101
IMPULSIVE TSUNAMI GENERATION BY RAPID BOTTOM DEFLECTIONS AT
INITIALLY UNIFORM DEPTH
P. A. TYVAND1, T. MILOH2 and K. B. HAUGEN1
1
Department of Agricultural Engineering
Agricultural University of Norway, 1432 Aas Norway
2
Faculty of Engineering Tel Aviv University,
Ramat Aviv 69978 Israel
Abstract
An analytical small-time expansion is developed for impulsive tsunami generation at
initially uniform depth in two and three dimensions. The flow is due to impulsive bottom
deflections that are rapid, with duration shorter than the gravitational time scale for the
given depth. The bottom deflection is assumed separable in space and time. A third-order
small-time expansion is given. The first and second-order solutions are general, while only
the gravitational effect is included. The solutions are given as Fourier transforms and Green
function integrals. The solutions are integrated numerically for the simple case of a rising
block.
1. Introduction
The present paper is concerned with tsunami generation due to impulsive bottom
deflections which has a shorter duration than the gravitational time (h/g)1/2. Here h is the
ocean depth, which is assumed initially constant, and g is the gravitational acceleration. For
typical ocean depths of 3000 meters, this means that the significant bottom motions must
have stopped after 15 seconds. This constraint may apply to tsunamis generated by
earthquakes, but not to tsunamis due to landslides or volcanic eruptions.
The present work is only concerned with the evolution of the wave shape during a rapid
bottom deflection, not with the subsequent wave propagation. After the rapid bottom
motion has ceased, the wave will evolve as a Cauchy-Poisson problem with a rigid bottom.
Traditional numerical modelling of tsunami generation due to rapid bottom motion has
often taken the surface deflection to be the same as that of the bottom [1]. This piston-type
hydrostatic modelling is not supported by Laplace's equation. Analytical descriptions of
tsunami generation and propagation have been made on the basis of Laplace's equation ([2],
[3], [4]).
Detailed analyses of the flow during a rapid impulsive generation process are still
lacking. Tyvand and Storhaug [5] and Miloh, Tyvand and Zilman [6] have developed
Green functions for the initial flow for various bottom topographies in two and three
dimensions. This covers only the first-order theory in a small-time expansion. It has the
disadvantage that it disregards nonlinearity as well as the early gravitational flow during
the generation. In the present work a third-order small-time expansion will be developed,
which accounts for early nonlinearity and early gravitational effects.
A. C. Yalçıner, E. Pelinovsky, E. Okal, C. E. Synolakis (eds.),
Submarine Landslides and Tsunamis 101-109.
@2003 Kluwer Academic Publishers. Printed in Netherlands
102
2. Formulation of the problem
We consider inviscid flow in an incompressible fluid, which is initially at rest with constant
depth h. The gravitational acceleration is g. The free surface of the fluid is subject to
constant atmospheric pressure. An impulsive flow due to rapid bottom deflections starts at
time t=0. From Kelvin's circulation theorem the flow will be irrotational and obey
Laplace's equation:

The velocity field is the gradient of the velocity potential (x,y,z,t). The surface
elevation is (x,y,t). We apply dimensionless variables with unit of length h, unit of time
(h/g)1/2, unit of velocity (gh)1/2. The z axis points vertically upwards, while the x, y plane is
in the undisturbed free surface. We will investigate the early surface disturbance generated
by a rapid deflection (x,y,t) of an initially horizontal and impermeable bottom. The
deflection starts impulsively at t=0, which gives the initial conditions:
(x,y,0,0)=0,
(x,y,0)=0
(2)
The bottom deflection is assumed to be separable in space and time:
(x,y,t) = Z(x,y) T(t)
(3)
Apart from this separation constraint, the functions Z(x,y) and T(t) can be quite
arbitrary. But in order for our small-time expansion to be valid we must put three additional
restrictions on (x,y,t):
(i) The maximum value of  must be small compared with the constant unit depth.
Otherwise nonlinear effects would be too strong to be captured by a second-order theory.
(ii) The forced bottom velocity  / t must be negligible for times greater than one.
More precisely: T'(t) must be approximately zero for t>t0 where t0 is somewhat smaller
than one.
(iii) The forced bottom motion should start with a nonzero velocity: T'(0) 0. If this is
not the case, the orders of approximation cannot be classified the way this is done in the
following.
Surface conditions:

Kinematic:

Dynamic:
Bottom condition:

Kinematic:
/t +  = /z,
/t + (1/2)2 + z = 0,
 /t +  = /z,
z=(x,y,t)
(4)
z=(x,y,t)
(5)
z = (x,y,t)
(6)
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We develop an asymptotic series approximating the surface elevation (x,y,t). We seek
to separate each term in the time and space-dependencies in order to get an asymptotic
series with n terms:
n
(x,y,t)= H(t) i (x,y) Ti(t)
(7)
i=1
Here H(t) denotes the Heaviside unit step function, which is defined equal to zero for t
0 and equal to one for t>0.
Each term is governed by the given bottom deflection (x,y,t)=Z(x,y)T(t) and the
boundary conditions. The kinematic free-surface condition (4) will now be linearized. This
is compatible with full second-order theory, and includes the third-order gravitational flow.
This linearized kinematic condition makes the velocity potential separable to each order:
n
(x,y,z,t)= H(t) i (x,y.z) Ti’(t)
(8)
i=1
Here the time dependencies are the derivatives of the time dependencies of the
corresponding surface elevations so that each i(x,y,z) meets the requirement:
i (x,y) = [i (x,y.z)/ z] z=0
(9)
Boundary value problems to each order
With the restrictions we have put on (x,y,t), the asymptotic series (7) may be useful up
to t=t0. Below the boundary value problems for the space dependent parts of the velocity
potentials are listed along with the corresponding surface elevations. Laplace's equation is
valid to each order. Also listed are the differential equations the time dependent parts must
satisfy. There are three possible agents driving the flow to each order: The forced bottom
motion, nonlinearity at the free surface, and gravity. The two former agents are active in
second order, and their time dependencies are in general different. Therefore the secondorder problem must be split into two sub-problems (subscripts 2B and 2S).
First-order bottom contribution


z=0 = 0

z] z=-1 = Z(x,y)
1 (x,y) = [1/ z] z=0
T1’(t) = T’(t)



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Second-order bottom contribution
B

Bz=0 = 0

Bz] z=-1 = [ (Z(x,y)  1)] z=-1
1B (x,y) = [1B/ z] z=0
T2B‘(t) = T(t) T’(t)

Second-order surface contribution
S

Sz=0 = - 12 /2

Sz] z=-1 = 0
1S (x,y) = [1S/ z] z=0
T2S’’(t) = (T’(t))2
Third-order gravity contribution


z=0 = - 1

z] z=-1 = 0
3 (x,y) = [3/ z] z=0
T3’’(t) = T(t)

On the time factors
Let us summarize the conditions for the time factors to each order:
T1(t) = T(t), T2B(t) = (1/2) T(t)2, T2S‘’(t) = (T’(t))2, T3’’(t) = T(t)
(10)
We have here taken into account the following constraints for an initially flat bottom:
Ti(0)=T(0)=0. In this note we do not give examples of integrating up the different
functions Ti(0) when T(t) is given. But we note that T2S(t) is governed by a differential
equation, which will usually give a solution different from the explicit formula for T2B(t).
This shows the necessity of working with two second order sub-problems.
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3. General solutions in two dimensions
In the present work we give only results in two dimensions. There are two different exact
methods for constructing exact solutions. Below we show the complete results according to
the two-dimensional Fourier transform method. First we show the Green function solutions
for the linearized first and third order problems:
Green function approach
First-order contribution:

1(x) =  Z(x’) sech (((x-x’)/2) dx’
(11)
-
Third-order gravity contribution:

3(x) =  Z(x’ ) [((x-x’)/2) tanh (((x-x’)/2) - 1]
sech (((x-x’)/2)
dx’
(12)
-
Fourier transform approach
First-order contribution:

1(x) =(2 )-1  sech (k)  (ik) eikx dk
(13)
-
Second-order contributions:

2B(x) = - 2S(x) = (4 )-1  (k+K) sech (k+K) tanh (k)  (ik)  (iK) ) ei(k+K)x dK dk
(14)
-
Third-order gravity contribution:

3(x) = - (2 )-1  k tanh (k) sech (k)  (ik) eikx dk
(15)
-
In the equations above, Z is the position dependent part of the bottom deflection, and 
is the corresponding Fourier transform:

 (ik) =  Z(x) e-ikx dx
-
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In Figures 1-4 we present some results from numerical evaluation of these Fourier
integrals.
We consider a rising block of dimensionless width L:
Z(x) = , x < L/2
Z(x) = 0, x > L/2
(16)
Figures 1 and 2 show numerical results for the case L=10. This block is wide enough to
produce an almost flat region in the middle. In Figure 1 the first and third order solutions
are shown. The third-order solution shows that the early influence of gravity vanishes in the
middle of the block. Gravity is important only near the edges of the rising block. Figure 2
shows the second order elevations, which are equal in magnitude but with opposite sign.
Figures 3-4 show the corresponding results for a narrower block: L=2. There is no
longer a flat middle region to first order. Gravity is not merely an edge effect, but it will
influence the rising heap of fluid as a whole. The negative third-order elevation will in fact
have largest amplitude above the middle of the block. From Figures 2 and 4 we see that the
second order elevations, being equal but with opposite sign, will have their zero-elevation
points slightly outside the block. The block displacements at the bottom are included in
Figures 1 and 3 as dotted lines. The first-order curve alone takes care of the mass flux due
to the rising block.
Figure 1. First and third order contributions to the surface elevation for a two-dimensional, rectangular
bottom deflection. The width is ten times the water depth.
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Figure 2. Second order contributions to the surface elevation for a two-dimensional, rectangular bottom
deflection. The width is ten times the water depth.
Figure 3. First and third order contributions to the surface elevation for a two-dimensional, rectangular
bottom deflection. The width is twice the water depth.
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Figure 4. Second order contributions to the surface elevation for a two-dimensional, rectangular bottom
deflection. The width is twice the water depth.
We note that the second-order elevations due to the bottom effect and the free-surface
effect are equal but with opposite sign. In general they do not cancel each other because
their time dependence will be different. In three dimensions these two second-order
elevations will not be equal in magnitude.
4. Concluding remarks
The scope of the present work has been to improve the modelling of a special type of
tsunami generation. The initial build-up of surface elevation by rapid bottom deflections on
initially uniform water depth. The simplest possible model for the surface displacement is
to let it be equal to the bottom displacement. This piston-type hydrostatic modelling is
physically inconsistent, but still quantitatively acceptable if the gradient in the bottom
displacement is small. A more consistent model, which smoothes out the surface
deflections is given by impulsive Green functions. This is equivalent to our first-order
theory. It shares the disadvantage with the hydrostatic piston model that the surface is
assumed to be completely at rest after the bottom motion has ceased. Our theory is a type of
third-order small-time expansion that accounts for the early nonlinearities as well as early
gravity effects in a consistent way. This will produce a deflected surface that remains in
motion after the generation process stops. The later wave propagation can be described as a
Cauchy-Poisson problem with nonzero initial velocity as well as nonzero initial elevation.
The results presented here are only two-dimensional. The corresponding theory in three
dimensions will be published elsewhere.
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References
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Mech. 60, 769-799 (1973).
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5. P.A. Tyvand and A.R.F. Storhaug, Green functions for impulsive free-surface flows due to bottom deflections
in two-dimensional topographies, Phys. Fluids 12, 2819-2833 (2000).
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110