数理解析研究所講究録
第 1645 巻 2009 年 14-21

14

Numerical Simulation of Tsunami Generation
鹿児島大学工学部

海洋土木工学科

柿沼 太郎 (Taro Kakinuma)
Dept. of Ocean Civil Engineering,Kagoshima University

1. INTRODUCTION
In numerical computations of tsunamis due to submarine earthquakes, we usually assume
that the initial displacement of water surface is equal to the permanent shift of sea bottom,
after which we start calculation of tsunami propagation using a shallow-water or longwave model. Similar calculation is performed considering multi segments also whenseabed-deformation areas change their location. Tsunamis are, however, generated withtime. For example, in case of “creeping”, where the seabed defornation proceeds slowly, it
is not difficult to imagine that the profiles are different between water surface and sea
bottom because of propagation during generation of tsunamis.
Outstanding studies have been done focusing attention mainly to initial profiles oftsunamis under some assumptions. Linear analytical solutions of tsunami generation due toseabed deformation were derived by Sano and Hasegawa (1915), Syono (1936), Ichiye(1950), Takahashi (1942), Momoi (1962), Kajiura (1963), Honda and Nakamura (1951),
etc., while a weakly nonlinear solution was shown by Hammack (1973). Hydraulicexperiments were also performed by, for example, Hammack (1973) and Matsuyama et al.
(1995), whose results were compared with those of numerical calculation by Nakayama(1983) and Matsuyama et al. (1995), respectively, using boundary element methods withvelocity potential. Hwang and Divoky (1970) reproduced the field-scale phenomena oftsunami generation and propagation due to the 1964 Good Friday earthquake through ahorizontally two-dimensional model with the shallow-water assumption. Ohmachi et al.
(2001) used a three-dimensional grid system for computations of water motion, where
upwelling currents as sources of tsunami generation were given on the sea bottom.
When a deformation area of seabed has a complex topography and its deformation
velocity changes in space-time, the initial tsunami profile, which affects calculation resultsof tsunami propagation and runup, can be evaluated by considering velocity and pressure
fields accurately. In this study several generation processes of tsunamis due to seabeddeformation have been numerically simulated without assumptions of hydrostatic pressure
and long waves. Moreover, several cases where the uplift area of seabed changes its
position are treated in the present paper. Aida (1969) performed horizontally twodimensional linear calculation to investigate directional characteristics of tsunamis due to adislocation progressing mainly along the long axis of seabed fault. We target upliftdislocation movement along the short axis of seabed-deformation area, resulting in tsunami
eaithquakes or tsunamigenic earthquakes to increase the tsunami potential remarkably.

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2. NUMERICAL MODELA three-dimensional VOF model, which was named STOC-VF and installed in a numericalsimulator to smdy storm surges and tsunamis, i.e., STOC (Kakinuma and Tomita, 2005), isapplied to incompressible-fluid motion. Goveming equations are the continuity andReynolds-averaged Navier-Stokes equations, which are solved with a fmite difference
method. The Poisson equation on pressure is also solved to evaluate not only hydrostatic
but also dynamic pressure. The medium porosity is considered to describe smooth shapesof the sea bottom and smicture faces, around which the water velocity can be represented
more accurately in numerical computations.
When a seabed uplift or subsidence occurs in veiy deep water, it is efficient to use
numerical cells whose size is larger than the seabed shift or water surface displacement.
The numerical model has been improved to take into account temporal change of mediun
porosity, such that the seabed shift is described by temporal change of porosity inside eachcell in the neighborhood of the seabed, while the water surface displacement is evaluatedby solving an advection equation of a VOF fUnction, which means that elevations of bothsea bottom and water surface can be treated also when their movement stays inside eachcell. Accordingly, the grid system is fixed without changing the boundary shape ofcomputational domain throughout numerical calculation of tsunami generation, propagation, and mnup.
In the computations of this paper, the initial water is still with air cells on the water
surface. All fixed boundaries including the sea bottom and side walls are assumed tosatisfy the slip condition. The fluid density is spatially uniform and temporally constant,i.e., $\rho=1.0^{x}10^{3}kym^{3}$ . The total viscosity
is equal to $1.0x10^{-6}m^{2}/s$ , where any
disturbance of a shorter scale than the cell size is neglected.
$\nu_{\epsilon}$

3. MODEL VERIFICATION IN UPLIFT CASES
Computational results are compared with the corresponding experimental data obtained byHammack (1973). The area where $-b\leq x\leq b$ uplifts uniformly. The uplift shift of seabed,, is given as$\zeta$$\zeta(t)=\zeta_{0}$

(l-e

)

when

where $\alpha>0$ . The ratio
is equal to 12.2, where
following two cases are treated in the x-z plane:$b/h_{0}$$t\geq 0$$h_{0}$,(1)is the initial water depth. The(a) Impulsive motion$\zeta_{0}/l_{4}=0.2$

and$t_{c}\sqrt{gh_{0}}/b=0.069$,(2)(b) Transitional motion$\zeta_{0}/h_{0}=0.1$

where$\zeta(t_{c})=2\zeta_{0}/3$

The grid widths

and$t_{c}\sqrt{gh_{0}}/b=0.39$

,

(3)

.
$\Delta x$

and$\Delta y$

are equal to$0.2h_{0}$, while$\Delta z$

is $0.05h_{0}$ , where the number

16(a) Impulsive motion(b) Transitional motion

Fig.1 Time variations of water surface displacement over the center of seabed-uplift area

Fig. 2 Defimition sketch of calculation domain

Fig. 3 Initial tsunami profiles, which
depend on the initial water depth$h_{0}$of cell along the y-axis is one. The time-step interval changes automatically depending on
the Courant number.
The results of water surface displacement over the center of uplift area, where $x/h_{0}=0$ ,are shown in Fig. 1. In the case of impulsive motion, the present calculation evaluates thewater surface displacement accurately, including the short-period oscillations, whose
generation should have not only linear but nonlinear mechanisms as well, for the solutionof Hammack’s linear theoiy does not show harmony with the experimental data. Also in
the transitional motion, the present calculation shows good results, as well as thecalculation performed by Nakayama (1983) using a boundary element method (BEM).

4. TSUNAMI GENERATION DUE TO SEABED DEFORMATIONWHOSE RISE VELOCITY IS CONSTANT
The calculation domain is shown in Fig. 2. The ghd widths
andare equal to 1 km,
while
is 50 , where the number of cell along the y-axis is one. The time-step interval
is equal to 0.5 .
Figure 3 shows water surface profiles of tsunamis due to a uniform uplift of seabedwhen $t=20s$ , where the initial water depth
is 500, 1,000, or 4,000 . The seabeddefonnation occurs inside the area where 30 km
s60 km at the constant rise velocity of$\Delta x$$\Delta z$$\Delta t$$\Delta y$

$m$

$s$

$h_{0}$$\leq x$$m$

17Fig. 4 Water velocity around one end $(x=30 km)$ Fig. 5 Initial tsunami profiles, whichof the seabed-uplift area
depend on applied models

0.15 $m/s$ while $0s\leq t<20s$ and stops when $t=20s$ . In the numerical computations, theporosity of the lowest cells changes from 1.0 to 0.94 inside this deformation area.
According to Fig. 3, we can recognize that the “initial profile” of tsunamis, whose
generation depends on the initial water depth, is not always the same as the permanent shiftof seabed.

Figure 4 shows velocity vectors over the seabed around one end of the defonnation
area when $t=20s$ , where $h_{0}=4,000m$ . If the water depth is deep in a tsunami-generationarea, the horizontal velocity over an edge or a slope of the seabed-deformation area showsa vertical distribution, which cannot be represented through a shallow-water model without

dispersion terns.
Figure 5 shows water surface profiles obtained using the present model (3D) and anonlinear shallow-water model (SW) at the time when $t=20s$ . The seabed uplift is thesame as that in the cases of Fig. 3. The initial water depth is equal to 4,000 $m$ . In thecalculation with the SW, the temporal derivative of seabed elevation,, isconsidered inside the seabed-deformation area. In Fig. 5, the water indicated by $A$ , whichis evaluated through the $3D$ , is lifted up to the area calculated by the SW, such that the
SW overestimates the potential energy of initial tsunami. The wave length estimated by the
SW is shorter than that by the $3D$ , resulting in the difference ofwater surface slope.
$h_{0}$$\partial z_{bo0\propto n}/\partial t$$B$

5. TSUNAMI GENERATION DUE TO SEABED DEFORMATION
WHICH CHANGES rrs OCCURRENCE PLACE
Tsunami earthquakes, which were named by Kanamori (1972) and have been studied by
Pelayo and Wiens (1992) etc. mainly in seismology, can generate tsunamis of a larger
heigt than that estimated using only seismic-wave data obseived before the tsunamis
reach coasts. It is to clear occurring mechanisms of both tsunami earthquakes, which aretypified by the 1896 Meiji Sanriku earthquake, and resulting tsunamis that is important forprotection against disasters. In this study the seabed deformation due to tsunami
earthquakes is classified into the folloWing groups in the view of fluid mechanics throug
tsunami-generation processes.

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Fig. 6 Time variation of water surface profile due to two-stage uplifts in different areas ofseabed. The top faces of gray areas indicate the shapes of seabed. The second stage
occurs while 50 $s\leq t<70s$ .
(I) Seabed deformation without interaction with water motion

Type $A$ : Long-duration deformation in a fixed area
Type : Defornation changing its occurrence place
Type : Deformation generating water motion affected by surroundings
Type $D$ : Deformation generating significant compression water waves$B$$C$(II) Seabed deformation with interaction with water motion

Type : Deformation due to land slides
Type : Deformation due to inelastic ground movement
Type : Deformation as extravasation
$E$
$F$

$G$

(III) Seabed deformation with multiple aspects

In order to evaluate seabed shifts and water surface displacements of Types $E,$ , and ,it is necessary to solve the motion of both sea bottom and sea water simultaneously, for theseabed deformation is interacted with the water motion.
In the present paper, several cases belonging to Typeare treated. The seabeddeformation of Type is subdivided into the following three types.
$F$

$G$

$B$

$B$

Type B-l: Multistage deformation in different areas
Type B-2: Deformation changing its occurrence place continuouslyType B-3: Multistage deformation changing its occurrence place continuously in
different areas

Figure 6 shows the time variation of water surface profile for a two-stage-deformationcase of Type B-l. The seabed inside the area where 50 km
km uplifts while$s\leq t<20s$ , after which the seabed inside the area where 40 km $\leq x<50$ km uplifts while 50$s\leq t<70s$ . The rise velocity of deformation is 0.15 $m/s$ and the permanent shift of seabed,$\leq x\leq 60$

$0$

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Fig. 7 Time variation of water surface profile due to a seabed uplift changing its
occurrence place continuously. The top faces of gray areas indicate the shapes of seabed.
The total width of uplift area, , is equal to 20 km and the progress duration of seabed
uplift, , is equal to 70 .
$R$

$s$

$\tau$

Fig. 8 Relation between height of stably propagating tsunamis,, and progress speed ofseabed uplift,. The rise velocity $W$ is equal to 0.15 $m/s$ . The results are shown forcases of different total width ofuplift area, .
$\eta_{\max}$$R/\tau$$R$, becomes 3.0 $m$ . The initial water depth

is equal to 4,000 . Tsunamis can grow due
to succeeding stages inside different areas of seabed defonnation. In the present case thealmost stably propagating tsunami which has experienced both uplift stages is about twice
the height of that in the corresponding one-stage-defomiation case when $t=200s$ . If there
are two areas of dislocation, the tsunami height depends on both place and timing of thesecond-stage deformation of seabed.
Figure 7 shows the time variation of water surface profile for a case of Type B-2. The
seabed-uplift place moves from its start point , where the uplift starts when $t=$ Os, to theend point $(x_{0}-R)(R>0)$ , where the uplift starts when $t=\tau(s)$ , at the constant speed of. The rise velocity $W$ is equal to 0.15 $m/s$ throughout the seabed-deformation area, after
which the uplift stops when the seabed shift becomes 3.0 , such that the seabeddeformation at the end point $(x_{0}-R)$ stops when $t=r+20s$ . The initial water depth ,the start-point position , the total width of seabed-defornation area, , and the progress
duration of seabed deformation, , are equal to 4,000 $m,$ $60$ km, 20 km, and 70 ,respectively. In this case the tsunami which propagates towards the negative direction of xaxis at the time when $t=200s$ is more than twice the height of that in the corresponding
one-stage deformation case, where the seabed-uplift area is fixed, for the seabed-uplift area$\delta$$h_{0}$$m$$x_{0}$$R/\tau$$m$$h_{0}$$R$$x_{0}$$\tau$$s$20

chases this tsunami to be effective in growing the wave. On the other hand, the seabeddeformation area escapes from the generated waves traveling towards the positive directionof x-axis, resulting in a narrow width of uplift area which concems the tsunami growth,
while the wave length on this side becomes long.
Figure 8 shows the relation between the wave height of stably propagating tsunamis
which have been generated and grown by seabed uplifts of Type B-2,
, and the$R/r$$W$
progress speed of seabed deformation,, where the rise velocity
is equal to 0.15
$h_{0}=4,000m$
and. It should be noted that Fig. 8 was obtained using the nonlinear shallowwater model without dispersion terns, i.e., the SW model, although the $3D$ model wasapplied to obtain Figs. 6 and 7. The growth rate of tsunamis on the progress sides becomeshigh especially when the progress speed of seabed deformation, $R/\tau$, is close to the longwave celerity of tsunamis in the initial water, i.e., approxin$ately\sqrt{gh_{0}}$ , which is indicatedwith a dashed line in Fig. 8.
The tsunami heigt
depends on both and , as well as $W$ and , such that hugetsunamis, which camot be predicted using only seismic ground waves, can appear when
the seabed deformation occurs due to dislocation of a shallow slide angle or a slow rapture
velocity, plastic movement belonging to Typeor (e.g. Tanioka and Seno, 2001),
magma intrusion into sedimentary layers (Kanamori et al., 1993), etc.
$\eta_{\max}$$n\vee s$$\eta_{\max}$$R$$h_{0}$$\tau$$E$$F$

6. CONCLUSIONS
Numerical computations of tsunami-generation processes due to seabed deformation in thevertical two-dimension were performed using the three-dimensional model for incompressible fluids, resulting in accurate initial tsunami profiles, which affect following
propagation and runup. The seabed displacement is described by the temporal change ofporosity inside numerical cells around the sea bottom, while the water surface elevation is
calculated through the VOF method, which means that the present model is able to treat
both sea-bottom and water surface elevations also when their change stays inside each cell.
The initial profile of tsunamis, whose generation is dependent on the initial water depth,
is not always identical with the permanent shift of seabed. If there are several areas orstages of dislocation, the tsunami heigt depends on both place and timing of thefollowing-stage deformation of seabed. The growth rate of tsunami on the progress side ofseabed deformation becomes high especially when the progress speed of seabeddeformation is close to the celerity of tsunami.
Precise analyses of tsunami generation based on fluid mechanics shall contribute todiscovery of unheeded types of seabed deformation including those of tsunami earthquakes.

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