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*Corresponding author (email: [email protected])

• RESEARCH PAPER • September 2012 Vol.55 No.9: 1554–1564

doi: 10.1007/s11430-012-4388-1

Numerical analysis of wave hazards in a harbor

JING HuiMin1,2*, ZHANG Huai2, YUEN David A2,3,4 & SHI YaoLin2

1 College of Mechanical Engineering and Mechanics, Ningbo University, Ningbo 315211, China; 2 Key Laboratory of Computational Geodynamics, Chinese Academy of Sciences, Beijing 100049, China; 3 Department of Geology and Geophysics, University of Minnesota, Twin Cities, MN 55455-0219, USA;

4 Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455, USA

Received July 28, 2011; accepted December 22, 2011; published online April 5, 2012

Resonance may occur when the periods of incoming waves are close to the eigen-periods of harbor basin. The amplified waves by resonance in harbor will induce serious wave hazards to harbor structures and vehicles in it. Through traditional theoretical approaches, the eigen-periods of harbor basin with regular shapes can be obtained. In our study, we proposed a numerical model to simulate the behavior characteristics of the harbor waves. A finite difference numerical model based on the shallow water equations (SWE) is developed to simulate incoming tsunami and tidal waves. By analyzing the time series data of water surface wave amplitude variations at selected synthetic observation locations, we estimate the wave height and arrival time in coastal area. Furthermore, we use frequency spectrum analysis to investigate the natural frequencies from the data recorded at the synthetic observation stations.

numerical simulation, harbor, wave hazard, frequency spectrum analysis

Citation: Jing H M, Zhang H, Yuen D A, et al. Numerical analysis of wave hazards in a harbor. Sci China Earth Sci, 2012, 55: 1554–1564, doi: 10.1007/ s11430-012-4388-1

Tsunami is an oceanic gravity wave generated by submarine earthquakes or other geological processes such as volcanic eruptions or landslides [1]. Tsunami earthquakes are defined as shocks of marine origins, which generate extensive tsu-namis but relatively weak seismic waves [2, 3]. They most likely take place in trench regions with large tectonic dislo-cation and young folded crustal belts [2]. Most tsunamis are caused by large shallow earthquakes which characterize the active plate boundaries. Recent tsunami events (e.g. Suma-tra 2004, Java 2006, Solomon Islands 2007, Chile 2010, and Japan 2011) demonstrated the need of providing worldwide accurate and timely tsunami warning [4]. Improvement in the availability of sea-level observation and advances in numerical modeling technique with realistic situation are highlighting the potential of tsunami warning based on nu-

merical models. Numerical tsunami propagation and inun-dation models are well developed and benchmarked [5–7]. However, they are facing a challenge to run in real-time, which is partially due to the computing power limitations of efficiency and speed, also due to a lack of detailed knowledge on the earthquake rupture physical details [8].

Tsunamis are usually small and barely noticed in deep oceans, but they become large and cause damage when they approach coast or harbor. A harbor is the area where the water becomes shallower. And the shallower the water, the greater the interaction happens between the wave and the seabed, which causes dispersion and attenuation, including wave speed and wavelength decreasing, wave height in-creasing, dissipation, reflection, wave-breaking and etc. There are wave height amplification effect in some harbors with varied water depth and geometry of the harbor basins. For example, the wave height of Okushiri tsunami in 1993 was general 5m in most areas while up to 30 m in some

Jing H M, et al. Sci China Earth Sci September (2012) Vol.55 No.9 1555

particular coastal regions. The Chile tsunami in 1960 formed a huge tsunami in Japan because the natural periods of some harbor basins coincided with that of tsunami which had crossed the Pacific Ocean and led to resonance. The coastal height along the harbors is generally low, making it extremely vulnerable to incoming waves even with a height of only a couple of meters. In particular, many economical-ly important cities are located in the coastal areas, for ex-ample, around the South China Sea and in Taiwan regions, and would suffer devastating disasters.

The behaviors of waves in coastal regions, such as run-up on beaches or resonance in bays, are mostly studied by coastal engineers [9–12]. There are two types of coastal wave prediction models widely used in coastal engineering [13]. They are the energy balance model and the mass and momentum conservation model. The energy balance model is well established, which is generally used for predicting the deep ocean wave from climate changing model, such as the well-known WAM3G model [14, 15]. Its merit is its capabilities to simulate the effects of time-varying wind field conditions in coastal engineering. The mass and mo-mentum conservation model is most widely used for coastal engineering, and stands out in its richness and variety of numerical techniques, including the models based on fol-lowing typical equations: mile-slope equations, parabolic equations, nonlinear shallow-water equations, higher-order Boussinesq equations, and the Green-Naghdi theory, re-spectively. Although these two types of models are well developed, they are not appropriate to simulate the tsuna-mis.

Here we consider the characteristics of tsunami wave or standing wave during its propagation in a harbor. We pro-pose numerical models to simulate the generation and propagation of tsunami and investigate the waves’ charac-teristics in a harbor in order to evaluate the attendant haz-ards. The initial source of tsunamis was generated in the form of elastic half-space theory, which was summarized by Okada based on previous works in 1985 [16]. Two-dimensional shallow water equations (SWE) are used to simulate the propagation of tsunami across the deep oceans. The SWE is applicable when the wave heights are much smaller than the depths of water, which in turn are much smaller than the wavelengths. Two-dimensional models are nor-mally used for tsunami propagation simulations, as these models provide an adequate solution. On the other hand, three-dimensional models require an excessive memory and a long computational time, and hence are rarely used [17]. Our simulation model is based upon the finite difference method, which is also employed by many well-known mod-els, such as MOST [18], TUNAMI [19], and COMCOT [20]. The time series of wave height are recorded at the synthetic observation stations. We can estimate the wave height and travel sequences in coastal area caused by means of analyzing the time series data at selected observation sites.

It is important to emphasize that earthquake generic tsu-nami in this paper is to be a source of disturbance. Tsunami waves generally include waves with a wider range of fre-quencies; we adopted this assumption without taking the geophysical significance into consideration. Taking the co-seismic seabed dislocation from earthquake as disturb source, different magnitudes of earthquakes will generate waves with different wave length and frequencies. Thus we consider a variety of magnitude earthquakes in our work. Furthermore, we also considered the incoming plane waves on the boundary near the ocean as a supplement. In such cases, the source of incoming waves can be adjusted ac-cording to requirement in order to investigate waves of par-ticular frequency. Combining these two kinds of sources, we can investigate waves in a wider range of frequencies as well as waves with particular frequency independently. Be-cause of wind, wave hazards are more frequent and the fre-quencies of wind are generally regular. Earthquake tsunami combining with plane incoming waves simulations satisfy the extension and profundity of investigation simultaneously.

1 Governing equations for tsunamis

Tsunami simulation has been conducted mostly in 2-D space on the basis of the long-wave approximation. Simula-tion based on the 2-D linear long-wave (LLW) equation is efficient and widely used for tsunami propagation analysis [21]. It is usually employed to derive Green functions for the estimation of slip distribution along seismic fault or of initial tsunami distribution [22, 23]. It is also used in tsuna-mi hazard assessments from large earthquakes [17, 24].

For conventional tsunami simulation, 2-D tsunami mod-els derived from the Navier-Stokes equations have often been employed [25]. Detailed formulation is attached in the appendix I in this paper. For general cases, the mass con-servation equation which is also called the continuity equa-tion in shallow water equations is

B B[( ) ] [( ) ] 0.h

h h u h h vt x y

(1)

Here, h(x, y, t) is the height of the ocean free surface, hB(x, y) is the height of the ocean’s bathymetry, and u and v are the velocities in x and y directions, respectively.

The momentum conservation equation is

2 ,

DUU g h

Dt (2)

where

U is velocity vector and

is the angular velocity, respectively.

In general, this type of shallow water equations can be solved by adopting the finite difference method with high precision schemes, such as upwind total variation diminish-

1556 Jing H M, et al. Sci China Earth Sci September (2012) Vol.55 No.9

ing (TVD) scheme [26]. Multigrid method may also be uti-lized to obtain a better proficiency in numerical computing [27, 28]. In recent years, finite-volume method is becoming more and more popular for strongly nonlinear hyperbolic problems, if the convection term predominates over the other terms, especially when the waves break upon as they are arriving at the coastlines [29]. In our program, finite difference method has been used to solve the SWE numeri-cally. This method is also employed by many well-known models, such as COMCOT [20], TUNAMI [19] and MOST [18].

First, we simplify the equation into a succinct and linear-ized form. Taking h(x, y, t) as the height of the free surface of the ocean, hB(x, y) as the height of the bathymetry of the sea bottom, we define H(x, y) to be the average water depth, and (x, y, t) to be the vertical displacement of water sur-face above the average water surface as shown in Figure 1.

Figure 1 Three-dimension coordinate system for SWE.

We get

B ( , ) ( , ) ( , , ).h h x y H x y x y t (3)

Apparently, B ( , ) ( , ) ,h x y H x y C in which C is a con-

stant. Then the partial derivatives of h(x, y, z) are

( , , ) ( ( , , ) ) ( , , ),

( , , ) ( ( , , ) ) ( , , )

( , , ) ( ( , , ) ) ( , , ).

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

, (4)

Assuming that change of the model variables around their average fields are small compared to their average values, we linearize them by neglecting the product term of water level and horizontal velocity. The following linearized form of the SWE is obtained as

,

,

( ) ( )0.

ug fv

t xv

g fut y

Hu Hv

t x y

(5)

The leap-frog scheme is often used for the propagation of waves, where a low numerical damping is required with a relatively high accuracy. Staggered grids, where the mesh points are shifted with respect to each other by half an in-terval, are used to evaluate centered differences with accu-

racy in 2 2( ).x t Figure 2 shows the points have been

used in the staggered numerical scheme, where x, y, t are the grid sizes in x direction, y direction and time interval, respectively.

By using central difference method for horizontal veloc-ity and forward difference method for water level with the staggered numerical points, we get finite difference dis-cretization in staggered leapfrog scheme of SWE as

Figure 2 The staggered points for leap-frog numerical scheme.


1 1 11 12 2 2

1 1 1, , 1, , ,

2 2 2

1 1 11 12 2 2

1 1 . 1 , 1, , ,

2 2 2

, , 1, 1,1, ,

1 1( ) ,

1 1( ) ,

1 1( )

2

n n nn ni j i j

i j i j i j

n n nn ni j i j

i j i j i j

n ni j i j i j i jn n

i j i j

u u g fvt x

v v g fut y

H H

t x

1 1

2 21 1

, ,2 2

, , , 1 , 1

1 1

2 21 1

, ,2 2

1

2

0.

n n

i j i j

n ni j i j i j i j

n n

i j i j

u u

H H

y

v v

(6)

In 2-D simulation, long-wave approximation is employed and it does not allow calculation of the vertical velocity. The tsunami generation process due to the sea bottom up-lifting is not usually calculated but given as a source. In our simulation, we assume that the initial tsunami distribution is identical to the sea bottom deformation caused by the earthquake. This method is also implemented in numerous 2-D tsunami simulations [30]. In our work, the tsunami’s source is introduced by synchronizing the variation of ba-thymetry hB with respect of the sea bottom co-seismic dis-placement during the big earthquakes, as shown in Figure 1.

2 Numerical simulation of tsunami

The numerical simulation model is composed of bathymetry data, scenarios of the wave sources, and wave propagation simulation packages. We use the high resolution bathymetry data of the Pohang New Harbor in South Korea. The com-puting domain is a region of 80.5 km by 100 km. We in-

clude the 90 m resolution topography data (SRTM3) on the land and 1 km resolution bathymetry data (SRTM30) in the seabed, which have been shown in Figure 3. The Shuttle Radar Topography Mission (e.g. http://www2.jpl.nasa.gov/ srtm/.) is the digital topographic database of elevation data on a near-global scale. The higher resolution data are inter-polated to the coarser grid point where both kinds of data exist. This helps us to improve our model’s resolution from different data resources of different resolutions. Especially we use the high precision data with the grid size of about 50 m by 50 m in the harbor area.

The initial tsunami source is generated by Okada co- seismic displacement calculation [16]. The two-dimensional SWE tsunami propagation simulation is conducted around the Pohang harbor area. We set initial tsunami source from east direction with magnitudes equal to Mw6.0 (case A), Mw6.5 (case B) and Mw7.0 (case C), respectively.

When the tsunami or ocean waves from far-field arrive at the sea shore, most of them can be approximately consid-ered as plane waves, or they can be assumed as plane inci-dent wave sources. Our numerical models can produce a superposition of a variety of waves with different wave-lengths and frequencies. The range of frequency covers that of the most of the harbor waves. The ordinary devastating waves in the harbor, such as swell, their periods are about tens to hundreds of seconds. By adopting the plane incident wave source as a supplement to the tsunami source model, we can examine the wave with particular frequency. The potential earthquake tsunami can cause different wave height and frequencies in a harbor, which has been dis-cussed in our previous work [32]. In this paper, we assume that the boundary incident wave height is 1.0 m, and the plane incident wave function is given by

cos ( cos sin ) ,xz A K x y T (7)

where A is the amplitude for the wave, θ is the propagation direction, λ is the wavelength, 2 π /xK is the wave

number, 2 π f is the angular velocity, and f is fre-

quency, respectively. Parameters inputted for this simula-

Figure 3 The figure showing the land, seabed bathymetry, and the interpolated data. (a) The topography data on the land of SRTM3; (b) the bathymetry data of SRTM30 in the sea bottom; (c) the interpolated topography data of our computational domain in a region of 80.5 km by 100 km.


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tion are A = 1 m, = 1000 m, =180°, 270°, 270°, which indicate the waves from different directions. The value of f is given according to the periods of incident waves. In the plane, incident wave’s periods are 20, 30, 50, 200, 300, and 400 s, respectively.

Wave propagation problem can be solved by using the governing equations in section 2 of different kinds of boundary conditions. A number of boundary conditions have been proposed in literature [32–34]. Solid reflective boundary condition is imposed on all sea-land boundaries so as to act as a reflective surface, whereas open radiation boundary condition is imposed on the boundaries with ad-join water to allow the waves to radiate out without reflec-tion. The grid size is 50 m, resulting in a total number of 4 million grids. The time step is 0.05 s. The simulated results are summarized in the form of snapshot and displayed in next section. The results are saved at interval of 60 s and animations are made from these data by the visualization package of Amira (e.g. http://www.amiravis.com/). The time series of water evaluation and the spectrum analysis results at selected observation locations are also analyzed in section 4.

3 Results and discussion

The validity of our linear model has been discussed in pre-vious work [31]. We use the commercial software Amira to visualize the numerical simulation results [35]. By visualiz-ing the results from the numerical simulation, we get the animations that illustrated the reflection, refraction, diffrac-tion, and interaction of the waves. Pictures made from the animations are shown in Figure 4, which describe the char-acteristics of the tsunami waves’ propagation in our compu-ting domain. In order to show wave propagation character-istics in the harbor area more clearly, some of the pictures are zoomed in.

In the coastal area, the water depth varies from a couple of meters to few centimeters, which changes the character-istics of the waves and results in the changing of wave di-rection, speed, and frequencies. They occur when the waves

interact with seashore or harbor medium. When refraction occurs, the decrease in wave speed will also be accompa-nied by a decrease in wavelength. It can be observed that the wave crests become nearer to each other. At the same time, the wave height increases dramatically to devastating surge. In such cases, even the wave height is small in the far field ocean. In Figure 4, the refraction is shown clearly. For example, in the up middle part of the sea where the water becomes shallower, the wave velocity decreases greatly. There are some selected observation locations in the harbor as shown in the inset of Figure 5. We recorded the water surface amplitude (water depth plus bathymetry) with re-spect of time at these synthetic checking points, and applied frequency spectrum analysis to obtain the wave’s frequencies.

Figure 5 shows the location of the computing domain ((a) and (b)) and the synthetic observation points (c). Among these recorder points, point1 is located at the outside of the harbor, which is the reference record point in our numerical simulation. Take Mw6.5 earthquake tsunami model as an example, time series data of water surface amplitude and the frequency spectrum analysis results are shown in Figure 6. The Maximum wave height and wave frequencies at re-corder points from different magnitude of earthquake mod-els are listed in Table 1. Wave frequencies are chosen based on the frequency spectrum analysis results from Figure 6. The final wave height is overlaid by several waves with different frequencies.

Wave records on synthetic check point 1 in Figure 6 show that it is not affected by the harbor structure, of which the spectrum contains most of the incident waves’ frequen-cies. The records from these synthetic observation sites in the harbor are greatly affected by the harbor structure. Some waves of different frequencies are amplified or weakened respectively. These phenomena are clearly shown at syn-thetic check points 2 and 6. At these two check points, res-onance effects are obviously shown. Taking synthetic check point 6 as an example, in Figure 5, point 6 is located in the innermost part of the port and thus wave height amplifica-tion is the most obvious. Using different source models, we obtained that the same sensitive frequency is 0.002 Hz. While spectrum analysis of the record point 1 indicates that

Figure 5 This figure showing the location of the harbor in southeast of South Korea. The selected observation locations are plotted in the inset.


Figure 6 Time series data of water evaluation and the spectrum analysis result. The order of the figure is according to the points in Figure 5.


Table 1 Maximum tsunami and sensitive frequencies on recorder points

Num. of the point Case A Case B Case C

Max wave height (m)

Sensitive frequency (Hz)

Max wave height (m)


Max wave height (m)


1 0.074

0.008 0.007

0.187

0.009 0.008 0.007 0.004 0.001

0.449

0.007 0.005 0.001 0.004

2 0.042 0.004 0.008

0.188 0.004

0.517 0.002

3 0.030

0.009 0.008 0.003 0.002

0.119

0.003 0.002 0.009

0.309

0.002 0.001

4 0.039 0.008

0.105 0.008 0.007 0.003

0.270 0.001

5 0.019

0.008 0.002 0.001 0.005 0.015

0.107

0.004 0.002

0.297

0.001 0.002 0.005

6 0.029 0.002

0.15 0.002 0.004

0.543 0.002

7 0.023

0.008 0.002 0.019 0.001

0.096

0.002 0.003 0.006

0.302

0.001 0.002 0.005

8 0.022

0.011 0.015 0.019 0.002

0.082

0.002

0.355

0.001 0.002 0.005

the advantages frequencies of the incident wave do not in-clude 0.002 Hz, so we can point out that this particular fre-quency is actually the resonant frequency of the harbor at this point physically.

The frequency spectrum analysis results listed in Table 1 show that, for the same incident wave, the same or similar wave’s frequencies are obtained on different synthetic record points. These frequencies are the predominant frequencies of the coming incident wave. Obtaining different wave fre-quencies on each synthetic record point should be the ef-fects of discrepancy of water depth or geometry of the har-bor, which is also related to the location of the record point itself. Furthermore, the wave’s frequencies at the same syn-thetic record point generally satisfy multiple relationships reciprocally.

Numerical simulation using earthquake generic tsunami as wave source can gain waves with different wave length and frequencies. Figure 7 shows the spectrum of tsunamis generated by different magnitude earthquakes in numerical simulations.

As we mentioned above, the synthetic record point 1 is located outside of the harbor. The wave record is not af-fected by harbor structure, which can be considered as inci-dent wave equally coming from the open sea from the east direction to the harbor. Figure 7 is obtained by frequency spectrum analysis of the time series data of wave height on synthetic check point 1 in different models. In the three cases, the frequencies of incident waves cover a very wide range

Figure 7 Spectrum analysis results of incident waves in different models, respectively corresponding to cases A, B, C from top to bottom.

and are enough to contain most of the frequencies of harbor waves.

As a supplement of initial wave sources generated by tsunami, taking plane incident waves as the initial wave


sources, we investigate the amplification effects on some particular wave frequencies at the synthetic record points near the harbor shore. In our simulation, the periods of plane incident waves vary from 20, 30, 50, 200, and 300 to 400 s, respectively. The ratio of maximum wave height on other points located inside the harbor to that on point 1 re-flects the amplification effects which are shown in Figure 8(a). Furthermore, we also investigate the effects of waves from different directions to the wave height in the harbor and the results are shown in Figure 8(b).

Figure 8 shows the amplification effects on the sites in-side the harbor. It is obvious that those at points 2, 5, 6 and 7 are among them. Figure 8(a) is the results when the period of the incident wave was changed, and waves with period 200 s were amplified especially, which means this period is close to the intrinsic period of some points inside the harbor and the resonance frequency may be around 0.005 Hz. This is close to the results we obtained from the first type of wave source scenario. Figure 8(b) shows that the amplifica-tion degree is different when the wave direction is changed and the eastern incoming wave is amplified the most obvi-ously.

4 Discussion and conclusions

Resonance effect is one of significant causes of wave haz-ards in a harbor. It occurs when the frequency of incident waves is the multiple or close to the intrinsic frequency of the harbor basin. We call these frequencies as sensitive fre-quencies. The amplification of wave height caused by reso-nance would result in serious disaster. Through traditional theoretical methods we can only obtain the eigen-periods of basins with regular shapes and average depth, but rather than the realistic cases. In this study, we proposed a numer-ical simulation method to investigate the resonance effects in the harbor. First we applied the numerical tsunami mod-

els with pseudo-earthquake tsunamis as initial wave sources to simulate the wave behaviors in the harbor and investigate the characteristics of waves with different wave lengths and frequencies during the propagation processes inside the harbor. Since tsunamis generated by earthquakes cover a wide range of waves with different wave lengths and fre-quencies, through changing the magnitude of the pseu-do-earthquake, we can obtain incident waves containing most frequencies of harbor waves. Then frequency spectrum analysis is applied to the time series data of wave height or amplitude to obtain the sensitive frequencies at the synthetic record points. We found that different locations of the har-bor have different eigen-periods because the discrepancy of the harbor’s geometric and water depth can be fully ana-lyzed by the frequency spectrum analysis. Comparing with analysis result of the time series data at the reference record point outside of the harbor, we can determine whether the sensitive frequency is the resonance frequency in the harbor or the predominate frequency of the tsunami. Furthermore, the simulations of plane incident waves are supplements to the tsunamis simulation, which can be applied to investigate the resonance effects of waves with particular frequencies or directions.

We use numerical simulation to investigate the wave hazards in a harbor and combine tsunami and plane incident wave assumptions to satisfy the wide scope of frequencies of incident waves. This method is feasible and efficient. Frequency spectrum analysis method is applied to the time series data to obtain the sensitive frequencies in the harbor and to study the resonance effects. Comparing the results of numerical simulations with historical wave observation data in particular region is able to evaluate the potential wave hazard in the harbor or its nearby areas. From these results, we also can determine where resonance is likely to occur when waves with particular frequencies come to the harbor and make some improvements of harbor construction ac-cordingly.

Figure 8 The ratio of maximum wave height on record points inside the harbor and on point 1. (a) The results when the periods of the waves are 20, 30, 50, 200, 300 and 400 s; (b) the results when the direction of the incident waves changed.


We thank Prof. Kwang Ik Kim for providing high precision bathymetry data of Pohang New Harbor and fruitful advices on this study. Careful reading of the manuscript and constructive comments by anonymous re-viewers were very valuable for revision of the paper. This work was sup-ported by the National Natural Science Foundation of China (Grant Nos. 40574012 and 40676039), National Basic Research Program of China (Grant No. 2008CB425701), National High-tech R& D Program of China (Grant No. 2010AA012402) and K. C. Wong Magna Fund in Ningbo Uni-versity.

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Appendix I Derivation of the Shallow Water Equation

The motion of a fluid is described by the following 3-D NS equations, where we take the z-axis in the vertical direction and the x and y axes in horizontal space in Cartesian coor-dinates.

Navier-Stokes equation for incompressible flow can be expressed as

(1) Mass conservation,

( ) 0;Ut

(A1)

(2) Momentum conservation,

22DU

U p U fDt

, (A2)

where U

is velocity (u, v, and w are velocity components along the x, y, and z axes, respectively), p is pressure, is

density,

is angular velocity, is the kinematic viscosity

coefficient for seawater [36] and f

is other body forces.

Applying the constant density assumption, and then inte-grating the mass conservation equation in the z direction,


( , , , ) ( , , ).u v

w x y z t z w x y tx y

(A3)

When a fluid has a well-defined boundary across which there is no relative motion, then the boundary must be a material surface that retains its parcels. This is equivalent to saying that the flow must either turn to be parallel to the boundary, and/or it must move perpendicular to the bound-ary at exactly the same speed, as the boundary itself is moving in that direction. The mathematical expression for this is called the kinematic boundary condition. By defini-tion the boundary location F(x, y, z, t)=0, the kinematic boundary condition is

0, at 0.DF F F F F

u v w FDt t x y z

(A4)

The kinematic boundary condition at the sea surface z=h(x, y, t) is given by

( , , , ) .h h h

w x y h t u vt x y

(A5)

Similarly, at the bottom z= hB(x, y), it has

( , , , ) .B BB

h hw x y h t u v

x y

(A6)

Substituting eq. (A6) into (A3)

( , , ) .B BB

h h u vw x y t u v h

x y x y

(A7)

Combining eqs. (A3), (A5) and (A7)

[( ) ] [( ) ] 0.B B

hh h u h h v

t x y

(A8)

Equation (A8) is the mass conservation equation that is also called the continuity equation of Shallow Water Equation.

Since the viscosity of seawater is characterized by a very small value of 106 m2 s1 we consider that offshore tsunami

motion is nonturbulent [37], and for gravity ,f gk

the momentum conservation equation is given by

2 .

DUU p gk

Dt (A9)

In shallow water theory, the effects of vertical shear on hor-izontal velocity are negligible. Consequently, the vertical motion of water particles has no effect on the pressure dis-tribution that means hydrostatic pressure approximation, i.e.

2 2( ), with 1.

p

g Oz

(A10)

Integrating this equation with respect to z, i.e.

( , , ).p gz p x y t (A11)

And applying constant pressure on wave surface, i.e.

0( , , )p x y h p we got the expression of p,

0( ) .p g h z p (A12)

Applying constant density assumption,

0[ ( ) ] .p g h z p g h gk

(A13)

Substituting eq. (A13) into (A9),

2 .DU

U g hDt

(A14)

Equation (A14) is the momentum conservation equation of Shallow Water Equation.

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