "

[", ",

o -r]1

[*. - oo {][r-'ro

= K, sinh sob - "ß("*'b

([)"*'

- ik*)Alex'*(Æ)'3D

a 1 3.5.10

a+K.b,

3

( 0)

T ,9.

T r9.

-ik

1, = 0r1r... rN

- cosh s'b), [ = 0r1r...,N,

( 0)Kz+P

24

"*,0 - (-t)'Eo e-ik

+Goe

* u,' å "-'*Í''"[, - o-ß-] * I' Er "-t*Ío'"[o,.r0.. -

* I, cc .' nÍ r) " [o-îo-

*am)

k+c l.nt¿

+ cosh K¡a t* ]

* o^r."'*l*) " [r.oå]

* I' Ar "'*10'" [o*î0," *

sQ sinh soo * Pm S ("ottt

a2

1 mT

bx'*(9)'zÞ. - ( o)lR e

21

r'*(9)'ZD

m

[- *, . o- i][e-K'b - (-1)'"]

ßr rnlI

"tf1+pßmm ßr I

iOK '+ ,-:i;j:1þ sinh K1a(l+rQzJso

( -1)r

"r'*(Tl'r -T' io Î'rr, tg. L ,9. ( 1+i0z ) "rl

Tqg+ DgI - 1)e

2 mTT 2

1

sbe.

(+ )

(o)

[", . 0," i][t - ,-r)^ .-*'o

]-ik=e 3

+

where

rt*(S) t3Þ

L

Ao r'*(4r)'3b

[-, - o- i][,-1)' .K" - t]k ,e( o) I

means

e

"Qb- (-1)'),L= 0,1,...,N

Toç. = ,p(cosh sob - (-1)-) * 0,,,,, S sinh sob, [ = 0r1r...rN

Pp I ,r = 1r...,N 3.5.11ntf

TrL

IN

II

f'*"ans

[(-1)r'--1]

N

I9=tlf-

m^bl^Xnr TT

and Fç Ilm 3.5.t2

25

(2) ContinuitY of volume flux at x = -a, 0 { y < b

Using the expressions 3.2'10 and 3'4'l-8, equation 3'5'2 may be

written as

- eo t!o) "-' nl o)

'' *' ( v- t) + c¡ tlo) ei kl o) a+Kr(v-b)

Ir

e.icrQ= t

eEl([)

-ik a/[)2

t costrJ wb

vr

Ut

s1n1

b

b

( [) trrbr.(

r) ["2Lk A

2

* oo[- i*, sinh Kla cosh so(v-b). *# cosh K1a sinh 'o{v-u)]

iK¡ cosh K1a cosh ss(Y-b) - sinh Kla sinh ss(Y-b)

os

e

Il,oo = ê

K.Y

lil = cos

+ vQ s1n

0 t¡2so ghz

. ni,

t, å $ c-tlo sinh so (v-b) * rl

FQ i &I (-r)l cosh so (y-b)

+Fo

hghz

where

wíth

ffi{-n',"( 0)

-ik K3v ... ¡ ¡(o)03

t( ß) ["o"3[

(o)lk a*Ky3

9"eZ +

a â' 3e

e.iAe9.= t

(r)k a &v

b+1l0 sin Uv

b3. 5. 13

3.5.14a

3.5.14b

3. 5. 15

)

The Galerkin equations, resulting from

tJ2 0 b o=otvr =ñ .Fîñ;

tu- a

ts2 e buq =;ñ;;( ç '

Itú hz iz(-a,Y)dY =m

t) hs üs(-a,Y)dYm

0o

mTyb

ate

m+ 1rn, sin ry , m = 1,...,N, 3.5.16

26

( o)-ik a

21

Ks-KzKbe't.

L

K b II- no k!o) e

+ co kto) ea1

K3+K2

e

(0)IJ

-ik( l)

(l)r ik+IGte 2

K-b -K-be' -e

[*, . "- f]fr-rlo "*,0 -r]

[*,-"of][{-r)r.",o-.]

tk

Er-I e. ( [)R2

1

*3*(b

)'

, r k(e) I

. o, ;fu [- ir, sinh Kra ï,0 . * É cosh rra Tro

]30

*ro tK2-sz30

- o 9'sinhK,ar Iso ghz ' 20 J

rli Kr cosh K1a TL-ro

*?*t Lt¡\,b

(r)- ik+TfuE . ( r)

K3

(o)ik a 2Kb[e t -t]

+lDr+## r,p +lFr r+ rl

1F"3

l, !1.19rì J - r(rol b e-inÍo)n + ¿ ¡(o)hz (1+i0s) I ¡!r o 3

3

.à 1

[*, - " f][(-1)r

"*.0 t])x'*(F)'3Þ

(Note that if h3 = l:2, the term wiEh coefficíent

where

Ku(eK'b - "o"h sob) - so sinh s'b, l, = 0,1,"',N

- ( o)Ele rs -g þlt20

3.5.t7

-t nl o) a + K, b

e ),

T

TrQ

P

K, sinh sob - "l ("*ob - cosh spb), [ = 0r1,. " ,N

and, defining

Trp = "l sinh soo * um S(cosrt tgb - (-1)-)'

Toç" = sg(cosh "tb - (-1)-) * U- + sinh s'b,

.0 = 0r1, . .. ,N

- Eo k!o) e( 0)

-ik ,e 1

27

[*, . ,,", l]["",o - (-1)-]

-Kz+UI oì

+ co t!o) "tuì 'a

IIIT.2b,l

fmfib

K2+ (2

u!o'[u,"îo- --ieL

ET

]["-.,0 - (-t,*]MTT

bm

]) ,- = 1,...,N. 3.s.18* I, Ar "t

*1 r)

" n!e) [u^îo-

*

1

x2+(2

Þ2

- E I "-tnl'', r.!'r[r - v u l*.nt¿Lmnìl

'oî*]['-îo- *

n!'' I

^

tn)

* o ?'coshKraT IsoBhz ' 40l

eT

3Tor*lFrDrI

n!*'[t . oi]

,o(et 1+vunlmm

([)k a

7 IVQ¡nl

. . < ltGO et*t aI <rt+ k 2

- o, ¿fu;u [i*, "i.,n Kla r30

+Fo iK1 cosh Kla T¡oe b)2

so thzsinh Kra T- ¿1{)

12

"å*(T)

o r¡2 (-t)ls[ ghz

"rr*(T)'

Þ3h2 ffi{-n!"'.-'nlo)u r'*(S)'

3Þ

1

[*, . u- i][t - ,-r)'" "-*,0 ]

I o)+ lo t!o)

"t uì 'a 1

[-, - ,- i][{-r)'" .*,0 - t]*:.(i)

* o", å .' *!')

"

IuI ,t9.

(3) Continuity of Elevation at y = 0, -â ( x < 0.

If expressions 3.3.5 and 3.4.I5 are to be consistent with equation

3.5.3, the coefficienÈs must satisfy the equation

28

Bs cos psd cosh K1x + ys sinh K¡x

Cs i cos psd sinh K1x + \[s cosh K¡x

@

I@

T Cf sin pOd

- f .[nx l,nxIcos ptdlcos T * Yt "tt " I+ B1

+9.= t

(o)Eoe

ik x*K, b +

"Y[ g,rx-.-_s-rrt-tan¿pÎ d a

-¡k(r)

19.

.1.nxcos - a

2

æ ..([)I no "'*'

x

2=t

-tk( 0)

+Goe *-Iqb + [2 Gr lxe9=t

+ Do cosh sob cosh Klx + ¡o sinh K¡x

D[ cosh s'b 1,nxcos:+a

. .0nxsln -

d

+ Fo cosh sob sinh Krx + ì'l o cosh K¡x

+

+

io

n119.

I eF cosh s b

. l,nxs1n--2 a ne

Yo=itanPod

i0at9 (1+i0r) l't¡ *P

.Cnxcos -

a

Pld, Lez

3.5.19

3,5.20

3.5.2L

2=t

with

tan+

+

-iOK'no = ffi; tanh s6b

i0 9'¡t = ï1.ïõr-) "fotanh s'b, LeZ

If the weighting functions are chosen to be

fi)q = cosh K1x + 1o sinh Krx

m'lTx mlfxw = cos 5 ç ¡ sin -,

m = 1r...rN,mg'ma'

3.5.22

29

then the Galerkin equations, found from

Iw^ Èt(x,0)dx = *rnËr(x,o)dx, fl = 0,1,...,N,

ate

r

Bo

* I cr sin pod

= Eo ê

+Goe

+ Focosh S^b

4Kr

cosh se*lDn

(1 - yone)2K1a + (1 + Yorìo)sinh 2K1a

+ (yo+no)(r - cosh 2Kra)]

- co i frp It.ro-no)2t<,a + (Yo+no)sinh 2K¡a

+ (l+yorìo)(r - cosh 2Kra)]

* I Br cos pQ ufu, [,*, - ,, E no)Trr + (YoKr - vo flr.o]Ia

*"*(E)'LttaY[ T I

Ltt

I )

+ (YoK, . # vo fl r.o]

r,t 1 [r¡orr - it!o) )fro + (Kr - ikloh, ) t, ](klo) )t*r', L'

-"'o ffi1 [rnor'

* it!o) )T'o + (Kr + it!o)n"tJ

+ I Er &[tnor,

- ít.12) )T'r * (K, - itfrerno)Tro]

* I cr aft [rnrx, * it!e))T.o + (Kr * it!r)no)ïoo]

{ r-nfi)2t<ra + (1+n;)sinh ZKra + 2no(1 - cosh 2 K¡a)

2no sinh 2K1a + (r+¡f ) (1 - cosh zrra) ]

*?*(E)' [,*, - nono f)rsr * (rìoKr - no flr.o]b

- cosh sOb

- I re **øþ [,"!,t¡

aa

ta

+ nqKr)fro +( + ngnoKr)Tot 3.5.23

where

=Do

t¿=1-e

30

( Q)

' I cosh K¡a

sinh K1a

-tk( Q)

-ik

T

Trg

Trp

ToP

"8=e

5T

To

2

I

. . ( [)lk e

= 1 - e- --t - cosh K1a

. . ( g)

= et *' " sinh K1a, .Q, = 0r1r... rN

0

= (-1)^ sinh K1a

0

=f -(-1)^ coshKla, L= 1,...,N;and

and

Bo [,*, - yor- Tlrr", * (K'Yo - n", f,lru*]

- i co äËþ [,*,ro

- n- T)ïs- + (Kr - Yor- i,t.-]la

* u-; "o" n*afr * v.n-] * I' Br cos Pru[n-'¡", * rot*]

* r^Tsin n-a[r #] * i' .o sin pouln,.ro," - ft t*1

[,*, - n'n,,, T)Ts. + (Krn' - r,,, T".',]co sh snb

+Fo

+ Eo e

[,*rno - n," f)ts- + (Kr - ror. Trt.-]

* o^L cosr, s^u[t.nå1 * I' Dt cosh soo[n-'o- * ntt"n]

* I' Fl cosh sob r,r* - lll-r[- l- it{ror ]F (-1)'" '*

(r<!ol )'-(T)

(o)

Krb 1 mfT

a1-

ï - tn!o'][t

- (-1)- "-ik

12

z

1([)2

a*lEr

+Goe -K, b

(klr) ,'-,7)'nm

1 ["T. 'nI'][1 - (-1)' .t nl o'"

]

<[t

(t!or l'-cffl'

I[" i * itf2e) ][t -

(-1)'" "'*,8*Icr

(E j r'-rryr', ú = 1r...rN 3.5.24

31

where

r^¡i th

, Lf m.

(4) Zero Normal VelociËy at the Boundary x = 0, 0 ( y < b.

Expression 3.4.18, substituted into equation 3.5.4, yields

rnil = +# [(-1)r'--11

&!r

3.5.25

3.5.26

3.5.27

i ro * $ "i"n so(y-b) + i FQoQ cosh so(y-b)=09. e

æ

- ¡o k!o) "-K'

( v'b) I Et( [) tTrJ-k 2 cos

bvl sin

bQ=t

+Gok(o)2

K^(v-b)e' + Gr k!r)

["o" T - "o "i" S] = o,

with vl as defined in 3.5.L4a and

0s = iK¡

,* =íÆ,9'ez*

Application of 3.5.26 to

æ

¡e

rl,l)

0

*^ir(Ory)dy = 0¡ fl = 0,1,...,N,

W=cos mTIy +V sin mfiy, [ = 1r... rN,m m

1_

0

bb

yields the Galerkin equations, which are

lU ,, - J*'b cosh sgu) - rrJK'b "i't sgu]

[U "-! b sinh sob - K z{t - e- K'b costr sou}]

NIhll= o

N+[

I =o

Ouz 1

"r gn " x7-";

1

* rot(ro)b - co u!o' u*{t - "-'*'o] - I uo u!r)

Fr oþ

*7-"i

I

t.-K9,¡r

b It"b

TT- (-1)

Ér*( )b

[*' - uo ][.-"'o-(-r)o]=o'+ [ cp r<f;)rr*(4)"zo

9"n

b

2 2* vQ

3.5.28

L

32

and

oï. * +É

sQ sinh sO b+vm ff (costr

"tb - <-rl'l]

["q ("o"r,

"Qb - (-1)-) * u- + "int sou]

FrN

I=oe

+ %

- Eo k!o)

uL

1

*1*(Y)"Kr+V-m

mlT

b

MTb

]l"o'- (-r)-] - '- t old I

m]1"*'o- c-rl'] * .- å o!-'l

r!ß) [u*îo-

* unî*] = o,

1-vzm

o)1 1+v2-Go

+

xï*

EQ n!*'[

mïl 2)

Kz-vm(

b

I v îo- - "oî*] + i f G[m

m = 1r...rN. 3.5,29

33

(5) Zero Normal VelocitY at the Boundary, y = -d, -a < x < 0.

Use of 3.3.9 in the equation 3.5.5 gives the relationship

- Bopo sinh K¡x + i CoPo cosh K1x

- oÏ,

Br '*+. ßnx

s]-n -

fa

MTTX

_=CoSlr

cr iPQhx

COS

- = U.

a

m = 1r"'rN

æ

TL 3.5.30

3.5.31

3.5.32

3.5 .33

e

The Galerkin equations are produced from the integration

t"A) i1(x.-d)dx = Q, m = 0,1,.'.,N,I m

where the weighting functions are chosen to be

tOg = cosh K¡x

w

The resulting equations are

-Bs Po4Kr [r-"o"t2Kr"] -iBtt*

)'I

a

+ iCo sinh 2K1a + ZKta + IcrPo4Kr

2(-1) cosh K1a -- 1

e(-1) sinh K1a = 0;

. ,N,

*l*c

ipQKr

t<2* (&)'1A

and

u, ffi [,-t,'cosh K¡' - r]- I'Br t *# t*

. arffi (-r)'sinh K¡a * c- iP- 9r= o, R = 1,'

where I is as defined in 3.5.25.ntl

34

(6) Continuity of Volume Flux at y = 0, -å ( x < 0.

For completion, the equations 3.4.L6, which ensure Èhat lhe condition

hrvl(x,0) = hzvz(x,O), -a { x < 0

is satisfied exactly, are Presented again here:

a s ( t+i+ 1+ þz c s intr s ob

[go sin pod - Cs cos psd] 3 '5.34)Do

Dl

FQ

0

CI pt st ( 1+iQ2 )

= - trsl'-( 1+i0z )zrrt)sinrrsob

a

lut sin pgd - Ct cos POd],

!' = 1-r, .. ,N 3"5,35

.r0 - -CIi s (t+

0 K 1+i02 S[Bs cos pod + C¡ sin ped] 3.5.36

0sirùr s b

0

uJ2 a

t ghr l'n lBt cos

9,= I,...rN

p1d + C[ sin POdJ,

3.5 .37

where

CI =Þltz[(t+ior)2- o2]

(1+i0r)

The linear simultaneous equations described above may be solved

for the unknowns &, Bg, C[, Dl, El, Fl, Gt, (1' = 0,1,"',N) by

inverting Ehe (ZtI+7)x(Ztl+Z) complex matrix whose elements are defined

by the equations derived in this section. The convergence of the method

is tesÈed by checking that the residuals of each equation become smaller

as N is increased, as shown in Chapter 6'

35

CHAPTER 4

AN EXTENSION TO THE MODEL: AN ADJOINING CHANNEL

4.L The EquaÈions

The model in the previous chapter can be extended to include the

presence of tidal forcíng in a second semi-infinite channel, adjoining

the juncÈion region and occupying the area x > 0, w2 ( Y < ws ' I'üilliarns

(Ig72) (and, subsequently, Buchwald and !üi1liarns (1975)) has considered

the case ,r= o, *, = b in his earlier analysis which neglected coriolis

as well as frictíon. Here, the two semi-infinite channels, not necessarily

the same width, could be as depicted in Figure 4.Ia, if ", * b' or in

Figure 4.1b, if tr, b, with both cases allowing for 0 < wt < b' The

analysis which follows is carried out for the situation in Figure 4'la,

but that for Ehe case of Figure 4.lb does not differ substantially from

the presentatíon below.

The area is now divided into four regíons with a depth discontinuity

at the junction of two adjacent regions, indicated by broken lines in the

figure. Once again, the solution is sought to the equations

v2 + \'et = o, i 4.1.1

where

Çt I 432

The solution must satisfy a radiation condition in each of the

semi-infinite channels, as well as the following bc'undary conditions:

urL =,:l

36

b Region 3 Region 2

Region 1

b Region 3 Region 2

I 4 r^rl

lReg on

Region4

w2

I^¡ 3

t(a)

(b)

ItI ¡

I"¡ 3

t{2

r

Region 1

The rectangular resonat.or-channel system, with

two connecÈing channels. The case of ws ( b

is shown in (a), while we Þ b is depicËed in (b).

Figure 4.1

37

ur(o,y) = ur(-a,y) = o, -d < y < o

vz(x,b)=0, -a<x<0

v:(xrO) = v3(xrb) = 0, x ( -a

vq(x,wz) = v,*(xrws) = 0, x Þ 0

v1(x,-d)=0, -a<x<0

and matching conditions

ÇzGa,y)

h2u2(-a,y)

h2u2 (0 ,y)

r,z(0 ,y)

ç 1 (x,0)

hrvl (x,0)

= 6¡(-a,y), 0 < Y < b

= h3us(-a,y), 0 < Y < b

0, ws<Y<b

haua(O,y), wz < y < \¿s

0, 0<y<tz

= 64(0,y), w2 { Y < eI3

= r,2(xr0), -a < x < 0

= hzv2(xrQ), -a < x < 0.

The u. (x,y) and v. (x,y) may be found from the relations 2.3.4

4.2 The Solutions

The expressions for 4, (x,y), u, (x,y), tj (x,y), j = 1,2'3 ' are

exactly the same as those containing the unknown coefficients, found in

Chapter 3. The solution in region 4 is found analogously to 6g(x,y)

in Section 3 "2, so Èhat

-ik(o) *10)x-rçyq,*(x,y) = óù e

' '*+K¡Y n Âo "t

æË^t+ ) 4el=r

ui'&(y-'r)] ,

4.2.r

uÍr)*["." #,r-wz) - 1çft;# nlo'

38

where

and

u+(xry)

and

K4

( o)k

4

- ( r)k4

v/t=vI3-vl2

The velocities maY be written as

k

ik ( e)vrr IlnLK

ilfu {t klo) "-'ulo)**Ko' - Âo tlo)

(#l)

(#)

r/"

( 0)k x- K¿Y

4

4.2.2

4.2.3

4.2.4

4.2.5

4.2.6

4 .'¿ .1

= {*i+9"ez

e

ie*

o=l' 4

oo

¡I

Ío'. ¡-ulo) "o" fltr-",) + 0 # * "i,,

&(y-'r,]),

2 . .0nsln -hTI

(v-tr) .v,* (x ry) Ao

(i)

( ii)(11r1

( iv)

(v)

(vi)

ezG^,y) = 6s(-a,Y),0< Y< b

h2u2(-ar!) = h3ug(-ary), 0 < Y < b

6r(x,o) = c,z(xr0), -â ( x < 0

v1(xr-d) = 0, -a< x< 0

h¡v1(x,0) = h2v2(x,0), -a < x < 0

62(o,y) = e,r(o,Y), I,tz < Y < r{3

0, ws<Y<b

huuu(O,y), ¡¿z < y < rI¡

0, û < y < tz

¡x z4

e +

As was Èlie ¿ase in Chapter 3, some of the conditions listed in

section 4.1 still remain to be satisfied. These are

(vii) h2u2(0,y) = 4.2.8

39

As before, a Galerkin technique is used to find an approximaÈe

solution to satisfy Èhese equations. The algebraic equations resulting

from application of the Galerkin process to (i) - (v) are those found in

the previous chaptef, namelyr 3.5.10 and "11,3.5.17 and .18,3.5.23 and '24,

3.5.32 and .33 and 3.5.34, .35, .36, .37. Conditions (vi) and (vii) are

nohr considered.

Condition (vi). ContinuitY of Elevation at x = 0, I,{2(Y(1^7,

SubstiÈuÈion of the expressions 3.4.15 and 4.2.1 into equation

4.2.7 yields

æ

il=o

+ Eo e

The weighting

-K, ( v- b)+ Eg Ar-

b

Dt cosh sO (y-b) + t t, l1çþsinh so(v-b)

ßr rt"+]ræ

I cosL=t

æ

+ Gs eK'( v'b) + [l= r

&v- nyb

+ ßr s].nGl cosIL b

IJ

= ô "*ot + Âo e-K.Y + I

9=t(v-tz )

+9.e2,

chosen for the evaluation of

Ee "i" ff<v-'r) ]oo

4[ 9,ncos -\dl

where

0 wr -(l)II.Iõ;' ffi *u

b .([)ñxz

4.2.9

4.2.r0

and is as defined by 3.5.27.

ßr

IeÎ

0ß

functions which are

tL) irto,¡dy = f tÁ) iu{o,y)ay, n = 0,1,...,Nm

W,

lÐ =eo

K,Y

mlTv1+ mlTy

m

bb

ate

V,

1,0 = cos ßmsan , f, = 1r"'rN'

40

The Galerkin equation for m = 0 produced from this integration is

N N

I D[

+ Eo Idr e

+

11KzT tQ - ,eTzp. I F

2KcT-2 I s

eTrl

9"= lr...rN

+.Q=o

U

1+iS2K

%

"rIl=o

Krb

*i-"; s2

2

[{*,. *o f}r,o . {f -ßo*,}r.o]

e

1-K*"of* " "o

["'*' *'

- "'*"n^

I IEr

9.r

+ G9

+ A

t<2+(2

e

b)'

= ôì ñä ["t"'*K')w' - "(*o**'t*']

[{'., - q f}',,. {f . oo *,}r*o ]

"(*r- Ko)w, _

"( *r- Ko)w,

I I*7*(T)"

* I4 -fu [*,. t, #]ß-1¡r"K"' - "*"1, 4.2.r12 w!

(Note that if hz = h,*, the terrn with coefficient Âo is Âotr),

where

e "*'*' cosh s t

e

Í, = 0r1,,.. rN

T = a*, *t cosh so t,

2

T2

Trn

= "*r*, "irrh , t3

= a*, *,

"o,cos

l,nwg _bTol = "*'*' ,i'

K- w^e ' 'san

- "*r*, sinh sot,

ry "*,*,

I

9"nw2b

9.¡rwz

b

and l =rJ -b, i=2,3.

4l

The Galerkin equations for m = 1r...rN are

+ sgTD2QN

Ioo9=o '- Isoro,o

ß* "rïr.o ]Torl*MTT

b

ß-"0 Ïo, o ]

Tøoe - ß" Trort +MT

b+

["0 rorr

ßln

N

+

[=o

+ Eo e

+Goe

+Gm

1o /:l-

'î"(T)"i F

e

0

It+iqJ

Krb

-K, b

1

[- r*, * ß-T ){" K2

"o" ff - "-*, *,

"or ffÌ

ßrrrzÌ{eK'*' sin ff - "*'*'"ir, ryt]

r2*(T)2D

. {i - ß* Kr}{.-*'*'"ir, ry - "-*'*'"i" fff ]

* n"'[f r-ofr ]4..'3-wr¡ * ,frtr+oz]{sin + - sin ryr]

* I'En +#, [ru - *ßoß*]ï",0 - {'- [ßnßn,]T"rr * {'ßn, * ußr}T"rs

+ {[ß,,' * *ß[ ]T"ot

1

r'*(S)'2D[,*, - ß", T]{eK'*' "o" Sb - "*'*'"o. ffr

[tt.e;lZcÌ^r3-rd2) . rfttr-ofr]i"i., ry - sin *t

{"o"S-cosryt]

+{ MT

-+b

-ßm 2mlf

b

* I'Gs +i* [,u. mßoß',]T"rr - {' * t'ßnß^}Turp * {*ß,,, - ¿fu}Tu.r

+ {1.ß_ - *ßl }Tuop

42

*l*tïl' [,*- - 0", Tl{eKo*' "o" ff - "*o*, "o" fff

tff . ß","uÌ{"Ko*' "ir, ff - "*o*'"i' i"t]

1cL

+

*Ao

* I"4 1

+

l- **- * ß,. TÌ{e-Ko*' "o" ff - "-^o*' "o" ff}

{6r ß,,, TÌ{(-1)r "o" ff - cos ryr]9-r

-+I^7 I

4w [tt-g.r"]"o" Se o Erlsinry]

.mTtb ß-K,* Ì {"-*o *'

" ir, ry -

"-*o *'

"t" fff ]

r$l'-r&l'D rrrt[try - ß-6r #]{(-r)r "i.' ff - sin fft

+

{ß +m

AL ,0=1 ,N, 4.2.12

where

ro, I = cosh tr t,

TD4 r

. mfiId as1n Jr - cosh sl t2 s]-nmThlZ

b

mTWZ

bTo"g. = sinh sOt, "o" $b - sinh s't, cos

Trrg. = cosh sot, "o" ff - cosh sot, "o" Sg. IIllThl r III'lTht rs]-n -b"J - slnh spf2 sln --5: .Q, = 0r1r...rN= sinh sO t,

Tr.rP

TBrg -- sín

= s1n

= cos

= sin

lnw¡b

mTfüt g

b

Î,ftrsb

lnw¡b

cos

cos

cos

sin

Ll¡vzb

mfihtZb

l.,nwzb

Lr¡wzb

cos

cos

cos

sin

lll'lTlrl Z

b

[-t¡w zb

mTl"ü72

b

mll\dZb

mTlsl a

-5- - srn

.Q,tn¿.-1* - srn

IIIlIht ¡J¿ - cos

IfllTW a-5* - srn L = l, N

Tnr9.

TBoQ

L is such that mL\¡t I

N

I9=r2fl-

and l" denotes

b4.2.t3

43

Condition (vii). ContinuitY of Volume Flux at x = 0, i{2 { Y < vrs

Using 3.4.18 and 4.2.5 in equation 4.2.8 gives

î0o2.-_lo

oo _ fr sinh so(y-b) *

o=lo Fr op cosh so(y-b)

2

- Eo k!o) .-K'(v'b) I I-([)R2 cos

@

E

l=t

with v[ as defined in 3 .5.L4a

Urb

Ar-VQ s 111

\,t2<y<w3 t

b

+ co kto) "K'

( v'b) cr r.!r) ["." + W+ v[ sln

0, 0 < y < tz and r,.t3 < Y < b

h¡+ ( l+iôz) ít', Ît.lõ-¿ I ô kto)

"to " * Âo tÍ-o) .-*o t

. oi, 4 o1.o'["." #,r-wz) - Kr "i" &(v-"']]

oo

II+

b

and

0[

Kß

as defíned in 3.5.27

=e-Éî+t, ,t..2*

4.2.14

4.2.15

With the weighting funcËions

il,lo = €K:(y.b)

W =cos +v sin m]ïY , ß = 1r"'rN,mlTy

m m bb

the Galerkín equations, trhich are produced from

hzüz(o,y)dy = r,0* haür+(0ry)dy, f, = 0,1,...,N,fo

,-o

are

vz

44

oi.oo * * ù["0t, - "-*'b "osh

sob]- K2e-Kzb sinh ,oo]]

. oi,

FQ or ffi¡"re-K'b sinh sob - x'{t - "-*'b co"h "oo}]

* ¡o t!o)u - [ Er t!ß) ttJ [*'. uo f]['-.'" - (-1)ß]

- co k!o)ù[t - e-z*'o]* I.o u!o' "fu [-, - uo f]["-." - (-1)e]

= |; ffi e-K"b {-, n!" nft, ["t"'+K2)$'3 - "(*o**'r*' ]

* Âo r.!o) ¡fr; ["t "'' K¡ ) w¡ - .( *" *''*' ]

* ¡ Âo r!r) ry [*,. *o ff-rlc.K"t - "*t']'

hz = hq , Èhe term with coef f icient Áo is Âow,);

2.16

(Note that if

and

o¡, to

* ["o t"o"t stb - 1-t)'o] * u- T sint' soul

[*, * u., T][""'o - ,-r,"'] * u- å n!"0

[* no t!o)

]["-",0 - (-r)'] - ... ] u!'"' [t . ",î]

sinff-"-*o*,"t"ryÌ]

r.ù2 1

45

rn

gn'z o .fft'Il .2

"t'* (-Ë- I

N

IQ=o

0r

'l'*(T)'"r-

s inh s* f{"o"t, "tb - (-1)''i]IFr b+u

1 L-v2m2Ki.(Tl

* cs k!o) Kz-v

* I'ur n!o'I î0', - uoî*]v

mT

b

mlI- I' cs r!e)

[r,"î0"' * VQ

m

h',=J h2 rf.<ff> [t*- - u., T]{eKn*' "o" ff - "*o*' "o" ff}

t 1l L

k( o)

[r*,' . u- T

-{v

+ {i . v-Ka}teKo*' si" ff - t*o*' "t" ry}]

- Âo tlo)

-K4

Â, uÍ*"' zrrftr - v-KL] cos io. {v- * rc"} st" T"l

Ì{e "o"ff-.-*o*, "o"ffÌ1

*l*(T)'Ko

+ {vnKa - llt"

+

+ k l+ÂrT,, (ryf-ëfD f./f

lrt - v.Kr flitt-t>o sin ff - sinfft

ry-"."ry,]im

Lr Imïïb

+"Q

)i(-rl cosI^I I

is as defined ín 4,2.L3

is as defined in 3.5'I2'

m = 1r...rN, 4.2.17

where L

reand m

46

4.3 Determination of ô

The equations in the previous section contain the factor ô which

represenEs the amplitude and phase of the input wave in region 4 relative

to the input in region 3. Since, in collecting data, one cannot

distinguish between input and output hraves, the value of ô cannot be

specified but must be included as an unknown and determined as a solution

to the system as a whole. Thís can be achieved if data is available

in Èhe two regions.

The elevations at t\,üo positions (X3 ,Y3 ) and (Xa ,Y,*) , in regions

3 and 4 respecËively, are related by the equaEion

-i(Þ6s(xs,Y3) = ô e 6+(Xa,Y'*)

qrhere ô and Q may be determined from measured tidal data.

expressions 3.2,9 and 4.2.L, this equation may be replaced by

approximation

, *i o) *. 1 o'*.

+K3 Y3e i "K'Y' + Ao e-i kì

( r)

Us ing

the

*Ifo e-l k X

( 0)

f LnY.

lcos ;- + Pg

x4+K4Y4 * Âo "

"'"+]^ {kO¿E 4^-io0e J

I( o)

ik Xo-KoYo

(Y,*-wz) - Eg.

( r) . Lt¡sln -t¡ I

(Y.-")] Ì

.4.3. 1.lq eik x4 Lr

Iì7 I4

L

cos

This equarion, with the 8(N+1) equations 3.5.10, .11; 3.5.L7,

.18; 3.5.23, .24; 3.5.32, .33; 3.5.34 - .37i 4.2'LI,'L2; 4'2'16, 'L7

are solved for the 8N+9 unkno$tns ô, &, Bl, Cl, Dg, El, FQ, Gl, q,

!, = 0r1,... rN, bY using a complex matrix inversion routine. The

results are presented in Chapter 6.

47

CHAPTER 5

TT{O NUMERICAL MODELS

Analytic models have a number of importanE attributes which make

their development worthwhile. Although they simplify the features of

the region of study to enable a solution to be found, they can throw

a greaÈ deal of light on the important factors governing the fluid moËion

in the region. They also provide a guideline against which to comPare

numerical models. These latter must be developed for more realistic

quantitative analyses, sincer excePt in cases r'rhere perturbation techniques

may be applicable, analytic solutions can only be found for systems which

are apEly described by linear equations and have simple geometrical

boundaries and dePth Profiles.

The restriction of linearization is noE as serious as the other

tv/o; many models, including those of Platzman (1958) and Heaps (1969)

have used the linear equations to good effect. Sometimes¡ âs, for

example, for residual circulation studies (see Nihoul and Ronday (1976))

or for areas which are quite shallor¿ so that the surface elevation is

comparable with the mean depth (see Flather and Heaps (1975))' the

inclusion of the non-linear terms in a numerical model is essential'

However, if the water is deep enough to justify the omission of these

terms, and yet, Èo gain a realistic view of the behaviour of the system,

complicated boundaries and depth conÈours need to be taken into account,

a linear numerical model can be quite useful'

48

5.1 A Linear Finite-lifference Numerical Model

Assuming the time dependence factor "-t* , the linear equations

governing tidal motion are those given by 2,2.I, that is,

(-íur+r/h)u - f v = - * Foâx

(-it¡+r/h)v+fu=-c+"ðy

ð

ã; 3tn.ri = iur6dy

(tru ) +

5"1.la

5.1.1b

5.1. lc

The numerical model uses finite-difference approximations to

these equations. To determine the appropriate finite-difference forms,

a two-dimensional rectangular grid is superimposed on the region of study,

the boundaries thus being approximated by horizontal and vertical straight

line segments. The grid points lie aE the intersection of the lines

drawn parallel to the x- and y-axes; the grid spacing in the x-direction

is Ax and that in the y-direction is Ay'

The grid is composed of elements; within each element are a Ç- '

a u- and a v-point arranged in a staggered fashion, as shown in Figure

5.1. This configuration has been used by many auÈhors, including

Platzman (1958), Leendertse (L967) and Ronday Qglø), because of the

simple form the coastal boundary conditions take. If E is evaluated

at a grid-poinË with coordinates (xry), u is evaluated at (x-ax,y)

and v at (x,y-Ay); the depth, h, is specified at the position (x,y)'

Each element is identified by an ordered pair, ([,j), with 1 < !, < m

and 1 < j < n, l, increasing in the positive x-dírection, j increasing

in the positive y-direction. The corresponding values of (, u, v, h are

denoted by Ç0,, , tQ, j , tl, j and no,, Figure 5'1 shows the ([, j)tt't

element and surrounding grid-poi.nts. Each element is fr¡rther labe1led

according to r¿hich one of 12 classes it belongs. These classes identify

49

Aut- ,, ¡ *,

r

,o-r,, ¡ttQ,

,

A¡t*- ,,,

L

x4Q't,J'' ttl, j - ,

At[,¡*t

-l

xx

A

eeu[+t,

¡

^y

Itß, ,

x6Q, j -,

J

u[+1,¡-t

+--- A X "-+

Figu re 5.1 The (1.,j)th element and surrounding grid-points'

50

the manner of allocation of values to 6, u and v and are set out in

Table 5.1. From this table it can be seen that the coasÈal boundaries

are always approximated in such a manner as to ensure that, if a land

boundary is parallel to the y-direction, it passes through a u-point,

and, if parallel to the x-direction, it passes through a v-point.

As far as is practicable, centred finite-difference approximations

to 5.1.1 are used. Consider, for example, equation 5.1.la which rnay

be writÈen

{-io + r(x-Ax,y)h t(*-A*,y)}u(x-Ax,y) - fv(x-Àx,y)

ca6ðx

(x-Ax,y) 5.L.2

Now, using Taylor series expansions

6(x,y) = 6(x-Ax,y)

6(x-2Ax,y) = E(x-Ax,Y) - o" * (x-ax,y)

* a* ff{*-ax,y) . ry $t*-o*,y) + o(ax3)

and

so that

..' ry S t*-o*,y) + o(Ax3),

6(x,y) - E(x-2Ax,Y) = 2Lxff C"-n",v) + o(Ax3),

that is, if the ([,j)ttr element is being considered,

ff t*-n*,y) = *. rto,, - Çe-r, j ) * o(axz).

the best approximation to v(x-Ax,y) isIn Èhe same mannert

+ vl-t ,i +r,i +1\(up

,¡+ vQ + tl_r,j) + 0(Ax2,Ay2),

5L

but, to avoid the inversion of a large

explicit system is developed, so that

3mn x 3mn complex matrix, an

the approximation used is

v(x-Ax,y) = 4&n- r,, * tl- 1, j+r ) + O(AxrAy2).

Hence, 5.1.2 maY be written as

tQ, j tq, jÌrQ,, - 4fko-r,. * t[-r,,*r)

__ gzL,(ee,j - ea-r,j) * o(Ax,Ay2), t'1'3a

is the friction parameter value at the grid point associated

(ttris parameter is discussed in Section 5'4) and

{-ic,r * r h

where

with

ttQ

tl, j

rJ

h

In the same manner, the finite-difference aPproximations to

equations 5.1.1b and c are found to be

tr.

tg, j-It[, j

9., j+ 4fGto,,

4(hn,¡ * ht-r,i

hr, j, & = 1'

)v

+

) e.+r

+ tl , i ' t )

i+ 1

i-n expllcit form, the

{-it¡ + r

h

h

= ---Þ-- ( 72AY 'o1, ¡

1e O (Ay, Âx2 ) 5.1.3b

*'n.o-r,r to*r,, - nro,j uÎ, j t . fttttrp,¡*r t[,j*r - htl, jtr,

,]and

where

it¡ Ç2 + o(Ax2,Ayz), 5. 1. 3c

t[, j {

L"(hg, j

hQ, , ' j

These equations are real.ranged to give '

eqrrations for the evaluation clf t[, j , t[, i and

points : -

L^ at all interior'X' i

52

tl, j to- t,,v^ -h,¿- I . j + I tl-r,, ){tr

'[, j = f¡

"e,j -'g-t,!

t[- ,, , t!- t, ¡ *l

+ 2ioAx eg- ,, , \

t!- t, ,

2Lxe

{(-ir¡ + r )to, ,

+ t[, j - , )

h I

5. 1 .4a

5 . 1.4b

tl, j tQ, j

- Yf(vo-r,. * tl-r,¡*r)Ì

tr,, = - {-io o t.,ro,, nJot, .}-r{\f(tq, ¡

5 . 1.4c

The first equation is obtained by rearranging 5.1.3c after replacing

L by L-L, the second by rearranging 5, 1.3a and the third by rearrangement

of 5.1.3b. These equations are explicit since, on calculation of any of

the unknohrns on the left-hand side, all the quantities on the right-hand

side are known provided that Èhe s\¡/eep through the grid follows increasing

values of !, and, for each 9" , increasing values of j . The equations

5.I.4 must also be evaluated in the order given above.

The appropriate forms of these equations for non-interior points

are given in Table 5.1.

5.2 The EVP Method

The solution, using the equations 5.1.4, is found by the EVP

method described by Roache (tglz, p.L24). From Table 5.1, it can be

seen that, for elements labelled 3 and 7, the value of tl,, is calculated

according to equation 5.L.4a; the desired result of such a calculation is,

naturally, zero, since the associated grid-point lies on a land boundary.

Also, for elemenÈs labelled 11 ar'd 12, Çg,, is calculated according to

5.1.4b, the desired result being some knoqm input value along an oPen

* ft t'0, 6l,r-r)]

1

2

3 J2,tt

,!lt

tQ

Çe,,

v

53

=Q

provisionally as signed

=Q

tQ, , in 5.1.4

tQ,, as in 5.L.4a

not calculated

=Qtg, j

tr, , l.Í

not calculated

ltr,

,=Q

(tr{estern open boundary)

7

I

r

Çe

ÇaÌ.".l/,

zzÞ ÇQ,,

tr =Q

rQ,, as in 5.L.4a

l.I

tI

9

ee

av

not calculated

Ç2 not calculated

Çe

=Q

provisionallY assigned

as in 5.1,4c

10. (I,rlestern open boundarY)

II

¡I

AI

I

tQ

Ça

provis ionallyas s igned

is given

as in 5 .L .4cv2

5

tg

x

4

(I

I

I

AI

I

t.Q

vI

T

^

(I

I

It(I

11 Eastern open boundarY)

t[, ,

ÇQ,,

tR, j

as in 5.1.4

l2 Eastern open boundarY)

t[, , as in 5.1.4

tl , j

= Q

the linear numerical model and

allocation of values Èo 6, u

land, indicates a solid

boundary, while >

^ a v-Point.

un provisionallYx¡ t assigned

Çg,, is given

Vn =Qxrt

6 tr, ,

eo, ,

as in 5.L.4

to, ,

The classes of elements for

their associated methods of

and v /ZZZ indicates

boundaryand--- anoPen

u-point, x aÇ-Pointand

AÇs

TABLE 5.1:

54

boundary at the EasÈern extremity of the region of interest' Hence'

it is necessary that those values (called starting-values) designated

as "provisionally assigned", namely, ee,j io elements labelled 1 and

5andUninelementslabelledgandl0,shouldbesuchthatasweePx¡ I

through the grid produces the correct end-values for elements labelled

3, 7,11 or 12 on calculation of 5.L,4" The correct starting values

are determined by finding the end-values produced by specific provisional

starting-values .

Foraconsistentschemewithauniquesolutionthenumberof

starÈing values, sây K, is the same as the number of end-values' The

t\^ro Sets of values are numbered in increasing order as they are encountered

in the scheme. For any seË of starting-values {sO, a = 1,"'rK}, the

s\deep through the grid produces a corresPonding set of end-values '

t€ . o¿ = 1,...,KÌ. Since Lhe equations 5 'l'4 are linear' there is a

0-

símple linear relation between the "o and the €o' namelY

=where :

:€-o

A

where

gríd

As €-0

+

(s, ,... ,s *)t

(Grr...ra^)t

is the end-vector Produced bY s=0

is a KXK complex matrix whose columns are generated by

form fu = (ôul) where ô*l is the Kronecker

A:g * :0, then 2p = 1l - 5o , where *. is thestarting-vectors of the

delta. Thus, if I =

l,th column of A.

once these guantities have been determined, the correct startlng-

vector, :*, is determined bY

*s;t=[r(€ -€o)

desired end-vector. Hence, (f+1) s\¡7eePs through the

correct starting-values which are then used in a final

the values of Ç, v and v throughout the whole system

e is the

produce Èhe

to deÈerminerun

55

5.3 StabiliËv Cons is tency ánd Converqence

The concepts of stability, consistency and convergence are discussed

in all books which deal with the numerical analysis of finite-difference

methods.Roache(:-g72,P.7andp.50)givesaninformativediscussion

of these features yet keeps them in their propef perspective with regard

to an analysis of a finiËe-difference approximation (FDA) to a set of

partial differential equations (pl¡).

Consistency is simply the requirement that, as Ax, Ày + 0, the

truncation errors (as evident in equations 5.1'3) rnust approach zero,

so that the FDA aPProaches the PDE.

As Ax, Ay + 0, the discrete solution must approach the continuum

solution, that is, the soluÈion to the FDA must converge to the solution

of the pln. This is usually hard to Prove as the FDA is used solely

because the solution to the PDE is not knor¿n. Linear initial-value

problems may use Lax's Equivalence Theorem to relate consistency and

stabiLity to convergence. However, no analogous theorem exists for

schemes such as the EVP method, which have no explicit time dependence

and so cannot be classed as initial-value problems, nor for non-linear

schemes. Ilowever, the FDA can be used to solve a simil-ar, but more

simple, problem for which there is an analytíc solution, and a comparison

belween the two can be used as a guideline as to the likely convergence

of the FDA in more general problems. Probably the best Èest of convergence

of the FDA is a comparison with field data, if adequate information is

available.

An FDA is stabLe if the difference between its theoretical solution

and its actual numerical solution remains bounded. This difference arises

because of round-off errors.

56

5.3. 1 Srability

To deEermine the usefulness of the finite-difference model set

up in Section 5.1, the error arnplification properties of the scheme

need to be analyzed. This examination can be carried out by means of a

discrete perturbation analysis (Roache (L972), Noye (1978))which, although

lacking the methodical formulation of the conunonly used von Neumann

stability analysis, has the advantage of providing a round-off error

bound, rather than just the reassuring informaEion that (for a time-

stepping scheme) repeated progressions through the grid will not increase

the error wiEhout bound.

In the analysis which follows, it is assumed Ehat both the depth,

h, and the friction parameter, r, are constanË. The simplified finite-

difference equations are

t[, j ug. r,, - a{vo- 1, j +1 - vQ- 1, j } + ôtÇl- ,, j 5.3. la

Çe ,r-r,, - ôl{U to,, - \l(wo-r,. * tl-r,r*r)} 5.3.lb

tQ,J = - ß riàf(,rÎ,- rQ,j.r) + ôlr(60,, - qr,j-r)Ì 5.3.lc+

cr, = Ax/Ay

ß=-ir¡+r/h

6t = 2Ax/B

62 = 2Ly/g

ô, = 2it¡Ax/h.

where

If the magnitude of anY error in ee,,

it can be seen that

¡a.ro,, I = lAtp,,*, I = lg-t

1S lAql,, I

ô;' aEl,, I

then, from 5 .3. lc,

57

arid so, from 5.3.la,

lAk*,,¡ | * zcrlß-tô;t A ç0,,1* lô, ¡ Çr,rl

Finally, using these error bounds and 5"3.lb, the uPper estimate of the

error in Çp.* t, ! is

la fu*,,¡ | * {t + zoô'ôãt * ô,16rßl * lfß-'lo'o;'}lA çt,¡ I

or

la Çl*r, ¡ | {r+ foltr+q21-%a*zaz

. # (ax)2 [1+02]n'] l¡ 60,, I5.3.2

where the notation of the analytic model is used, namely,

o--flw

0 = r/t¡h

The expression 5,3.2 indicates Èhat the scheme is unstable from

the point of view that an error inÈroduced at any point is increased

at each stage of the progression through Èhe grid. However, Èhe largest

e:-tox occurs at ttre end-boundary in the x-direction (1, = m), and the

error at any interior point is smaller than this end-error. Hence, by

lirniting m , the number of grid-steps in the x-direction, the round-off

error at L = m (and so for '1, < m) can be kept within a desirable

range.

Howeverr 5.3.2 also shows Èhat Ëhe error is smaller for smaller

values of Ax and C[ so that some comPromise musE be made between having

a small value for Ax and a sma1l number of grid-steps in the x-direction.

Smaller values of cl can be achíeved with larger values of Ay, but

this must increase the truncation error (as evident by 5.1.3) and, once

58

again, some compromise must be made between having small round-off error

propagatiori charact.eristics and having an acceptable truncation error.

If an acceptable value of Ax results in a grid which does not

cover the region of interest, double precisíon can be used so that a

useful number of significant figures can be retained at the end boundary

of a larger grid; hor^rever, this increases the computer memory and time

required for calculations and so limits the usefulness of the model.

Nevertheless, it has been found that, for regions which are not too

extensive, the amplification of round-off errors does not limit the use

of equations 5.1.4 in describing tidal proPagation.

The results of the application of these equations to the Gulf of

Carpentaria, Australia, are given in Chapter 6. The model values used

ate

Àx=13kmIo=j

h>5rn

lOl * .5 (using a latitude of r2L""s),

so that the error amplification, given by

la Er*r, ¡ |1.5314 Çt,,1

is not too restrictive, and the maximum error in the end-condition, using

m = 25, is - 10-s, which is acceptably smal-l.

5.3.2 Consistencv

The Taylor expansion approach used to obtain the finite-difference

equations may be reversed for an analysis of the schemets consistency.

Thus, for example, each variable in equaÈion 5.1-4a may be expanded about

the point (x,y), corresPonding to the posiÈion where tr-r,, is calculated,

to yield

59

a

âx(hu) + f, rn"r = io6 - (ax),[å" *. å *. ## .',* #]

¡2dv6f 2 a3h n ð3vlá"ãF.ä#l +o(ax3,ay')- (Ay)2 4

Hence, it can be seen that, as Ax , Ay + 0, 5.L.4a approaches equation

5.1.lc. Similar results are obtained when the process is applied to the

other two equatíons, indicating that the FDA given by 5.1.4 is indeed

consistent r¿ith Èhe PDE 5.1. 1.

5. 3. 3 Convergence

The convergence of the system ís tested by using the equations

5.I.4 to find the solution for the exact situation as is modelled analytically

in Chapter 3. The results, indicating satisfactory convergence, are

presented in Chapter 6.

5.4 The Friction Parameter

The form chosen for the friction parameter is that given as 41.5

in Appendix 1, that is

-=9-L,,'3lrc2'mt

with C, the Chèzy coefficient and V* some estimate of the maximum

magnitude of the velocities. Using a value of .030 for Manningrs rl'

[â.r a2n

Lç ¡-;'ahãt+ +

5.4.1

The value of V- has been modelled in two r^tays:-

(i) v is constant over the whole region and is chosen as anm

estimate of the mean value of Ëhe maximum rnagnitude of the velocities as

given by the analyEic model for the dominant component (if it exists)

and (ii) vr' varíes with the grid point according to

60

Input from analytic model gives

open boundary values and, ifT = TD, the first estimates

of r .ru'v

NO

YES

EVP l4ethod

NO

Has iterationconverged?

YES

+Store values of rrr rr, on

a file Ëo be used if t # Tn

PrinL Re lts of

6, u, v

Flow chart for the linear numerical model,

indicating iterative calculation of the

frictíon paremeter ' TD is the dominant

tidal period'

Read values of ru rt"off the friction filecreatedby T=To

t.tu'v

FIGURE 5.2:

v = {(u*ttÎ, j 2' l)2 (t. )'zÌ

6L

+

)2 + (vï, .)'\h ,I x, J

v,

v i (u*9,

5 .4.2*tr,,

where

"* = \{vx2, i [, j

*v*[-t,i+r

fv*l, i +r

*v*l- t, j

*u*[, j' t

Ì

Ì2ri

= |{s* f u*2,i l+t,iu* {u*

[+ 1, j - 1

the *ts indicating that the values used are those obtained as a previous

EVP solution. the values Vmu!, j , urrl,, "t. thus found by iteration

as shown in Figure 5.2, If a dominant tidal component exists, the

iteration of the friction Parameter is carried out only for Èhis

component. The values are then stored on a file to be used for any

other component in accord with 41.6 '

Henee, in general

r h t = U .OO744ln_413

vuÎ, ¡ *t9,,t[, j tQ, j

-413, h-r = U .00744 h vtQ, j tÎ, j tl, i *tl, i

where u*"0,, ''*t[,, are either constant or as given by 5.4.2' and

u=1

1.5{if the component is a dominant tidal component

if the component is not dominant.

5.5 A Non-linear l"lode1

A non-linear model has been developed to provide a comParison

with the tínear model. The relevant eguations of motion and continuity

are those given as 2.1.1 in Chapter 2, omiËting all the external inf I'uences

62

accumulaled in the term I . These equations, written in component

form with the noÈation of Chapter 2 are

ðt 1¡+z).fr f(tr+z)ul ¡| trn*zlv1 = Qð +

v

5.5.la

5.5.lb

wíth À a constant, associated with a constant value of Manning's n'

The grid is composed of elements, exactly as described in section

5.1. An explicit, forward-time, centred-space finite-difference aPProx-

imation to the equations 5.5.1 is used, following along much the same

lines as that used by Flather {og72). The approximations are found in

the same nannei ;Ë :hat given for equation 5"I"2 in Section 5'1' The

notation is the same as used in that section with the addition of a

superscript to designate the time level at whictr differenE quantities

are used in the calculations ' Thus 'i:i

denotes Z evaluated in

the (L,j)th element at time t = nAt, At being the time increment'

The approximations to equations 5'5'1 may be written

P.uP*v+-fv=-dt dx dY

ðZcax - À gçgz+vz)k

(¡+z)43

av. + ug * ylJ * i àz +- v(ts2*v2)h , 5.5.1c-a. -ðx 'âv ' u = - gÚ

ø.rg v\u rv /

where Ëhe form A1.3, as discussed in Appendix 1, has been chosen for Y,

that is

,( = X/ç¡*z)at ,

zl n+1),j

At2L",i n)

,j(tr*z( ") ) l.+ r, i

u[n)+1, j

v[*_{<*r.1,i., q:]., - {r,*z(")) 2,in) 5.5.2a

n+ 1)

,ju[ u[

63

4:l{'[lì,, - u[n),j

At6

At4Ax

4'ì {'[,]., - u[

n)- l, j

LI

- ^t

R(")tl, j

n)

, j" ttI

n)

,juI

1+ At fv) {,i

n+ I ),i

At-2ñ.en) 4ri:ÌÌ

{41ì,, - '[]1,, ]

5.5.2b

5.5.2c

5.5.3a

5. 5. 3b

v[n+ 1)

,j 4tÌ _^rñ4Ax "l

n+1),i

lì{4:1., - 4lì.'} - ^'d;l 4:Ì

l;" #*{'[:;" -'[:;]Ì] '

v[Atñ

where

-^Ëf4

q n)

R( ")tl, j

R( n)

tl, j

, = 4{hg,, .'lil

= äiu[:l . u[TÌ,, * u[

no- ,,, * t[]Ì,, ]+

, = 4{nQ,, * rli', * no,, -, * r;.n)

¡ j' l

n)+1,j-1

n)

¡j'l

Ì

+ u[

qlÌ = å{v[:l . u[:]., * v[]),,,., * u[]1,, ]

= À{(n*l;t;o *,}-"

icu["] )' . (q:',)']*

= ^tt -¡ , t-"' {(q,ì1'z . rv[]',r'r' .

l4odified forms of these equations are required if the elemenÈ is

adjacent Ëo a boundary. The classes of elements are the same as those

for the línear model and are given, wiÈh the appropriate finite difference

forms associated with Ëhem, in Appendix 4'

64

5.6 ConsisÈencv. Convergence and Stability

The analysis of the consistency of the non-linear FDA is carried

out in the same manner as for the linear scheme. By expanding each

variable in the equations 5,5.2 as a Taylor series about the position and

time at whích the first quantity on the right-hand-side of each equation

is evaluated, it can be shown Èhat the FDA differs from the PDE, 5.5.1,

only by a truncation error of 0(A*2 ,Ly'rAxAy,Àt ). As Ax, Ay, At -+ 0,

it can be seen that the FDA approaches the PDE and so is consistent'

For the area in which the models Í/ere applied, it was considered

unlikely tha¡ the non-linear advecËion terms would greatly influence the

elevation of any fundamental frequency (though they would probably have

a greater effect on the velocities and on any harmonics, as shovm by,

for example, Flather (L972), and Flather and Heaps (1975)). The friction

paremeter in the linear model vras an adapted form of the frictionat term

in the non-linear model. Hence, the convergence of the non-linear FDA

has been determined by a comparison of the results of the linear and

non-línear models as shown in the nexÈ chapter'

The likely srabilíty of a non-linear system is assessed by an

investigation of the appropriaÈe linearized problem:

zl zín)+1, j -4ll ]-H{u[,]., -4n+ 1)

,i

n+ I ),l

n),j

n)

4n

{hAÈ-ñ

-RÂt

n)

,jÌ 5.6. la

4 u[ 4n),i 2Lx

Atz{z(,

+1)1, j

n+ I ),j

u[

zl Ì

. rft4 n),j

n)- 1, i +1

+ 4lì., * 4 u[]ì,, ]

Ì

5.6.lb+

4 4n)tJ

n+1),j

- *r 4,ìAt ¡-(n+t) o(

ñ gt"o,j - og

n+ 1)

' j' 1

- rfru[i;" * up

with R and h constants.

( n+1)+1, j

+n+ 1)+1, j - I

+ 4, ilìt 5. 6. lc

65

Using a von Neumann analysis (see, for example, Roache (L972)),

the Fourier components of the solution for each Z, lJ, v may be written

,l:l ,,ro:Ì ,4:ì ) =(A' ,B' "l k"Av

¡"i z( 1k*ax+j k"ay)

( cnAxk.'

x 5.6.2

where At rBo ,Ct are the amplitude functions at time

wave number in the x-direction, for any component, k,

in the y-direction and i = vq.

Defining 0* = k* Ax

0 =k Avvv

Ar= -:- s1n UAxx

e

sin 0

t = nAt, kx is the

the wave number

0

tAß= Ay v

€ =RAt

ô =f At cos 0 cos 0x v

and using 5.6.2, the equaÈions 5.6.1 may be rewritten as

which can be rearranged into the form,

A

n+1B

=A'-ihoBt-ihßc"

= (1Æ)n" - igcr An*l + ð c'

= (l-€)Cn - giß An*l - ô B'*t

n+ IC

n+ IA

n*1B

n+1c

tlG

n+

where

66

-íha -ihß

-iga 1-c-ghcr2 -ghoS+ô

I

Ç=

{ igcrô

-ießÌ

{ -ghoß

-51 1€-ghcr2 ) Ì

i l-e-ghß2

+ ghcrgð-ô2]

The characteristic equation of this matríx is

-À3+arÀ2+ a"X+ â3=0,

where

a' = 3 - 2e - ô2 - gh(o'*g') + ghaBô

dz = - 3 +48 - e2 + ô2 + (1€)gh(o2+92¡

- ehoßô

as = ( L<)2

For a strongly stable scheme, lfl < 1 for all 0

the case of / = 0 , 5.6.3 may be factorized as

so that the eigenvalues are

ÀI = 1-€

I2r 3=Dt{o2-(L<)}Y'

where

D=1-; !@'*ø').

From these values , it is f ound t'hat I i I < 1 if

e<2

5.6.3

0vx

11-e-À){À2 + À[-z+€+gn(o2+ß2)] + (1-€)]= o,

For

and 7.+(o.2+82)<r

67

or S.rand + < 1.

(fnis second condition may be slightly over-rèstríctive depending on

whether or not (0* = trf 2, 0" = tr/2) satisf ies the co¡rdition

l7.+ (a2+s2']" gtr(az+ß2)' )

This is a subcase of the conditions determined by Flather (L972) '

An explicit expression for the eigenvalues, À, cannot be found whea

R # 0 r f # O but an analysis is possible usíng the Routh-llurr¡itz

criteria set out in Appendix B in Leendertse's (1967) paper' The analySis

is given in detail by Flather and the resultíng conditions which ensure

stabílíty are

(n +lf l)lt < z

and5.6,4

RAr2

1ìi^.' 1{*+* f tntl'

f c* * lr l) + f ta.l'{ro# . urL,J . t

68

CHAPTER 6

APPLICATIONS TO THE GULF OF CARPENTARIA

6. I The Gul f of Carpentaria

The rnodels developed in the previous chapters have been applied

to Ehe Gulf of Carpentaria and the adjacent waters. This Gulf is located

in the North-Eastern parÈ of Australia, between latitudes 10oS and 17oS

and longitudes 135"E arrð l42oB. It has a roughly rectafigular geometry,

a surface area of about 1931000 kmz and a relaÈively smooÈh bathymetry.

It is a shallow area, the greatest depths being about 70 m, the nearest

deeper hrater occurring either East of Torres Strait or in the Timor

Trench. Figure 6.1 shows the overall geography of the area and the

depth contours in metres (after Rochford (1966))'

6. 1. 1 Tidal Measurements

The amplitudes and phases of the four main tidal components at

various places are given in the AusÈralian National Tide Tables, L978.

These components are

Solar Diurnal (r¡) wittr period 23'9 hours

Lunar Diurnal (Ol) wittr period 25.8 hours

Solar Semi-diurnal (Sz) with period 12'0 hours

and Lunar Semi-diurnal (Mz) wittr period 12'4 hours'

However, all measurements are taken eiEher very close to the mainland or

on islands. The value of such measurements in providing a comparison

for tidal models is questionable, since the data ís collectedilin the midst

of the very coastal features most likely to exert anomalous effects on the

phase and amplitude of the tide" (HendershoÈt and Munk (1970))'

69

FIGURE 6.1: The Gulf of CarpenBaría and adjacent waters.

The depth contours are shown in metres (after

Rochford (1966)).

\tII

I

I,

N"I

\Ð

0070

50

0 ¡

60

s

Þo

I

II

ItIIII

III

4

\I

70

6.L.2 Tidal Studies

An outline of the tidal features of the Gulf is given by Easton

(1970), though his co-range lines are not very informative. He states

thatttthe presence of a cenlral amphidromic point is suggested by the

diurnal amd semi-diurnal components; further nodal points occur probably

near Karumba and Groote Eylandt." Cresswell's (1971) study suPports

the suggestion that "lhe diurnal r^rave travels clockwise around the

perimeter of the Gulf, pivoting on some as-yet-unknown amphidromic point

within the Gulf."

ltilliams Q972) has studied the response of the Gulf to tidal

forcing by means of two analyt.ic models, one without the effect of the

Coriolis force (also published in Buchwald and I,Iilliams (1975)) and the

other including rotation. Both of these studies neglected the effect

of dissipation of energy by bottom friction and the Presence of tidal

forcing through Torres Strait.

Calculations by Mi1ler (L966) indicate thaE approximately 102 of

the toËal lunar tidal flux out of the deep oceans enters the Arafura sea

and is dissipated in the vicinity of the Gulf of Carpentaria, so that

the inclusion o.E sùme energy dissipation mechanism would aPPear to be

almost mandatory. According to Teleki et al (1973), "the bottom friction

should be of considerable amplitude for the entrainment of the fine grain

size sediments found in the Southern part of the basin." Bottom friction

is the mechanism chosen to model dissipation of energy ín this thesis.

Torres Strait, complicated by its array of islands, shoals and

atolls, is very shallow in comparison with the Gulf and, for this reason!

tr{illiams (lrg72) considers the Strait as a land barrier. However, during

certain periods of the year, there is subsÈanÈial water movement through

Ëhe Strait into the Gulf (see Newell (1973)) and so its effect on the tides

in the Gulf is considered in the model developed in chapter 4.

7T

FIGURE 6.2: The boundary approximation for the analytic

models, showing the regions into which the

area is divided.

P.N .G.

N

EYLANDT

HT

ARAFURA SEA

JUNCTION Wed

( region 3 )( reg ion 2,

RESO NATOR

( region 1 )

GULF OF

CARPEN TAR IA

EDWARD

\ PELLEW

P

IPA

N.T.

OLD.

CALEDON

PORT

KARUMBA

72

l,Iilliams (1972) includes a depth discontinuity between Èhe junction

and the resonator but uses a constant depth of 91.5 m in the channel

(regíon 3) and junction. Frorn Figure 6.1, it can be seen that a better

approximation would be a depth discontinuity between the channel and the

junction.

Figure 6.2 shows the rectangular resonaLor-channel system which

has been used in this analytic study of the Eidal propagation in the Gulf

of Carpentaria. The dashed lines indicate the common boundaries of the

regions into which Èhe area has been divided and also the discontinuity

in depths as modelled in Chapters 3 and 4.

6.2 The Analy tic Model of ChapÈer 3

The values of the various constanÈs used in Chapter 3 are

a=468km

b=390km

d=468km

h1 =55m

h2=60m

hg = 91.5 m

Í =-3.1 x 1g-s"-t (corresponding Èo latitude I24oS).

The friction parameter used is that given as 41.5 in Appendix 1,

that is

8ç=-'31r

Using n = .030, this may be written as

v

_g_ç2

vm

1t.

J.oo744 h.

J

u3 j = 1 ,2,3, 6.2.L

73

where the value of v is found, by trial and error from the ¡42 tide,mj

to be

-l 6.2.2v = .35 ms , j = I1213,Ill

J

With these parameter values, the assumption that the inprt wave

may be approximated by the form 3.2.8 which neglects all modes other than

the Kelvin ü/ave, must be justif ied.

Consider any comPonent

-ß x+iu x6o(x,y) =oo e Q 'e

Y(y)

where Y(y) is a sinusoidal function of Y,

% is a constant,

r Irk;-' = ill * tßn with Bg > 0, ilt > 0.

All such componerits decay exponentially as x becomes less negative.

To neglect the components with 9" > L, it is required that

e uGa,v)-L'Y << 1 and aa 1r

ÇL 6 .2.3

where (L-a) is the effective length of the input channel (1, > a). Now,

ßo (a-L) < O and if lßo {r-1,) i >> 1, rhen 6.2.3 follows. Since, for the

above parameter values, it is found that

ß <<9t<32<

it is sufficient that L-a >> 1/ßr for it to be possible to neglect the

Poincar! wavês. In fact , Ilßt ^' 151 km for the ylz tide (this being

the dominant tide in the area) and the effective length of the input channel

is greater Èhan 500 km. Hence, it may be considered that the form 3,2.8,

0

74

that is( o)

6o (x,y) ik=g

x- K¡ y

is a good approximation to the tidal input for the Gulf of Carpentaria.

It would not be expected, though, that the results present an accurate

picture of tidal propagation farther up-channel, away from the channel/

junction boundary.

The tide aË Jensen Bay (see Figure 6.2) is used as the reference

point for Èhe scaling of the resPonse as described in Section 3.2.

6.2.L Convergence of the Galerkin Method

Using the above parameter values and the N72 tide, the convergence

of Èhe Galerkin method is tested by checking that the residuals of each

of the conclitions in Section 3.5 become smaller as the value of N is

increased (see Appendix 2).

(1) The errors in Condition 3.5.1, Çz(-a,Y) = 6s(-a,Y), 0 ( y < b

are presented in Tables 6.la and 6.lb" The percentage error is calculated

according ro rhe rarío Lll;rl , wheo A - llerl - lr,rl l, ot. Lltre(ez) ¡¿hen

[ = larg(6a) - lrg(62)l The errors in both the amplitude and the phase

can be seen to become acceptably small as N is increased. The largest

errors occur aÈ the two ends, Y = 0 and y = b, though the error at y = b

is less Ehan L% for N = 6. The larger error at (-a,0) could be accounted

for by the erroneous nature c¡f the condition on the velocities at this

corner (see SecÈion 3.1).

Q) The convergence for the condition 3.5.2, hzvz(-a,y) = h3u3(-a,y),

0 < y < b, is much faster than for the previous condition, as can be seen

by Table 6.2. The percentage error is calculated according Ëo the ratio

L/l1zuzl , where ^

- lttrl"rl -trl"rll; a zero is entered in the table if

A < .005 and Èhe percentage error is less Èhan '057"

75

TABLE 6.la: The error in the amplitude in condition 3.5.1,

Çr(-a,y) = 6,(-a,y), 0 < y { b' using the Galerkin

t.echnique, for increasing values of N'

r= | lc, I _ le,l I

N=BN=6N=4N=2vlb

Ao/ AAA

040

.009

.006

.005

003

.001

002

004

0

0

.002

6.20

1.28

.85

.61

.4L

.23

0

0

.2L

.28

.44

.046

.016

.005

.003

.006

.003

.oo2

.004

.002

.002

.005

7 .20

2.t6

.7r

.38

.69

.33

.23

.46

.18

.28

.62

8.75

2.17

t.44

1 .03

"94

.52

.80

.2r

.75

"03

1 .00

LL .47

.47

3.L7

2.93

1 .01

.94

1.88

1.56

.35

1 .04

2 .08

.056

.016

.011

.008

.007

.004

.006

.002

.007

. 001

.009

0

Llto

2/L0

3/to

4/to

5lro

6/Lo

7 lLo

8/ 10

9 /ra

1

.07 5

.003

.024

.023

.008

.008

.015

. 013

.003

.009

.019

76

The error in Èhe phases in condition 3.5.1,

Çr(-ary) = 6r(-a,y), 0 < Y < b, using the

Galerkin technique, for increasing values

of N. [ = lerg(e) - e'e(qs)l

TABLE 6. lb:

vlb N 2 N 4 N 6 N I

A 1¿ dol A A

o/

0

Llto

2lLO

3lr0

4lro

5 /LA

6/ Lo

7 /r0

8/L0

9 lLo

1

.028

.005

.007

.008

.004

0

0

0

.004

0

.013

.37

.06

.09

.11

.06

0

0

0

.05

0

.L7

.015 20 .01 .13 007 09

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

004 .06

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.010 .L2 .008 09 006 o7

77

The error in the amplitudes of condition 3.5.2,

hrur(-a,y) = h.ur(-a,y), 0 < y < b, usíng the

Galerkin technique, for increasing values of N

[ = | In.url - lt'r"rl I

TABLE 6.2:

vlb

0

rlt0

2/L0

3/ro

4/ro

51rc

6/LO

7 /L0

8lL0

9 /L0

1

N=2 N=4 N=6 N I

7" A 7" Aol

A

0

.006

.005

. 014

. 010

.005

.o2L

.024

.006

.023

.044

0

0

0

.09

.08

0

.16

.i7

0

.L4

.27

0

0

0

0

0

0

.008

0

0

0

.02

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.05

0

0

0

13 01

0

0

0

0

0

0

0

0

0

0

09

0

0

0

0

0

0

0

0

0

0

.01

0

0

0

0

0

0

0

0

0

0

06

78

TABLE 6.3a: The error in the amplitude in condition 3.5.3,

6r(x,0) = Çr(x,O), -e < x < 0' using the

Galerkin Èechnique, for increasing values of N

r=llerl -le,l I

N=8

i/A AAA

o/

.003

.002

. 001

.004

.004

.001

.004

.005

.004

.001

. 011

.022

.066

.46

.31

.15

.61

"78

.24

1. 58

8.84

4.54

.58

2.74

4.28

IO.2L

.006

. 001

.006

.003

.005

.005

.006

.007

.003

.004

.003

.o27

.080

.73

.11

.87

.43

1. 15

1.38

2.46

10.48

3.01

5.50

.70

5.50

L2.33

. 010

.004

.008

.010

.001

.013

.009

.007

.020

0

.032

.020

.1_04

r.37

.55

t.24

L.82

"20

3. 59

4.06

t2.65

L6.95

0

8.74

4.r9

16.05

3.78

3.12

.83

2.75

6.92

10.36

9. 99

9.L7

56 "77

28.75

L2.27

5.59

24.96

.029

.023

.006

.016

.033

.035

.020

.006

"044

.060

.043

.028

. 163

0

rlL2

2lL2

3l12

4lt2

5 /L2

6lt2

7 /12

8lL2

9 /L2

N 6N 4N 2

*ra

TOI12

LLIL2

1

TABLE 6.3b:

79

The error in the phases in condiÈíon 3'5.3,

6r(x,o) = Çr(xrO), -â { x < 0, using the

Galerkin technique, for increasing values of N

[ = lArg(Ç") - erg(er)l

0

LlL2

2/L2

3/ 12

4/L2

5l L2

6lt2

7 /L2

8/12

9/L2

n/ L2

I2ú/

1

N 2 N 4 N=6 N 8

Ao/ A

t/ Ao/

.02L

. 010

.004

. 018

.029

.019

.008

.383

.004

.010

.019

.031

.009

1.03

.51

.18

.92

r.44

1. 35

.39

16.16

.05

"t4

.25

"4L

.13

.005

0

.005

.002

.008

.oLl

.008

.2t9

.084

.027

.013

.010

.005

.24

0

.24

.10

.40

.86

.40

7 .60

1. 13

.34

.17

.13

.07

.002

.001

0

" 004

.002

.007

.002

. 161

.035

.004

.013

0

.004

.10

.07

0

.18

.08

.37

.11

6 .06

.47

.05

.16

0

.05

.001

. 001

. 001

0

.008

.004

.002

.092

.006

.009

.006

.004

0

05

05

.06

0

.13

.19

11

3.60

08

.11

o7

.05

0

I9 /ro

.0012

.0029

.0025

.0014

8/ 10

.0061

.0021

.0009

.0009

7 /LO

.0059

.0015

.0005

.0007

6/LO

.0020

.0014

.0009

.0005

5 /L0

.0021

.0009

.0005

.0003

4lto

0041

.0012

.0003

.0001

3/LO

.0033

.0005

.0006

.0001

2lro

.0006

.0012

.0003

.0002

1/ 10

.0023

.0001

.0003

.0003

0

N=2

N=4

N=6

N=8

.0038

.0012

.0006

.0003

vlb

.0189

.0106

.0073

.0056

TASLE 6.4: Values of l,tr(O,y) I

increasing values of

Values ofincreas ing

calculated using the Galerkin technique, for

N

ltt (*,-¿) I

æ

1

.0101

005 7

.0039

.0030

IT/L2

.0017

.0011

.0013

.0010

LO /T2

.0027

.0018

.0001

.0005

9 l12

.0037

0

0007

.0001

8/L2

.0025

0011

.0002

.0002

7 /12

.0005

.0005

.0004

.0003

2lt2 3/L2 4/L2 5 /L2 6 /12

.0013

.0005

.0003

.0002

.0021

.0007

.0002

0

.00 18

0

0003

.0002

0007

.0006

.0001

0002

" 0006

.0003

.0003

.0001

Llt2

.0013

0003

0

0

0

.0014

.0004

.0002

.000 I

*/tal

N=2

N 4

6

8

N

}]=

TABLE 6.5values of N

calculated using the Galerkin technique, for

81

(3) Tables 6.3a and b indicate the error in condition 3.5.3,

Çr(x,O) = Çz(x,O), -â ( x < 0. The residuals of this condition are

also seen to decrease as N is increased, however, a comparison with

Table 6.1 shows that the convergence is slower than for condition 3.5.1.

Except near x = -a, there is error only in the 3rd decimal place when

N = g, but the percentage error is still hígh; this is because of the

smal1 amplitude region associated with the arnphidromic point as seen in

Figure 6.5, which shows the co-amplitude and co-phase lines for the N12

tide. Once again, the largest error in the amplitude occurs at (-a,0)

(4) Table 6"4 shows the error in condition 3.5.4, u2(0,y) = 0,

0 < y < b. satisfacÈory convergence is obtained as N is increased.

(5) The error in condition 3'5'5, v1(x,-d) = 0, -a < x ( 0'

is shown in Table 6.5. This also sholnls satisfactory convergence as the

value of N is increased.

The condition hlvl(xr0) = hzvz(x,0) is satisfied exactly in

Section 3.4 and Èhe error htas correspondingly found to be zeto.

6 .2.2 Corrver Us ing the Collocation Method

since the mathematicalmanipulationusing the collocation method

is less work Ehan for the Galerkin technique, the same situation was

prograrEned, using this simpler meÈhod, Èo comPare the rates of convergence'

The results for condiËions 3.5.1 and 3.5.3 are shown in Tables 6'6 and

6.7 respectively. I^Ihereas the errors do decrease as N is increased,

the rate of convergence is slower than for the Galerkin technique' The

zero entríes in these tables correspond to chosen collocation points.

The figures in brackets in Tabl-e 6.7 for N = 2 are calculated

according ro rhe ratio L/lerl instead of LllÇzl since Ehe latter

ratio gave an error of greater than IOO7", distorting the indication of

accuracy.

82

TABLE 6.6: Errors in the ampliËudes in condition 3.5.1'

Çr(-a,y) = E,(-a,Y), 0 < Y < b' as calculated

using Collocation, for increasing values of N

r=llrrl-lrrll.

vlb

0

rlL0

2/LO

3/10

4lL0

5 lL0

6l 10

7 /L0

8/Lo

e /rc

I

N 2

0

.084

.104

.082

.040

0

.025

.031

.023

.010

0

N=4 N=6 N=8

lr d A /"

0

.076

.032

.o23

.028

0

.016

.008

.006

.007

0

0

9.98

4.09

2.98

3 .53

0

L.92

.89

.66

.78

0

0

.050

.019

.013

.016

0

.009

.004

.003

.004

0

0

6.67

2.56

r.7I

1 .93

0

1.08

.49

.36

.43

0

0

.022

.02L

. 014

.006

0

.004

.004

.003

.002

0

0

3.01

2.8L

L.79

79

0

42

.51

"37

.t7

0

83

TABLE 6.7: Errors in the amplitude in condition 3.5.3,

4r(x,0) = Çr(x,O), -a ( x ( 0, using the

Collocation method, for increasing values of N

r = I lr,l - le,l I .

^ø/

A 7 Ao/

7"

.014

.043

.093

002

008

017

0

0

0

0

0

0

0

L.26

4.06

27 .t4

15 .08

19. 16

0

30

0

0

0

0

0

0

.003

.006

.008

.010

.018

.034

.056

001

0

0

0

0

0

0

.16

.38

0

1.18

2.t3

0

L9 .44

16.98

0

8.28

11.36

0

12.02

12.63

4r.63

200.48(66.51)L}t.24(50.0)63.32

30.93

3.82

7 .98

0

0

0

.96

0

.004

.008

0

. 018

.o24

0

.026

"052

0

.090

.r28

0

0

.51

0

0

3.63

6 .48

0

57.30

35.90

0

24.34

26.74

0

*la

N 2 N=4 N 6 N=8

A

0

rl12

2/12

3/12

4 /12

5/L2

6l12

7 lt2

8/L2

e /L2

LO/T2

LLI 12

1

0

.007

.026

.047

.056

.042

0

.026

.r43

.201

.2L3

.L52

0

84

The comparison of Table 6.1 with 6.6 and Table 6.3 with 6.7 jusÈify

the use of the more comPlicated Galerkin technique '

6.2.3 The F Response of the Gulf

lrrilliams Qg72), using a Gulf \^ridth of a = 480 km, found the

resonant periods of the Gulf to be 7.86 hrs, 10.35 hrs and 16.0 hrs' His

frequency-response curve is based on the amplitude at Karumba and is shown

in Figure 6.3. It displays a broad maxímum over the periods 15 ' 5 hrs

to L7,0 hrs raÈher than a resonance peak. On the basis of this figure,

he uses a period of 11.8 hrs for the semi-diurnal tide rather than 12.4 hrs.

since bottom friction will tend to damp out oscillations, and in

order Èo find a Gulf width which produces a resPonse which agrees with

the observed resonance oscillations of 10.6 hrs and 16.0 hrs (Uelville and

Buchr¿ald ( 1976) ) , the frequency resPonse curves lvere determined for several

different Gulf widths. These curves rePresènt the amplitude at Karumba

in response to a unit amplitude at Jensen Bay. The results for a = 468 km,

520 km and 546 km are presented in Figure 6.4. Each curve shows a marked

resonance near 8.5 hrs, the peak values being

22.2 m at 8.3 hrs for a = 468 km

26.3 m at 8.5 hrs for a = 520 krn

24.6 m at 8.6 hrs for a = 546 km'

Melvi1le and Buchwald ( Lglù indicate that there is some evidence

of resonance activity at a period of abouÈ 8.0 hrs. using Figure 6'4,

the width of the Gulf was chosen as 468 km since this gives the best

agreement with observed resonant frequencies as well as a low arnplitude

for the period of L2.4 hrs.

The co-amplitude and co-phase lines are shown for Èhe l4z tide in

Figure 6.5 and for the K¡ tide in Figure 6 ' 6 '

.Doo

lUofÞ=fL

8,0

6.0

lr'0

2.0

85

712 172227TIDAL PERIOD (Hours)

FIGURE 6.3: The frequency response at Karumba according to

l{illíams (L972), with a = 480 km'

zIktrjJtrJ

0

86

I

I

I

a=546 km

a= 52O km

a= 468 km.

I

I

I

I

I

I

IItI

I

t

t

\

II

tj

II

II

II

Â,t

I

t

II

,

,III

II II

IIII I,

II/

ahpo

u.lo? 2.0:iù

zIktlJJUJ

lr.0

3.0

0I

07 17 19 21 23I ll

T]DAL

r3

PERIOD

15

(Hours)

FIGUR-E 6.4: The frequency response at Karumba for various Gulf widths.

FIGURE 6.5:

B7

The co-amplitude and co-phase lines for the Mz tide

according to Èhe analytic model of Chapter 3. The

amplitude,

-s

is shown in centimetres and the

phase , in hours.

roo

o

I

\e\\\\

80 40

-- -6.

//

I

I

ß{\

I

20

/\

40

---/

/

I

\\

lor\2\ 2.51

I

1.5

a

9-1

88

FIGURE 6.6: The co-amplitude and cc-phase lines for the K1 tide

according to the analytic model of Chapter 3 '

65

IItI

II

I\tIt\\

40

\\

t\\

1'-'Ittt,tl

t. I

-t----' 12

o

\\I

I

IIIII\

\\

o

\I \II 30\\\

20

19

o

t

\'I

40

\ 8

10

Þ

17

89

6.3 The Analy tic l{odel of Chapter 4

As well as the constants specified in the previous section,

the values

w1 = 156 km

wz = 234 km

ha=10m

v = .70 msm

4

were chosen for the model which considers the effect of tidal forcing

from Torres SÈrait.. In the manner described previously, the effective

length of the input channel is required to be greater than 57 km. If

the channel is extended out to the islands and reefs on the Eastern

side of Torres Strait, the channel length may be considered as greater

than 100 km, so that, once again, the Kelvin !{ave is a reasonable

approximation Èo the inPut wave in this channel.

The method described in ChapÈer 4 is applied to the region

depicËed in Figure 6.2, so that the second connectíng channel occupies

theregion wz<y<b.

The reference point chosen for evaluation of condition 4.3.1

is !'Iednesday Island.

6.3. 1 Convergence using the Galerkin Technique

The resíduals of the conditíon (i) to (v) in Section 4,2 showed,

as would be expected, the same convergence as Èhose in Section 6.2.1 and

so the results are not presented here.

The errors in condition 4.2.7, ez(O,y) = çu(O,y), \úz ( y < b are

shown in Table 6.8. The residuals of this condition appear to be smaller

Ëowards the centre of the channel for N = 4 than for N = 6; this is

due to the manner of choosing points for presentation in the table

I

90

vlb

6/Lo

7 /L0

8/ 10

e lr0

1

N=2 N=4 N 6

l-\o/ A /" A /"

.007

.005

.004

.007

.o23

.63 006 .52

0

.05

.08

.68

.003 .29

.t2

.14

.24

.51

43 0 001

.35 .001 .003

59 001 .003

L.99 .008 .006

TABLE 6.8: The errors in the amplitudes in condition 4.2.7,

Çr(O,y) = 6u(0,Y), wz < ! < b, using the Galerkin

technique, for increasing values of N' A - I lerl lru I

TABLE 6.9: The errors in the Condition 4'2'8, hzuz(O'y) = h,,t4(0'y) '

wz < Y < b, using the Galerkin technique' for íncreasing

values of N. [ = ltt, 1", I -lttu l"u ll

/"/"A7"

5L.7 4

4.76

5.39

7 .99

11. 5

2.784

.243

.287

"455

.624

49.57

I.T7

8.0

4.85

6.09

2.600

.061

.426

.27 3

.335

57 .90

36.81

L2.09

14. 13

34.12

6lL0

7 /ro

8/ 10

elL0

1

3 .011

I.962

.649

.762

r.929

N=6N=4N=2vlb

A

91

TABLE 6.10: The values of lur(O,y) l, O < Y < wz

using the Galerkin techníque, for

increasing values of N

vlb N=2 N=4 N=6

0

rlL0

2/Lo

3lL0

4/t0

5/LO

.0059

.0046

.0021

.0033

.0102

.0211

.0053

.0001

.0058

.0004

.0088

.0044

.0043

.0021

.0025

.0051

.0039

.0024

92

(spacing of 39 km across the channel, the same as in the previous tables).

In fact, the error is comparable for N = 4 and N = 6, the convergence

being very slow. However, the error at the channel wa1ls is smaller

as N increases, so that the overall error may be considered to decrease

wiÈh increasing N

The convergence for part of condition 4.2.8, that is , hz,¿2(O,y)

= hr+u¡+(Ory) r n2 € y < b, is slow, as shown in Table 6.9, though the

error is less than 10% away from the sides of the channel for N > 4.

Once again, the higher errors for N = 6 are a little misleading as the

overall error is similar to that for N = 4. The large error at Y = Íñ2.

is probably caused by the condition on the velocities at this corner

which is simílar to the condition at the corner (-a,0). The errors at

Èhis junction, rdz < y < b¡ are generally highet than for the other matching

conditions. This is possibly due to the large relative change in depth ,

being 827" aE this boundary buË on1-y 347" at the boundary between region 2

and region 3. However, the other part of condiÈion 4.2.8, u2(0,y) = 0,

0 < y 1 þr2: shows satisfactory convergence as N is increased as indicated

by Table 6.10. The percentage fÍgures in Tables 6.8, 6.9 are

calculated wlth reference to the values in Region 2.

The case of N = 8 for these conditions is not shovm as 1t re-

quires over 200K words of Central Mernory on t'he comPuter '

6.3,2 Convergence using the Collocation Method

Because of the slow convergence of condition 4.2.8 using the

Galerkin technique, it was decided to try CollocaÈion for comparison, âs,

intuitativelyrbetter results mey be expected from the latter method for

this, virtualty, two-in-one condiÈion. However, as for Èhe results in

SecÈion 6.2, the errors were larger and the convergence slower when

Collocation \das used.

93

FIGURE 6.7: The co'amplitude and co-phase lines for the M2 tide

according to the analytic model of Chapter 4'

1.57.5 o

94

FIGURE 6.8: The co-amplitude and co-phase lines for the K1 tide

according to the analytic model of Chapter 4.

105

0

tIII1

t

t7III

I

IIII

/0

o

\\

I\\

III /

\\\ 2\

IIII

IlII

-tt'

III

///

\ \

_ _19. \ \

ItII,I

\40

o

I

\ \\\\

1819

95

The co-amplitude and co-phase lines for the M2 and K1 tides,

for the model of Chapter 4, are shown in Figures 6.7 and 6.8 respectively.

6.4 The Linear Numerical Model

The boundary configuration of the numerical model is a closer

approximation to the coastline than that for the analytic model' This

can be seen in Figure 6.11. This figure shows the coastal boundary

approximaEion used for the numerical models and also the position of

the open boundaries at which the tidal inputs are specified. I^lith

reference to Table 5"1, the element labels associated with this config-

uration are sho\{n in Figure 6.9. The depths, assigned at Ç-points,are

shown in Figure 6.10. Ax is taken to be 13 krn and Ay to be 39 km'

The boundary approximation shown in Figure 6.11 was found to

be the one which gave the closest results to the observed tidal phenomena'

The seemingly poor approximation on the !üestern side of the Gulf is

consistent with Teleki et alts (1973) observation that "most of Linnnen

Bight, between Groote Eylandt and the Edward Pellew Group, is a shallow

area where the bays and river mouths remain choked with sediment most

of t.he year. Thís is a low energy coast.rr An idea of the islands,

shoals and sand or mud banks in the area may be obtained from Aus

ctrarL 410. The area to the south-East, near Karumba, is modelled as

being wider and shallower than it is in reality. This is to try to

account for the dissipation in the l'/Iz tide. There is a long sand-

bank in this area, shornm on Aus Chart 410'

The input along the open boundary for Ehe numerical model is

obtained from the analytic mode1, there being no data available across

the input channel. It could be possible to determine input values

from co-tida1 and co-range charts as given by Easton (1970), but such

96

data would be inÈerpolations on diagrams which are themselves obtained

by interpolation and extrapolation, and hence the input is not likely

to be very accurate.

The convergence of the model is tested by modelling the exact

system described in Chapter 3 and comparing the outpuÈs for the yI2

tide with that obtained from the analytic model. The appropriate form

of the friction parameter, 5.4. 1, uses ,r,, = .35 ms-I. The results

for the amplitudes are given in Table 6.11 and for the phases in Table

6.12.

The largest discrepancies occur at the corner x = -a, where the

analytic model incurred the largest errors in the matching conditions,

and in regions affected by Ehe amphidrornic points (compare with Figure

6.5), particularly in the bottom right-hand corner of each tab1e. Away

from these regions, the results are in good agreement, the maximum error

in the ampliÈudes being about 3% and for the phases about 57" íf. the phase

is larger than 2 hours. Sometimes the percentage error is larger than

this for phases smaller than 2 hours, but the maximum absolute error is

comparable to that for the larger phases, being about 18 minutes.

Hence, accepting the fact that amphidromíc points are singular

regions in which any linear depth-inEegrated model is likely to be

inaccurate (see Nihoul (f977)), the otherwise favourable agreement of

the numerical model rnrith Èhe analytic model indicates that the solution

provided by the linear numerical model is likely to be convergent to the

true solution for the situation of a more complicated boundary and bottom

topography.

The results, incorporating the input from Torres strait, are

shown in Figure 6.11 for the Nlz tide and in Figure 6.12 for the K1

tide. The results for the linear numerical model which uses the

iterated form of the friction parameter, as given by 5.4'2, are shown in

88888888888888888888888886 11

6123

3

3

3

3

3

3

3

3

3

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6

6

6

6

6

2

6

6

6

6

6

6

6666666666666666666666666666666666666666666666666666

9

9

9

9

10 6

6

2266666666665666666666444

44444884444

1666666666666

6

6

6

1

6666666666

665 6

6

I

6

6

2 6

56666r2222 \o\¡

6

2

6

2

6

2

6

2

6

24444444444t+44444444444

FIGI]RE 6.9: The element labels for the linear numerical model associated

with the boundary configuration in Figure 6'11'

I

*******.**Jr*******}L*******

65

65

70

50

50J

t

¿

65

60

70

30

30¿

+

s

60

55

65

50

30

&

10

10

+

60

55

65

50

45

35

40&

30

20

50

55

65

50

50

50

45

20

35

20

J

50

55

60

55

50

50

50

40

40

25¿

-*

40

55

60

55

55

55

55

50

50

25

¿

J

40

55

57

55

55

55

55

55

50

35$

¿

40

55

57

55

55

55

55

55

50

40

20

40

55

55

55

60

55

55

58

55

45

25

40

55

55

50

60

60

60

60

55

50

30

40

55

55

50

60

60

60

62

55

50

30

40

50

55

55

65

60

60

65

55

55

30

40

50

55

55

65

60

60

65

55

55

30

40

50

55

60

65

65

60

65

55

50

30

5

40

50

55

60

60

65

60

66

55

45

35

5

40

50

55

60

65

65

60

65

55

45

35

5

J.

40

45

50

60

70

65

60

60

50

40

35

5

30

t+5

50

60

70

60

60

54

45

40

30

5

20

40

45

55

70

60

60

50

45

40

30

5

10

30

40

50

60

55

55

45

40

35

25

5

10

20

30

40

50

50

50

40

35

25

25

5

9

10

20

30

40

45

40

30

20

20

15

5

9

10

10

20

30

30

30

20

15

15

10

5

10

10

&

ú

&

J.\o@

FIGURE 6.10: The depths in m, specified at Ç-pointsrfor the numerical model with

configuration shown in Figure 6^11. The *'s shor¡ land, and the

depths correspond to the bathymetry shown in Figure 6.1.

ÊÞtËFt4

IHH

\o\o

P rd0¡ c)Ëo50ÉFrÞãOlJEFl o< gtH.0J rt Fto o F.H.Þ f o oHO 3ÞF1 ^Oo x o.o(,¡ù O ÉtÉ\4 '-lJv afÉ H.:'- Ðoc"0J cj0¡:'p5ão É ÞEo€ H

rfH.*rrOrr!' n ÉoÈts. u)oÊ.Þ 0jH.0' I ^o Ho.H.Þ< Þrt at clP. P.:' ãÞoov

0c FJlJHO F.FrÞø50ÞÉoFtP.uÊ¡.rrl-lû

JH.i oV'ÊãB0qoNH. FI< H.rÌooP.ÞÞo.

HO0lØcräo ooc< ÈP.oo<POrl. Ð

od

525525

.359

.363

.209

.2L8

.095

. 113

x-Distance from Input Boundary (kn)

572

t.l-641. 150

Ll221. 106

1. 0491.031

.949

.933

.830

. B17

.688

.683

520

1.0951.083

1.050r.036

.97 8

.962

.883

.868

.770

.7 58

. 638

.632

. ¿+83

"482

.324

.326

. 180

.184

. 071

.083

.011

.060

468

.948

.940

.903

. 891

837822

753739

.655

.643

.537

.532

.398

.396

.254

.255

.r22

.IzL

.027

. 031

.041

.062

4l-6

.737

.735

.692

. 685

.645

.624

.567

.555

.489

.480

.395

.392

28028L

t57158

.052

.038

.069

.060

.LL4

. 118

374

.486

.497

- t+37

.440

.388

.383

.339

.331

.287

.282

.225

.225

.140

.t44

.053

. 051

.081

.067

. 156

.148

200198

3L2

.256

.297

L792TL

.t2L

.138

.087

.087

.o72

.066

042046

.014

.017

.092

.083

.L7 7

"]-67

.248

.2t+O

.288

.284

260

275326

.zLL

.252

.180"204

.t7L

.178

.L62

.t57

"L42.t28

.t47

.133

.L96" 185

.267

.258

. 330

.324

.367

.363

208

.505

.537

.467

.489

"443.455

"4L9.422

.37 4

.370

.300

.27 6

.256

.237

.278

.266

.336

.328

.394

.388

.430

.426

156

.734

.756

.703

.717

.67 5

.683

.64r

.643

.572

.570

4L4376

319297

.323

.311

.37 6

.368

.432

.427

.467

.463

104

.900

. 910

.870

.877

842845

808808

753759

J

52

.970

.97 5

943946

.915

.9L7

886887

.858

. 859

J

J

&

0

.991

.991

.965

.965

940940

.916

. 916

899899

&

39

117

195

273

351

429

507

585

663

741

819

É:¿

,ôil

à¡..qtÉÉ

oÊq

EoH

l+{

(¡)oË(ü+J(D

. r'lâI>r

.034

.07 5

x-Distance from Input Boundary (km)

572

1. 38t.29

1

1

4942

1. 651.60

L.821. 80

1. 982.OL

2.L42.22

2.282.43

2

2

3967

2.493.01

2.583.7r

2.794.87

520

1

1

3323

1

1

4638

I.621" 56

L.79t.77

1" 95L.97

2.092.L7

22

2L36

2 "282.56

2.302. 88

2.243.73

2

5

3247

468

r.231. 11

1. 39r.29

1

1

5750

L.7 5r.7 2

r.921.93

22

0613

2.L42.29

2

2

1545

L.992.7 2

.66t+.62

9. 137.23

4t6

1 08

4I

r.261. 13

1. 501. 39

L.721. 65

1.911.89

2.042.08

2.O72.2r

1.912.25

.752.07

9

87064

o439

9

I

374

.66

.42

1.00.77

1

I3515

1. 68L.52

1. 951. 85

2.O72.05

1. 95z.o5

.761. 31

9

9

8958

9.289.03

9.028.81

3t2

11.9011. 61

L2.3911.96

.70

.Q7

1. 65.90

2.47I .87

22

8115

10.0511. 56

9.389.52

9.269.27

9.L69. 13

9 .039 .00

260

9.4t9.53

8.939.20

8.308. 70

7.V88. 19

7 .537 .87

7

87203

8.378.65

8. 899.07

9

9

0919

9.119.18

9.059. 11

208

8.408. 53

8. 148.31

7

I9008

7 .737 .90

7

7

6681

7 .797 .99

II

2649

8.768.96

9.029.17

9

9

IO2L

9.O79.18

156

8.018.09

7 .877 .97

7

7

7687

7

7

6878

7.637.72

7 .747 .90

8

8204r

8.728. 93

99

0118

9.109.24

99

1023

104

7.797 .83

7.727 .78

7 .677 .73

7.637 .68

7

7

5863

J

J

52

7

7

6467

7.627 .65

7

7

6063

7.587 "6r

7

7

5659

.L

0

7 .5L7.5L

7.527.52

7.537 .53

7

7

5454

7 .557 .55

J.

39

117

195

273

351

429

507

585

663

74L

819

Êj¿

¡Iàh¡{qt

'ÚÉ

oÊ4

Éoþ

r+t

c)oÉ(ü+J(,

'F{âI>.

Fltr,tr{r{

Its:

HOo

ÞFtlrlÕÉ o !f0ãtsoão€Flo0r0¡ts.p¡ Þ Hoo0tH.OJ fFÚIH <Or1ÞË ^H.oxoo(I)e F¡É< 3t-vQ Ér-r O. 5- OO(t-11Êj €5t-t-.ifoÉÞÞoø

- OOtí- !]. ø

r-t g¡5€^

H.O cf F.I oÞÈ0r Flt.Þ

='.o 0¡ (,FIÊi FJ (tJ

11< q¡ vts.rrÞ.Þ F"o0qOHrrrOHB !f dÊroo5úirlÀÉ F5. lJP.OÉÞ

o|-.or 3grFll\)0qÉH.Ê ó<3P.ooo.tFlo

H.eÊlocrÊr0qO PH.oãoo5ÊÈ!' o úoH<

FIGURE 6.11:

101

The co-amplitude and co-phase lines for the lqz tide

according to the linear numerical model which uses the

non-iterated form of the friction parameter. The open

boundaries are indicated, with a X represeriting the

position of a C, eríd point.

12I 2

\\\

IIIIII

x

I,

I

1

xo0l

x

It

IxI

¡

II

I

II

.ilx

xI

,,I,II

1s\\

I\

2

/

o

,II

.10

III

\\

f

/

,r

\

\8.5r

\\

3tII

,r-

tl,II

\\\

31

tt

II

\\\

40

\10

II

I

11

L02

FIGURE 6,L2: The co-amplitude and co-phase lines for the K1 tide

according to the linear numerical model which uses

the non-iterated form of Ëhe friction parameter.

,II

I

I

I

o

5

1qI

o

//

/12

I\ I

I

I

II

ItII

20

I\ \f8_\_s

*

/f

/

o

\\

\\!" \

II

I

II!I

¿-

20

2',|

103

FrcuRE 6.13: The co-amplitude and co-phase lines for rhe þr2 tide

according to the linear numerical model which uses

the iterated form of the friction parameter.

100 2

8lI

I

IIII

oo

./o

ItI

1OO

\rlt,

4

--"-

o

\

o8¿/

't2tI

II

2-'

80

I,

\\Iì

O¡I\

III

\

II

I

\\I

-'i,

4tIII

I\lta, i

40

II

o

o

III

Lo__ _

t\10.5

10

3

104

FIGURE 6.L4: The co-amplitude and co-phase lines for rhe K1 tide

according to the 1ínear numerical model which uses

Ëhe iterated form of the friction parameter.

10

t7

IIII

II

o

0

Jt

IIt

\lo.,

//'ì

I

ItII

o

//f/

IIII

Io

t

III

\\

\16

,,a

121

18

2

20

1

105

Figures 6.13 and 6.14. For the N12 tide, which has the dominant velocity

component in the region studied, the friction parameter converged at each

velocity grid-point after 11 iterations. For both linear models, the

maximum error in the end-condition, after the EVP solution had been deter-

mined, \¡Ias 2 x 10 5

6.5 The Non-linear Numerical Model

The labels ín Figure 6.9 and the depth array in Figure 6.10

are also used for the non-linear model.

The value of À , in the fríction Parameter form of 5.5'3, is

taken to be .00878, corresPonding to a value of .030 for Manning's n.

The time increment rrhich satisfies the stability condition 5.6.4 is taken

to be 120 secs for the Yt2 tide and 239 secs for the K1 tide, the

number of time-steps per tidal cycle being roughly the same in both

cases. As for the linear model, the input daEa was obtained from the

analytic resrrlts. The model took 13 tidal cycles to iteratively converge'

that is, Íf p ts the number of sÈeps per tídal cycle'

trfl¡* tt - ,Íll I . .001, .c = 1,...rmi j = 1,.-.rni 6'5'1

when v/p = 13. The output during the 14th tidal cycle is stored on a

random-access file, using the Random Mass Storage package on the computer,

and later Fourier-analyzed to obtain the fundamental frequency (and

harmonics, if desired) using the efficient program detailed in Ralston

and Ìlilf (1960).

At first íÈ was thought that the long model time required for

iteration convergence mey be due to the method of modelling the advection

terms, but removal of these terms from the equations had no effect on

the time for convergence. This is reassuring from the poinÈ of view thaE

106

the advection terms should not have a very large effect on the results

(see Flather and Heaps (1975)).

The iteration convergence time seems to be dependent more on

the type of area being modelled than on the manner in which the PDE is

approximated by the FDA. The model of Morecambe Bay developed by

Flather and Heaps (1975) takes 11 tidal cycles to iteratively converge

in the manner of 6.5. 1.

The long model-time for convergence in this study is probably

due to two factors:-

(i) the large model region

and (ii) the input wave has to svreep around Èhe corner into the

resonator region and, because of this, the transient motion associated

with the wave and its reflection may take a while to die ar¡ray"

The truncation convergence, that is, the property of convergence

discussed in Sections 5.3.3 and 5.6, is investigated by comparing the

results for the NIz tide with the results obtained from the linear

numerical model ¡¿hich used the iterated friction parameter" The

agreement between the two, shown by a comparison of Figure 6. 13 with

6.15, is exceller¡s, The only discrepancy is in the South-Eastern

corner where depths are only 5 m and the advection terms are likely to

be c.rf more importance than elser¿here in the Gu1f. Such a comparison

shows not only the convergence of the non-linear mode1, but also the

value of the simple linear model,

The result for the K1 tide is shoqm in Figure 6.16. This

shows favourable agreement with Figure 6.14, any differences arising from

the fact that Ehe linear model accounts for interaction with the 142

component, through the manner in which friction parameter is modelled,

while Èhe non-linear model does not.

107

FIGURE 6.15: The co-amplitude and co-phase lines for the l,l2 tide

according to the non-linear numerical model.

ItIttII

oo

B

I

.la

10

0

\\It2

arf

o

\\

o

t

4

,I

12t

80

\II

4tI,I

2\

\\

I

\\ \

I

.tI

III

\l

/zI-

IIII

\ \\

10

III

o

I

\

4010.

10

B

3

108

FIGURE 6.16: The co-amplitude and co-phase lines for the K1 tide

according Èo the non-linear numerical mode1.

o

¡

I

40

I

l7

II

101

IIII

5\\\

I

f

--t'l

It

\l

Itt

o.,

It

fI//

II

II

\\

/I

:.16

I21 I

II

/12

50

4\

18

\-----20

109

TABLE 6.1-3: Details of the requirements of each model

on the CDC CYber 173 ComPuter.

MODEL TIME( secs )

CENTRAL MEMORY(r words)

LINEAR ANALYTIC

Torres Straít closed

Torres Strait open

LINEAR NUMERICAL

simple frictioniterated friction

NON-LINEAR NT]MERICAL

basic program - 14 cycles

Fourier analysis

53

72

6.4

2L.8

1024.t

5.4

100.2

140

60

60

54

60

110

6.6 The Prosrams

The programs for all the models described were written in Fortran

IV and run on the CDC Cyber 173 computer at the University of Adelaide.

The results for the analytic models hrere obtained using N = 6. Table

6.13 indicates the time required (in secs) Èo run each program and the

length of Central Memory needed(in Kilowords).

6.7 The Response of the Gulf to Tidal Forcins

The tidal response predicted by the models is now discussed in

more detail.

6.7.L The Semi-diurnal Response

The semi-diurnal response of the system is investigated with

reference to the l'Íz tide which has a period of. I2.4 hours.

The response predicted by the analytic model of Chapter 3 is

shown in Figure 6.5. It features three amphidromic points, one in the

junction which is clockwise and two in the resonator, Ëhe one near

Karumba in the South-East being clockwise, the other anti-clockwise. This

agrees fairly well with the results of Williams (tglZ) wittr the difference

that the features that he suggests should appear in his results (namely,

that the amphidromic point in the juncËion should be nearer to the

resonator than to the boundary y = b, and the existence of Èhe amphi-

dromic point near Karumba) in fact do appear in Figure 6"5'

Figure 6.7 shows the N2 tidal response when flow through Torres

Strait is allowed. This flow forces the three amphidromic points to

contract to one, frear Karumba. Apart from this, the response is si¡nilar

to that of Figure 6.5. This result is analogous to the effect of allowing

flow through the StraiLs of Dover in a North Sea model (see Nihoul and

111

Ronday (1976)) where the overall response is similar when the Strait is

opened or closed but the position of the amphidromic point is changed

when the Strait is opened. The phase distribution varies accordingly.

Figures 6.11, 6.13 and 6"15 show the response as predicted by

the numerical models which more accurately approximate the boundaries

and the depth contours. The three amphidromic points reaPPear. That

the first analytic model agrees so well in this respect with the numerical

models is deemed to be completely fortuitous. It may be personal inter-

pretat.ion by interpolation since the region inside the .2m contour contains

a very narrohr region of amplitudes less than .lm in both analytic models.

The three amphidromes which appear in the numerical model (and thus the

features suggesÈed by williams (1972))are likely to be due to the bottom

topography which causes this region of small amplitudes to bend away from

the deeper water to the North-East of the resonaÈor. The linear model

which uses the simple friction parameter agrees well with the other

numerical models. It, in fact, gives a more accurate response at Karumba,

predicting an amplitude of .18m, while the othet two models predict an

arnplitude of .4m (the measured value is .17m).

All rnodels predict the peak amplitude of the Gulf to be at the

point (0,0) as r¿ell as high amplitudes in the South-hlest of the Gulf.

The amplitudes and phases for setrected positions,as given in

the Australian National Tide Tables for 1978, are shown in Table 6.L4.

Keeping in mind that these observations are made at sites which may be

subject to local influences not capable of resolution in the model (one

grid element represents a surface area of 2028 kmz), the following

observaËions may be made:

(1) The movement of the tide around the perimeter of the resonaÈor

agrees with observaÈion in that the tíde at l{eipa lags behind that at

Karumba and the tide in the Northern region around Caledon Bay lags behind

Ehat at Port McArthur i.n the South.

,L12

TABLE 6.L4: The ampliÈudes and phases for the ylz and

K1 tides at selecEed sites, as given in the

Australian National Tide Tables, 1978'

LOCATION 142 Kr

JENSEN BAY

},IELVILLE BAY

CALEDON BAY

PORT LAI{GDON

PORT McARTIIUR

I(ARUMBA

T{EIPA

WEDNESDAY IS.

PHASE(trrs )

AMPLITUDE(m)

PHASE AMPLITUDE

7"5

I 3

0

2

6

2

1

3

9

9

1

6

5

1

.91

.80

.50

.26

.4r

.t7

.36

.40

5"0

5.8

2.7

23.2

23.6

22.O

14.4

14. 1

.32

.26

,2

.15

.41

.91

.46

"56

113

(2) The amplitude response at Inleipa is too high in all models.

Thís site is located near a relatively steep bottom slope and a better

representation of the bathymetr:y in this area may improve the results.

The amplitude along the perimeter in the Southern half of the Gulf is

generally in good agreement with data in the analytic results but too

high in Èhe numerical results. This is surprising, as a closer aPProx-

imation to the boundaries and bottom topography should give more accurate

results.

Using the input from an analytic model with a different Gulf

width produced no significant change in the response of the numerical

models. Increasing the value of the fricEion parameter will decrease

the arnplitudes in the South of the Gulf, but Èhis has a detrimental

effect on the amplitude of the diurnal tide and is hardly justifiable on

the results of the analytic model. It is probable that the input provided

by the analytic model is sÈill not accurate enough. This could be improved

if the reference point for the scaling of the response could be chosen away

from the mainland or islands. Ilowever, no data is available at such sites

and so this could not be tested. The reference points were changed to

other mainland sites but Jensen Bay and I'lednesday Island gave the best

results.

6.7 .2 The Diurnal Resoonse

The response of the system of chapter 3 for the K1 ride is

shown in Figure 6.6. There is a single amphidromic poinÈ about which the

rotation ís clockwise, in keeping with observations. I^Iith the opening

of Torres Strait, the location of the amphidrome moves South from just

outside the resonator to just inside. This is shown in Figure 6.8. The

bathymetry introduced in the numerical model moves the amphidromic point

further South-EasÈ. This is in agreement with lüilliam's (L972) suggestion

tL4

that the amphidrome lies 240 km in a direction East-South-East of the

position, towards the centre of the channel-junction boundary, predicted

by his model, As the amphidrome moves further into the resonator, the

phases in the l^lestern and Southern parts of the Gulf change accordingly,

providing better agreement l¡ith observed values.

Comparison with Table 6.14 shows Èhat the amplitude at l'leipa,

Karumba and Port McArthur is too small while the agreement in phase is

quite good. The phase and amplitudes at Port Langdon and positions

north of this agree quite well with observations'

The results of the Sz tide were similar to those of the Nl2

ticle, and the 01 tide similar to the K1 tide; hence the results are

not presented here.

Since no data is yet available away from the rnainland or islands,

there is no conclusive evidence as to the quantitative accuracy of the

models in the interior of the Gulf. However, there is good agreement

amongst the models, givíng an indication as to the main features of the

response of the Gulf to tídal forcíng. The discrepancies with regard to

specific observed tidal phenomena rnay possibly be due to lack of detail in

coasEal boundaries and in the bottom toPograPhy near the coast' As mentioned

in Chapter 1, the water motion in the Gulf is influenced by a variety of

faclors and it may be necessary to incorporaEe some of these, for example,

a horizontal density gradient from South Èo North, or the effect of winds,

to more accurately predict the Èidal motion in the Gulf.

The results of the urodels in this thesis could be used in more

LocaLízed studies with a grid refínement such as used by Ranrning (L976),

giving a better approximation Èo coastal boundaries and bathymetry' Thus,

if a model of the Gulf included more detail of Lir¡nen Bight ' a more accurate

study of this area could be obtained by using a finer grid and inputs from

some outer boundary to the East of GrooÈe Eylandt, Ehe inÈeraction between

115

Èhe coarse and fine grids being account.ed for in the manner described

by Rarmning. The fine grid model would, of necessity, be a non-linear

model because of the importance of the advection terms in shallow coastal

areas

llhereas it is possible, by including features and refinement.s as

described above, to improve the models in this thesis, it is considered

that one of the factors limiting the accuracy of the numerical models may

still be the ínput along the open boundary. If the reference locations

of the analytic model could be more ideally chosen, € more accurate input

for the numerical models could be obtained.

116

CHAPTER 7

CONCLUSION

Two analytic models to determine the tidal propagation in a

resonator-channel system have been developed in this thesis. They are

based on the two-dimensional depth-inÈegrated equations of continuity and

momentum conservation which govern fluid flow. The rnodels have been

applied to the Gulf of Carpentaria, Australia. The second model accounts

for flow through Torres Strait and shor¿s the ínfluence of this Strait on

the position of amphidromic points and the subsequent effect on the phase

distribution in the Gulf. There is little change in the amplitude

response of the Gulf with the inclusion of this second channel.

As well as giving a good indication of the general features of

the tidal response of Ëhe Gulf, these models are useful in providing a

comparison for the two numerical models which are developed to approx-

imate the boundaries and bottom topography more accurately. The second

analyÈic model also provides the input along the open boundaries for the

numeríca1 models since there is ínadequate measured data available.

The two numerical models use finite-difference approximations to

the two-dimensional equaÈions, the first being linear and the second,

non-linear. The results of the linear model agree very well with those

of the non-linear model, indicating the usefulness of linear schemes

r¡hich model Èhe friction parameter judiciously. They also indicate, as

would be expected, that the advection lerms are not important in Èhe

interior of the Gul-f .

LL7

Table 6.13 supports the usefulness of the linear model, which,

as well as providing good results, has a much smaller running time

on the computer than the non-linear model. It, thus, would be an ideal

model Èo provide inputs for more localized studies utilizing a finer

grid resolution. These localized studies would use the non-linear

numerical- model to give better quantitative agreement with the data

avail-able near the mairrland.

r18

APPENDIX 1

The esentation of BoEËom Friction

Integration of the general three-dimensional equations over depth

introduces the surface and bottom stresses (see, for example, Dronkers

(1964),Nihoul (1975)). The generally accepted formula used for the

latter is (see Groen and Groves 0962), Nihoul (1977))

Iu = - tI" * YPllqllq

where Ju and L are the bottom and surface stresses, respectively,

m and y are empirical constants and g i" the horizontal velocity

at some reference heíght above the bottom (henceforth termed the ilbottom

veloclÈyt') or the depth-averaged velocity and p is the densfty of

Ëhe fluid. This fott"f" includes a stress exerted on the bottom

even aÈ times when q = 0. However, Ín tidal models where Èhe ef-

fects of wind are neglected, I" t" taken to be zero.

(a) The Non-linear resentation kll

Since, in two-dimensional models, the value of the bottom velocity

is not kuown, non-linear equations use the form

Fo(= tolott) = kllqllq A1. 1

-1where H 1s the water depth and k has dimensíons m ' Ín rnks unlts'

As a ffrst estimation, k-rnay be given a consÈanÈ value, equi-

valent to considering a reglon of constant dePth' Thus

k=y/h

wh-ere h 1s the depth of undisÈurbed water and y ís dimensionless'

Taylorfs (1920b) study of dlssÍpaËion 1n the Irish Sea used y = '002'

119

Giace (1936,1937)

attempted to determine appropriate values for Y by using measurements of

tidal eleva¡ions along the coasts of the Bristol and Englj-sh Channels'

He found values of Y ranging from .0014 to .0041 (average value of

"0026) for various sections of the Bristol Channel, and from.0024 to

.OZL (average value of .0093) for various sections of the Engl-ish Channel.

The larger values for the English Channel l^tere associated with large apparent

phase differences betqreen the current and t.he frictional st.ress and, according

to Bowden and Fairbairn (1952), are "probably less significant than the

Bristol Channel results." Bowden and Fairbairn give a value of Y = .0018

when using the mean current in a hrater depth of abouÈ 19rn in their invest-

igation off Anglesey, whereas, in a later paPer (1956), they find an average

value for ] of .0024 when using the current at a specified height above

the bottom, Ëhe depths ranging from 12m to 22m'

Numerical models ofËen use (see, for example, Flather (1976))

k=y/H

v¡ith H = ft * Z , the toËal depth of water. Values for Y which are

conrnonly used lie in the range

.0024 ( y ( .0030

(see Dronkers (L964), Nihoul and Ronday (1976)).

A perhaps more realistic formula considers the roughness of Ehe

bottom material. It is a combination of the de Chèzy and the Manning

forrm¡lae (see Dronkers (1964, p.156)) which were originally developed for

the study of channel flows:

n=õä^L.2

L20

whererl6

1 .003 R' Yzc=-m n

R is the hydraulic radius (usually aPProximated by H for a

shallow sea)

n is Manningts roughness coefficient which varies with position

according to the roughness of the bottom

maEerial.

tlang and Connor Q975) give .025 < n ( .040 and their subsequent

calculations of the bottom friction parameter for different depEhs

(1 < h ( 10Om) and different types of bottom material yield values of k

in the range .0013 - .0095. Harleman and Lee (1969) use values of n

as lorv as .020.

Using R = H, AL.z can be written

u = X/n43 41.3

where À depends on n.

In channel or river flow studies where Ehe region of interest may

be djvided into a seríes of one-dimensional secÈions, the value of n may

be varied in each section unÈi1 the results obtained agree with observations

This is the approach of Harleman and Lee (1969). However, such systematic

variation of n, in the case of a two-dimensional shallow sea model, is

not always feasible and À is usually given a constant value. A1'3

is used by Teubner (1976)rwho considers a value of tr corresponding to

n = .030, and Prandle (1978)rwho uses ll = ,O25.

Leenderts e (1967 ) says that when the bottom roughness has a

considerable influence on water movemenÈ "the parameter C has to be

found in an iterative manner by comparing results with actual field

measufements. I' He obtains

C = 19.4 9-n[0.9H]

-1s

t2L

experimentally from computations of his Haringvliet mode1. However,

as previously stated, such experimerrtal evaluation of c is not always

feasible, especially when Èhere is a paucity of field data available'

some, more sophisticated, models return to the definition which

employs the bottom velocity. using vertical velocity profiles adjusted

to observations, the reference velocity can be expressed in terms of the

depth-averaged velocity. Thus,Nihoul and Ronday (L976) quote Ronday

(L976) as using

0¡Y=

ç¡.2r+e-a o':lr'

where cis is a constant and z0 is a roughness length. This can be

obtained from a velocity profile of the type due Eo vori Karman (see

Dronkers (1964, p.156)). Dronkers gives zo = '03d' where d is a

scale for the height of the irregularities of the bottom'

The non-linearity of the friction term kll qll q provides one

mechanism for the interaction between different tidal constitutents and

for the generation of harmonics. This effect of the quadratic 1aw has

been studied by, amongst others, Dronkers (1961) for the case of a two-

dimensional mono-periodic tide, and by Le Provost (1976) for the case of a

multi-periodic tide. Le Provostrs investigaÈion of the components of

the friction in the English Channel lead him to conclude that "for a

first approximation, the M2 component could be studied alone uPon a

given atea, but that to study a secondary \¡/ave, 52,N2,K2, it is necessary

to consider their propagation together with the Mz comPonent""; a

simulation taking together YI2 and sz or ulz'sz and N2 gave a

better representation of the componenE l{2 ." His analysis depends on

the presence of a dominant tidal component. However, such a dominant

componentmaynoËalwaysexístrandrinsuchacase'acompletepictureof

122

the Ëidal motion can probably only be obÈained by considering the whole

tide, a Fourier analysis of the results providing the componenÈs if these

are required. The main deterrenÈ against such studies is not only the

lack of adequate data along open boundaries, but also, if such information

did exist, the data over about 15 or 29 days would be needed as input for

a model (see Defant (1961, p.304)). This is necessary to account for

the effects of such factors as Èhe semi-monthly inequality and the contrast

between the spring and the neap tides. Then' not only would the model

have to be run long enough to converge numerically, but it would also

have to be run for a further period to provide the necessary output. Time

and cost obviously preclude such a study. One of the advantages of a

linear representation of the frictional force (in linear equations of

motion) is that the complete tide may be estimated from the suPerPosition

of the solutions for the individual constitutents.

(b) The Linear resentation of kll q

Many models tave linearízed the quadratic bottom friction law for

Èhe sake of simplicitY, taking

q or A1.4

In MKS units, r has dimensions *"-1 This linear representation is

essentíal for analytic methods of solution which do not rely on perturb-

ation or iterative techniques. The second exPression of 41"4 is

sometimes used in linear numerical models which still retain explicit time

dependence,

Most studies take r to be a constent, for example, Heaps (1969)

uses r = .0024 rns-l while Flather (1972) uses a value of .0014. Real-

istically, the value of r will not be constanÈ, but will vary with

pos ition.

Io =ågrIF

-b

t23

The linear expression q can be related to the quadratic law

ftCffC The LorenEz approximation for r , in the one-dimensional case,

is found by equating Ehe dissipatíon over a tidal cycle given by each

of tl¡e expressions (see Proudman (1953)). Thus, it is found

th

where q = U cos trtt.

Ir==-JlT

It=-'3n

YU

åt"*

Harleman and Lee (1969) use

where C is the Chèzy coefficient which can vary with position and t*,. is

an over-all estimate of the maximum velocity. Dronkers (1964, p.191) gives

the two-dimensional version of this,

8s'=ËÉu_ Al.s

where V_ is the mean value of the maximum magnitudes of the velocities,

if it may be assumed that V- does not vary greatly" If C is given

a constant value, 41.5 may be used as an estimate of r in an analytic

analysis of a tidal region.

Dronkers (1961), produces a lineatízed form of the quadratic

friction term for the tr¡o-dimensional case of a mono-periodic tide and

shows that, not only the magnitude of Èhe velocity, but also the relative

phases of the two velocity cornponents should be taken into account.

Us ing

u = U cos(6t+Cr)

v = V cos(ot+ß)

he finds, on neglecting the harmonics which arise from the quadratic law

L24

u(u2+v2 )!' = xu. r #

v(u2+v2)/"=À.r-r#

where À and m are quite complicated functions of U,V,o and $. This

form is only useful in idealized studies when (for example, if only Kelvin

waves are considered) exact values for À and m may be found. However,

use of this linear form is usually precluded by the fact that foreknowledge

of UrVrOt and ß are required and these are, in fact, unknowns of the

sys tem.

prandle (1978) uses a non-linear Èidal model in conjunction \^tith

a linear model for secondary effecÈs (.such as those of wind) and so is

able to relate, at each grid-point, the linear friction coefficient K(=;)

of the dominant constituent (say Mz) to tire non-linear law' He finds

K at a u-velocity grid point by rninimizing € with respect Èo K' wheft

)r0

2

e2 dr

where T is the period of the NI2 tidal constituent,

and Ce is a frictional coefficient corresponding to Il = .025.

An analogous expression is used at a v-velocity grid-point.

However, for obvious reasons, this approach cannot be adopted in

linear tidal models and the best approximation to the friction coefficient

is probably given bY 41.5.

The effecÈ of a dominant tidal constituent on other components

is easily accounted for with a linear friction representaEion' Jeffreys

(1976) shows thaÈ, for two tidal velocity consËituenÈs Ulcos t!1t and

u2cos (s2t, with lJz/Üt < \' the frictional comPonent with frequency (¡J1

is

,, = *.{ uÎ "o"r-Dlt = T ut cos 01t,

125

while the component at frequency tr2 is

cos (r)2t cos (t)2t

Hence, if r1 is a frictional coefficient for a dominant Eidal constituent'

the appropriate linear friction factor for any other constituent is

F,=+{u'u, =?u,n

1 5 r A1. 6Tz

Garrett (I972) states that this result is readily extended to Èhe

two-dimensional case when "tidal ellipses (are) of the same eccentricity

for all constituents being considered." This interaction of constituents

is not so easily taken into account with a non-linear friction law.

(c) Comparison between the two representations

Although, according to Nihoul and Ronday (L976), "it is now

connnonly admitted that a quadratic law must be used", Durance (1975)

justifies his use of the linear law thus: "Although there is evidence to

suggest that the boÈtom friction does depend quadratically on the velocity

near the bottom, there is no direct relationship betweem the near-bottom

velocity and the mass transport, and in some situations they can be in

opposite directions. In addition, Èhe bottom friction coefficient is

likely t.o depend on position because of both the general bathymetry and the

variation in bottom roughness." The investigation of velocity profiles

by Johns Qg|6) and NihouI (1_97 7) would seem to substantiate that, about

the time of tide reversal, the validity of the quadratic law is questionable;

however, Nihoul shows that Ehe discrepancy does not affect the results

significanÈ1y.

Flather (1972) compares the results of a linear scheme to the

results of a non-linear scheme applied to the computation of the Mz tide

in a rectangular sea 65m deep. He finds Iaxge differences in the NI2

L26

amplitudes obtained by the two models, but says that this is "probably

due largely to the choice of friction parameters" (he uses r = .0014 and

y = .0025). Noye and Tronson (1978) show that with a judícious choice

(through trial and error) of the value of the linear parameter, good

agreement can be obtained between the results produced by the two models.

Apart from simplicity and the possibility of superposition of

solutions, the linear friction coefficient has the advanËage of being

more easily able to include the effect (shown by Jeffreys (1916) and

Le provost (1976)) of a dominant tídal constituent on the other constituents.

However, if the non-linear effects in shallow l¡¡ater are of special interest,

it seems more important, in some cases, to include a non-linear friction

law than to include the non-linear advection terms (see, for example,

Flather and Heaps (1975)).

In some cases, as in Leendertsers (1967) Haringvliet study

(where the maximum depth is 13rn), the bottom roughness influences the v/ater

movements to a considerable extent and it would be expected that a quadratic

law r¿ould be essential. Even for the non-linear representation, careful

estimates of the friction parameter are necessary and usually have to be

found in an iterative anner, comparing computed results with actual

field measuremenEs.

No matter which form is chosen for the representaEion of the

bottom fricËional force, no maÈter how complicated the analysis used to

obtain it, there is always some empirical facLor associated with it; and

it seems that the justification for any choice of parameterization lies

solely in the accuracy of the resulÈs obtained by the model.

r27

APPENDIX 2

The Galerkin and Collocation Methods

The exact solutíon to a differential equation and its boundary

conditions cannot always be found and an approximate solution must be

sought, either by analytical methods or by numerical methods. An

analytical approximation may be obtained by the Method of l^leighted

Residuals (see Finlayson (1972)). The two techniques discussed here, the

Collocation and the Galerkin methods, belong to this class.

The classic approach of these two methods ís to find an infinite

series of, for example, trigonometric functions which satisfy at least

some of the boundary conditions exactly, and to Proceed to solve for a

finite number of unknown coefficients in the series by approximately

satisfying the differential equation and any remaining boundary conditions.

The Galerkin technique used in this manner is described in detail by

Fletcher (1978).

However, if it is possible to find a solution to the differential

equation, Shuleshko (196la, 196Ib) has shown that better results are obtained

if the approximation method is applied to the boundary condition rather than

to the differential equatíon. This is the technique used here.

Consider the Helmholtz equation

(v'+x2)e = o A2.l

with its boundary conditions

M e [6(x,y)] = 0o (*,v) along s p 1r... rPp

where M_ are linear differential operators and "o is Ehe pth portion

p

of the boundary. The solution to the differential equation may be written as

6(x,y)

t28

æ

Ç, n (x,Y)

P

¡ nI n=0

where ã, ^ are unknown coefficients and each term, 4, n ,Jn

satisfies 42.1 exactly. It is also requíred that

in the series

ejn] -0p= Q along S A2.2

n=0

For some boundary conditions it may be possible to find a simple

relation between the tj' such that A2.2 is satisfied exactly; but, if

not, an approximate solution may be found by truncating the infinite

series; in which case

P

MtIo j="t

æ

P = 1r"-rP

Mp [6(x,y)] - Qo(x,y) =r (x,y), along s P 1r...,P

p p

whereP

7=\5L

j =t

N

I",n=O

w =6(x-x

Çjn

,Y - Yrro )

n

and r is the residual, or the error in the pth boundary condition'p

It is expected that the residual be sma1l in some measure and that the

effect of obtaining a new solution with N increased should cause a

reduct.ion in r in some average sense along s . Inlhen t, = 0, f.otp '---e P P

all p, the exact solution has been found'

The meEhod proceeds Lo solve for the â, - by imposing an ortho-

gonality condition on the ro (x,Y),

Jsp

ro(xry)Ø,ro(xry)ds = O, f, = 0r...,N for P = 1r"''P A2'3

nrhere ds is a line increment along "o and 'r,o

function. If , as N + -r{rrro} is a complete.set,

is the exact soluÈion. If

is some chosen weighting

rhen r =O and ¿p

np np

where ô(x,y) is

are points along

t29

the Dirac de1!a function and (*no ry,,o ) (n = 0r.. .rN)

S , use of 42.3 is called the Collocation Method. Ifp

np M te. l,p Jn

then use of A2.3 is called the Galerkin technique.

Application of 42.3 results in a set of linear simultaneous

equations which may be solved for the "j '

The advantage of Collocation is its simplicity; no integration

or mathematical manipulation is required to set up the simultaneous

equations. However, the accuracy of the solution depends on the position

of the Collocation points (see Shuleshko (196la)). Chapter 6 shows thaÈ

the overall accuracy and convergence using the Galerkin method is better

than that obtained by Collocation. Since the mathematical manípulation

for the Galerkin method need to be done only once, even if physical constants

are varied, it is the Galerkin method which has been adopted for the analytical

approximations in Chapters 3 and 4.

IÀ) = j € {1,...,P} for î = 0,. N

130

APPENDIX 3.

Evaluation of the Intesral Form for (" (x.v)

In Chapter 3, er(x,y) is expressed in integral form by 3.4.13'

that is,

Çz(x,Y) = * [- II

e-ilx {Àgo +iKrfo]).2+Kz

I0(À)dÀ

+ .-i"(**u) {go[À cosh Kra - iKr sinh Kle]

f s[À sinh K1a - iK1 cosh K1a]tSft oi

{Às+i4r}anæ

In

e

æ

[ {-r)"

x2NT

0( À) dr

æ

I (-rrnn= 1

a(

-i ÀxIJ-

f

+

n=1

with

0( À)

and

This may be written as

-t ¡\( x+a)e

À0 sinh s( -b * " (1+i ) cosh s( -b

0 _S 1+i z) sinh sb

-¿(x<0

2s =\2

Çz(x,y) = Q{-,21T' ^uo T+-Lo+

A3. 1

L3.2

and is evaluated

(n>0) isan

- ur2

2

iÀx

Lr')]'+I

r"r,

where I,rrr(n > 0) ís an integral associated with e

using a contour closed in the uPPer half-plane; and

integral associated with "-i À(x+a) and is evaluated using a contour

closed in the lornrer half-Plane'

For

131

r"o' the poles of the integrand are given byru

I iK1, (s

o

)2

2

4 -('J l'.I

x2t(+l '.)nfi

d

Yz

= Kz)

43.3a

A3.3b

43. 3c

sln

À

À

í(Kî so)+ X

'[a,r*iq,r]/' ,(" 0I r¡2t-

Lel,',

r/"

1kÍro)

À = 1 k(2e)

where the aPProPriate value of s is

chosen for A3.3b,c is such that trn(k

lz

defined as shown and the branch

( c)

(s

2

1Lrb

g>0)

) > 0, Lr 0

The poles assocíated with l,rr,,I"n (n > 1) are given by

À 1 S=

À

À

+ k( o)

2

+0¡ 1,>0

I

i

{-K, xe' -f -i0Kr sinh s +s

zI

( 1+i0z sinh s¡b( t+i ) cosh s -b

k(2

Noting that In(1K ) < O, 43.2 may be written as a sum of residues

IUo

ILo

2rí

2trí Res(-iKr) + ne"(t!o) ) n""(tta) ) )+9.

+K

( (

2 U S0

( o)-ik x+ K, ( v- b) + Ai- b -([)

g-n k,It rr-ik x trrll.

]J+ô eoo

ôol e cosb b

e

2rí nes ( iKl ) + Res,-n(o) ¡ + Res(- . ( [)Ki )

l.I22

9

(

{K x +f i0K sinh s ( t+i cosh s -b))+s

2trí e 2l0zr + S 1+ þz sinh s ob0

ie ++€

oo

k( 0) x-Kr(v-b)

9.Ï, ."0 .'nirr.["o, + .,.iõ,, * ulo"t" +]]

L32

Un

III = zTtí ttes(tlo)) *

æ

TL

e-nes(r.!e)) + a*."(T). ,*""( T))

2ntt e

+

( o)IK x+Kr(V"b)

t Ît

ôno

I[-

ônl wb

. .,.qrot # nlo' "t,,+]-i k

þex

2 OS

nn0l_1 -

'a. n'lTX /sin

- sl-nh s rY-b) +

anS

n(1+i0z)"o" lII cosh s"(v-b)1

ob+ 2 [(nn A/a)z - s¿n

( t+i6r ¡ , lsinh s bn

nTlX/r\cos

- slnh s (y-D/

an1S (1+i0z)sin 4E cosh s.(y-b)1.nrt0t, n+íf 2 [(nn 0/a)' ( 1+i0z )'l sinh s b

tfn

=-27

sn

I(

f

i{n""(-táo)) * [t Q= r

nes(-r.Ío') * zn."(- T). ,*"" g))Ln

z" i{eL n.

i kÍo) x- K.( v- b)e

enI e

. . ( Q)l Kz x cos UJ - , 9, . =b u{Q)b ( 1+i02 ) r.r 2

W.tie

+ srnb

rm0[-r - d "irU sinh s (v-b)

an+ s (1+i0z ) cos ü

"o"¡ 5An

(y-b ) 1

+çÞ 2l( / a)' s' t+iþz )' l sinh s bnn n

n1T0 nll x

- ccs

AJsinh sn (v-b) i s (t+i0z) . filTX . ¿srn-coshsrY-b)l

n n

+ if2 [(nnn 0/a)2 - s2 (r+i0z)21 sinh s b

n

where €nQ,ônl (n > 0, .Q, > 0) are linear combinations of the gn and fn'

theit actual form being immaterial'

Thus, using 43.2 ,

133

(.2(x,y) = nt{(Ï")"t

-1 o) *' *, ( v= b)u^, )

I o)-ik.'x+K.(y"b)

e€ . (Ï,no

. -1, (ì, e

t ll

).ik Lry

bx

2

)cosh K

Lrb . t rlKz sln W

b

x cosh s ¡)

cos.l

. -1,(Ï, u"o)" nla)'.["'" T. t,-Îõt # u!'', "" +]+go ieK sinh K x sinh s -b) + s (t+

0 K +S 1+ þz) I sinh s¡b0

ieK cosh K x sinh s _¡) + s (t+ito K +S 1+i0z ls nh sob

I

)sinh K x cosh s -b(+ fo

ó nrO ntxL-:. ;_ s1n

-sinh s,r(y-b) * ",r(1+i0z) cos S cosh s.(y-b)J

Ie"n= I

[ (nn0/ a)2 snz(1+iQ2)2 s inhsbn

-nn0 nTTx[; cos

-sinh s,,(v-b) - i "n(l+iÔz)sin

S cosh sn(v-b)l-1 i

n

f 2 - s2( t+i4r¡'1 sn

inhsbn

Ifn [ (nrO/a)

where it has been assumed that the order of summation of the series

associated with €rr[ , ôr,I may be interchanged '

This expression may be rewritten as

6r(x,y) = Eo "'*)"'*-K'(v'b)

+ Go ejuio)*+K2(v-b)

(r) r'rrb #uto)sinry]

A3 .4

oo

+lEL9[= t

æ+[%!=l

"ikzx cos

cos LTIIb

+-*(1)*fe2L

sinh Krx cosh s¡(Y-b) +

"tÎõJ # uf'o' "t" T]

["o"t, Krx cosh so(y-b) . çffi sinh Krx sinh so(y-b)*Do

+Fo

+

i0KS9 1+i0 2

cosh K1x sinh sr{v-t)]

@

I2

.i9.=t'

Dt ["'" T eosh so(y-b) ffiJ ,l "'" T sinh sprr-o)l

Ft ["r" T cosh so(y-b) . dt 'tn

"o" [rï-

"ir,¡ "e{v-u)]A3. 5

134

wirh

(Do ,Fo )

(to,ro)

f-l i so(t+i4rt0 K +S 1+i02 ) I sinh sob

(go,fo)I 0

CIi so ( 1+iQ2)

[ ( l,nO sf(1+i$r)zl sinh "Qb

+(g[,fp), 9-e za

_ ( 0)K2

( t)2

Sg

"I

K2

(Ti

1 K2

,{

9"ez

ffirr+io,)Ì

k -(ilj",u'=*{.;

%

X+ 2

2

^,,\,"Xrl={+

(r)2 I

-sh2 (t+îõz)

l.*Í

The Classes of Elements for the Non

135

-Linear Plodel and their Associated

APPENDIX 4.

Finite-oi f ference Equations

1

2

3

Z,

u[

I n+ 1)Zp','¡' ' as in equation 5.5.2a

u[,;" = u[,ì #'[,] t'[ïì,, -u[

as in equation 5.5.2a

= u[:;" - o

- at n( ^)ul'ltQ,., *'

'

0

n)- 1, j

Atã--\/2LY '[n), j +1

n+1)rJn+1),j

Ìn)

{u[

n)

-u[n) i

4. /s<

n+ I )¡l

not calculated

n+ 1)

,j

A

n+ I )

,j

* ^r

f q"l # s{z["*" -r[]i]ì I

v[

-(n+l)"Q, :

u[];' '

vI

zln+1),j

=Q

r[,;" , u[ not calculated

5_(n+l)tQj as in equaÈi ot 5 .5 .2a

u[ =Q

v[n)

,j

Ar

At 4:;" {u[:ì,, -u[,1 ]7t1 v[,a

l

n+1)ttn+l),j 2Ax

4, ìt - Ar R(n)tr, ,4Lv

- Arf

u[, ]., -u[:ì. ,] v[

4l;" - rl e{z['*"-t[']ll]

6

l

Z,n+ 1)

,jn+l),j

uI v[n+1),j

{, n+l),jn+ I ),j

not calculated

= u[:;" = Qu[

AS in equations 5.5.2

136

8n+1),Jn+1)tt

n+ I ),j

Z; uI not calculated

V; 0

(trrlestern open boundary)

(n+l)zQ,

¡= Re{z e-i.(n+1)At t

1n.J

where Z. gives the1n.

amplitude änd phase of the

elevation at y = jAY along

the open boundarY.

n)

Q, i

9I

I

I

xI

AI

I

I

II

xI

^

uIn+1)rjn+l),j

not calculated

4:l åi 4:;" rulTì,, -u[, ] i

#'[]ì {'[:]*,-u["]., ] - n.

vI

4

R( 4:ì

as f or: e lement 9

n+ 1)

:J

10. l (Llestern oPen BoundarY)I

t zÍ"]t) = Re{Z. "-t-(n+r)at}rj

./../>/ t[,;t) t'ot calculated

v!"*t) = o't, j

l2 (Eastern open BoundarY)

-it"r(n+1)^te.

- arf ";:,.')- it e{z['*"-r["]Jl]

= à{u["*]l.4ii:Ì",i

z

Ì

1n.J

Z: as for1n.

t

elements 9

Z.ln'

J

and 10///////

,l:;') = ne{2f,,.

uIn+ I )rJ

+Atf

1n

4:ì *'[:ì {'[:]-u[]ì,, ]

*q:ì t4:Ì.,-u[:] ] - a, n

n) Ar

( n)to,

,

uIn)

{ Ì

vÍ "l')Xt J

0

q i 2Lx eo(og

n+1),j

-2l9-n+ 1)- l, j

t37

11. (Eastern open Boundary)

,;",;') = ne{21o. "-i-(n+r)4t1,

I

II

xI

AI

I

Z'. as in element 12l-tl.

J

uI uIn)n+ I )

'j

At2Ñ.

n) n)uI tu[ -u[

n)_ l, j ]

#q:l t4:1.,-u[:1.,] - a, R( ")tQ, j

u[n)

4:ì

+Atf q n) # ,{rl"*"-t[]ill]Att^"v[

n+ 1)

,J

n+ 1)

'jvI n) q

an elevation grid-Point

land

an open boundarY.

t4,l-u[]ì,, ]

Atö'[]l r'[]Ì*,-u[, ì., I - o. *l;],

- At f õolî.r)- it c{z['*"-r[,i]ìl

4:;') = à{u[":".u["i]| t

A indicates a V-velocity grid-point

x

138

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I^IILLIAMS, N.V ' |.]jlÐ z The Application of Âesonato¡s and 0ther Methods

to Problems in }ceanogtaphg' Ph' D' Thesis' U' of N'S'I^l'

The Admiralty Charts and Èhe Australian National Tide Tables, published

by the llydrographer, R'A'N' were also used'