139
TSUNAMI WAVE EXCITATION BY A LOCAL FLOOR DISTURBANCE
I.T. SELEZOV
Department of Wave Processes
Institute of Hydromechanics, Nat. Acad. Sci.
8/4 Sheliabov Str., Kiev 03680, Ukraine
Abstract
This paper presents the modeling of tsunami wave generation by a bottom displacement
in three-dimensional settlement taking into account elastic floor. The corresponding initial
boundary value problem is solved and analyzed. The IBV-problem based on the shallow
water wave model and the problem of nonlinear-dispersive waves over an excitable bottom
are discussed and first order asymptotic approximations describing the problem are
presented. A new type earthquake wave model is presented and analyzed. The existence
of the solutions of jump type is established which can lead to triggering phenomena.
Undeterminacy to state the initial boundary value problems for tsunami generation is
discussed.
1. Introduction
Generation of tsunami waves is a problem of great importance and many researches have
been focused on this problem for a long time [2, 7, 9, 12]. Up to now there are no perfectly
clear explanations of the earthquake triggering mechanism and as a consequence there are
no perfect models to formulate initial conditions. Nevertheless, in any case the problem of
tsunami wave generation is stated as initial boundary value (IBV) problem with the given
initial conditions corresponding to some idealized situations.
Considering the problem to state initial boundary value problem for tsunami wave
generation shows that this problem is undeterminate. The main reason is the
undeterminancy of earthquake triggering mechanism in general, including underwater
earthquakes particularly, in spite of having been investigated for a long time [1-3, 7-20]. As
a result, till now there are no completed satisfactory achievements and explanation of the
causes of this phenomenon. As far as a triggering mechanism is concerned it is difficult to
make this problem to perfectly determine, in the mathematical sense.
For example, in 1998 three huge tsunami waves of 10 m elevation run up on lagoon in
Papua, New Guinea. Tsunami waves were generated by the underwater earthquake due to
underground shocks. When should these appear? With what time intervals between them?
The question arises how to state the IBV-problem? Later on we would present some
particular models from this point of view.
A. C. Yalçıner, E. Pelinovsky, E. Okal, C. E. Synolakis (eds.),
Submarine Landslides and Tsunamis 139-150.
@2003 Kluwer Academic Publishers. Printed in Netherlands
140
It was noted by Braddock et al [2] and this situation has not changed that the actual
bottom motion is not completely understood. For example, seismic investigations show that
actual flow conditions in the mantle are complex and undeterminate varying from cracks of
whole mantle to the layered structure [4]. As has been noted by Papanicolaou in [11], the
energy localization in random elastic media is possible due to such causes as the mode
conversion, the transfer of energy from compression to shear waves, and polarization.
Brevdo [3] considered an elastic layer in 3-D and instead of a convenient purely normal
wave mode treatment investigated the asymptotic behaviour of wave packet. On this basis,
a possible resonant triggering mechanism of certain earthquakes is shown due to localized
low amplitude oscillatory forcing at resonant frequencies. Considering the energy budget of
deep-focus earthquakes suggests that they may be slip-sliding away [8], leading to
undeterminate triggering mechanisms. One possible triggering mechanism can be initiated
by the re-polarization of elastic waves at an interface where the perfect matching can be
violated by enough strong tension stresses [20].
A new earthquake model of wave type has been proposed in [18]. It is based on the
hydrodynamic flow of geomaterial along the ray tube of tectonic stream and on the
evolution of the medium damage governed by a kinetic equation for the damage ratio
[0,1] . The analysis of dispersion equation shows that wave propagation soliton-like
disturbances can be expected for the narrow wave beams in the vicinity of the critical
Morse points. In the case   1 the singular degeneration takes place and this leads to the
jumps in structural parameters which can cause earthquake. The triggering time can not be
established exactly if the above presented procedure is followed.
This paper considers the problem of excitation of the surface gravity waves in ocean
due to the underwater source. The concentrated source is placed at the interface between
water and elastic half-space. The original problem is essentially simplified and can be
useful for analysis of tsunami waves. The IBV-problem based on the shallow water wave
model and the problem of nonlinear-dispersive waves over an excitable bottom are
discussed and first order asymptotic approximations describing the problem are presented.
2. Wave Generation by a Source on Elastic Floor
Corresponding initial boundary value (IBV) problem is stated for the fluid of finite depth in
1  r, , z  :r[0, ), z0,1 over an elastic half-space in  2 when at the interface z  0 a
source is switched on at the initial time t  0 which sharply increases up to the maximum
and then exponentially decreases with time.
The motions of fluid and elastic solid are governed by the potential flow equations for
incompressible inviscid fluid and by the elastodynamic equations for isotropic
homogeneous medium respectively
  1 
2
2
2
 1
2
t
2

2 1
 z2
 1
z
0
in
0
1 ,
1
 uz
 2
 t at z  0
at z  1 ,  z
(1)
(2)
141
2 
  

2  u
2
 u  0 1    u  2
t2

2

 

 u u
  02  2  r  r
r
 r
 ur
u
 z  0
z
r
2 ,
in
A0

u

r 
   02 z   1  
F t  H t 
z
t

r , z0,

z 0
 1
 uz
u
1 
 uz 
 ur  r  0
t
t
t
under t  0 ,
u r , uz 0
under z    ,
at
(3)
(4)
(5)
(6)
(7)

where  2  1/ r  /  r (r  /  r ) ; 1 is the velocity potential; u is the elastic displacement

vector, v  (ur , u z ) ; H (t ) is the Heaviside function;  r  is the delta-function; F (t ) is the
excitation function. The equations (1)-(7) are written in dimensionless form according to
the formulas
r* 
gh
1
r
z
 
u
h
, z*  , t*  t
,  *  , u *  ,  1* 
,  ,
R
h
R
h
h
R
R gh
0 
ce
,
cs
gh
1 
ce
, 2 
gh
cs
,

0 g h
.
G
where R is a characteristic horizontal scale.
It is necessary to find the potential 1 as the solution of the Laplace equation (1) and
the radial and vertical displacements u r , u z as the solutions of the Lame equation (3),

u  ur , 0, u z  , satisfying the conditions (2) at the free surface z  1 and at the interface
z  0 , the boundary conditions (4), (5) for normal and shear stresses  zz , rz at z  0 , the
initial conditions (6) and the regularity conditions (7).
The problem is solved by using the Laplace transform in time


0
0
1L   1 e  pt dt , u rL   u r e  pt dt

u zL   u z e  pt dt
0
(8)
The values (8) are presented in the form


 1L   ( p, s, z ) J 0 ( sr )d s u rL   U ( p, s, z ) J 1 ( sr )d s
0
,
0
,

u zL   W ( p, s, z ) J 0 ( sr )d s
0
(9)
142

 (r )
Taking into account the formula
  s J 0 (rs) ds and the expressions
r
0
(8), (9) reduces the problem (1)-(7) to the boundary-value problem for amplitudes , U, W
 ''  s 2  2   0 ,
'
(10)
2 2
'
2
  p    0 at z  1 ,   p  W at z  0 ,
(11)
 02 W ' '  s  U '  s 2 2 W  s  U '  p 2 2  22 W
(12)
  02 s 2 2 U  s  W '  U ' '  s  W '  p 2 2  22 U ,
(13)



'
U  s  W  0,

 02  2s U   02 W'   p    A0 s FL (p) at z  0 (14)
The resultant equation describing the free surface elevation is given in Eq. 15

A0 2
s J 0 (rs )
L
L
  F ( p)


p  f (s)
0
0
ds
(15)
where
f (s) 
0 
s
2 s e s
s  p 2  (s  p 2) e  s ,
s (1   2 ) (  02   02 c1 c 2  2)  (c 2  c1 ) (  02 1  2  s 2 (  02  2))

c 2 1  c1  2
p 2  sth s
2
s  p  th s
  p2
.
Using the Cagniard approach [8] the exact analytical solution for the epicenter elevation
can be obtained. For arbitrary values of r (radial coordinate) calculations are carried out on
the basis of a numerical Laplace transform inversion. The approach developed is based on
the expansions of desired functions with respect to the orthonormal system of FourierBessel functions
143
Figure 1. Free surface elevation generated by the excitation function F (t )  t 2 exp( at ) for a  1, 2, 3 :
a) at the epicenter r / R  0 ; b) over the edge of seismic center r / R  1 .
f (t) 

 c n J  (k n e   t )
n 1
c n 
,

J  (k n )  0 ,
m
2
(1)
 k n 


2
J  1 (k n ) m  0 m!(m    1) 2 

(16)
2m  
F(( 2m  2  ) )
(17)
and on the Tikhonov regularization procedure to improve the series convergence

f (t) 
 c n  n  n (t)
n 1
n 
,
(18)
1
1   k 2n
(19)
144
Figure 2. Free surface shape at different times:
gh
R
t  0.5; 20; 50 .
The results of calculations are presented in Figs. 1 and 2. The free surface elevation
depending on time at two places: r  0 (epicenter) and r  1 (the edge of seismic center),
for different values of a are presented in Fig. 1. The parameter a characterizes the
sharpness of the excitation impulse, so that increasing a increases the impulse sharpness,
as well as the sharpness of the free surface response, but in this case the magnitude
decreases due to decreasing the time to transmit the energy from seismic center to the free
surface.
Solid curves in Figure 1a correspond to exact solution, while dotted curve shows the
results of numerical inversion for evaluation of the exactness of numerical transform
inversion. The calculations were also carried out for the free surface elevations at different
ratios of the propagation velocities of shallow water waves and shear waves in solid,  1
and  2 , and for different relative depths.
Approximate analysis of tsunami wave generation in water of the variable depth can be
carried out on the basis of shallow water wave equations. In this case the corresponding
IBV-problem is essentially simplified. At the same time, this approximate model is
applicable with sufficient exactness at some distances from an epicenter. The
corresponding IBV-problem is stated as follows




 2 1
  H (r )  1 
 0,
t2
1
t 0
 0 (r ),
 1
t
t 0
r  [0, a ] ,
 0,
t  [0, ) ,
(20)
1  0 under r  0
(21)
145
 2 
2
2
t 0
 2 2
t2
 0,
 2
t
 0,
r  ( a, ) ,
t 0
 0,
t  [0, ) ,
 1
1 r a  2 ,  r
(22)
r a

 2
r
r a
.
(23)
On the basis of (20)-(23) the effect of the initial bottom elevation ( H 0 ) which varies as
H (r )  H 0  (1  H 0 )r 2 on the tsunami wave generation has been investigated in [15, 17].
3. Excitation of nonlinear water waves
The investigation of nonlinear-dispersive effects during tsunami wave propagation is a
problem of great importance [12]. The problem of particular interest is the excitation of
nonlinear water waves by a bottom surface. The original problem of nonlinear water wave
propagation over inhomogeneous moving bottom is stated as follows
 2 
2 
0
 z2
in 
 
1
t        z

at z   
 2   2
  t 
 x  ( )  0
2
2
z 
at
 
 

 ( t     )     H   z

at z   H ( x, y )   ( x, y, t ) ,
(24)
25)
(26)
(27)

where  2 and  are horizontal operators. In (24)-(25) nondimensional values are used
according to the formulas (asterisks are omitted): ( x * , y * )  ( x, y ) l , ( z * , H * )  ( z, H ) / h0 ,
 *   /  0 ,  *   / a ,  *   g h0 / g l a , t *  t g h0 / l ,
where l and h0 are the characteristic
are the amplitudes of free surface and bottom elevations,
length and depth, a and  0
respectively.
As we can see from (24)-(27), the nonlinear parameter   a / h0 , the dispersion
parameter   (h0 / l ) 2 and the parameter of nonstationary bottom state    0 / l are
responsible for the phenomena under consideration.
The problem (24)-(27) after some considerations and with the assumption    1 is
reduced to the following simplified form
146

2 
t
2


 0,
z

z   ,
      
     H  
,
 t
 z

t
z   ,
(28)
z H  .
(29)

Considering the case H  1 and using power series expansion     n ( x, y, t ) ( z  1) n
n 0
also simplifies the problem (28), (29) to a recurrence system of equations.
Assuming   1 ,   1 ,   O( ) and applying asymptotic analysis, this system is
reduced to the exactness up to the order O( ,  ,  ) of the terms in evolution equations.
 2 0
t2
 c02 (, )  2  0 
F    

F
  2  2 0  4
  0 
,
2
2 t
6
t
   2
  ,
t2 2
2

2
0  


 0
 
  H  0 
 2
2
t
t ,
2
 0
,
t
(30)
(31)
2
c 02 (, )  1   0   
(32)
In the dispersion-free case but of variable depth, H  const , the system (30), (31) is
reduced to the following equation


 2 0
 2
  H  0 

 t2
 t2
(33)
As we can see from the equation (30), the presence of moving bottom leads the
appearance of excitational force and changing the propagation velocity c0 .


4. A new earthquake model of wave type
There are several seismic models but two of them are in the interest of this paper. First of
all, the diffusion theory developed by Elsasser [6] is well known but not capable to explain
the migration of seismicity to large distances. Unlike this theory the model of Nikolaevsky
et al. [10] predicts propagation of tectonic stress disturbances which are like triggers to
initiate earthquakes. He considers the bending and compression of lithosphere plates
contacted with tectonic streams neglecting inertial forces in these streams.
We developed a new model to predict the possibility of solitary wave propagation. This
model is based on a geological concept of tectonic streams as hydrodynamic structures. The
tectonic streams appear at the boundaries of plates as a result of their interactions. These
streams are characterized by small velocities (1 cm/year), nonlinear effects, laminar flows,
147
and small Reynolds' numbers 1010 1020  . Typical behaviour of such tectonic streams is
observed in Carpathian region.
In a Cartesian coordinate system x1, x2 , x3   x, y, z  we consider 2-D flow of the
medium the state of which is characterized by the vector

Q t  u i , , p, 
(i  1,3)
(34)
where u  u1  ux , u3  uz  is the velocity vector,  is the density, p is the pressure,
 is the damage ratio of medium.

Vector Qt is presented as a superposition of undisturbed and disturbed states




Q t  Q 0 x, z, t    Q x, z, t '

(35)
corresponding to slow (tectonic) time t and rapid time t ' , so that t '  t . In the

expression (35) Q 0 is independent of t ' and it can be considered as a "frozen" background

field,  is a small parameter. The field Q is considered in the thin layer of a unit width
(thin ray tube).
Let us introduce the non-dimensional axial and transverse coordinates   l / l 0 ,   r / h ,
where  [, ] ,  [1,1] . Hereinafter it is assumed that the dynamic viscosity has a local
minimum on the axial line, so that  r  0 at   0 .
The governing equations are written as follows
D
  u k, k  0
Dt
k  1,3
,
D ui

 p , i  ij, j   g i3 ,
Dt
D
 R (p, * ) (1   )  
Dt
where
  0,
(36)
i, j  1,3
(37)
(38)
 
 
D

1
 ij  ij )1 / 2 is

 u   ,  ij is the tensor of viscous stresses,  *  (
2
Dt
t
the intensity of shear stresses. The system (36)-(38) includes the mass conservation law
(35), Navier-Stokes equations (37) and kinetic equation (38). The volume compressibility
will be ignored in the stress tensor  ij .
From the first law of thermodynamics for a unit mass flow, the following equation can
be obtained
148

DS  D u
1  
u
 g u 3   ext  tr (   u )  0
Dt
Dt

,
(39)
where S  E  p /  is the specific enthalpy, the value ext characterizes heat-exchange with
the surrounding medium
Now the closed system of equations can be written for the undisturbed state
corresponding to a steady laminar stream with the "frozen" value   * and negligibly
 
0
small value D Q ( x, t )  1 . Taking the uniform field as the simplest solution of undisturbed
Dt
state, the system of equations of the disturbed state is derived. Considering traveling waves
along the streamline yields the dispersion equation
F(k, , )  0 ,
(40)
where k is the wave number,  is the angular frequency,  is a perturbation parameter. .
Then the analysis is carried out in the neighborhood of critical Morse points (CMP). At the
isolated nondegenerated CMP Wm C 3 the following conditions hold
F
F
F
Wm 
Wm 
W 0
k

 m
det Jk, ,  Wm  0
,
(41)
(42)
where J is the Gessian of F ,  is a nonspectral parameter
F(k, , ) : C  C  C  C .
According to Morse' lemma the complex hypersurface in the vicinity of CMP has the
standart representation of dispersion equation in the following form
1 k 2   2  2   3  2  2   0 .
(43)
Now it is possible to pass to the configuration space k  i / s,   i /  , where s, are
linear combinations of l, t .
Pre-ruptured state takes place when (1   )  0, [0,1] , and in this case the system of
equations predicts solutions of jump type leading to triggering phenomena.
5. Conclusion
Brief review of tsunami excitation by earthquakes is presented showing undeterminate
data to state corresponding IBV-problems.
The IBV-problem for tsunami wave generation is stated and solved on the basis of
Fourier-Bessel expansions and Tikhonov regularization procedure. The results of
149
calculations are presented demonstrating the initial development and evolution of tsunami
waves. The dependence of tsunami generation on the sharpness of a source is analysed.
Evolution equations for propagation of nonlinear-dispersive water waves over an
excitable bottom are derived starting from the original 3-D statement.
A new wave model for triggering mechanisms of earthquakes is presented. The problem
includes the equation for the damage ratio  whose a critical value can essentially
influence the solution. The analysis is based on the critical Morse' point approach.
Acknowledgement: This investigation is supported by the Research Project INTAS (Grant
99 - 1637) and SFFR Project of Ukraine (Grant № 01.07/00079).
References
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