The tsunami generated by the Great East Japan Earthquake (GEJE) caused serious damage to the coastal areas of the Tohoku district. Numerical simulations are used to predict damage caused by tsunamis. Shallow-water equations are generally used in numerical simulations of tsunami propagation from the open sea to the coast. This research focuses on viscous shallow-water equations and attempts to generate a computational method using finite-element techniques based on the previous investigations of Kanayama and Ohtsuka (1978). First, the viscous shallow-water equation system is derived from the Navier-Stokes equations, based on the assumption of hydrostatic pressure in the direction of gravity. The derived equations have the horizontal viscosity term hereditary from the original Navier-Stokes equations. Next, a numerical finite element scheme is shown. Finally, tsunami simulations of Tohoku-Oki are shown using the above-mentioned approach. Our main concern in this chapter is how to set the boundary condition on the open boundary.
- the Great East Japan Earthquake (GEJE)
- Viscous shallow-water equations
- Navier-Stokes equations
- Finite-element method
The coastal areas of Tohoku district suffered serious damage from the tsunami caused by the 2011 off the Pacific Coast of Tohoku Earthquake that occurred on March 11, 2011 [1, 2]. Numerical simulations are used to develop disaster-prevention measures to deal with such tsunami disasters. They are also used to predict potential future tsunami disasters, to design disaster-prevention facilities such as coastal breakwaters and levees and to predict tsunami attacks immediately after an earthquake occurs [3, 4]. The Central Disaster Prevention Council  prepares a basic disaster prevention plan and participates in the determination of important disaster-prevention matters. If the Tonankai-Nankai earthquake occurs, it may cause considerable damage. Consequently, the council has been performing numerical calculations to predict wave height and arrival time when tsunami reaches the coast.
In this chapter, to let it be self-contained, the viscous shallow-water equations are again derived from the Navier-Stokes equations in which the hydrostatic pressure in the direction of gravity is assumed. In the numerical analysis of tsunamis, the viscosity term is often omitted or simply added [6, 7]. In this study, however, a computational method that does not omit the viscosity term is adopted, that is, because it enables more rigorous analysis to be performed and we intend to include the viscosity term in future stress analysis of tsunamis. The approximation scheme  is given below and simulation results  for a Tohoku-Oki model are again presented for self-containing. Our main concern in this chapter is how to set the boundary condition on the open boundary, which is completely changed to a new one.
2. Viscous shallow-water equations
The tsunami generated by the GEJE caused serious damage to the coastal areas of the Tohoku district. Numerical simulations are used to predict damage caused by tsunamis. Shallow-water equations are generally used in numerical simulations of tsunami propagation from the open sea to the coast. This subsection focuses on viscous shallow-water equations and attempts to generate a computational method using finite-element techniques based on the previous investigations of Kanayama and Ohtsuka .
In the numerical analysis of tsunami, a viscosity term is often omitted or simply added [6, 7]. In this subsection, however, a computational method that does not omit the viscosity term is adopted, that is, because it enables more rigorous analysis to be performed, and the authors intend to include the viscosity term in future stress analysis of tsunami. Recently, the authors have found an interesting book  that deals with sound mathematical topics related to the viscous shallow-water equations.
2.1. Derivation of the viscous shallow-water equations
As shown in Figure 1, Kanayama and Dan  considered the shallow-water long-wave flow in which the wavelength is sufficiently long relative to the water depth . First, the viscous shallow-water equations are again derived to let this chapter be self-contained. Detailed derivation comes from the study of Kanayama and Ushijima . Orthogonal coordinates [m] are used, where
Eqs. (1)–(3)  are integrated with respect to
The fixed density of the layer is represented by
Note that /2 is often used for the definition of . Here, however, we follow the definition in  for convenience.
The computational domain is the two-dimensional (2D) polygonal region
2.2. Finite-element approximations for the viscous shallow-water equations
The finite-element approximation  is as follows. First, Eqs. (6) and (7) are multiplied by test functions and then integrated over the computational domain
In the above, is determined from Eq. (12). From the obtained value and , the value is determined by Eq. (13). Here, and are approximate values of and respectively, at the node
It is well known that integration by parts is used for the viscosity term in (13).
Because of the presence of the basic boundary conditions, Eq. (12) holds for all nodes except for the nodes on
In general, tsunami is excited in the following two ways. The first one is to consider the tsunami excitation as the initial condition of the water surface, for which we do not have sufficient input information in such an artificial tsunami of Hakata Bay . The second one is to consider it as the boundary condition of the water surface as in the next subsection. In the setting of Hakata Bay, a computational domain is not so wide that the above approach may be the only way. It is also noted that 50 [m] at the open boundary for the later Tohoku-Oki case may be too high. In the computation, the tsunami arrived at Oshika Peninsula after 20 min, and the highest wave height reached 15 [m]. These numerical results should be checked more carefully with data on the open boundary.
3. Computational examples
3.1. Tohoku-Oki models
Now, numerical results  for a Tohoku-Oki model are again shown for self-containing. When not specifically defined, the same physical values as for Hakata Bay were used. Regarding the mesh data, the total number of nodes is 56,562, the total number of elements is 113,013, and the ocean floor ordinate is set to become deep gradually from
Figure 2a shows the ordinates of the water surface (water level) after 18 [min]. Since the computational domain is not wide, it looks that there is a reflection from the northern boundary in Figure 2a. This artificial reflection can be removed by suitable boundary conditions on
Next, new numerical results for the second Tohoku-Oki model are shown. When not specifically defined, the same physical values as for Hakata Bay were used. Regarding the mesh data, the total number of nodes is 563,100, the total number of elements is 1,123,178, and Figure 3 shows the ocean floor ordinate, which is constructed from 30-arc sec interval grid of JTOPO30, provided by the Marine Information Research Center. We set the initial conditions at ζ0 (
Figure 6 shows the water level change at the three points Miyako, Soma and Choshi in Figure 4. Solid curves show observed data  and dashed curves show numerical results. Though the arrival time of the first wave is almost same, the wave height is small compared with observed data at the points of Miyako and Soma. Results may be improved by changing initial and boundary conditions.
4. Concluding remarks
In this study, the viscous shallow-water equations have again been derived from the Navier-Stokes equations in which the hydrostatic pressure in the direction of gravity is assumed. Tsunami propagation is then simulated by a finite-element computation. In the study of Kanayama and Dan  using the Hakata Bay model, in which the tsunami-generating area was taken to be the epicenter of the West off Fukuoka Prefecture Earthquake (2005), tsunami propagation from the open sea to the coastal area might be produced. This is an example in which the conventional open boundary condition Eq. (10) is still valid. In this chapter, however, in the second Tohoku-Oki model, the open boundary condition is changed from Eq. (10) to Eq. (16) to overcome artificial reflection. In addition, we have reconstructed a system in which the tsunami arrival time and the wave height at the time of tsunami attack can be obtained from numerical results. Since our analysis takes the viscosity term into account, this study can also be considered to be a preliminary study for future planned stress analyses of tsunamis (see Figure 7). We still want to check whether the hydrostatic pressure is always the dominant force in the case of tsunami attack compared with other forces including the viscous force.
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