*3.1.1. Cubic expansion of water through heat-induced evaporation*

In this section, the process of tsunami generation due to a submarine volcanic eruption, is discussed considering a phreatomagmatic explosion, where the seawater touches high temperature magma in the seabed neighborhood, after which the water evaporates instanta‐ neously with explosive increase in its volume, lifting the water over the water vapor bubble. We will develop a model for tsunami generation due to a submarine eruption with phreato‐ magmatic explosion, and introduce a submarine explosive index concerning the relationship between the submarine phreatomagmatic explosion, and the resultant initial tsunami height.

Remember the following characteristics of water: the mass, and the density, of 1.0 mol of liquid water, are around 18.0 g, and 1.0 g/cm3 , respectively, such that 1.0 mol of liquid water, occupies a volume of 18.0 ml. Conversely, 1.0 mol of vapor, assumed to be an ideal gas, occupies a volume of 22,700 ml at the standard temperature and pressure (STP), where the temperature, and the pressure, are 0.0°C, and 1.0 bar, i.e., 1.0 × 105 Pa, respectively. Thus, when liquid water transforms to vapor at STP, the volume of the vapor becomes 22,700/18.0 ≈ 1.261 × 103 times as much as that of the liquid water.

If the pressure is *p* (Pa), then the volume of a gas at temperature *τ* (°C), *V*, is evaluated by *V* = *V*0 (105 /*p*) (1 + *τ*/273) according to Boyle‐Charles' law, where *V*0 denotes the volume of the gas at 0.0°C.

Consequently, when liquid water with a volume of *V*w at STP, transforms to vapor with a volume of *V* at *τ*(°C), and *p* (Pa), the volume expansion ratio between liquid water, and vapor, is

$$
\alpha = V \;/\; V\_{\text{w}} = 1.261 \times 10^{8} \left(1 + \tau / 273\right) / p. \tag{1}
$$

#### *3.1.2. Volume expansion ratio of water through a phreatomagmatic explosion*

Immediately after magma touches water, the following equation explains their interface [30]:

$$(\left(\tau\_{\rm m} - \tau\_{\rm i}\right) / \left(\tau\_{\rm i} - \tau\_{\rm w}\right) = (\rho\_{\rm w} c\_{\rm pw} k\_{\rm w} \,/\rho\_{\rm m} c\_{\rm pm} k\_{\rm m})^{1/2},\tag{2}$$

where *τ*<sup>i</sup> is the temperature at the interface between magma and water; *ρ, c*p, and *k* are density, isopiestic specific heat, and heat transfer coefficient, respectively; the subscripts m, and w, denote the variables of magma, and water, respectively. The general values of these parameters are as follows [26]:

[The general values of the parameters for magma]

Density *ρ*m = 2400 kg/m3

in a slower motion, with a smaller volumetric flow rate, in the falling body. This is the reason why the tsunami height is lower in the cases where the submarine landslide occurs than in the

In this section, the process of tsunami generation due to a submarine volcanic eruption, is discussed considering a phreatomagmatic explosion, where the seawater touches high temperature magma in the seabed neighborhood, after which the water evaporates instanta‐ neously with explosive increase in its volume, lifting the water over the water vapor bubble. We will develop a model for tsunami generation due to a submarine eruption with phreato‐ magmatic explosion, and introduce a submarine explosive index concerning the relationship between the submarine phreatomagmatic explosion, and the resultant initial tsunami height.

Remember the following characteristics of water: the mass, and the density, of 1.0 mol of liquid

a volume of 18.0 ml. Conversely, 1.0 mol of vapor, assumed to be an ideal gas, occupies a volume of 22,700 ml at the standard temperature and pressure (STP), where the temperature, and the

If the pressure is *p* (Pa), then the volume of a gas at temperature *τ* (°C), *V*, is evaluated by *V* =

Consequently, when liquid water with a volume of *V*w at STP, transforms to vapor with a volume of *V* at *τ*(°C), and *p* (Pa), the volume expansion ratio between liquid water, and vapor,

Immediately after magma touches water, the following equation explains their interface [30]:

/*p*) (1 + *τ*/273) according to Boyle‐Charles' law, where *V*0 denotes the volume of the gas

( ) <sup>8</sup>

== ´ + *V V*/ 1.261 10 1 / 273 / . *p* (1)

 t

transforms to vapor at STP, the volume of the vapor becomes 22,700/18.0 ≈ 1.261 × 103

, respectively, such that 1.0 mol of liquid water, occupies

Pa, respectively. Thus, when liquid water

times as

(2)

**3. Tsunamis due to a submarine eruption with a phreatomagmatic**

cases where the landslide occurs above the offshore water level.

**3.1. Submarine explosive index concerning tsunami generation**

*3.1.1. Cubic expansion of water through heat-induced evaporation*

water, are around 18.0 g, and 1.0 g/cm3

much as that of the liquid water.

*V*0 (105

is

at 0.0°C.

pressure, are 0.0°C, and 1.0 bar, i.e., 1.0 × 105

a

w

*3.1.2. Volume expansion ratio of water through a phreatomagmatic explosion*

**explosion**

48 Tsunami

Temperature *τ*m = 973 K

Isopiestic specific heat *c*pm = 1.2 × 103 J/kgK

Heat transfer coefficient *k*m = 1.2 W/mK

[The general values of the parameters for water]

Density *ρ*w = 1000 kg/m3

Temperature *τ*w = 273 K

Isopiestic specific heat *c*pw = 4.2 × 103 J/kgK

Heat transfer coefficient *k*w = 0.61 W/mK

We substitute these general values into Eq. (2), and obtain the temperature at the interface, *τ*<sup>i</sup> , as

$$
\pi\_i = \text{ } \text{649.0 K } = \text{ } \text{376.0} \text{°C.} \tag{3}
$$

This value is larger than the spontaneous nuclear generation temperature of water, which is approximately 583 K at 1.0 atm. Note that the temperature increases by around 10.0 K as the pressure is increased by 2.0 MPa [26].

*3.1.3. Relationship between the still water depth and the volume expansion ratio for seawater near the seabed*

The water pressure at a submarine crater in still water, *p*, is defined as

$$\mathbf{p} = \rho\_{\mathrm{w}} \mathbf{g} \\ h = \text{ } 9800 \text{ } h \text{(Pa) (unit length in meter)}, \tag{4}$$

where *h* denotes the still water depth at the crater location, and the gravitational acceleration *g* equals 9.8 m/s2 . Substituting the value of *τ*<sup>i</sup> shown in Eq. (3), and the value of *p* given by Eq. (4), into *τ*, and *p*, in Eq. (1), respectively, leads to

$$
\alpha = V \,\, \vert \, V\_{\text{w}} = \,\, 30,600 \,\, \vert \, h \text{(unit length in meter)}, \tag{5}
$$

where the water surface displacement is assumed to be much smaller than the still water depth. Eq. (5) determines the relationship between the volume expansion ratio of water over the crater, and the still water depth at the crater location; for instance, *α* ≈ 10.2 when *h* is 3000 m, while *α* ≈ 6.1 × 102 when *h* is 50 m.

#### *3.1.4. Relationship between the submarine volcanic explosion and the initial tsunami profile*

Assume that a circular crater (indicated with "A" in **Figure 18**) with a radius of *r*, appears at the horizontal seabed, with the seawater (B) over the crater then being vaporized to expand vertically in an instant (C), such that the initial tsunami profile becomes a cylinder (D) with a height of *η*0. E in **Figure 18** indicates the still water surface, where the still water depth is *h*. Although in case with a seabed rise, the initial tsunami height decreases, as the seabed‐rise speed decreases, and also as the still water depth increases, as described by Kakinuma and Akiyama [31], these effects on tsunami generation are neglected for simplicity. Thus both the shape, and the size, of the cylinder D, are the same as that of the cylinder C, such that the volume of these cylinders, *V*, equals *πr*<sup>2</sup> *η*0, where *η*0 is the initial tsunami height.

**Figure 18.** A schematic for tsunami generation due to a submarine phreatomagmatic explosion. "A" denotes a crater with a radius of *r*, at the seabed; "B" the original water with a volume of *V*w, before vapor transformation; "C" the generated vapor with a volume of *V*; "D" the initial tsunami profile with a radius of *r*, and a height of *η*0; "E" the still water surface, where the still water depth is *h*.

By substituting *V* = *πr*<sup>2</sup> *η*0 into Eq. (5), we obtain:

$$W\_{\rm w} = \pi r^2 \eta\_0 h / \, 30,600 \, \approx 1.0 \times 10^{-4} \, r^2 \eta\_0 h \text{(unit length in meter)},\tag{6}$$

where *V*w is the original volume of the seawater (B), before vapor transformation. Eq. (6) indicates that the cylindrical initial tsunami profile with a radius of *r*, and a height of *η*0, is generated when a submarine eruption, transforms water with a volume of *V*w at the seabed neighborhood, to vapor, where the still water depth is *h*. If we know all the values of *V*w, *r*, and *h*, then we can evaluate the value of the initial tsunami height *η*0. Therefore, the original volume of seawater, which transforms to vapor through a phreatomagmatic explosion caused by it touching high temperature magma, i.e., *V*w, is a submarine explosive index concerning tsunami generation.

where the water surface displacement is assumed to be much smaller than the still water depth. Eq. (5) determines the relationship between the volume expansion ratio of water over the crater, and the still water depth at the crater location; for instance, *α* ≈ 10.2 when *h* is 3000 m, while *α*

Assume that a circular crater (indicated with "A" in **Figure 18**) with a radius of *r*, appears at the horizontal seabed, with the seawater (B) over the crater then being vaporized to expand vertically in an instant (C), such that the initial tsunami profile becomes a cylinder (D) with a height of *η*0. E in **Figure 18** indicates the still water surface, where the still water depth is *h*. Although in case with a seabed rise, the initial tsunami height decreases, as the seabed‐rise speed decreases, and also as the still water depth increases, as described by Kakinuma and Akiyama [31], these effects on tsunami generation are neglected for simplicity. Thus both the shape, and the size, of the cylinder D, are the same as that of the cylinder C, such that the

**Figure 18.** A schematic for tsunami generation due to a submarine phreatomagmatic explosion. "A" denotes a crater with a radius of *r*, at the seabed; "B" the original water with a volume of *V*w, before vapor transformation; "C" the generated vapor with a volume of *V*; "D" the initial tsunami profile with a radius of *r*, and a height of *η*0; "E" the still

( ) <sup>2</sup> 4 2

where *V*w is the original volume of the seawater (B), before vapor transformation. Eq. (6) indicates that the cylindrical initial tsunami profile with a radius of *r*, and a height of *η*0, is generated when a submarine eruption, transforms water with a volume of *V*w at the seabed neighborhood, to vapor, where the still water depth is *h*. If we know all the values of *V*w, *r*, and *h*, then we can evaluate the value of the initial tsunami height *η*0. Therefore, the original volume of seawater, which transforms to vapor through a phreatomagmatic explosion caused by it

, <sup>−</sup> *V* = *�r h* ≈ × *r h* (6)

 / 30,600 1.0 10 unit length in meter η

*η*0 into Eq. (5), we obtain:

w 0 0

*η*0, where *η*0 is the initial tsunami height.

*3.1.4. Relationship between the submarine volcanic explosion and the initial tsunami profile*

≈ 6.1 × 102

50 Tsunami

when *h* is 50 m.

volume of these cylinders, *V*, equals *πr*<sup>2</sup>

water surface, where the still water depth is *h*.

η

By substituting *V* = *πr*<sup>2</sup>

On the other hand, if we assume that the seawater over a crater is vaporized, becoming a half sphere with a radius of *R*, where the sphere center coincides with the crater center at the seabed, then the sphere volume *V* is 2*πR*<sup>3</sup> /3. If we assume also that the initial tsunami profile is a half sphere, with a radius of *R*, then we obtain

$$W\_w \approx 6.8 \times 10^{-5} R^3 h \text{(unit length in meter)},\tag{7}$$

where *V*w is the original volume of the seawater, before vapor transformation. Eq. (7) can be rewritten to

$$R \approx 24 \ (V\_w \, / \, h\text{)}^{1/3} \text{(unit length in meter)}.\tag{8}$$

According to an old document, the initial tsunami height *η*0 was around 9 m, owing to a submarine volcanic eruption in Kagoshima Bay, Japan, on September 9, 1780 [27], where the still water depth at the eruption location is about 200 m. In this case, we substitute both *R* = 9 m, and *h* = 200 m, into Eq. (7), resulting in *V*w = 9.9 m3 . It should be noted, however, that future work is required to know accurately both the profile, and the size, of the initial tsunami, for instance, by performing laboratory experiments using both high temperature material, and water.
