**3.2. Numerical simulation for propagation of tsunamis due to a submarine phreatomagmatic explosion in a bay**

#### *3.2.1. Examples of values of the submarine explosive index concerning tsunami generation*

The relationship between the crater radius *r*, and the eruption amount *V*e, is expressed by Sato and Taniguchi [32] as

$$r = \, 0.97 \, V\_{\varepsilon}^{0.36} \left( \text{unit length in meter} \right). \tag{9}$$

For example, the volcanic explosive index (VEI) introduced by Newhall and Self [33], is assumed to equal three, larger than two for a standard explosion, then the eruption amount *V*e is between 1.0 × 107 m3 and 1.0 × 108 m3 ; hence the crater radius *r* becomes approximately between 321 m and 736 m, based on Eq. (9). Thus we assume that the crater radius *r* is 700 m, in the present computation. Conversely, owing to the submarine explosion in Kagoshima Bay on September 9, 1780, the sea level rose by around 9 m over the explosion, as described above, such that we assume that the initial tsunami height *η*0 is 9.0 m. If the initial tsunami profile becomes a cylinder, as shown in **Figure 18**, then the value of submarine explosive index concerning tsunami generation, *V*w, is evaluated by Eq. (6) as 2.3 × 104 (unit length in meter) when the still water depth at the eruption location, *h*, is 50 m, while 4.5 × 104 (unit length in meter) when *h* is 100 m.

#### *3.2.2. Numerical model and calculation conditions*

Numerical simulation for tsunamis due to a submarine volcanic eruption in Kagoshima Bay, is generated using the shallow water version of the nonlinear wave model [34], where the computational program developed by Nakayama and Kakinuma [35] to simulate internal wave propagation, is partially rewritten to solve the set of finite difference equations for nonlinear surface waves. **Figure 19** shows the seabed level in the northern bay, where the still water level is described by *z* = 0.0 m. The shorelines are assumed to be vertical walls with perfect wave reflection, while the Sommerfeld radiation condition is applied at the open boundaries inside the sea area. The initial tsunami height *η*0 is assumed to be 9.0 m, as described above.

**Figure 19.** The seabed level in the northern area of Kagoshima Bay, Japan.

#### *3.2.3. Water surface displacements*

Water surface displacements are obtained through numerical calculation for the trial cases where the craters are located at Point ➀ and Point ➂, the locations of which are shown in **Figure 20**, as well as for the above‐described actual case on September 9, 1780, where the crater is located at Point ➁.

There are many submarine fumaroles in Kagoshima Bay, and an active volcano exists in Sakurajima, as shown in **Figure 21**. Sakurajima, where "sakura" means cherry, while "jima", or "shima", means island, in Japanese, was an isolated island prior to its violent eruption in 1914, when Sakurajima was connected to the Ohsumi Peninsula by the eruption with a large amount of ejecta. As shown in **Figure 20**, Point B is located in the West Sakurajima Channel, on the western side of which lies the most urban area in the prefecture, Kagoshima City.

concerning tsunami generation, *V*w, is evaluated by Eq. (6) as 2.3 × 104

meter) when *h* is 100 m.

52 Tsunami

*3.2.2. Numerical model and calculation conditions*

**Figure 19.** The seabed level in the northern area of Kagoshima Bay, Japan.

*3.2.3. Water surface displacements*

is located at Point ➁.

when the still water depth at the eruption location, *h*, is 50 m, while 4.5 × 104

Numerical simulation for tsunamis due to a submarine volcanic eruption in Kagoshima Bay, is generated using the shallow water version of the nonlinear wave model [34], where the computational program developed by Nakayama and Kakinuma [35] to simulate internal wave propagation, is partially rewritten to solve the set of finite difference equations for nonlinear surface waves. **Figure 19** shows the seabed level in the northern bay, where the still water level is described by *z* = 0.0 m. The shorelines are assumed to be vertical walls with perfect wave reflection, while the Sommerfeld radiation condition is applied at the open boundaries inside the sea area. The initial tsunami height *η*0 is assumed to be 9.0 m, as described above.

Water surface displacements are obtained through numerical calculation for the trial cases where the craters are located at Point ➀ and Point ➂, the locations of which are shown in **Figure 20**, as well as for the above‐described actual case on September 9, 1780, where the crater

There are many submarine fumaroles in Kagoshima Bay, and an active volcano exists in Sakurajima, as shown in **Figure 21**. Sakurajima, where "sakura" means cherry, while "jima", or "shima", means island, in Japanese, was an isolated island prior to its violent eruption in

(unit length in meter)

(unit length in

**Figure 20.** The point locations in Kagoshima Bay, Japan. Sakurajima, which was an isolated island, has an active volca‐ no. Point B is located in the West Sakurajima Channel.

**Figure 21.** The active volcano with an eruption on Sakurajima, Japan. The author took this photo in 2015.

**Figure 22.** The water surface displacements at Points A and B for different crater locations, i.e., Points ➀, ➁, and ➂. The point locations are indicated in **Figure 20**.

Shown in **Figure 22(a)**, and **22(b)**, are the numerical results for the water surface displacements at Point A, and Point B, indicated in **Figure 20**, respectively. The tsunami height for not only the first wave, but also the several following waves, is larger than 1.0 m at Point A, where it takes a longer time for the oscillation to attenuate because of multiple reflections of tsunamis at the bay head.

Although Point ➂ is more distant from Point B than Point ➁, the maximum tsunami height at Point B, near the highest population area, is larger in the case where the crater is located at Point ➂ than that in the case where the crater appears at Point ➁; for in the former case, the wave energy is large for the wave component approaching Sakurajima in a direction oblique, or parallel, to the seashore of Sakurajima, resulting in a tsunami traveling along the shoreline of Sakurajima toward the west.
