5.1 Estimate equations for the wave height and wavelength of generated long waves

consider a sever case due to the Proudman resonance. It is also assumed that the wavelength of long ocean waves, λ, is much larger than the still water depth h, that

Long-Wave Generation due to Atmospheric-Pressure Variation and Harbor Oscillation…

x = x<sup>0</sup> + DP + ΔL, respectively, where ΔL = CPΔt. The water body CDFE sketched in Figure 19, which is part of the raised water at the initial time, is relieved owing to

The parallelogram CDFE, which we call S0, shown in Figure 19, corresponds to the trapezoid S1 shown in Figure 20, where the height and the length of lower base of the trapezoid S1 are a (m) = �PΔL/DP/10,000 and L<sup>1</sup> = DP + ΔL, respectively, for side EF shown in Figure 19 is an isopotential energy level at t = Δt. The relieved water body S1 transforms to two long ocean waves, propagating in the positive and negative directions of the x-axis, where the wave height, the wavelength, and the absolute value of phase velocity, of the two long waves, are approximately a/2, L1, and C,

Through the recovery of low pressure after t = Δt, the relief of water body is repeated, such that the long ocean wave, propagating in the positive direction of the x-axis, is overlapped by other long waves generated continuously. Consequently, the wave amplitude H and wavelength λ of the long wave traveling in the positive

An enlarged illustration of part of the water surface profile shown in Figure 18. Side AB of the low-pressure profile shown in Figure 17, is above side DC of the water surface profile at t = 0.0 s, after which side AB comes

An aggregation of water columns, S0, relieved owing to the recovery of low pressure (a), and the corresponding

H ¼ �ð Þ PLP=DP =20; 000 mð Þ, (8)

λ ¼ DP, (9)

gh p . When t = Δt, the positions of points A and B become x = x<sup>0</sup> + ΔL and

is, <sup>h</sup>/<sup>λ</sup> <sup>≪</sup> 1, such that CP <sup>¼</sup> <sup>C</sup> <sup>¼</sup> ffiffiffiffiffi

DOI: http://dx.doi.org/10.5772/intechopen.85483

respectively.

Figure 19.

Figure 20.

93

trapezoidal water body S1 (b).

above side FE at t = Δt.

the recovery of the low pressure during Δt.

direction of the x-axis are estimated by

As mentioned in Section 1, long waves due to atmospheric-pressure variation can cause large harbor oscillation, resulting in hazards including the damages of fish boats and the inundation of houses, such that it is necessary for fishing cooperatives, town offices, etc. to prevent such hazards. If a simple method to predict the generation of serious long ocean waves is available, then they can make provision against meteotsunamis, several hours before. In this section, we propose equations to estimate both the wave height and wavelength of coming long ocean waves, using the measured or GPV data of atmospheric pressure, without derivation, integration, or complex numerical calculation.

It is assumed that the distribution of atmospheric pressure p above the outer sea is trapezoidal at the initial time t = 0.0 s, as shown in Figure 17, where the profile for a low-pressure case is illustrated, after which the atmospheric-pressure wave propagates stably at a constant phase velocity, in the positive direction of the x-axis.

The water surface is assumed to rise 1.0 cm owing to the pressure decrease of 1.0 hPa, and then the initial profile of water surface is also trapezoidal as shown in Figure 18. The maximum value of water surface displacement is P/10,000 (m) for x<sup>0</sup> + DP ≤ x ≤ x<sup>0</sup> + DP + LP at t = t0, where P (Pa) < 0 is the minimum pressure value of the atmospheric-pressure wave shown in Figure 17, and LP is the distance where its pressure value hardly shows variation.

After the initial condition shown in Figure 17, side AB of the low-pressure profile moves at a constant phase velocity, resulting in a gradual recovery of atmospheric pressure from the low-pressure condition. The moving velocity of point A, where the pressure recovery starts, that is, the phase velocity of the atmosphericpressure wave, CP, is assumed to equal the phase velocity of long ocean waves, C, to

Figure 17.

The initial atmospheric-pressure distribution of a low-pressure wave. After the initial time, the pressure wave propagates at a constant phase velocity in the positive direction of the x-axis.

#### Figure 18. The initial water surface profile due to the initial atmospheric-pressure distribution shown in Figure 17.

Long-Wave Generation due to Atmospheric-Pressure Variation and Harbor Oscillation… DOI: http://dx.doi.org/10.5772/intechopen.85483

consider a sever case due to the Proudman resonance. It is also assumed that the wavelength of long ocean waves, λ, is much larger than the still water depth h, that is, <sup>h</sup>/<sup>λ</sup> <sup>≪</sup> 1, such that CP <sup>¼</sup> <sup>C</sup> <sup>¼</sup> ffiffiffiffiffi gh p .

When t = Δt, the positions of points A and B become x = x<sup>0</sup> + ΔL and x = x<sup>0</sup> + DP + ΔL, respectively, where ΔL = CPΔt. The water body CDFE sketched in Figure 19, which is part of the raised water at the initial time, is relieved owing to the recovery of the low pressure during Δt.

The parallelogram CDFE, which we call S0, shown in Figure 19, corresponds to the trapezoid S1 shown in Figure 20, where the height and the length of lower base of the trapezoid S1 are a (m) = �PΔL/DP/10,000 and L<sup>1</sup> = DP + ΔL, respectively, for side EF shown in Figure 19 is an isopotential energy level at t = Δt. The relieved water body S1 transforms to two long ocean waves, propagating in the positive and negative directions of the x-axis, where the wave height, the wavelength, and the absolute value of phase velocity, of the two long waves, are approximately a/2, L1, and C, respectively.

Through the recovery of low pressure after t = Δt, the relief of water body is repeated, such that the long ocean wave, propagating in the positive direction of the x-axis, is overlapped by other long waves generated continuously. Consequently, the wave amplitude H and wavelength λ of the long wave traveling in the positive direction of the x-axis are estimated by

$$H = (-PL\_P/D\_P)/20,000 \text{ (m)},\tag{8}$$

$$
\lambda = D\_{\rm P} \tag{9}
$$

Figure 19.

5. Simple method to estimate long waves due to an atmospheric-

5.1 Estimate equations for the wave height and wavelength of generated long

As mentioned in Section 1, long waves due to atmospheric-pressure variation can cause large harbor oscillation, resulting in hazards including the damages of fish boats and the inundation of houses, such that it is necessary for fishing cooperatives, town offices, etc. to prevent such hazards. If a simple method to predict the generation of serious long ocean waves is available, then they can make provision against meteotsunamis, several hours before. In this section, we propose equations to estimate both the wave height and wavelength of coming long ocean waves, using the measured or GPV data of atmospheric pressure, without derivation,

It is assumed that the distribution of atmospheric pressure p above the outer sea is trapezoidal at the initial time t = 0.0 s, as shown in Figure 17, where the profile for a low-pressure case is illustrated, after which the atmospheric-pressure wave propagates stably at a constant phase velocity, in the positive direction of the x-axis. The water surface is assumed to rise 1.0 cm owing to the pressure decrease of 1.0 hPa, and then the initial profile of water surface is also trapezoidal as shown in Figure 18. The maximum value of water surface displacement is P/10,000 (m) for x<sup>0</sup> + DP ≤ x ≤ x<sup>0</sup> + DP + LP at t = t0, where P (Pa) < 0 is the minimum pressure value of the atmospheric-pressure wave shown in Figure 17, and LP is the distance where

After the initial condition shown in Figure 17, side AB of the low-pressure profile moves at a constant phase velocity, resulting in a gradual recovery of atmospheric pressure from the low-pressure condition. The moving velocity of point A, where the pressure recovery starts, that is, the phase velocity of the atmosphericpressure wave, CP, is assumed to equal the phase velocity of long ocean waves, C, to

The initial atmospheric-pressure distribution of a low-pressure wave. After the initial time, the pressure wave

The initial water surface profile due to the initial atmospheric-pressure distribution shown in Figure 17.

propagates at a constant phase velocity in the positive direction of the x-axis.

pressure wave

integration, or complex numerical calculation.

Natural Hazards - Risk, Exposure, Response, and Resilience

its pressure value hardly shows variation.

waves

Figure 17.

Figure 18.

92

An enlarged illustration of part of the water surface profile shown in Figure 18. Side AB of the low-pressure profile shown in Figure 17, is above side DC of the water surface profile at t = 0.0 s, after which side AB comes above side FE at t = Δt.

#### Figure 20.

An aggregation of water columns, S0, relieved owing to the recovery of low pressure (a), and the corresponding trapezoidal water body S1 (b).

respectively. When LP is the moving distance of side AB, Eq. (8) corresponds to the prediction equation shown by Hibiya and Kajiura [3], using the method of characteristics. The parameters P, LP, and DP can be evaluated according to the observed or GPV pressure data for the wave profile of an atmospheric-pressure wave.

Conversely, if we observe the time variation of atmospheric pressure at several offshore sites, to obtain the recovery rate of pressure p, that is, rP, which is defined by ∂p/∂t, the estimate equation for H is

$$H = (r\_P L\_P / C\_P) / 20,000 \, (\text{m}),\tag{10}$$

atmospheric-pressure waves. As DP is decreased, the wavelength of the

height and wavelength of severe meteotsunamis, using observed or GPV

6. Numerical calculation for harbor oscillation in harbors of various

Meteotsunamis can be amplified to be heavier through harbor oscillation, as well as shoaling. In this chapter, we discuss oscillation in harbors of various shapes, by applying the numerical model based on the nonlinear shallow water equations, that is, Eqs. (1)–(3). The Coriolis coefficient f and horizontal eddy viscosity coefficient

computational domains, where the harbor of a horizontally rectangular shape, we call, an I-type harbor. A train of incident regular waves, the wave height of which is 0.2 m, enters the computational domain through its leftward boundary and then

The target harbors are model harbors of various shapes, as well as an actual bay, where the horizontal shapes of the model harbors are I-type, L-type, C-type, and

propagates inside the harbor, leading to harbor oscillation.

/s, respectively. Illustrated in Figure 22 is an example of

Shown in Table 1 are the numerical results of wave amplitude H, wavelength λ, and wave period T, for the generated long waves propagating through x = x<sup>1</sup> indicated in Figure 18, in comparison with the corresponding estimated values of both H and λ from Eqs. (8) and (9), respectively. The estimated values show good agreement with the corresponding computational data, such that the proposed estimate equations are available to predict the approximate values of both the wave

meteotsunamis decreases, but their wave height increases.

are defined in Figure 17; rP is the recovery rate of atmospheric pressure.

Values of pressure parameters Numerical results obtained

DOI: http://dx.doi.org/10.5772/intechopen.85483

using Eqs. (1)–(3)

400 0.160 0.16

400 0.080 0.08

400 0.040 0.04

DP (m) rP (Pa/s) LP (m) P (Pa) H (m) λ (m) T (s) H (m) λ (m) 12,500 1.00 50,000 400 0.080 12,500 400 0.08 12,500 100,000 200 0.080 0.08

Long-Wave Generation due to Atmospheric-Pressure Variation and Harbor Oscillation…

1.00 100,000 400 0.160 0.16 25,000 0.50 50,000 400 0.040 25,000 800 0.04 25,000 100,000 200 0.040 0.04

0.50 100,000 400 0.080 0.08 50,000 0.25 50,000 400 0.020 50,000 1600 0.02 50,000 100,000 200 0.020 0.02

0.25 100,000 400 0.040 0.04

The wave amplitude H, wavelength λ, and wave period T of the generated long ocean waves, at x = x1 indicated in Figure 18, obtained using the numerical model based on the nonlinear shallow water equations, that is, Eqs. (1)–(3), as well as the estimated values of H and λ through Eqs. (8) and (9), where DP, LP, and P

Estimated values from Eqs. (8) and (9)

atmospheric-pressure data.

Ah are 0.0 s<sup>1</sup> and 30.0 m<sup>2</sup>

6.1 Numerical calculation conditions

shapes

95

Table 1.

for rP corresponds to �PCP/DP (Pa/s), according to Figure 17. It is noted that Eqs. (8)–(10) can be also applied to high-pressure cases, where the positive value P is the highest value of atmospheric pressure.

### 5.2 The validation of predicted values through the estimate equations

Several results through the proposed estimate equations, that is, Eqs. (8) and (9), are compared with the corresponding numerical results obtained using the numerical model based on Eqs. (1)–(3), for the one-dimensional generation and propagation of meteotsunamis. The still water depth h is assumed to be uniformly 100.0 m. In the numerical computation, the distribution of atmospheric pressure at the water surface is changed gradually from zero to a low-pressure distribution as shown in Figure 17, resulting in a water surface profile as shown in Figure 18. After obtaining the initial steady state, side AB shown in Figure 17 moves at CP <sup>¼</sup> <sup>C</sup> <sup>¼</sup> ffiffiffiffiffi gh p . The Coriolis coefficient, seabed friction coefficient, and horizontal eddy viscosity coefficient are zero in Eqs. (1)–(3) for simplicity.

Figure 21 shows the numerical calculation results of water surface displacements at x = x<sup>1</sup> indicated in Figure 18, where P = �400 Pa and LP = 100,000 m; DP = 12,500, 25,000, and 50,000 m. The decrease in water surface displacement η for t < 7,000 s is due to the decrease in atmospheric pressure before the initial time, while the increase in η for t > 11,000 s is caused by the propagation of the

#### Figure 21.

The numerical results of water surface displacements at x = x1 indicated in Figure 18, obtained using the nonlinear shallow water model, for various values of DP, where P = �400 Pa; LP = 100,000 m; DP = 12,500, 25,000, and 50,000 m.


Long-Wave Generation due to Atmospheric-Pressure Variation and Harbor Oscillation… DOI: http://dx.doi.org/10.5772/intechopen.85483

#### Table 1.

respectively. When LP is the moving distance of side AB, Eq. (8) corresponds to

Conversely, if we observe the time variation of atmospheric pressure at several offshore sites, to obtain the recovery rate of pressure p, that is, rP, which is defined

for rP corresponds to �PCP/DP (Pa/s), according to Figure 17. It is noted that Eqs. (8)–(10) can be also applied to high-pressure cases, where the positive value P

Several results through the proposed estimate equations, that is, Eqs. (8) and (9), are compared with the corresponding numerical results obtained using the numerical model based on Eqs. (1)–(3), for the one-dimensional generation and propagation of meteotsunamis. The still water depth h is assumed to be uniformly 100.0 m. In the numerical computation, the distribution of atmospheric pressure at the water surface is changed gradually from zero to a low-pressure distribution as shown in Figure 17, resulting in a water surface profile as shown in Figure 18. After obtaining the initial steady state, side AB shown in Figure 17 moves at CP <sup>¼</sup> <sup>C</sup> <sup>¼</sup> ffiffiffiffiffi

The Coriolis coefficient, seabed friction coefficient, and horizontal eddy viscosity

Figure 21 shows the numerical calculation results of water surface displacements at x = x<sup>1</sup> indicated in Figure 18, where P = �400 Pa and LP = 100,000 m; DP = 12,500, 25,000, and 50,000 m. The decrease in water surface displacement η for t < 7,000 s is due to the decrease in atmospheric pressure before the initial time,

while the increase in η for t > 11,000 s is caused by the propagation of the

The numerical results of water surface displacements at x = x1 indicated in Figure 18, obtained using the nonlinear shallow water model, for various values of DP, where P = �400 Pa; LP = 100,000 m; DP = 12,500,

5.2 The validation of predicted values through the estimate equations

H ¼ ðrPLP=CPÞ=20; 000 mð Þ, (10)

gh p .

the prediction equation shown by Hibiya and Kajiura [3], using the method of characteristics. The parameters P, LP, and DP can be evaluated according to the observed or GPV pressure data for the wave profile of an atmospheric-pressure

wave.

Figure 21.

94

25,000, and 50,000 m.

by ∂p/∂t, the estimate equation for H is

Natural Hazards - Risk, Exposure, Response, and Resilience

is the highest value of atmospheric pressure.

coefficient are zero in Eqs. (1)–(3) for simplicity.

The wave amplitude H, wavelength λ, and wave period T of the generated long ocean waves, at x = x1 indicated in Figure 18, obtained using the numerical model based on the nonlinear shallow water equations, that is, Eqs. (1)–(3), as well as the estimated values of H and λ through Eqs. (8) and (9), where DP, LP, and P are defined in Figure 17; rP is the recovery rate of atmospheric pressure.

atmospheric-pressure waves. As DP is decreased, the wavelength of the meteotsunamis decreases, but their wave height increases.

Shown in Table 1 are the numerical results of wave amplitude H, wavelength λ, and wave period T, for the generated long waves propagating through x = x<sup>1</sup> indicated in Figure 18, in comparison with the corresponding estimated values of both H and λ from Eqs. (8) and (9), respectively. The estimated values show good agreement with the corresponding computational data, such that the proposed estimate equations are available to predict the approximate values of both the wave height and wavelength of severe meteotsunamis, using observed or GPV atmospheric-pressure data.
