**2. Resonant and amplifying mechanisms of the meteorological tsunami**

#### **2.1. Proudman resonance**

of the negative (positive) pressure anomalies. The sea level also rises accordingly with the horizontal convergence of the wind stress, for example, around the front or the coastal area. Such basic mechanism to set the wave up is similar as the storm surge. But the period of the storm surge is as long as tidal motion (approximately half or 1 day), while that of meteorological tsunami is generally much shorter than tidal motion. The intensity of the sea level deformation by severe storm is typically as large as several 10 cm or higher than 1 m including wind stress. But the intensity itself of the forcing to meteorological tsunami is much smaller than that related to storm surge: an order of several centimetres in typical cases. The multiple mechanism of the resonance at the sea interface enlarges the amplitude of the ocean long waves, and the wave will possibly become higher than 3 m with flooding and destruction of the dikes. The primitive mechanisms on meteorological tsunami are founded by geophysicists in the earlier and middle twentieth century [2–4]. In recent decades, however, the integration of the scientific findings throughout severe case studies and the advances in field observation, numerical modelling and data assimilation have revealed the significance of the multiscale mechanisms on atmospheric

In the present chapter, we briefly introduce resonant mechanisms as propagating meteorological tsunamis. Next, we show the area where destructive meteorological tsunamis have been reported. After that, we present the various kinds of the weather conditions resulting in meteorological tsunamis. **Figure 1** indicates the typical scale of meteorological phenomena and oceanographic motion. The temporal scale of meteorological tsunami typically matches the meso-β-scale motion in the atmosphere such as squall line, internal gravity wave, gravitational flow and so on. However, the genesis of those motion is influenced by both smaller- and largerscale motions. The organization of turbulent plume will generate the cumulus convection, and

cluster of the convective cells will form the meso-β-scale system.

**Figure 1.** Scale of the various meteorological phenomena, and oceanographic motions.

and oceanographic processes.

14 Tsunami

One of the most important resonances was derived by Proudman (1929) [4], which is a coupling of the ocean long wave and the atmospheric disturbance. Let *U* the propagation speed of the pressure disturbance, and *c* = (*gH*) 1/2, the phase velocity of the ocean long wave dependent on the ocean depth *H* and the acceleration of the gravity, *g*. The Froude number Fr is defined as the ratio between the atmospheric motion and the ocean wave propagation, Fr = *U/c*. Assuming that the friction, Coriolis, eddy viscosity and advection terms can be negligible (i.e. onedimensional (1D) linear wave), the normalized sea level change becomes

$$\frac{\eta}{\eta\_s} = \frac{1}{1 - Fr^2} \tag{1}$$

in equilibrium, where = − the sea level change under static balance with the intensity of the pressure anomaly *ΔP* and the density of water *ρ*. Equation (1) implies that sea surface can be deformed unlimitedly as the pressure disturbance moves the ocean with the same speed of the phase velocity of the long wave (i.e. holding Fr ~1).

When Fr ~1, the work due to the atmospheric pressure gradient force supplies wave energy at the same phase continuously, and the magnitude of the sea level deformation will be proportional to the travelling distance of the pressure disturbance on the sea interface. If *U* > *c*, that is, Fr > 1, that is, supercritical propagation over very shallow water, a wake forms similar to that passing the vessel with high speed. On the other hand, the subcritical propagation such that *U* < *c*, the ocean long wave will propagate ahead of the atmospheric disturbance, with the wave length becoming longer and longer instead of enlarging the amplitude.

The effect of the Proudman resonance typically ranged from two to five times of the stationary state. When the atmospheric disturbance travels longer than 1000 km keeping Fr =1 ± 0.10, the resonant effect can be higher than 10 times of the stationary state. Hibiya and Kajiura (1982) [5] derived the formula under the condition of Fr ~1 as,

$$\frac{\eta}{\eta\_s} = \frac{1}{2} \frac{X\_f}{W} \tag{2}$$

where *Xf* represents the fetch of the pressure disturbance, and *W* is the width of the pressure disturbance. For example, *Xf* = 500 km, *W* = 50 km yields the amplification ratio *η*/*η*s = 5.0.

#### **2.2. Shoaling and refraction**

Similar to seismic tsunami, meteorological tsunami is also amplified by shoaling and refraction. Linear theory can be used as the first approximation to calculate the wave height as the wave moves across an ocean and undergoes wave shoaling and refraction. The formula can be written as (e.g. Synolakis, 1991 [6])

$$\frac{\eta\_{\rho}}{\eta\_{o}} = \left(\frac{D\_{o}}{D\_{\rho}}\right)^{\frac{1}{4}} \left(\frac{B\_{o}}{B\_{\rho}}\right)^{\frac{1}{2}} \tag{3}$$

where *ηo* = wave height (crest to trough) at the original point, *ηp* = wave height (crest to trough) at any point, *Do* = water depth at source point, *Dp* = water depth at any shoreward point, *Bo* distance between wave orthogonals at a source point of water and *BP* = distance between wave orthogonals at any shoreward point of water. The ratio between *Bo* and *Bp* indicates the refraction, and the ratio between water depth indicates the shoaling, a part of Green's law.

#### **2.3. Resonance of the edge wave**

The resonance due to continental shelf is also significant in amplifying the meteorological tsunamis. For example, Greenspan (1956) [7] derived the surface elevation across the continental shelf from the linearized shallow water equation. Let the beach slope *α* = tan *β*, and *U* = velocity of pressure disturbance along the edge, the frequency of *n*-th edge mode can be described as

$$\mathbf{k}\_n = \left(2n+1\right)\mathbf{g}\,\alpha/U^2\tag{4}$$

Giving the pressure distribution as a Gaussian function moving along the shelf edge, the amplitude of the *n*-th edge wave depends (with the wave number of *kn*) on the intensity of the pressure disturbance *P0*, the moving speed of the system, horizontal scale of the disturbance *a* and the distance from the coast line *y* as

$$\mathbf{A}\_{n}\left(k\_{\pi}\right) = -\frac{2}{U} \left(-\frac{2\sqrt{\pi}UP\_{0}a}{\rho}\right) k\_{\pi}^{2} \exp\left(-\frac{a^{2}k\_{\pi}^{2}}{4}\right) \Bigg\int\_{0}^{\alpha} L\_{n}\left(2k\_{n}y\right) \exp\left\{-k\_{n}y - \frac{\left(\boldsymbol{\mathcal{V}} - \boldsymbol{\mathcal{V}}\_{o}\right)^{2}}{a^{2}}\right\} dy \tag{5}$$

where

1 2 *f*

*X W*

Similar to seismic tsunami, meteorological tsunami is also amplified by shoaling and refraction. Linear theory can be used as the first approximation to calculate the wave height as the wave moves across an ocean and undergoes wave shoaling and refraction. The formula can be

1 1

4 2 *<sup>p</sup> o o o pp D B D B*

æ öæ ö = ç ÷ç ÷ è øè ø

where *ηo* = wave height (crest to trough) at the original point, *ηp* = wave height (crest to trough) at any point, *Do* = water depth at source point, *Dp* = water depth at any shoreward point, *Bo* distance between wave orthogonals at a source point of water and *BP* = distance between wave orthogonals at any shoreward point of water. The ratio between *Bo* and *Bp* indicates the refraction, and the ratio between water depth indicates the shoaling, a part of Green's law.

The resonance due to continental shelf is also significant in amplifying the meteorological tsunamis. For example, Greenspan (1956) [7] derived the surface elevation across the continental shelf from the linearized shallow water equation. Let the beach slope *α* = tan *β*, and *U* = velocity of pressure disturbance along the edge, the frequency of *n*-th edge mode can be

> ( ) <sup>2</sup> <sup>n</sup> k 21 / = + *n gU*

a

Giving the pressure distribution as a Gaussian function moving along the shelf edge, the amplitude of the *n*-th edge wave depends (with the wave number of *kn*) on the intensity of the pressure disturbance *P0*, the moving speed of the system, horizontal scale of the disturbance

0

¥ æ ö æ ö ì ü ï ï - =- -ç ÷ ç ÷ - í ý - è ø è ø ï ï î þ

*n o*

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

*UP a a k y y <sup>k</sup> <sup>k</sup> L ky ky dy U a*

ó ô ô õ

n 2

*n n nn n*

(4)

h

h

represents the fetch of the pressure disturbance, and *W* is the width of the pressure

= 500 km, *W* = 50 km yields the amplification ratio *η*/*η*s = 5.0.

<sup>=</sup> (2)

(3)

(5)

*s*

h

h

where *Xf*

16 Tsunami

disturbance. For example, *Xf*

**2.2. Shoaling and refraction**

written as (e.g. Synolakis, 1991 [6])

**2.3. Resonance of the edge wave**

*a* and the distance from the coast line *y* as

p

r

0 2

2 2 <sup>A</sup> exp 2 exp <sup>4</sup>

described as

$$L\_{\pi}(s) = \exp(s) \left(\frac{d^{\ast}}{ds^{\ast}}\right) \left(s^{\ast} \exp(-s)\right) \tag{6}$$

For fundamental edge mode (*n* = 0), the maximum wave height becomes maximum with the half-pressure radius of 80–90 km. This implies that the meso-β- or γ-scale disturbance, the typical wavelength of the meteorological tsunami, is able to amplify the wave by edge mode rather than meso-α-scale disturbance such as hurricane.

#### **2.4. Harbour resonance**

The resonance in the harbour or the inlet is the conclusive effect of how much sea level oscillates inside them. Assuming that the bay shapes rectangular open channel with uniform sea depth (*H*), and the incident direction of the ocean wave is parallel to that of the axis of the bay. The period (*T*) for eigen oscillation of the bay is

$$T = \varepsilon \frac{4L}{n\sqrt{\text{g}H}} \tag{7}$$

with *L* = axis length of the bay. The coefficient *ε* represents the correction coefficient for the bay mouth effect due to the generation of scattering waves near the bay of the mouth [8] as

$$\varepsilon = \left[ 1 + \frac{2B}{\pi L} \left( 0.923 + \ln \frac{4L}{\pi B} \right) \right]^{\frac{1}{2}} \tag{8}$$

where *B* = the width of the bay mouth. For more details, refer to Ravinobich (2009) [9] for the various shapes of the inlets.
