**3. Computational examples**

#### **3.1. Tohoku-Oki models**

Now, numerical results [9] 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 *h* = –10 to *h* = –1,000 [m]. We set the initial conditions at ζ0 (*x*) = 0 and *Ui*<sup>0</sup> (*x*) = 0, and the boundary conditions o (,) = 50 [m] (0 < *t* < 60 [s]) at the tsunami-generating area and o (,)=0 [m] at other areas (see **Figure 2**). It is noted that 50 [m] at *Γ*<sup>o</sup> may be too high. In our computation, the tsunami arrived at Oshika Peninsula in **Figure 2** after 20 min and the highest wave height reached 15 [m]. These numerical results should more carefully be checked with boundary data on *Γ*o.

In the above,

and Ushijima [13, 14].

the open boundary.

**3.1. Tohoku-Oki models**

o

**3. Computational examples**

+ 1 is determined by Eq. (13). Here,

the following symbols are also used:

,

66 Tsunami

+ 1 is determined from Eq. (12). From the obtained value and ,

and ,

, , *i i j j U U ν ν dx x x* <sup>W</sup>

, , <sup>ˆ</sup> <sup>ˆ</sup> , . *<sup>n</sup> nn n <sup>i</sup> <sup>i</sup> ik k ik k k k U UU U* = = å å f

Because of the presence of the basic boundary conditions, Eq. (12) holds for all nodes except for the nodes on *Γ*o and Eq. (13) holds for all the nodes except for the nodes on *Γ*c. However, on the boundary *Γ*o, an approximation method is adopted in which the advective term of Eq. (13) uses Tabata's upwind approximation [9, 12]. A mathematical justification for the approximation scheme for linearized equations related to the above scheme was given by Kanayama

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 [9]. 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

Now, numerical results [9] 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 *h* = –10 to *h* = –1,000 [m]. We set the initial conditions at ζ0 (*x*) = 0 and *Ui*<sup>0</sup> (*x*) = 0, and the boundary conditions

(,) = 50 [m] (0 < *t* < 60 [s]) at the tsunami-generating area and o

 f

æ ö ¶ ¶

It is well known that integration by parts is used for the viscosity term in (13).

respectively, at the node *k* after *n* time steps, and Δ*t* represents the size of time steps. In addition,

, the value

are approximate values of (,) and (,)

(15)

(,)=0 [m] at other

ç ÷ <sup>=</sup> ç ÷ ¶ ¶ è ø <sup>ò</sup> (14)

**Figure 2.** A computational model of Tohoku-Oki [9]. (a) Contour map of *ζ* [9]. (b) Time histories of *ζ* [9].

**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 *Γ*o. Details are later mentioned. **Figure 2b** shows the water level change at the two points A and B in **Figure 2**. The wave height at the point B is higher than the point A after about 1500 [s]. When the tsunami reaches coastal areas, the water depth becomes shallow and the wave height becomes high.

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 (*x*) like **Figure 4** based on the data of Fujii et al. [15] as a source of tsunami. **Figure 5a–f** shows the ordinates of the water surface (water level) every 8 [min] from after 8 [min] to after 48 [min]. In our previous paper [9], since the computational domain is not wide, it looks that there is a reflection from the boundary *Γ*o (see **Figure 2a**). This artificial reflection can be removed by changing boundary conditions from (10) to (16) as follows:

$$\begin{aligned} U\_n(\mathbf{x}, t) &= \mathcal{L} \frac{\sqrt{|\mathbf{g}|} \| \mathbf{f} \|}{H} \mathcal{L} \left( \mathbf{x}, t \right) \\ U\_t(\mathbf{x}, t) &= \mathbf{0} \end{aligned} \quad \text{on } \Gamma\_o, \tag{16}$$

where *Un* and *Ut* denote the normal component of velocity and the tangential component of velocity, respectively, and *c* is a constant. We used (*c* = 0.9) in this chapter.

**Figure 3.** The second computational model of Tohoku-Oki.

**Figure 4.** Contour map of initial water level.

ordinates of the water surface (water level) every 8 [min] from after 8 [min] to after 48 [min]. In our previous paper [9], since the computational domain is not wide, it looks that there is a reflection from the boundary *Γ*o (see **Figure 2a**). This artificial reflection can be removed by

( )

<sup>ï</sup> <sup>=</sup> <sup>þ</sup>

*H on*

ü

ý G

denote the normal component of velocity and the tangential component of

(,) , ,

z

= ï

(,) 0

*g h U xt c xt*

*t*

*U xt*

velocity, respectively, and *c* is a constant. We used (*c* = 0.9) in this chapter.

o

(16)

changing boundary conditions from (10) to (16) as follows:

*n*

**Figure 3.** The second computational model of Tohoku-Oki.

**Figure 4.** Contour map of initial water level.

where *Un* and *Ut*

68 Tsunami

**Figure 5.** (a) Contour map of after 8 min, (b) contour map of after 16 min, (c) contour map of after 24 min, (d) contour map of after 32 min, (e) contour map of after 40 min, and (f) contour map of after 48 min.

**Figure 6** shows the water level change at the three points Miyako, Soma and Choshi in **Figure 4**. Solid curves show observed data [15] 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.

**Figure 6.** Time histories of *ζ*.
