Numerical Evaluation of Tsunami Wave Hazards in Harbors along the South China Sea
Huimin H. Jing 1, Huai Zhang1, David A. Yuen1, 2, 3 and Yaolin Shi 1
1Laboratory of Computational Geodynamics,
Graduate University of Chinese Academy of Sciences
2Department of Geology and Geophysics, University of Minnesota
3Minnesota Supercomputing Institute, University of Minnesota
Contents
1. Introduction
2. Numerical experiments
2.1 Governing equations
2.2 Finite difference scheme
2.3 Numerical Model
3. Results and conclusions
1. IntroductionThe probability of tsunami hazards in South China Sea.
The Manila Trench bordered the South China Sea and the adjacent Philippine Sea palate is an excellent candidate for tsunami earthquakes to occur.
The coastal height along the South China Sea is generally low making it extremely vulnerable to incoming waves with a height of only a couple meters.
Historical earthquakes in South China Sea and its adjacent regions
The results of the probabilistic forecast of tsunami hazard show the region where the wave height is higher than 2.0m and between 1.0-2.0m with a grid resolution of around 3.8km (Y. Liu et al. 2007).
In order to investigate the wave hazard in the harbors, simulations in higher precise are needed……
L < 2hL < 2h L > 20hL > 20h
Deep water waveDeep water wave Shallow water waveShallow water wave
DispersiveDispersive
Extended Boussinesq equationsExtended Boussinesq equations
Shallow water equationsShallow water equations
Conventional Boussinesq equationsConventional Boussinesq equations
Weakly DispersiveWeakly Dispersive Non dispersiveNon dispersive
2. Numerical experiments
Dδ= <<1
L
is the basic parameter in the theory of shallow water model:
L
DhB(x,y)
h(x,y,t)
z,w
y,v
x, u
Ω
δ
2.1 Governing equation
2
Navier-stokes equation for incompressible flow
Mass conservation ( ) 0
Momentum conservation, 2
; densit
:
,
( )
Ut
DUU p U f
Dt
��������������
y U; velocity vector,
; angular velocity,
; pressure, μ;viscosity parameter,
,
p
; other body forces.f��������������
Derivation of the Shallow Water Equation
Derivation of the Shallow Water Equation For mass conservation equation
Navier-stokes equation for mass conservation
( ) 0
Applying constant density assumption,we get
0, i.e. 0
Ut
u v wU
x y z
-----(1.1)
Integrating (1.1) in z direction,
( , , , ) ( ) ( , , ) -----(1.2)
Definition of boundary location: ( , , , ) 0
Then the kinematic boundary cond
u vw x y z t z w x y t
x y
F x y z t
ition is, 0 0.DF
at FDt
On the water surface ( , , ), ( , , ) 0 -----(1.3)
Substituting (1.3) into the kinematic boundary condition,
z h x y t F z h x y t
( , , , ) -----(1.4)
Similarly, at the bottom ( , ),
( , , , ) -----(1.5
B
B BB
h h hw x y h t u v
t x y
z h x y
h hw x y h t u v
x y
)
Substituting (1.5) into (1.2), ( , , ) ( )-----(1.6)
Combining (1.2), (1.4) and (1.6),
[( ) ]
B BB
B
h h u vw x y t u v h
x y x y
hh h u
t x
[( - ) ] 0 -----(1.7)Bh h v
y
Derivation of the Shallow Water Equation For mass conservation equation
2
Navier-stokes equation for incompressible flow
2 ----- (2.1)
For non-viscosity fluid 0, and for gravity
Then we get
DUU p U f
Dt
f gk
��������������
����������������������������( )
2 2
2 ----- (2.2)
Hydrostatic assumption:
( ), with 1
DUU p gk
Dt
pg O
z
��������������( )
----- (2.3)
Derivation of the Shallow Water Equation For momentum conservation equation
0
Integrating equation (2.3) at z direction
( , , ) ----- (2.4)
Assuming on the suface z=h(x,y,t) ( , , ) ,
p gz p x y t
p x y h p
,
0
0
( ) ----- (2.5)
Applying constant density assumption,
[ ( ) ] ----- (2.6)
Substituting (2.6) into (2.2),
p g h z p
p g h z p g h gk
��������������
2 ----- (2.7)DU
U g hDt
( )
Derivation of the Shallow Water Equation For momentum conservation equation
Since the harbor area is not very large, neglecting the
Coriolis force term ( 2 U ) in shallow water equation
we get the following governing equation.
u u u hu v g 0
t x y x
v v v hu v g 0
t x y y
h
B Bh h u h h v 0
t x y
Governing equation
2.2 Finite difference scheme
We simplify the equation into a linearized form
In our program, leap-frog scheme of finite difference method has been used to solve the SWE numerically and to simulate the propagation of waves. The leap-frog algorithm is often used for the propagation of waves, where a low numerical damping is required with a relatively high accuracy.
Simplify the equation into a linearized form
Let H(x,y) be the still water depth, and (x,y,t) be the vertical displacement of water surface above the still water surface.
Then we get
It follows from the definition
where C is a constant.
( , ) ( , ) ( , , )Bh h x y H x y x y t
( , ) ( , )Bh x y H x y C
Taking the partial
derivatives of h(x,y,t)
( , , ) ( ( , , ) ) ( , , )
( , , ) ( ( , , ) ) ( , , )
( , , ) ( ( , , ) ) ( , , )
0
0
( ) ( )0
h x y t x y t C x y t
t t th x y t x y t C x y t
x x xh x y t x y t C x y t
y y y
ug
t xv
gt y
Hu Hv
t x y
We linearized the equation by neglecting the product terms of water level and horizontal velocities to obtain the following linearized form of the SWE.
Simplify the equation into a linearized form
The leap-frog scheme is used to solve the SWE numerically and to simulate the propagation of waves.Staggered grids (where the mesh points are shifted with respect to each other by half an interval) are used.
Staggered Leap-frog Scheme
1 11 12 2
1 1 1, ,, ,
2 2
1 11 12 2
1 1 . 1 ,, ,
2 2
1 1 1, 1, , , 11 2 2 2
, , 1 1 1, , ,
2 2 2
1 1( ) ( ) 0
1 1( ) ( ) 0
1 1 1( ) ( )( ) ( )(
2 2
n n n ni j i j
i j i j
n n n ni j i j
i j i j
n n ni j i j i j i jn ni j i j
i j i j i j
u u gt x
v v gt y
H H H Hu u v
t x y
1
21
,2
) 0n
i jv
Discretization of staggered leap-frog scheme
Model components a. the actual topography bathymetry data b. the wave propagation simulation packages c. scenarios of the wave source
Considering that we haven’t the high precise actual bathymetry data of the harbors along South China Sea, we use the data of the Pohang New Harbor in the southeast part of South Korea instead.
2.3 Numerical model
Topography of Pohang New Harbor
Comprised topography data
The data of the compute area is comprised by topography data on the land (SRTM3, with a grid resolution of around 90m ) and bathymetry data of the seabed (SRTM30, with a grid resolution of around 900m ) .
In our numerical model, the grid size of the computational area is about 50m while the time step is 0.05s.
cos ( cos sin )
; ;
2* / ( )
2* * ; ;
;;
z A Kx x y T
A amplitude wavedirection wavelength
Kx angular wavenumber or circular wavenumber
f angular velocity f frequency
Scenario for plane wave source When the tsunami comes from far field, the incident waves near the harbor area can be approximately considered as plane waves. Plane wave function is as following:
Supposed the wave height is about 1m in the far field ocean, and carried on our simulations on the actual bathymetry data with the wave propagation packages.
Animations of the different sources The reflection, diffraction and interference phenomenon
of the waves are illustrated by the animations.
Water surface elevation track recorder points The time series data of the water level vary with time have
been recorded.
3. Results and conclusions
The results of point source case (from the east)
The results of plane wave case (from south direction)
:
cos ( cos sin )
2* *
2* /
;
;
;
(or circular wavenumber).
Plane wave function
z A Kx x y T
f
Kx
wavedirection
A amplitude
f frequency
Kx angular wavenumber
The results of plane wave case (from east direction)
:
cos ( cos sin )
2* *
2* /
;
;
;
(or circular wavenumber).
Plane wave function
z A Kx x y T
f
Kx
wavedirection
A amplitude
f frequency
Kx angular wavenumber
The results of plane wave case (from north direction)
:
cos ( cos sin )
2* *
2* /
;
;
;
(or circular wavenumber).
Plane wave function
z A Kx x y T
f
Kx
wavedirection
A amplitude
f frequency
Kx angular wavenumber
Water surface elevation track recorder
Recorder Points 1 2 3 4 5 6 7 8 9 10
Water Depth /m 3.568 3.251 4.857 14.922 21.928 21.076 16.739 25.291 55.232 14.321
Maximal wave height (E) /m
4.647 8.089 7.089 2.736 1.713 1.635 1.781 0.858 1.692 2.283
Maximal wave height (S) /m
3.486 3.180 4.202 1.787 0.927 0.673 0.388 0.231 0.230 0.251
Maximal wave height (N) /m
0.828 1.432 0.736 0.832 0.435 1.160 2.265 0.649 1.015 1.745
Maximal wave height at the track recorder points
By doing comparisons in the cases with different incident waves at the same point we get the effects of wave direction.
By doing comparisons in the same incident wave case at different recorder points we get the effects of water depth.
1. The numerical simulations can be conducted to evaluate the reasons of harbor hazards and investigate the effects of different incident waves.
2. The direction of incident waves affect the wave hazard in a harbor.
3. The wave height in the coast area would be 7-8 times higher than it is in the ocean.
4. Water depth is the significant factor which affects the wave height.
3. Conclusions
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