Some features of the landslide mechanism Some features of the landslide mechanism of the surface waves generation in real basinsof the surface waves generation in real basins

Institute of computational technologies SB RAS

Novosibirsk, Russia

Beisel S., Chubarov L., Khudyakova V., Shokin Yu.

Novosibirsk, Tsunami-2009

Previous studies:•very simple bathymetry: linear slope + flat bottom,•Watts’s law of motion.

Questions:•Nonlinearity, dispersion – when and where?•Which NLD model is better?

Methodology:•Numerical modelling (SWE, Full model for potential flow)•Laboratory experiments

Shokin Yu.I., Fedotova Z.I., Khakimzyanov G.S., Chubarov L.B., Beisel S.A. Modelling surfaces waves generated by a moving landslide with allowance for vertical flow structure// Rus. J. Numer. Anal. Math. Modelling. – 2007. – Vol. 22. – №1. – pp. 63–85.

Novosibirsk, Tsunami-2009

Now:From model bathymetry to «real» one

• The «reality» during the wave propagation• The «reality» in the law of landslide motion

From 1D simulation to 2D simulation • 2D effects in the law of landslide motion• 2D effects during the wave propagation

Novosibirsk, Tsunami-2009

Area of interest:Mediterranean Sea, Israeli coast

Novosibirsk, Tsunami-2009

( ) ( , )bt slz h x h x t ( )btz h x – bottom

( , )slz h x t – landslide

0 0( , ) ( ( ))sl slh x t h x x x t

0 0

2 21 tanh ( ( ) / 4) 1 tanh ( ( ) / 4)

cos cos( , )1 tanh / 4 1 tanh / 4[ ( )][ ( )]

[ )][ )]( (c c

sl

x x t L x x t Lx t

L Lh h

Novosibirsk, Tsunami-2009

2

2sl a g b d

d sV F F F Fdt

Slide:

2 ( 1)sin ,t

d

gVu

C TW

0

1sin ,

m

a gC

00

( ) ln cosh ,t

s t st

,

0

2

0 a

us t 0

0

,tu

ta

const

New, 2009:

Watts, 1998:

( ) 22

0 02( )

1( ) ( ) ( ( ))sin ( )

2

r

l

x t

sl w w sl w sl d w

x t

d S dSC V Wg h x x x t x dx C TW

dt dt

22

2

1( ) ( ) sin

2sl w w sl w d w

d S dSC V Vg C TW

dt dt

( ) tan ,bth x x

V – volume,W – width,T – thickness,L – length,

sl – landslide density,

w – water density,

sl

w

– specific density,

wC – added mass coefficient,

– drag coefficientdC

aF – added mass force, gF – gravitational force,

bF – buoyancy force, dF – fluid dynamic drag force,

( )x

Laws of motion

Novosibirsk, Tsunami-2009

2.1 Approximation of Watts’ law for linear slope:

3. Approximation of Watts’ law for real slope “real” in motion law:

1. Watts’ law for linear slope

2.2 Approximation of Watts’ law for real slopeconstant in motion law:

2

0( )2

ts t a ( )tu u

( ) ts t u t ( )c stopx x

02 , ta const u const

( ) ( ) ( ) ( )i i

c sl i i sl ix x

x h x x h x 0 0 ( ( )),ca a x ( ( )),t t cu u x

Different laws

0

1.0,

1.5,

17900.0,

39500.0

( 0.5 ),

5000.0,

25

stop

d w

stop

x

C C

x

x

L

T

4. New law for real slope

+

+

Novosibirsk, Tsunami-2009

Different laws. Free surfaces

Novosibirsk, Tsunami-2009

Different laws. Mareograms

Novosibirsk, Tsunami-2009

( )

1 0 0

( )

( ( )) sin ( ) cos ( )r

l

x t

sl f

x t

I h x x x t x C x dx

22

1 22

1( ) ( ) ( )

2sl w w sl w sl w d w

d S dSC V WgI W I C T

dt dt

New law with friction

( )

2 0 0

( )

( ( )) ( )r

l

x t

f sl

x t

I C h x x x t K x dx 3

2'' '( ) ( ) 1 ( )bt btK x h x h x

– bottom curvature

*tanfC

* – friction angle

2

2sl a g b d n

d sV F F F F Fdt

nF – dynamic friction force

Novosibirsk, Tsunami-2009

Different friction angles

Linear slope: Real slope:

Novosibirsk, Tsunami-2009

Linear slope: Real slope:

Different friction angles. Mareograms

Novosibirsk, Tsunami-2009

Different submergences

Linear slope: Real slope:

* 2

* 1

Novosibirsk, Tsunami-2009

Different submergences. MareogramsLinear slope: Real slope:

Novosibirsk, Tsunami-2009

0 20000 40000 60000 80000

X, м

-1600

-1200

-800

-400

0

Гл

уби

на,

м

0

2

4

6

8

Уго

л н

акл

он

а, г

ра

дус

ы

3D reliefs

Novosibirsk, Tsunami-2009

Free surfaces

tt = 40 = 40 secsec tt = 120 = 120 secsec tt = 300 = 300 secsec

tt = 500 = 500 secsec tt = 860 = 860 secsec tt = 1000 = 1000 secsec

t_stopt_stop = = 850850 secsec

Novosibirsk, Tsunami-2009

Free surfaces: 1D vs 2D

- 1D - 2D

tt = 40 = 40 secsec

0 20000 40000 60000 80000

X, м

-4

0

4

Ам

плит

уда,

м

tt = 120 = 120 secsec

0 20000 40000 60000 80000

X, м

-4

0

4

Ам

плит

уда,

м

tt = 300 = 300 secsec

0 20000 40000 60000 80000

X, м

-4

0

4

Ам

плит

уда,

м

tt = 500 = 500 secsec

0 20000 40000 60000 80000

X, м

-4

0

4

Ам

плит

уда,

м

tt = 860 = 860 secsec

0 20000 40000 60000 80000

X, м

-4

0

4

Ам

плит

уда,

м

tt = 1000 = 1000 secsec

0 20000 40000 60000 80000

X, м

-4

0

4

Ам

плит

уда,

м

Novosibirsk, Tsunami-2009

Thank you!