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Generation and Propagation of Tsunami by a Moving Realistic Curvilinear Slide Shape with Variable Velocitie

Written By

Khaled T. Ramadan

Submitted: November 30th, 2011 Published: November 21st, 2012

DOI: 10.5772/50687

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1. Introduction

Tsunamis are surface water waves caused by the impulsive perturbation of the sea. Apart from co-seismic sea bottom displacement due to earthquakes, sub-aerial and submarine landslides can also produce localized tsunamis with large and complex wave run-up especially along the coasts of narrow bays and fjords. In recent years, significant advances have been made in developing mathematical models to describe the entire process of generation, propagation and run-up of a tsunami event generated by seismic seafloor deformation [1-3].

The case of particular interest in this chapter is the mechanism of generation of tsunamis by submarine landslides. When a submarine landslide occurs, the ocean-bottom morphology may be significantly altered, in turn displacing the overlying water. Waves are then generated as water gets pulled down to fill the area vacated by the landslide and to a lesser extent, by the force of the sliding mass. Submarine slides can generate large tsunami, and usually result in more localized effects than tsunami caused by earthquakes [4]. Determination of volume, deceleration, velocity and rise time of the slide motion make modeling of tsunamis by submarine slides and slumps more complicated than simulation of seismic-generated tsunami.

Constant velocity implies that the slide starts and stops impulsively, i.e. the deceleration is infinite both initially and finally. Clearly, this is not true for real slides, and a more complex shape of the generated wave is expected [5].

In this chapter, we concern about the tsunami amplitudes predicted in the near-field caused by varying velocity of a two-dimensional realistic curvilinear slide model. The curvilinear tsunami source model we considered based on available geological, seismological, and tsunami elevation. The aim of this chapter is to determine how near-field tsunami amplitudes change according to variable velocities of submarine slide. We discuss the nature and the extent of variations in the peak tsunami waveforms caused by time variations of the frontal velocity and the deceleration for the two-dimensional curvilinear block slide model and compares the results with those for the slide moving with constant velocity. It will show how the changes in the slide velocity as function in time acts to reduce wave focusing. Numerical results are presented for the normalized peak amplitude as a function of the propagation length of the slump and the slide, the water depth, the time variation of moving velocity and the deceleration of the block slide. The problem is solved using linearized shallow-water theory for constant water depth by transform methods (Laplace in time and Fourier in space), with the forward and inverse Laplace transforms computed analytically, and the inverse Fourier transform computed numerically by the inverse Fast Fourier transform (IFFT).


2. Mathematical formulation of the problem

Consider a three dimensional fluid domain Das shown in Figure 1. It is supposed to represent the ocean above the fault area. It is bounded above by the free surface of the ocean z= η(x,y,t )and below by the rigid ocean floor z=-Hx,y+ζ(x,y,t), whereη(x,y,t )is the free surface elevation, H(x,y) is the water depth and ζ(x,y,t) is the sea floor displacement function. The domain Dis unbounded in the horizontal directions x and y, and can be written asD= R2×[-Hx,y+ζx,y,t,ηx,y,t] . For simplicity, H(x,y)is assumed to be a constant. Before the earthquake, the fluid is assumed to be at rest, thus the free surface and the solid boundary are defined by z=0 and z=-H, respectively. Mathematically, these conditions can be written in the form of initial conditions:ηx,y,0=ζ(x,y,0) = 0. At time t>0the bottom boundary moves in a prescribed manner which is given by z=-H+ζx,y,t. The resulting deformation of the free surface z=η(x,y,t )is to be found as part of the solution. It is assumed that the fluid is incompressible and the flow is irrotational. The former implies the existence of a velocity potential ϕ x,y,z,twhich fully describes the flow and the physical process. By definition of ϕ, the fluid velocity vector can be expressed asu=ϕ . Thus, the potential flow ϕ x,y,z,tmust satisfy the Laplace's equation

2ϕ x,y,z,t= 0 wherex,y,zR2 × [-H,0 ]

subjected to the following linearized kinematic and dynamic boundary conditions on the free surface and the solid boundary, respectively

z=-H E2
 ϕt + g η=0 on

with the initial conditions given by

ϕx,y,z,0=ηx,y,0=ζ(x,y,0) = 0.(5)

So, the linearized shallow water solution can be obtained by the Fourier-Laplace transforms.

Figure 1.

Fluid domain and coordinate system for a very rapid movement of the assumed source model.

2.1. Solution of the problem

Our interest is focused on the resulting uplift of the free surface elevation η(x,y,t). An analytical analyses is to examine and illustrate the generation and propagation of a tsunami for a given bed profile ζ(x,y,t).Mathematical modeling of waves generated by vertical and lateral displacements of ocean bottom using the combined Fourier–Laplace transform of the Laplace equation analytically is the simplest way of studying tsunami development. Equations (1)–(4) can be solved by using the method of integral transforms. We apply the Fourier transform in (x, y)

F[f]=F k1,k2= R2fx,ye- i x k1 + y k2 dx dyE4

with its inverse transform

F-1F=f x,y = 1(2π)2R2 F k1,k2 e i x k1 + y k2 dk1 dk2E5

and the Laplace transform in time t,

£ [ g ] =G(s) = 0gte- s t dt

For the combined Fourier-Laplace transforms, the following notation is introduced:

F( £ ( fx,y,t )=

F- k1,k2 , s= R2e- i x k1+ y k20fx,y,t e- s t dtdx dyE6

Combining (2) and (4) yields the single free-surface condition

ϕtt x,y,0,t +gϕz x,y,0,t =0E7

and the bottom condition (3) will be

ϕzx , y , -H , t =ζ t( x , y , t )E8

The solution of the Laplace equation (1) which satisfies the boundary conditions (6) & (7) can be obtained by using the Fourier-Laplace transforms method.

First, by applying the transforms method to the Laplace equation (1), gives

F £ 2ϕx2+F £ 2ϕy2+ F £ 2ϕz2=0 E9

By using the propertydnfxn=(ik)nF-(k), Equation (8) will be

ϕ-zz k1,k2,z,s -  k12+k22ϕ- k1,k2,z,s = 0E10

Second, by applying the transforms method to the boundary conditions (6) & (7) and the initial conditions (5), yields

s2ϕ- k1,k2,0,s +g ϕ-z k1,k2,0,s =0E11
ϕ-z k1,k2,-h,s =s ζ- ( k1,k2,s)E12

The transformed free-surface elevation can be obtained from (4) as

 η- k1,k2,s = - sgϕ- k1,k2,0,sE13

The general solution of (9) will be

ϕ- k1,k2,z,s =A k1,k2,s cosh(kz)+B k1,k2,ssinh(kz)E14

wherek= k12+k22.The functions A k1,k2,s and B k1,k2,scan be found from the boundary conditions (10) & (11) as follows

For the bottom condition (atz=-h):

ϕ- k1,k2,-h,s z= -Aksinh(kh)+Bk cosh(kh)E15

Substituting from (14) into (11), yields

-Aksinh(kh)+Bk coshkh=s ζ- ( k1,k2,s) E16

For the free surface condition (atz=0):

ϕ- k1,k2,0,s z= Bk and
ϕ- k1,k2,0,s =AE17

Substituting from (16) into (10), gives

A= -g ks2 BE18

Using (17), Equation (15) can be written as

Bkcosh(kh)1+ gks2tanhkh=s ζ- ( k1,k2,s)E19

From which,

A k1,k2,s = -g s ζ- k1,k2,s coshkhs2+g k tanhkh , B k1,k2,s = s3ζ-( k1,k2,s ) k coshkh[s2+g k tanhkh] .E20

Substituting the expressions for the functions A k1,k2,s and B k1,k2,sin (13) yields,

ϕ- k1,k2,z,s =-g s ζ- k1,k2,s coshkhs2+ ω2coshkz-s2g ksinh(kz) , E21

where ω=g k tanhkhis the circular frequency of the wave motion.

The free surface elevation  η- k1,k2,s  can be obtained from (12) as

 η- k1,k2,s =s2 ζ- k1,k2,s coshkhs2+ ω2E22

A solution for ηx,y,t can be evaluated for specified by computing approximately its transform  ζ-(k1,k2,s) then substituting it into (20) and inverting  η- k1,k2,s  to obtainηx,y,t. We concern to evaluateηx,y,t  by transforming analytically the assumed source model then inverting the Laplace transform of η- k1,k2,s  to obtain  η- k1,k2,t  whichis further converted toηx,y,t  by using double inverse Fourier Transform.

The circular frequency ωdescribes the dispersion relation of tsunamis and implies phase velocity cp=ωkand group velocitycg=dωdk. Hence,cp=gtanhkHk ,  cg=12cp 1+2kHsinh(2kH).and

 k=2πλSince,kH0 , hence ascpgH, both cggHandvt=gH, which implies that the tsunami velocity H for wavelengths λ long compared to the water depth0tt1. The above linearized solution is known as the shallow water solution.

We considered three stages for the mechanism of the tsunami generation caused by submarine gravity mass flows, initiated by a rapid curvilinear down and uplift faulting with rise timex-, then propagating unilaterally in the positivet1tt* direction with timev, to a length L both with finite velocity x-to produce a depletion and an accumulation zones. The last stage represented by the time variation in the velocity of the accumulation slide (block slide) moving in the t*t  tmaxdirection with time α and deceleration tmax, whereαmin is the maximum time that the slide takes to stop with minimum decelerationy-. In the 0tt1direction, the models propagate instantaneously. The set of physical parameters used in the problem are given in Table 1.

Table 1.

Parameters used in the analytical solution of the problem.

The first and second stages of the bed motion are shown in Figure 2 and Figure 3, respectively, and given by:

  1. First stage: Curvilinear down and uplift faulting for x[ -50, 50]

where for ζdownx,y,t=0v t2S 1+cosπ50x 1-cosπ100 y+150 ,-150  y  -50,- ζ0v tS 1+cosπ50x,-50  y  50-ζ0v t2S 1+cosπ50x 1+cosπ100 y-50 ,50  y  150.

x[ 200, 300]E23

and for

ζupx,y,t=ζ0v t2S 1-cosπ50(x-200) 1-cosπ100 y+150 ,-150y-50, ζ0v tS 1-cosπ50(x-200),-50y50,ζ0v t2S 1-cosπ50(x-200) 1+cosπ100 y-50 ,50y150.E24


For these displacements, the bed rises duringζ0to a maximum displacement  t ≥ t1 such that the volume of soil in the uplift increases linear with time and vise verse in the down faulting.

Fort = t*= 200/v the soil further propagates unilaterally in the positive x- direction with velocity vtill it reaches the characteristic length L = 150 km at y=0.

Figure 2.

Normalized bed deformation represented by a rapid curvilinear down and uplift faulting at the end of stage one (t1tt*)

  1. Second stage: Curvilinear down and uphill slip-fault (slump and the slide) for ζdownx,y,t=ζ1downx,y,t+ζ2downx,y,t+ζ3downx,y,t

y[ -150,- 50]E26

where for ζ1downx,y,t=- ζ041 +cosπ50x 1-cosπ100 y+150  ,-50  x  0 ,-ζ02 1- cosπ100 y+150 ,0 ≤ x ≤ (t-t1) v,- ζ041+cosπ50x – t-t1 v 1-cosπ100 y+150 ,t-t1 v  x  t-t1 v +50,

y[ -50, 50]E27

and for

ζ2downx,y,t=- ζ021 +cosπ50x,-50 x  0 ,-ζ0,0  x  t-t1 v ,- ζ021+cosπ50x - t-t1 v,t-t1 v  x  t-t1 v+50 ,E28

y[ 50, 150]E29

and for

ζ3downx,y,t=- ζ041+cosπ50x 1+cosπ100 y -50  ,-50 x  0 ,-ζ02 1+cosπ100 y -50  ,0  x  t-t1 v ,- ζ041+cosπ50x- t-t1 v 1+cosπ100 y -50  , t-t1 v  x t-t1 v+50 ,E30

y[ -150, -50]E32

where for ζ1upx,y,t= ζ041 -cosπ50(x-200) 1-cosπ100 y+150 200  x 250 ,ζ02 1- cosπ100 y+150  ,250 ≤ x ≤ 250+(t-t1) v , ζ041+cosπ50x -250+t-t1 v 1-cosπ100 y+150  250+t-t1 v  x  300+t-t1 v ,

y[ -50, 50]E33

and for

ζ2upx,y,t= ζ021 -cosπ50(x-200),200 x  250ζ0,250  x  250+t-t1 v , ζ021+cosπ50x -250+t-t1 v,250+t-t1 v  x  300+t-t1 v ,E34

y[ 50, 150]E35

and for


 ζ041-cosπ50(x-200) 1+cosπ100 y -50  ,200 x  250 ,ζ02 1+cosπ100 y -50  ,250  x  250+t-t1 v , ζ041+cosπ50x-250+t -t1 v 1+cosπ100 y -50  , 250+t-t1 v  x  300+t-t1E37

The kinematic realistic tsunami source model shown in Figure 3 is initiated by a rapid curvilinear down and uplift faulting (First stage) which then spreads unilaterally with constant velocity v causing a depletion and accumulation zone. The final down lift of the depression zone and final uplift of the accumulation zone are assumed to have the same amplitudevt=gH.We assume the spreading velocity v of the slump and the slide deformation in Figure 3 the same as the tsunami wave velocity v=vtas the largest amplification of the tsunami amplitude occurs when ζ0due to wave focusing. The slide and the slump are assumed to have constant width W.

The spreading is unilateral in the x-direction as shown in Figure 3. The vertical displacement,tt* , is negative (downwards) in zones of depletion, and positive (upwards) in zones of accumulation. All cases are characterized by sliding motion in one direction, without loss of generality coinciding with the x-axis, and tsunami propagating in the x-y plane.

Figure 4 shows vertical cross-sections (through y = 0) of the mathematical models of the stationary submarine slump and the moving slide and their schematic representation of the physical process that we considered in this study, as those evolve for timet=t*. The block slide starts moving in the positive x-direction at timeL'=150 km and stops moving at distance L'while the downhill slide becomes stationary. We discuss the tsunami generation for two cases of the movement of the block slide. First, the limiting case in which the block slide moves with constant velocity v and stops after distance tmin=t*+L'/v with infinite deceleration (sudden stop) at timet2t*.

Second, the general case in which the block slide moves in time L' with constant velocity and then with constant deceleration such that it stop softly after travelingthe same distance t3 in time α which depends on the deceleration t2and the choice of timev(t).

The velocity vt=v t*tt2v-t-t2α , t2tt3in this case can be defined as


where αkm/sec and t3is the deceleration of the moving block slide. We need to determine the time L'that the slide takes to reach the final distance α and the corresponding decelerationt*=200v. This can be done by using the following steps:

Figure 3.

Normalized Bed deformation model represented by the accumulation and depletion zones at the end of stage two ( y=0 ) (a) Side view along the axis of symmetry atv(t)b) Three-dimensional view.

Figure 4.

A schematic representation of a landslide (bottom) travelling a significant distance L downhill creating a ‘‘scar’’ and a moving uphill displaced block slide stopping at the characteristic lengtht2.

  1. Choosing time t*t2t*+(L'/v)  as t*=t1+L/vand t1=50/vin which the slide moves with constant velocity v whereL* =(t2-t*)v.

  2. Getting the corresponding distanceL**=L'(t2-t*)v.

  3. Evaluating the remaining distanceL**.

Substituting L**=t3-t2v-1 2α (t3-t2)2 in the equation


When the block slide stops moving, then


Eliminating t3from equations (7.12) and (7.13), we get relation between t2 and deceleration α which further substituting in equation (7.13), we obtain theFor t2tt3.

velocity vt=v-t-t2α , the block slide moves witht2. Table 2 represents different values oft3 and the corresponding calculated value ofαandt2.

Table 2.

Values oft3 and the corresponding calculated values ofα and.

Figure 5 illustrates the position of the slides in the third stage for different choice of decelerationαmin. In this stage, L'is the minimum deceleration required such that the slide stops after traveling distancet2=t*. In this caset3=tmax=t*+(2L'/v)=59.51 min andα>αmin. For any othert*tt2, the slide moves with constant velocity with time α and with deceleration t3until it stops at time tmax which is less thant*t .

So, the stationary landslide scar forvtt*t2t*+(L'/v)and the movable block slide with variable velocity t*+(L'/v) t3t*+(2L'/v)for ζstat.landslidex,y,t*=ζ1x,y,t*+ζ2x,y,t*+ζ3x,y,t*and t*t2t*+(L'/v)can be expressed respectively as


Figure 5.

Slide block position against the instants of times ζblock slide x,y,t=ζ1x,y,t+ζ2x,y,t+ζ3x,y,tandζstat.landslidex,y,t*.

ζblock slide x,y,tis the same as (23) except the time parameter t will be substitutedbyS=t-t*v for t*tt2 , t-t*v-12α (t-t2)2 for t2tt3 ,.

Fory[ -150, -50], let ζ1x,y,t= ζ041 -cosπ50(x-(200+S)) 1-cosπ100 y+150  , 200+S  x 250+S ,ζ02 1- cosπ100 y+150  ,250+S  x  250+S+L , ζ041+cosπ50x -250+S+L  1-cosπ100 y+150 ,250+S+ L  x  300+S+L,be

the distance the slide moves during stage three, hence


y[ -50, 50]E44

ζ2x,y,t= ζ021 -cosπ50(x-200+S),200+S x  250+S ,ζ0,250+S  x  250+S+L, ζ021+cosπ50x -250+S+L,250+S+L  x  300+S+L,E45

and for

y[ 50, 150]E46

ζ3x,y,t2,t3= ζ041 -cosπ50(x-(200+S)) 1+cosπ100 y-50  , 200+S  x 250+S ,ζ02 1+ cosπ100 y-50  ,250+S  x  250+S+L , ζ041+cosπ50x -250+S+L  1+cosπ100 y-50 , 250+S+ L  x  300+S+L,E47

and for



Laplace and Fourier transforms can now applied to the bed motion described by Equations (21)-(24) and Equations (28) & (29). First, beginning with the curvilinear down and uplift faulting (21) and (22) forF£ ζx ,y ,tζ- k1,k2,s-∞e- i x k1 + y k20ζx,y,t e- s t dt d x d y .whereζdownk1,k2,s=, and

-150-5012 1-cosπ100 y+150 e-i k2ydy-50501+cosπ50xe-i k1x dx 00v t2S e-st dt E50

The limits of the above integration are apparent from Equations (21) & (22) and are done as follows:

+-5050e-i k2y dy-5050 1+cosπ50xe-i k1x dx 0- ζ0v tSe-st dt 
+5015012 1+cosπ100 y-50 e-i k2ydy-5050 1+cosπ50xe-i k1x dx 0-ζ0v t2S e-st dt E51
-150-5012 1-cosπ100 y+150 e-i k2ydy2003001-cosπ50(x-200)e-i k1x dx 0ζ0v t2S e-st dt E52

Substituting the results of the integration for +-5050e-i k2y dy200300 1-cosπ50(x-200)e-i k1x dx 0 ζ0v tSe-st dt  and +5015012 1+cosπ100 y-50 e-i k2ydy-5050 1-cosπ50(x-200)e-i k1x dx 0ζ0v t2S e-st dt into (20), yields

ζup E53

The free surface elevation  η- k1,k2,s = can be evaluated by using the inverse Laplace transforms of1coshkH( s2+ω2)ζ0v12(e-i 200 k1 -e-i 300 k1)ik1-11-50πk 2 ik150π2e-i300k1-ei200k1-(ei 50 k1 -e-i 50 k1)ik1+11-50πk 2 ik150π2e-i50k1-ei50k1×  given by Equation (31) as follows:

First, recall that( e i 150 k2 – e i 50 k2 )k2- 11-100πk 2 ik2100π2(ei50k2+ei150k2+4sin 50k2k2+( e-i 50 k2 - e-i 150 k2 )k211-100πk 2 ik2100π2(e-i150k2+e-i50k2)henceη-(k1,k2,t) becomes

 η- k1,k2,s E54
£-11 s2+ω2= sinωtω
 η- k1,k2,t E55

In case for η- k1,k2,t =, sinωtωcoshkHζ0 v2L e-i 200 k1 e-i 300 k1ik1-11-50πk1  2 ik150π2e-i300k1-ei200k1-ei 50 k1 e-i 50 k1ik1+11-50πk1  2 ik150π2e-i50k1-ei50k1will have the same expression except in the convolution step, the integral

become× ( e i 150 k2  e i 50 k2 )i k2- 11-100πk2  2 ik2100π2(ei50k2+ei150k2) +4sin 50k2k2+( e-i 50 k2 - e-i 150 k2 )i k2+ 11-100πk2  2 ik2100π2(e-i150k2+e-i50k2).instead oftt1.

Finally,  η- k1,k2,t is evaluated using the double inverse Fourier transform oft-t1tcosωτdτ= sinωtω-sinω(t-t1)ω

0tcosωτdτ= sinωtωE56

This inversion is computed by using the FFT. The inverse FFT is a fast algorithm for efficient implementation of the Inverse Discrete Fourier Transform (IDFT) given by

ηx,y,t E57

where  η- k1,k2,t is the resulted function of the two spatial variablesηx,y,t =1(2π)2-eik2y-eik1x η- k1,k2,t  dk1dk2,correspondingfm , n=1MNp=0M-1q=0N-1Fp , qeiMpm eiNqnp=0,1,…….,M-1 ,q=0,1,…….,N-1 , , from the frequency domain function fm,nwith frequency variablesm and n, correspondingx and y. This inversion is done efficiently by using the Matlab FFT algorithm.

In order to implement the algorithm efficiently, singularities should be removed by finite limits as follows:

1. As Fp,qthen p and q  has the limit

k1and k2E58

2. Ask0,implies k10 ,k20 and ω0 , then the singular term of  η- k1,k2,t  has the following limits

limk0 η- k1,k2,t =0 for tt1 , where t1=50v .and

3.As η- k1,k2,t , then the singular terms of limk10e-i 200 k1 e-i 300 k1ik1=100  have the following limits

limk10ei 50 k1 e-i 50 k1ik1=100 E60



Using the same steps,  η- k1,k2,t is evaluated by applying the Laplace and Fourier transforms to the bed motion described by (23) and (24), then substituting into (20) and then inverting limk20e- i 150 k2i k2 - e i 50 k2i k2=100,limk204sin( 50k2)k2=200 , using the inverse Laplace transform to obtainlimk20e-i 50 k2i k2 - e-i 150 k2i k2=100. . This is verified for η-k1,k2,t where η-k1,k2,sas follows:





t*=200 vE64
, then

Substitutingη-down(k1,k2,t)=-ζ04 coshkHe i 150 k2 e i 50 k2i k2- 11-100πk2  2ik2100π2(ei50k2+ei150k2) +-ζ0 coshkhsin50k2k2+-ζ04 coshkHe- i 50 k2 - e- i 150 k2i k2+ 11-100πk2  2ik2100π2(e-i150k2+e-i50k2) ×ei 50 k1 -1ik1+ ei 50 k11-50πk1  2 ik150π21-e-i50k1cosωt-t1+2vω2-k1v2 ωsinωt-t1+ik1vcosωt-t1-ik1ve-ik1t-t1v+e-i k1t-t1v-e-i k1 50+t-t1vi k1+11-50πk1  2ik150π2(e-i k1 50+t-t1v +e-ik1t-t1v) cosωt-t1 andη-upk1,k2,t=η-1upk1,k2,t+η-2upk1,k2,t+η-3upk1,k2,t into (34) gives η-up(k1,k2,t)=ζ04 coshkHe i 150 k2 e i 50 k2i k2- 11-100πk2  2ik2100π2(ei50k2+ei150k2) +ζ0 coshkhsin50k2k2+ζ04 coshkHe- i 50 k2 - e- i 150 k2i k2+ 11-100πk2  2ik2100π2(e-i150k2+e-i50k2) ×e-i 200 k1-e-i 250 k1ik1+ 11-50πk1  2- ik150π2e-i250k1+e-i200k1cosωt-t1+2ve-i 250 k1ω2-k1v2 ωsinωt-t1+ik1vcosωt-t1-ik1ve-ik1t-t1v+e-i k1250+t-t1v-e-i k1 300+t-t1vi k1+11-50πk1  2ik150π2(e-i k1 300+t-t1v +e-i k1 250+t-t1v ) cosωt-t1 forη-down(k1,k2,t). For the caseη-upk1,k2,t, η-k1,k2,twill have the same expression as (34) except the term resulting from the convolution theorem, i.e.


instead of


Finally,  η- k1,k2,t is computed using inverse FFT oft-t1-t*tcosωτe-ik1(t-t1v-τ)dτ=1ω2-k1v2ωsinωt-t1+ik1vcosωt-t1-e-ik1vt* ωsinωt-t1-t*+ik1vcosωt-t1-t*.

Again, the singular points should be remove to compute t-t1tcosωτe-ik1t-τvdτ=1ω2-k1v2 ωsinωt-t1+ik1vcosωt-t1-ik1ve-ik1t-t1v.efficiently

1. As ηx,y,t then  η- k1,k2,t  has the following limit

ηx,y,t E69

2. Ask0,, then the singular terms of  η- k1,k2,t  have the following limits

limk0 η- k1,k2,t = 0 t-t1t* 0 t-t1t* , where t*=200v .E70

3. As η- k1,k2,t , then the singular terms of limk10ei 50 k1-1ik1=50 ,limk10e-i k1t-t1v-e-i k1 50+t-t1vi k1=50 have the following limits

limk10e-i 200 k1-e-i 250 k1ik1=50 andlimk10e-i k1250+t-t1v-e-i k1 300+t-t1vi k1=50E72

Finally,  η- k1,k2,t is evaluated by applying the Laplace and Fourier transforms to the block slide motion described by (29), then substituting into (20) and then inverting limk20e i 150 k2i k2 - e i 50 k2i k2=100 ,limk20sin( 50k2)k2=50 and  using the inverse Laplace transform to obtainlimk20e-i 50 k2i k2 - e-i 150 k2i k2=100 . .

This is verified for η-block slide k1,k2,tandη-block slide k1,k2,s.


η-block slide k1,k2,tE74



Finally, t*+(L'/v) t3t*+(2L'/v)is computed using inverse FFT ofη-block slide k1,k2,t=η-1k1,k2,t+η-2k1,k2,t+η-3k1,k2,t.

Again, the singular points should be remove to compute η-block slide k1,k2,t=efficiently

1. As ζ04 coshkHe i 150 k2 e i 50 k2i k2- 11-100πk2  2ik2100π2(ei50k2+ei150k2) +ζ0 coshkhsin50k2k2+ζ04 coshkHe- i 50 k2 - e- i 150 k2i k2+ 11-100πk2  2ik2100π2(e-i150k2+e-i50k2) ×e-i (200+S) k1-e-i (250+S )k1ik111-50πk 2- ik150π2e-i(250+S)k1+e-i(200+S)k1cosω(t3-t*)+2ve-i (250+S) k1ω2-k1v2 ωsinωt3+ik1vcosωt3-e-ik1vt*(ωsinωt3-t*+ ik1vcosωt3-t*) +e-i k1250+S+L-e-i k1 300+S+Lk1+11-50πk 2ik150π2(e-i k1 300+S+L+e-i k1 250+S+Lcosω(t3-t*)then ηx,y,t2,t3 has the limit

 η- k1,k2,t2,t3E76

2. Asηx,y,t , then the singular terms of k0, have the following limits

 η- k1,k2,t E77

3. Aslimk0 η- k1,k2,t = 100 100+2Lζ0, then the singular terms of k10 have the following limits

 η- k1,k2,t and

limk10e-i (200+S) k1 -e-i (250+S )k1ik1=50 and limk10e-i k1250+S+L -e-i k1 300+S+Lk1=50.E78

We investigated mathematically the water wave motion in the near and far-field by considering a kinematic mechanism of the sea floor faulting represented in sequence by a down and uplift motion with time followed by unilateral spreading in x-direction, both with constant velocityk20, then a deceleration movement of a block slide in the direction of propagation. Clearly, from the mathematical derivation done above,  η- k1,k2,t depends continuously on the sourcelimk20e i 150 k2k2 - e i 50 k2k2=100 ,limk20sin( 50k2)k2=50 . Hence, from the mathematical point of view,this problem is said to be well-posed for modeling the physical processes of the tsunami wave.


3. Results and discussion

We are interested in illustrating the nature of the tsunami build up and propagation during and after the movement process of a variable curvilinear block shape sliding. In this chapter, three cases are studied. We first examine the generation process of tsunami waveform resulting from the unilateral spreading of the down and uplift slip faulting in the direction of propagation with constant velocity v. We assume the spreading velocity of the ocean floor up and down lift equal to the tsunami wave velocitylimk20e-i 50 k2i k2 -e-i 150 k2k2=100 .  as the largest wave amplitude occurs when  vdue to wave focusing.

3.1. Tsunami generation caused by submarine slump and slide- Evolution in time

We assume the waveform initiated by a rapid movement of the bed deformation of the down and uplift source shown in Figure 2.Figure 6 shows the tsunami generated waveforms during the second stage at time evolutionηx,y,t  at constant water depth ζx,y,t . It is seen how the amplitude of the wave builds up progressively as  vt=gH=0.14 km/secincreases where more water is lifted below the leading wave depending on its variation in time and the space in the source area. The wave will be focusing and the amplification may occur above the spreading edge of the slip.This amplification occurs above the source progressively as the source evolves by adding uplifted fluid to the fluid displaced previously by uplifts of preceding source segments. This explains why the amplification is larger for wider area of uplift source than for small source area. It can be seen that the tsunami waveformv=vt has two large peaks of comparable amplitudes, one in the front of the block due to sliding of the block forward, and the other one behind the block due to spreading of the depletion zone.

Figure 6.

Dimensionless free-surface elevation caused by the propagation of the slump and slide in the x-direction during the second stage with  t=0.4t*, 0.6t*,0.8t*,t*atH=2 km, t= 150 km, η/ζ0sec, (a) Side view along the axis of symmetry at v=vt(b) Three dimensional view.

3.1.1. Effect of the water depth H

Figure 7 and Figure 8 illustrate the normalized peak tsunami amplitudesH=2 km, L respectively in the near-field versus W=100 km, t*=200/v aty=0, the time when the spreading of the slides stops for ηR,max/ζ0km and forηL,min/ζ0 and L/H= 150 km,t=t*=t1+L/v.

Figure 7.

Normalized tsunami peak amplitudes, H=0.5, 1, 1.5 and 2at the end of second stage for different water depth  v=vtatL withW=100 kmand ηR,max/ζ0= 150 km,H=0.5, 1 , 1.5 and 2 km .

Figure 8.

Normalized tsunami peak amplitudes, t*=t1+L/vat the end of second stage for different water depth v=vtatL withW=100 kmand ηL,min/ζ0= 150 km,H=0.5, 1 , 1.5 and 2 km .

From Figure 7 and Figure 8, the parameter that governs the amplification of the near-field water waves by focusing, is the ratiot*=t1+L/v. As the spreading length L in the slip-faults increases, the amplitude of the tsunami wave becomes higher.

At L = 0, no propagation occurs and the waveform takes initially the shape and amplitude of the curvilinear uplift fault (i.e.v=vt.

The negative peak wave amplitudes are approximately equal to the positive peak amplitudes (L ) when W=100 kmas seen in Figures 7 and 8. The peak tsunami amplitude also depends on the water depth in the sense that even a small area source can generate large amplitude if the water is shallow.

3.1.2. Effect of the characteristic size

Figure 9 shows the effect of the water depth L/Hon the amplification factorηR,max/ζ0 =1,ηL,min/ζ0 =1) forηL,min/ζ0ηR,max/ζ0, with v=vtat the end of the second stage (i.e. ath ). Normalized maximum tsunami amplitudes for 19 ocean depths are calculated. As seen from Figure 9, the amplification factor ηR,max/ζ0 decreases as the water depth H increases. This happens because the speed of the tsunami is related to the water depth ( v=vt) which produces small wavelength as the velocity decreases and hence the height of the wave grows as the change of total energy of the tsunami remains constant. Mathematically, wave energy is proportional to both the length of the wave and the height squared. Therefore, if the energy remains constant and the wavelength decreases, then the height must increases.

Figure 9.

Normalized maximum tsunami amplitudes L = W = 10, 50,100 km and L=150 km, W=100 km for different length and width at t=t*=t1+L/vforηR,max/ζ0.

3.2. Tsunami generation and propagation-effect of variable velocities of submarine block slide

In this section, we investigated the motion of a submarine block slide, with variable velocities, and its effect on the near-field tsunami amplitudes. We considered the limiting case, in which the slide moves with constant velocity and stops suddenly (infinite deceleration) and the case in which the slide stops softly with constant deceleration forL = 150 km, W = 100 km andv=vt=gH.

3.2.1. Displaced block sliding with constant velocity v

Constant velocity implies that the slide starts and stops impulsively, i.e. the acceleration and deceleration are infinite both initially and finally. This means that the slide takes minimum time to reach the characteristic length ηR,max/ζ0 km given by t=t*=t1+L/v min. We illustrate the impulsive tsunami waves caused by sudden stop of the slide at distance v=vt in Figure 10.

Figure 10.

Normalized tsunami waveformsv=vt along the axis of the symmetry at L' = 150and their corresponding moving slide tmin=t*+(L'/v)= 41.65with constant velocity L'along y = 0, at time η/ζ0 fory=0, ζ/ζ0km and vkm.

Figure 10 shows the leading tsunami wave propagating in the positive x-direction during time evolution t*tt*+(L'/v)sec at h=2 km 0, 30, 60, 90, 120, 150 km respectively, where L'=150 represents that part of L (see Figure 4c) forW=100. It is seen in Figure 10 that the maximum leading wave amplitude decreases with time, due to the geometric spreading and also due to the dispersion. Att=t*, t*+0.2(L'/v), t*+0.4(L'/v),t*+0.6(L'/v), t*+0.8(L'/v),t*+(L'/v), the wave front is at x = 693 km and LE= decreases from 7.906 at LE to 5.261 at timev=vt=0.14 km/sec. This happens because the amplification of the waveforms depends only on the volume of the displaced water by the moving source which becomes an important factor in the modeling of the tsunami generation. This was clear from the singular points removed from the block slide model, where the finite limit of the free surface depends on the characteristic volume of the source model

3.2.2. Displaced block moving with linear decreasing velocity with time t

The velocity of the movable slide is uniform and equal t=tmin=t*+(L'/v)= 41.65up to time ηR,max/ζ0 as shown in Figure 4a, followed by a decelerating phase in which the velocity is given by


where vtkm/sec and  tmaxis the deceleration of the moving block slide.The block slide moves in the positive vt=v-t-t*α direction with time t*t tmax. wherev=vt=0.14αis the maximum time that the slide takes to stop after reaching the characteristic length x-km with minimum decelerationt*t tmax. Figure 11 shows the leading tsunami wave propagating in the positive x-direction during time evolution  tmax=t*+2L'/v min in case L'=150 (i.e. minimum magnitude ofαmin).

Figure 11.

Normalized tsunami waveformst=t* , t*+0.2(2L'/v),  t*+0.4(2L'/v),  t*++0.6(2L'/v),  t*+0.8(2L'/v),  t*+(2L'/v) and their corresponding moving slide t2=t*with variable velocity αalong y = 0, at timeη/ζ0 atζ/ζ0, v(t)km and t*t t*+2L'/vkm.

It was clear from Figure 10 and Figure 11 that at the instant the slide stops, the peak amplitude in case of sudden stop is higher than that of soft stop.

Figure 12 and Figure 13 show the effect of the water depth atH=2 km on the normalized peak tsunami amplitude L'=150 when the slide stops moving at length W=100instantaneously at L' = W = 10, 50,100 km and L'=150 ,W=100 kmwith infinite deceleration and stops moving softly atηR,max/ζ0 at the time L'with minimum deceleration fortmin=t*+L'/v.

Figure 12.

Normalized maximum tsunami amplitudes  L' when the slide stops suddenly at time  tmax=t*+2L'/v with different slide length v=vtand width ηR,max/ζ0and fortmin=t*+(L'/v).

Figure 13.

Normalized maximum tsunami amplitudes L when the slide stops softly at timeW with different slide length v=vtand width ηR,max/ζ0and for tmax=t*+(2L'/v).

It is clear from Figure 12 and Figure 13 that the waveforms which caused by sudden stop of the slide motion after they reach the characteristic length Lat time Whave higher amplitude than stopping of the slide with slow motion at timev=vt.This agrees with the mathematical relation between the wavelength and the wave height where the wave energy is proportional to both the length of the wave and the height squared.

3.2.3. Displaced block moving with constant velocity v followed by variable velocity L'

In this section, we studied the generation of the tsunami waveforms when the block moves a significant distance with constant velocity tmin=t*+(L'/v)then continue moving with variable velocity tmax=t*+(2L'/v) with constant deceleration until it stops at the characteristic length v(t)km. Figure 14 shows the tsunami waveforms at the times calculated in Table 2 when the slide reaches the characteristic length v=vtkm.

Figure 14.

Normalized tsunami waveforms v(t) along y = 0 at L'=150calculated in Table 7.2 withL'=150 km.

In Figure 14, the first waveform from the left indicates the shape of the wave at the time η/ζ0 in the limiting case when the slide stops moving suddenly. The last waveform indicates the wave in the other limiting case at the time t=t3 when the slide stop moving with minimum deceleration at the distanceL'=150. In between the two limiting cases, the slide begins moving with constant velocity a significant distance followed by a decelerating movement until it stop at the characteristic length t3=tmin=t*+(L'/v)= 41.65km at the timet3=tmax=t*+(2L'/v), see Table 2. It is seen how the peak amplitudes of the leading waves decreases gradually from 5.261 to 3.894.

3.4. Tsunami propagation waveforms

In order to compare shape and maximum height of tsunami wave at certain time for different decelerationL', we choose the timeL'=150. We study the case when the propagating waveforms resulting at the time difference between tmint3tmax and α (i.e. propagating of tsunami away from the slide) when the slide stops moving at the lengtht=tmax.

For the limiting caset3, there is no free propagation, while for the other limiting case“ the sudden stop”, there is free propagation between timetmax andL'. For the cases between the two limiting cases, the propagation time isαmin.

Figure 15 shows the shape of the tsunami propagation waveform attmin min (curves in black) for different deceleration tmaxand time tprop=tmax-t3 (time at which the slide stops).

Figure 15.

Normalized tsunami propagation waveformstmax=t*+(2L'/v)=59.51 along the axis of the symmetry at αatt3 withη/ζ0 km and y=0km.

As the wave propagates, the wave height decreases and the slope of the front of the wave become smaller, causing a train of small wave forms behind the main wave. The maximum wave amplitude decreases with time, due to the geometric spreading and also due to the dispersion.

Figure 16 represents the normalized peak tsunami amplitudes tprop.=0, 3.56, 7.14, 10.70, 14.28, 17.85 min and L'=150of the leading propagating wave in the far-field at time W=100 for the different deceleration ηmin/ζ0and time ηmax/ζ0 (time at which the slide stops) chosen in Figure 7.15.

Figure 16.

The normalized peak tsunami amplitudes t=tmax and αat time t3 for the different deceleration ηmin/ζ0and time ηmax/ζ0.

It can be seen in Figure 16 that theabsolute minimum peak amplitudes of the leading propagated waves at timet=tmax, after the block slide stops moving with different deceleration αand timet3, decreases gradually, while the maximum peak amplitudes increases progressively.


4. Conclusion

In this paper, we presented a review of the main physical characteristics of the tsunami generation caused by realistic curvilinear submarine slumps and slides in the near-field. It is seen that the tsunami waveform has two large peaks of comparable amplitudes, one in the front of the block due to forward sliding of the block, and the other one behind the block due to spreading of the depletion zone. The negative peak wave amplitudes are approximately equal to the positive peak amplitudes. We studied the effect of variable velocities of submarine block slide on the tsunami generation in the limiting cases, in which the slide moves with constant velocity and stops suddenly (with infinite deceleration) and the case in which the slide stops softly at the same place with minimum deceleration. It is seen that the leading tsunami amplitudes are reduced in both cases due to the geometric spreading and also due to the dispersion. We observed that the peak tsunami amplitudes increase with the decrease in the sliding source area and the water depth. We also investigated the more realistic case in which the block slide moves a significant distance with constant velocity v then continue moving with time dependence velocity v(t)and different constant deceleration until it stops at the characteristic length. It is seen how the peak amplitudes of the leading waves decrease gradually with time between the two limiting cases. In this case we demonstrated also the shape of tsunami propagated wave at certain time max t (time at which the slide stops with minimum deceleration). The results show that the wave height decreases due to dispersion and the slope of the front of the wave becomes smaller, causing a train of small wave forms behind the main wave. It can be observed that just a slight variation in the maximum and the minimum tsunami propagated amplitudes after the block slide stops moving with different deceleration tmax=t*+(2L'/v)and timeα, see Figure 16. The presented analysis suggests that some abnormally large tsunamis could be explained in part by variable speeds of submarine landslides. Our results should help to enable quantitative tsunami forecasts and warnings based on recoverable seismic data and to increase the possibilities for the use of tsunami data to study earthquakes, particularly historical events for which adequate seismic data do not exist.


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Written By

Khaled T. Ramadan

Submitted: November 30th, 2011 Published: November 21st, 2012