Overview of Tsunami

Abstract

When a large earthquake occurs in an offshore region, a tsunami is generated. The generation and propagation can be mathematically described based on equation of motion. This chapter visualizes and overviews tsunami generation and propagation to obtain an overall grasp of tsunamis and elucidate their fundamental nature. Section 2.1 illustrates the tsunami generation and propagation caused by a huge earthquake. We use a dynamic rupture scenario constructed for an anticipated huge earthquake in the Nankai Trough, Japan. Section 2.2 shows a simple model for the generation and propagation. While the generation is basically independent of gravity, gravity is the main force to move tsunami. Section 2.3 describes the fundamental properties of the propagation. Tsunami propagation depends on the wavelength and sea depth. Also, when the sea depth is very shallow, tsunami becomes to show nonlinear characteristics. Section 2.4 summarizes the main points of tsunami generation and propagation.

Appendices

Appendices

2.1.1 Appendix A: Simulation Method of Tsunami Generated by an Earthquake

There are various methods for simulating tsunamis. This appendix explains one of the methods for simulating tsunami induced by an earthquake. The method involves two steps (Fig. 2.21). In the first step, we conduct a simulation of seismic waves caused by an earthquake. Then, we conduct tsunami simulation across the sea using the seismic simulation result as input data.

Fig. 2.21

A flowchart of tsunami generation and propagation excited by an earthquake (same as Fig. 2.4)

First, we perform seismic wave simulation in 3-D space. A finite earthquake fault is divided into numerous small subfaults. Each small subfault is approximately represented by a point dislocation source. The point dislocation source is given by a stress change or stress glut ${\tau}_{ij}^s$ (see Chap. 4: Earthquakes). The equation of motion in the earth medium is

$$ \rho \frac{\partial^2{u}_i}{\partial {t}^2}=\frac{\partial }{\partial {x}_j}\left({\tau}_{ij}+{\tau}_{ij}^s\right), $$

(A.1)

where ρ is the density, u _i is the displacement, and τ _ij is the stress tensor. A constitutive law or the generalized Hooke’s law gives the relation between the displacement and the stress as

$$ {\tau}_{ij}={\lambda \delta}_{ij}{u}_{k,k}+2\mu \left({u}_{i,j}+{u}_{j,i}\right) $$

(A.2)

for an isotropic medium, where λ and μ are referred to as Lamé parameters. We numerically simulate the spatial and temporal distribution of the motion u _i(x, t) based on Eqs. (A.1) and (A.2) by using, for example, the finite difference method (the method is described in Sect. 4.3, Seismic Wave Simulation). We obtain the vertical velocity at the sea bottom ${v}_z^{\mathrm{bot}}={\dot{u}}_i\left(x,y,{z}^{\mathrm{bot}},t\right)$ where z ^bot indicates the location of the sea bottom.

In the second step, we use the velocity at the sea bottom ${v}_z^{\mathrm{bot}}\left(x,y,t\right)$ as source of tsunami. According to the incompressible fluid theory (see Chap. 5: Tsunami Generation), when the vertical displacement ${u}_z^{\mathrm{bot}}\left(x,y\right)$ occurs at the sea bottom, the sea-surface height is given by

$$ {\eta}_0\left(x,y\right)=\frac{1}{{\left(2\pi \right)}^2}{\iint}_{-\infty}^{\infty }{dk}_x{dk}_y \exp \left[i\left({k}_xx+{k}_yy\right)\right]\frac{{\tilde{u}}_z^{\mathrm{bot}}\left({k}_x,{k}_y\right)}{\cosh \left({kh}_0\right)}, $$

(A.3)

where ${\tilde{u}}_z^{\mathrm{bot}}\left({k}_x,{k}_y\right)$ is the 2-D Fourier transform of the vertical displacement at the bottom ${u}_z^{\mathrm{bot}}\left(x,y\right)$. Therefore, the sea-surface elevation Δη(x, y, t) caused by the sea-bottom displacement during a fractional duration Δt at the time t is given by

$$ \Delta \eta \left(x,y,t\right)=\frac{1}{{\left(2\pi \right)}^2}{\iint}_{-\infty}^{\infty }{dk}_x{dk}_y \exp \left[i\left({k}_xx+{k}_yy\right)\right]\frac{{\tilde{v}}_z^{\mathrm{bot}}\left({k}_x,{k}_y,t\right)}{\cosh \left({kh}_0\right)}\Delta t, $$

(A.4)

where ${\tilde{v}}_z^{\mathrm{bot}}\left({k}_x,{k}_y,t\right)$ is the 2-D Fourier transform of the vertical velocity at the bottom ${v}_z^{\mathrm{bot}}\left(x,y,t\right)$. The sea-surface elevation Δη(x, y, t) is added as η(x, y, t) = η ^∗(x, y, t) + Δη(x, y, t) where η ^∗(x, y, t) is the tsunami height distribution numerically calculated at each time step in the simulation. We use the 2-D nonlinear long-wave equations

$$ \frac{\partial \eta \left(x,y,t\right)}{\partial t}+\frac{\partial }{\partial x}\left[\left(h+\eta \right){v}_x^{\mathrm{av}}\right]+\frac{\partial }{\partial \overline{y}}\left[\left(h+\eta \right){v}_y^{\mathrm{av}}\right]=0, $$

(A.5)

$$ \frac{\partial {v}_x^{\mathrm{av}}\left(x,y,t\right)}{\partial t}+{v}_x^{\mathrm{av}}\frac{\partial {v}_x^{\mathrm{av}}}{\partial x}+{v}_y^{\mathrm{av}}\frac{\partial {v}_x^{\mathrm{av}}}{\partial y}+{g}_0\frac{\partial \eta }{\partial x}=0, $$

(A.6)

$$ \frac{\partial {v}_y^{\mathrm{av}}\left(x,y,t\right)}{\partial t}+{v}_x^{\mathrm{av}}\frac{\partial {v}_y^{\mathrm{av}}}{\partial x}+{v}_y^{\mathrm{av}}\frac{\partial {v}_y^{\mathrm{av}}}{\partial y}+{g}_0\frac{\partial \eta }{\partial y}=0. $$

(A.7)

The parameter η is the tsunami or vertical displacement at the sea surface, ${v}_x^{\mathrm{av}}$ and ${v}_y^{\mathrm{av}}$ are the horizontal velocity averaged from the sea bottom to the sea surface, is the sea depth, and g ₀ is the gravitational acceleration. We numerically calculate the spatial and temporal evolution of the tsunami η(x, y, t) based on Eqs. (A.5), (A.6), and (A.7) by using the finite difference method (see Chap. 6: Propagation Simulation).

2.1.2 Appendix B: Phase Velocity in Nonlinear Long-Wave Equations: The First-Order Approximation Method

One-dimensional nonlinear tsunami propagation in a sea with a constant depth h ₀ is described by the following equations:

$$ \frac{\partial \eta \left(x,t\right)}{\partial t}+\frac{\partial }{\partial x}\left\{\left[{h}_0+\eta \left(x,t\ \right)\right]u\left(x,t\right)\right\}=0, $$

(B.1)

and

$$ \frac{\partial u\left(x,t\right)}{\partial t}+u\frac{\partial u\left(x,t\right)}{\partial x}+{g}_0\frac{\partial \eta \left(x,t\right)}{\partial x}=0, $$

(B.2)

where η(x, t) is tsunami height, u(x, t) is horizontal velocity, and g ₀ is the gravitational acceleration.

The tsunami height and horizontal velocity are written as

$$ \eta \left(x,t\right)={\eta}^0\left(x,t\right)+{\eta}^1\left(x,t\right),\mathrm{and}\ u\left(x,t\right)={u}^0\left(x,t\right)+{u}^1\left(x,t\right) $$

(B.3)

where η ⁰(x, t) and u ⁰(x, t) satisfy the following linear equations:

$$ \frac{\partial {\eta}^0}{\partial t}+{h}_0\frac{\partial {u}^0}{\partial x}=0, $$

(B.4)

and

$$ \frac{\partial {u}^0}{\partial t}+{g}_0\frac{\partial {\eta}^0}{\partial x}=0. $$

(B.5)

We assume that |η ¹| ≪ |η ⁰| and |u ¹| ≪ |u ⁰|. In other words, we consider a situation in which the tsunami propagation is roughly described by the linear equations but also includes the nonlinear effects.

The linear equations of (B.4) and (B.5) give a wave equation for η ⁰ as

$$ \frac{\partial^2{\eta}^0}{\partial {x}^2}-\frac{1}{c_0^2}\frac{\partial^2{\eta}^0}{\partial {t}^2}=0, $$

(B.6)

where c ₀ is given by ${c}_0=\sqrt{g_0{h}_0}$. A plane wave η ⁰ = exp [−iω(t − x/c ₀)] satisfies Eq. (B.6) where c ₀ works as the phase velocity. Substituting η ⁰ = exp [−iω(t − x/c ₀)] into (B.5), we obtain

$$ \frac{\partial {u}^0}{\partial t}=-{g}_0\frac{i\omega}{c_0}{e}^{- i\omega \left(t-\frac{x}{c_0}\right)}. $$

(B.7)

Then, we suppose u ⁰ as

$$ {u}^0=\frac{g_0}{c_0}{e}^{- i\omega \left(t-\frac{x}{c_0}\right)}=\frac{g_0}{c_0}{\eta}^0=\frac{c_0}{h_0}{\eta}^0. $$

(B.8)

The η ⁰ = exp [−iω(t − x/c ₀)] and u ⁰ = (c ₀/h ₀)η ⁰ satisfy Eqs. (B.4) and (B.5). Substituting u ⁰ = (c ₀/h ₀)η ⁰ into Eq. (B.4), we obtain

$$ \frac{\partial {\eta}^0}{\partial t}+{c}_0\frac{\partial {\eta}^0}{\partial x}=0. $$

(B.9)

Substituting (B.3) and (B.8) into (B.2) gives

$$ \frac{\partial }{\partial t}\left(\frac{c_0}{h_0}{\eta}^0+{u}^1\right)+\left(\frac{c_0}{h_0}{\eta}^0+{u}^1\right)\frac{\partial }{\partial t}\left(\frac{c_0}{h_0}{\eta}^0+{u}^1\right)+{g}_0\frac{\partial }{\partial x}\left({\eta}^0+{\eta}^1\right)=0, $$

(B.10)

and we calculate as

$$ {\displaystyle \begin{array}{c}\frac{\partial {u}^1}{\partial t}=-\frac{c_0}{h_0}\frac{\partial {\eta}^0}{\partial t}-{\left(\frac{c_0}{h_0}\right)}^2{\eta}^0\frac{\partial {\eta}^0}{\partial x}-{u}^1\frac{c_0}{h_0}\frac{\partial {\eta}^0}{\partial t}\\ {}-{\eta}^0\frac{c_0}{h_0}\frac{\partial {u}^1}{\partial t}-{u}^1\frac{\partial {u}^1}{\partial t}-{g}_0\frac{\partial {\eta}^0}{\partial x}-{g}_0\frac{\partial {\eta}^1}{\partial x}.\end{array}} $$

(B.11)

Considering that |η ¹| ≪ |η ⁰| and |u ¹| ≪ |u ⁰| and neglecting the small terms containing |η ¹| or |u ¹| on the right-hand side, we approximate (B.11) as

$$ \frac{\partial {u}^1}{\partial t}\approx -\frac{c_0}{h_0}\frac{\partial {\eta}^0}{\partial t}-{\left(\frac{c_0}{h_0}\right)}^2{\eta}^0\frac{\partial {\eta}^0}{\partial x}-{g}_0\frac{\partial {\eta}^0}{\partial x}. $$

(B.12)

Using Eq. (B.9), we calculate (B.12) as

$$ {\displaystyle \begin{array}{c}\frac{\partial {u}^1}{\partial t}\approx \frac{c_0^2}{h_0}\frac{\partial {\eta}^0}{\partial x}-{\left(\frac{c_0}{h_0}\right)}^2{\eta}^0\frac{\partial {\eta}^0}{\partial x}-{g}_0\frac{\partial {\eta}^0}{\partial x}\\ {}=-{\left(\frac{c_0}{h_0}\right)}^2{\eta}^0\frac{\partial {\eta}^0}{\partial x}\\ {}=\frac{c_0}{h_0^2}{\eta}^0\frac{\partial {\eta}^0}{\partial t}\\ {}=\frac{c_0}{2{h}_0^2}\frac{\partial }{\partial t}{\left({\eta}^0\right)}^2.\end{array}} $$

(B.13)

As a result, we obtain

$$ {u}^1\approx \frac{c_0}{2{h}_0^2}{\left({\eta}^0\right)}^2. $$

(B.14)

Substituting (B.8) and (B.14) into (B.1), we calculate

$$ {\displaystyle \begin{array}{c}\frac{\partial \eta \left(x,t\right)}{\partial t}=-{h}_0\frac{\partial }{\partial x}\left[\frac{c_0}{h_0}{\eta}^0+\frac{c_0}{2{h}_0^2}{\left({\eta}^0\right)}^2\right]-\frac{\partial }{\partial x}\left[\left({\eta}^0+{\eta}^1\right)\left(\frac{c_0}{h_0}{\eta}^0+{u}^1\right)\right]\\ {}=-{c}_0\frac{\partial {\eta}^0}{\partial x}-\frac{c_0}{2{h}_0}\frac{\partial }{\partial x}{\left({\eta}^0\right)}^2-\frac{c_0}{h_0}\frac{\partial }{\partial x}{\left({\eta}^0\right)}^2\\ {}-\frac{\partial }{\partial x}\left({\eta}^1\frac{c_0}{h_0}{\eta}^0\right)-\frac{\partial }{\partial x}\left({u}^1{\eta}^0\right)-\frac{\partial }{\partial x}\left({\eta}^1{u}^1\right).\end{array}} $$

(B.15)

If we neglect the smaller terms on the right-hand side, we obtain

$$ \frac{\partial \eta \left(x,t\right)}{\partial t}\approx -{c}_0\frac{\partial {\eta}^0}{\partial x}-\frac{3{c}_0}{2{h}_0}\frac{\partial }{\partial x}{\left({\eta}^0\right)}^2. $$

This is rewritten as

$$ \frac{\partial \eta \left(x,t\right)}{\partial t}\approx -\frac{\partial }{\partial x}\left[\left({c}_0+\frac{3{\eta}^0}{2{h}_0}\right){\eta}^0\right]. $$

(B.16)

Comparing (B.16) with (B.9), we find that η(x, t) propagates with the phase velocity as

$$ c\approx \left({c}_0+\frac{3{\eta}^0}{2{h}_0\ }\right){\eta}^0. $$

(B.17)