*2.2.2. Calculation conditions*

will study several fundamental characteristics of tsunami generation caused by a landslide, or a sector collapse, on the basis of two‐dimensional vertical results, obtained through a numerical simulation using a moving particle semi‐implicit (MPS) model [24], where the falling body is assumed to be a fluid, or a rigid body, which moves down a slope with a constant gradient.

Second, we examine tsunamis caused by a submarine eruption. When a submarine explosive eruption occurs, volcanic products are blown out, as in case for an explosion from a subastral active volcano. Conversely, if some volume of magma is released out of a chamber, owing to a submarine volcanic eruption, ground subsidence occurs, leading to a creation of a caldera. Although both volcanic products, and a caldera, should cause tsunamis as mentioned by Egorov [25], and Maeno et al. [26], respectively, we take up a different tsunami source, peculiar to a submarine eruption, for discussion in this chapter, i.e., a phreatomagmatic explosion [27]. In the process with a submarine phreatomagmatic explosion, seawater contacts high temper‐ ature magma in the seabed neighborhood, after which the seawater evaporates with an explosive increase in its volume, resulting in a water surface displacement that generates tsunamis. A new index for submarine volcanic explosion, concerning tsunami generation, is developed by assuming the relationship between a phreatomagmatic explosion, and the resultant initial tsunami waveform. We specifically assume the value of this index, to generate a numerical simulation for tsunamis caused by a submarine volcanic eruption in Kagoshima Bay, Japan, where a submarine explosion with tsunami generation has been observed [28].

Numerical computations have been performed that represent tsunami generation due to a body, which moves down a slope. In the present calculation, the numerical model, developed by Iribe and Nakaza [24], based on the MPS method designed by Koshizuka and Oka [29], is applied to consider the furious water motion. The water surface level is determined using the spatial gradient of particle‐number density, to inhibit pressure disturbance at the water surface. No turbulence model is utilized for fluid motion, and both the elasticity, and the plasticity, of

In order to validate the applicability of the numerical model, several numerical results for water surface displacements, are compared with the corresponding experimental data. **Figure 4** depicts a setup for laboratory experiments, with an acrylic basin, where *h*off denotes the still water depth off of the slope, i.e., the offshore still water depth, and the water density is 1000

. The vertical steel gate, installed on a slope with a constant gradient of *β*, can be pulled up quickly, and smoothly, by operating two levers as shown in **Figure 5**, resulting in the

**2. Tsunamis due to a landslide or a sector collapse**

**2.2. Model validation for tsunami generation due to a falling fluid**

the falling bodies are neglected for simplicity.

**2.1. Numerical model**

38 Tsunami

*2.2.1. Experimental setup*

kg/m3

In the model computation, 17,000 particles are used to represent the above‐described initial condition, where the particle grid is 0.005 m for the MPS model. In the present calculation, the numerical results for water surface displacements in the case where the particle grid is 0.005 m, are in good agreement with the case where the particle grid is 0.01 m. The water density on both the offshore side, and the onshore side, of the gate is 1000 kg/m3 . The water on the onshore side of the gate, starts at the initial falling time, i.e., *t* = 0.0 s.

#### *2.2.3. Comparison between the numerical results and the corresponding experimental data for water surface displacements*

Shown in **Figure 6** are the numerical results for the water surface displacements, at the location for wave gauge (WG) 2 shown in **Figure 4**, in comparison with the corresponding experimental data. The slope gradient *β* is 30°, and 45°, in the cases shown in **Figure 6(a)**, and **6(b)**, respec‐ tively. The offshore still water depth *h*off, and the initial falling‐body height *h*s, are 0.1 m, and 0.15 m, in both the cases. The distance between the location for WG 2, and that for the gate, is 1.16 m. The experimental value for each case, is a mean value among values obtained through five runs of the experiment, with the same initial conditions. **Figure 6** indicates that, both the wave height, and the wave phase, of the first wave obtained using the MPS model, are in harmony with the experimental results.

**Figure 6.** The water surface displacement at the location for WG 2 shown in **Figure 4**. The falling body is water, with the same density as that for the offshore water, i.e., 1000 kg/m3 . The offshore still water depth *h*off is 0.1 m and the initial falling‐body height *h*s is 0.15 m.

#### **2.3. Tsunami generation due to a falling fluid**

Two‐dimensional vertical motion in a water basin shown in **Figure 7**, is simulated numerically, where the slope gradient *β* is 30°, and the distance between the slope foot, and the offshore vertical wall, is 3.5 m. The offshore still water depth *h*off is 0.1 m, or 0.2 m. Also in the following computation, the offshore water density is 1000 kg/m3 , and the particle grid is 0.005 m.

*2.2.3. Comparison between the numerical results and the corresponding experimental data for water*

Shown in **Figure 6** are the numerical results for the water surface displacements, at the location for wave gauge (WG) 2 shown in **Figure 4**, in comparison with the corresponding experimental data. The slope gradient *β* is 30°, and 45°, in the cases shown in **Figure 6(a)**, and **6(b)**, respec‐ tively. The offshore still water depth *h*off, and the initial falling‐body height *h*s, are 0.1 m, and 0.15 m, in both the cases. The distance between the location for WG 2, and that for the gate, is 1.16 m. The experimental value for each case, is a mean value among values obtained through five runs of the experiment, with the same initial conditions. **Figure 6** indicates that, both the wave height, and the wave phase, of the first wave obtained using the MPS model, are in

**Figure 6.** The water surface displacement at the location for WG 2 shown in **Figure 4**. The falling body is water, with

Two‐dimensional vertical motion in a water basin shown in **Figure 7**, is simulated numerically, where the slope gradient *β* is 30°, and the distance between the slope foot, and the offshore

. The offshore still water depth *h*off is 0.1 m and the initial

*surface displacements*

40 Tsunami

harmony with the experimental results.

the same density as that for the offshore water, i.e., 1000 kg/m3

**2.3. Tsunami generation due to a falling fluid**

falling‐body height *h*s is 0.15 m.

**Figure 7.** The target domain for computation, where the offshore still water depth *h*off is 0.1 m, or 0.2 m; the slope gradi‐ ent *β* is 30°.

Sketched in **Figure 8** are the initial positions of a falling body in Cases 1–4, where the initial level of the falling‐body bottom from the seabed, is 0.1, 0.2, 0.3, and 0.4 m, respectively, whether the offshore still water depth *h*off is 0.1 or 0.2 m. The initial shape of the falling body is a right triangle, where the height of its vertical front face is 0.1 m. The body on the slope starts at the initial falling time, i.e., *t* = 0.0 s. The falling body is assumed to be a fluid, or a rigid body. The fluid, and the rigid body, with the same density as that of the offshore water, i.e., 1000 kg/m3 , we call, a "light fluid" and a "light rigid body", respectively, while the fluid and the rigid body, with a density of 2600 kg/m3 , we call, a "heavy fluid" and a "heavy rigid body", respectively.

**Figure 8.** A sketch of the initial positions of a falling body in Cases 1–4.

**Figure 9** shows the numerical results for the water surface displacements at Point P, in Cases 1–4, when the falling body is a light fluid, and the offshore still water depth *h*off is 0.1 m. The distance between the location for Point P, and that for the offshore vertical wall, is 1.5 m, as shown in **Figure 7**. If the vertical distance between a particle, and its nearest particle below it, is larger than the particle grid, i.e., 0.005 m, then the upper particle is defined as a droplet, which is located over a particle at the water surface, around the horizontal location for the upper particle.

**Figure 9.** The water surface displacements at Point P (*x* = 1.5 m) for the different initial positions of a falling light fluid, with a specific gravity of 1.0. The offshore still water depth is 0.1 m.

Conversely, in **Figure 10** are the numerical results for the water surface displacements at Point P, in Cases 1–4, where the falling body is a heavy fluid, and the offshore still water depth *h*off is 0.1 m. The tsunami height is defined as the maximum value in water surface displacement at each location. In each of our cases, the tsunami height from the heavy fluid is twice as large as that from the light fluid. Thus, if both the initial position, and the volume, of a falling body are the same, the tsunami height increases as the density, i.e., the initial potential energy, of the falling body is increased.

**Figure 10.** The water surface displacements at Point P (*x* = 1.5 m) for the different initial positions of a falling heavy fluid with a specific gravity of 2.6. The offshore still water depth is 0.1 m.

However, it is in Case 4, where the falling‐body initial potential energy is largest, that the tsunami height at Point P, is the minimum value, whether the falling body is a light fluid, or a heavy fluid, as shown in **Figure 9**, or **10**, respectively. Conversely, the tsunami height at Point P, is the maximum value in Case 2, when the falling body is a light fluid, as shown in **Figure 9**, while in Case 1, when the falling body is a heavy fluid, as shown in **Figure 10**. How is this possible?

**Figure 9.** The water surface displacements at Point P (*x* = 1.5 m) for the different initial positions of a falling light fluid,

Conversely, in **Figure 10** are the numerical results for the water surface displacements at Point P, in Cases 1–4, where the falling body is a heavy fluid, and the offshore still water depth *h*off is 0.1 m. The tsunami height is defined as the maximum value in water surface displacement at each location. In each of our cases, the tsunami height from the heavy fluid is twice as large as that from the light fluid. Thus, if both the initial position, and the volume, of a falling body are the same, the tsunami height increases as the density, i.e., the initial potential energy, of the

**Figure 10.** The water surface displacements at Point P (*x* = 1.5 m) for the different initial positions of a falling heavy

fluid with a specific gravity of 2.6. The offshore still water depth is 0.1 m.

with a specific gravity of 1.0. The offshore still water depth is 0.1 m.

falling body is increased.

42 Tsunami

**Figure 11.** The simulation result for the particle motion in Case 3, where the falling body is a heavy fluid with a specific gravity of 2.6; the offshore still water depth is 0.1 m. The red points denote the falling‐fluid particles, while the blue ones the offshore‐water particles.

The particle motion numerical result for Case 3, is shown in **Figure 11**, where the falling body is the heavy fluid, and the offshore still water depth *h*off is 0.1 m. The red points denote the heavy‐fluid particles, while the blue ones the offshore‐water particles. **Figure 11** indicates that when the initial position of a falling body is high, the falling‐body group while moving down a slope, transforms and flattens, resulting in a flattened body rushing into the water, such that the volumetric flow rate of the falling body decreases, and pushes the water weakly. This is the reason why the tsunami height at Point P, is lower in Case 4, where the falling‐body initial potential energy is large, when the falling body is the heavy fluid, and also for the light fluid.

### **2.4. Tsunami generation due to a falling rigid body**

Shown in **Figure 12** are the numerical results for the water surface displacements at Point P, in Cases 1–4, when the falling body is a light rigid body, and the offshore still water depth *h*off is 0.1 m. The tsunami height at Point P, shows its maximum value in Case 4, where the falling‐ body initial potential energy is largest. This is not true when the falling body is a light fluid, or a heavy fluid, as described above.

**Figure 12.** The water surface displacements at Point P (*x* = 1.5 m) for the different initial positions of a falling light rigid body with a specific gravity of 1.0. The offshore still water depth is 0.1 m.

On the other hand, shown in **Figure 13** are the numerical results for the water surface dis‐ placements at Point P, in Cases 1–4, when the falling body is a heavy rigid body, and the offshore still water depth *h*off is 0.1 m. The tsunami height at Point P, is lowest in Case 4, where the falling‐body initial potential energy is largest. This is another challenge!

The particle motion numerical result for the Case 3, is shown in **Figure 14**, where the falling body is the heavy rigid body, and the offshore still water depth *h*off is 0.1 m. The red points denote the rigid‐body particles, while the blue ones the offshore‐water particles. The front face of the falling rigid body pushes the water, generating a large wave height plunging‐type wave, after which a turbulent tsunami propagates, with a bore, generated at its front face. Thus in both the generation, and the propagation, of this tsunami, the energy loss is larger, owing to the generation of splashes, the turbulence, and the bore. This is the reason why the tsunami height at Point P, is lower in Case 4, in which the falling‐body initial potential energy is large, when the falling body is a heavy rigid body. It should also be noted that the water surface profile for this tsunami, is forwardly inclined, such that the peak in the water surface displacement at Point P, appears earlier in **Figure 13**, in the case of the falling heavy rigid body, than in **Figure 12**, concerning the falling light rigid body.

the volumetric flow rate of the falling body decreases, and pushes the water weakly. This is the reason why the tsunami height at Point P, is lower in Case 4, where the falling‐body initial potential energy is large, when the falling body is the heavy fluid, and also for the light fluid.

Shown in **Figure 12** are the numerical results for the water surface displacements at Point P, in Cases 1–4, when the falling body is a light rigid body, and the offshore still water depth *h*off is 0.1 m. The tsunami height at Point P, shows its maximum value in Case 4, where the falling‐ body initial potential energy is largest. This is not true when the falling body is a light fluid,

**Figure 12.** The water surface displacements at Point P (*x* = 1.5 m) for the different initial positions of a falling light rigid

On the other hand, shown in **Figure 13** are the numerical results for the water surface dis‐ placements at Point P, in Cases 1–4, when the falling body is a heavy rigid body, and the offshore still water depth *h*off is 0.1 m. The tsunami height at Point P, is lowest in Case 4, where the

The particle motion numerical result for the Case 3, is shown in **Figure 14**, where the falling body is the heavy rigid body, and the offshore still water depth *h*off is 0.1 m. The red points denote the rigid‐body particles, while the blue ones the offshore‐water particles. The front face of the falling rigid body pushes the water, generating a large wave height plunging‐type wave, after which a turbulent tsunami propagates, with a bore, generated at its front face. Thus in both the generation, and the propagation, of this tsunami, the energy loss is larger, owing to the generation of splashes, the turbulence, and the bore. This is the reason why the tsunami height at Point P, is lower in Case 4, in which the falling‐body initial potential energy is large, when the falling body is a heavy rigid body. It should also be noted that the water surface profile for this tsunami, is forwardly inclined, such that the peak in the water surface

**2.4. Tsunami generation due to a falling rigid body**

body with a specific gravity of 1.0. The offshore still water depth is 0.1 m.

falling‐body initial potential energy is largest. This is another challenge!

or a heavy fluid, as described above.

44 Tsunami

**Figure 13.** The water surface displacements at Point P (*x* = 1.5 m) for the different initial positions of a falling heavy rigid body with a specific gravity of 2.6. The offshore still water depth is 0.1 m.

**Figure 14.** The simulation result for the particle motion in Case 3, where the falling body is a heavy rigid body with a specific gravity of 2.6; the offshore still water depth is 0.1 m. The red points denote the rigid‐body particles, while the blue ones the offshore‐water particles.
