149
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.
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 . 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 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 over an elastic half-space in when at the interface a source is switched on at the initial time 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
in , (1)at , at (2)
in , (3)
, , (4)
at (5)
under , (6)
under , (7)
where; is the velocity potential; is the elastic displacement vector, ; is the Heaviside function; is the delta-function; is the excitation function. The equations (1)-(7) are written in dimensionless form according to the formulas
where is a characteristic horizontal scale.
It is necessary to find the potential as the solution of the Laplace equation (1) and the radial and vertical displacements , as the solutions of the Lame equation (3), , satisfying the conditions (2) at the free surface and at the interface , the boundary conditions (4), (5) for normal and shear stresses at , the initial conditions (6) and the regularity conditions (7).
The problem is solved by using the Laplace transform in time
(8)
The values (8) are presented in the form
, ,(9)
Taking into account the formula and the expressions (8), (9) reduces the problem (1)-(7) to the boundary-value problem for amplitudes
, (10)
at , at , (11)
(12)
, (13)
at (14)
The resultant equation describing the free surface elevation is given in Eq. 15
(15)
where
,
.
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 Fourier-Bessel functions
Figure 1. Free surface elevation generated by the excitation function for a) at the epicenter ; b) over the edge of seismic center .
, , (16)
(17)
and on the Tikhonov regularization procedure to improve the series convergence
, (18)
(19)
Figure 2. Free surface shape at different times: .
The results of calculations are presented in Figs. 1 and 2. The free surface elevation depending on time at two places: (epicenter) and (the edge of seismic center), for different values of are presented in Fig. 1. The parameter characterizes the sharpness of the excitation impulse, so that increasing 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, and , 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
, , (20)under (21)
, , (22)
, . (23)
On the basis of (20)-(23) the effect of the initial bottom elevation () which varies as 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
in (24)at 25)
at (26)
at , (27)
where and are horizontal operators. In (24)-(25) nondimensional values are used according to the formulas (asterisks are omitted): , , , , , , where and are the characteristic length and depth, and are the amplitudes of free surface and bottom elevations, respectively.
As we can see from (24)-(27), the nonlinear parameter, the dispersion parameter and the parameter of nonstationary bottom state are responsible for the phenomena under consideration.
The problem (24)-(27) after some considerations and with the assumption is reduced to the following simplified form
, (28). (29)
Considering the case and using power series expansion also simplifies the problem (28), (29) to a recurrence system of equations.
Assuming , , and applying asymptotic analysis, this system is reduced to the exactness up to the order of the terms in evolution equations.
(30)(31)
, (32)
In the dispersion-free case but of variable depth, , the system (30), (31) is reduced to the following equation
(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 .
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, and small Reynolds' numbers . Typical behaviour of such tectonic streams is observed in Carpathian region.
In a Cartesian coordinate system we consider 2-D flow of the medium the state of which is characterized by the vector
(34)
where is the velocity vector, is the density, is the pressure, is the damage ratio of medium.
Vector is presented as a superposition of undisturbed and disturbed states
(35)
corresponding to slow (tectonic) time and rapid time , so that . In the expression (35) is independent of and it can be considered as a "frozen" background field, is a small parameter. The field is considered in the thin layer of a unit width (thin ray tube).
Let us introduce the non-dimensional axial and transverse coordinates, , where ,. Hereinafter it is assumed that the dynamic viscosity has a local minimum on the axial line, so that at .
The governing equations are written as follows
, (36)
(37)
, (38)
where is the tensor of viscous stresses, is 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.
From the first law of thermodynamics for a unit mass flow, the following equation can be obtained
, (39)
where is the specific enthalpy, the value 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 small value. Taking the uniform field as the simplest solution of undisturbed state, the system of equations of the disturbed state is derived. Considering traveling waves along the streamline yields the dispersion equation
, (40)
where 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 the following conditions hold
(41)
, (42)
where is the Gessian of , is a nonspectral parameter
.
According to Morse' lemma the complex hypersurface in the vicinity of CMP has the standart representation of dispersion equation in the following form
. (43)
Now it is possible to pass to the configuration space, where are linear combinations of .
Pre-ruptured state takes place when , and in this case the system of equations predicts solutions of jump type leading to triggering phenomena.