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Proceedings of the Sixth International Conference on
Asian and Pacific Coasts (APAC 2011)
December 14 – 16, 2011, Hong Kong, China
99
HINDCAST SIMULATION OF 2011 GREAT EAST JAPAN
EARTHQUAKE TSUNAMI
B.I. MIN, B.H. CHOI
Department of Civil and Environmental Engineering, Sungkyunkwan University, 300
Chencheon-dong, Jangan-gu, Suwon, Gyeonggi-do 400-746, Korea.
K.O. KIM
Korea Ocean Research & Development,
Ansan, 426-744, Korea
V.M. KAISTRENKO
Institute of Marine Geology and Geophysics Russian Academy of Sciences, Yuzhno-
Sakhalinsk, 693022, Russia
E.N. PELINOVSKY
Department of Nonlinear Geophysical Processes, Institute of Applied Physics, Russian
Academy of Sciences, Nizhny Novgorod, 603950, Russia
We proposed a rapid method for forecasting tsunami runup on the coasts combining 2-D
numerical model and 1-D analytical runup theory. In the first step, the 2-D numerical
simulations of tsunami generation and propagation are performed assuming impermeable
boundary conditions of a 5-10m depth at the last sea points (equivalent to a wall). Then
the time series of the water oscillations on the wall are used to calculate the runup
heights using the analytical integral expression following from 1-D theory. The
feasibility of this approach was validated against the disastrous 2011 Great East Japan
Earthquake Tsunami. We have demonstrated that the proposed approach is more
reasonable and rapid method of forecasting than complicated coastal inundation models
1. Introduction
Numerical modeling of the propagation of tsunami waves is an important tool for
forecasting tsunami heights and the risks posed to coastal populations. Previous
analyses of tsunami characteristics have used the shallow-water model with a no-
flux boundary condition at a depth of 5–10m on the shore (Titov and Synolakis,
1998; Choi et al., 2003). In this model, runup heights are determined using
simplified formulae of the 1-D, analytic theory of long-wave runup for a fixed
100
shape of an incident wave, whether a sine wave or a solitary wave (Choi et al.,
2002; Ward and Asphaug, 2003). The direct calculation of the propagation of
the tsunami wave from its source to the coastal zone using a single numerical
model results in low accuracy. As a partial solution, various nested methods with
different mesh resolutions in the open sea and the coastal zone have been
developed (Titov and Synolakis, 1998; Choi et al., 2003). Near the coast,
accurate computing of tsunami wave dynamics requires small grid steps of 10–
100 m. As a result, such numerical models are difficult to use in operational
practice. The primary purpose of this study is to develop a model that combines
numerical simulations for wave dynamics far from the coast with analytical
solutions for the wave runup. This new approach facilitates the rapid prediction
of a tsunami‟s characteristics upon its arrival at the coast to help prevent
tsunami-related disasters.
2. Analytical Approach to Compute the Runup Height through the
Height of a Tsunami Wave at the Wall
The popular approximation of the bottom profile is a plane beach combined with
a flat bottom (such an configuration is usually applied in laboratory modeling).
In this case the incident and reflected waves are easily separated on the flat
bottom, and the runup height of the monochromatic wave is (Shuto, 1972)
2
2
1
2
0)2()2(
2
/kLJkLJ
AR
(1)
where R is the maximal runup height, A is the incident wave amplitude, L is a
shelf width, k is the wave number of the incident wave, J0 and J1 are Bessel's
functions.
Figure 1. Schematic presentation of the coastal zone
With application to tsunami, the bottom profile is variable at any depth, and
a classic numerical model includes the wall on depth h on distance L from the
shore (Figure 1). As a result, numerical simulation allows computing the
oscillation of water level on the wall. To select the incident and reflected waves
101
from the wave records near the wall is not simple, and here we apply another
approach based on rigorous solutions of the wave equation.
The shallow-water equations in the linear approximation can be reduced to
the wave equation for the water displacement
x
xgh
x
t
)(
2
2
(2)
where g is the acceleration due to gravity and h(x) describes the bottom profile.
The general solution of (2) can be found using the Fourier transformation
dextx ti
),(
~
2
1
),(
(3)
where Fourier transform
~( , )
x
is the solution of ODE followed from (2)
0
~
~
)( 2
dx
d
xgh
dx
d
(4)
We will assume that
xxh
)(
everywhere. In this case an elementary
solution of (4) is presented through the Hankel functions.
)()()()(),(
~)2(
0
)1(
0
HAHBx
(5)
Here
gx/2
The equations (5) for x = L can be considered as a system of equations to
find an amplitudes of the incident and reflected wave.
),(
~
4)(
)( )1(
1
L
TTHi
A
(6)
),(
~
4)(
)( )2(
1
L
TTHi
B
Thus, the expressions (6) can be used to compute the spectral amplitudes of
the incident and reflected waves through the Fourier spectrum of the water
oscillations on the wall.
Let us consider now the wave runup on the same plane beach with no
vertical wall assuming that the incident monochromatic wave has the same
amplitude as in “wall” problem. In this case the bounded (on the shore) solution
(5) is
)()(2),(
~0
JAx
(7)
102
Eliminating in (7) from (6) we may calculate the wave field on a plane
beach versus the wave oscillation on the wall. In particular, the spectral
amplitude of the water oscillations on the shore is
),(
~
)()()()( )(2
),0(
~)2(
0
)1(
1
)2(
1
)1(
0
)1(
1
L
THTHTHTH TH
(8)
Using the Fourier transformation of (8) we may compute
),0( t
through
),( tL
. The kernel of this transformation contains the special functions and it is
too complicated to simplify the Fourier integral. If the wall is far enough from
the shore
)1( T
, using an asymptotic expression (6) the formulae (8) is
simplified to
),(
~
2)(
),0(
~)1(
1
L
TTHi
(9)
The expression (8) can be considered the runup ratio for a sine wave on a
planar beach. Using the Fourier transformation of (8), we can compute
),0( t
through
),( tL
. The inverse Fourier transformation in this asymptotic case is
expressed by the simple integral
Tt d
dLxd
Tt
t
tx
022
),(
)(
),0(
(10)
Formula (14) can be integrated by parts and transformed to
d
dLxd
Tttx Tt
2
2
0
22 ),(
)(),0(
(11)
In both formulas t = 0 corresponds to the wave approaching to the vertical
wall and in initial moment it is assumed that
0/)0,()0,( dttLxdtLx
. The developed analytical approach that
have be used in estimations of the 1983 and 1993 tsunami runup heights in the
East (Japan) Sea. A general description of these events are given in (Choi et al.,
1994, 2003) and the applicability of developed analytical approach was well
documented in (Choi et al., 2011 and Min et al., 2011)
3. Numerical Model
The finite-difference model (Choi et al., 2003) is constructed to simulate the
tsunami generation and propagation using the linear shallow-water equation with
a spherical coordinate system covering the north-east of Pacific Ocean area (Fig.
2); mesh dimension, 5040 x 6000, mesh size, 15 arc second (450 meter) and time
step, 0.5 sec.
103
Figure 2. Computational domain for numerical simulation of tsunami wave due to 2011-Earthquake
off the Pacific Coast of Tohoku, Japan and 3D projection of the initial water displacement of 2011-
Earthquake off the Pacific Coast of Tohoku, Japan (USGS, http://earth
quake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/finite_fault.php).
The initial surface profile of the tsunami can be estimated by the method of
Manshinha and Smylie (1971). According to our initial assessment in this case
(2011-Earthquake off the Pacific Coast of Tohoku, Japan), the determined initial
condition tend to over-estimate the tsunami heights because single fault
parameter represented the over 400 x 100km. Thus, we used that the earthquake
initial water displacement conditions are determined from USGS. The initial
surface water displacement are converted to displacement by removing the
instrument response and then used to constrain the slip history based on a finite
fault inverse algorithm. Figure 2. Shows the initial water displacement of 2011-
Earthquake off the Pacific Coast of Tohoku, Japan
The boundary conditions near the coast are “no-flux”: the normal
component of the particle velocity or flow discharge to the boundary is zero,
corresponding to a vertical wall at the last sea points. We previously used this
numerical shallow-water model in our earlier studies of tsunami waves in the
East (Japan) Sea (Choi et al., 2002, 2003).
Snapshots of the tsunami propagation computed in the framework of this
model, are shown in Figure 3. After approximately 5 hour since the tsunami first
occurred, the tsunami waves approached the southern Korean coast.
104
Figure 3. Snapshots of images of computed sea elevation for the 2011-Earthquake off the Pacific
Coast of Tohoku, Japan
4. The Distribution of Runup Height along the Eastern Japan Coast
Locations of post-tsunami runup surveys and spatial distribution of the maximum
heights at the Japan are shown in Figure 4. The highest run-up height has been
observed at Miyako on the eastern Japan. It reaches 40 meter according to
survey team. The displacement of the water on the shore was calculated using the
integral in Eq. (10) or (11) for the coastal locations shown in Fig. 5. The
amplification factor of the tsunami waves at near-shore is approximately 1.7 to
2.2. The observed maximal runup heights in these locations are given by red
circle in the map, water displacement on the wall are given by gray „x‟ and on
the shore are given by red triangle.
105
Figure 4. Location and runup wave heights of the measured points as of 5 May 2011, runup
measurement data provided by JSCE (http://www.coastal.jp/ttjt/)
Figure 5. Comparison of the computed and observed runup heights on the eastern Japan coasts.
106
5. Conclusion
Combining 2-D numerical models and 1-D analytical runup theory, we proposed
a novel and rapid method for forecasting tsunami runups on the coasts. In the
first step, the 2-D numerical simulations of tsunami generation and propagation
are performed assuming impermeable boundary conditions of a 5–10m depth at
the last sea points (equivalent to a wall). Then the time series of the water
oscillations on the wall are used to calculate the runup heights using the
analytical integral expression following from 1-D theory. We have demonstrated
that the proposed approach is a more reasonable and rapid method of forecasting
than complicated coastal inundation models.
Acknowledgement
This study was supported by Korea Ocean Research & Development Institute
(PE98671 & PE98562). Runup observation data provided by JSCE are greatly
acknowledged..
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