3.2 The long waves on the days when large harbor oscillation occurred in Urauchi Bay

The pressure profiles for atmospheric-pressure waves of patterns (b), (c), and (d) shown in Figure 6 are described for x � xc j j≤L=2 as

$$P(\mathbf{b}): P(\mathbf{x}, t\_0) = P\_0 \{ \mathbf{1} - \sin \left[ \pi (\mathbf{x} - \mathbf{x}\_c) / L \right] \} / 2,\tag{5}$$

$$P(\mathbf{c}):\ P(\mathbf{x}, t\_0) = P\_0 \{ \mathbf{1} + \cos\left[2\pi(\mathbf{x} - \mathbf{x}\_c)/L\right] \}/2 \tag{6}$$

$$\begin{aligned} \left[ \begin{array}{c} \text{\raisebox{-0.0pt}{ $\cdot$ }} \\ + \text{\raisebox{-0.0pt}{ $\cdot$ }} \end{array} \right] + P\_0 \{ \left[ \begin{array}{c} \text{\raisebox{-0.0pt}{ $\cdot$ }} \end{array} \right] \left[ \begin{array}{c} \text{\raisebox{-0.0pt}{ $\cdot$ }} \end{array} \right] \end{aligned} \right) \tag{6}$$
  $\begin{array}{c} \text{\raisebox{-0.0pt}{ $\cdot$ }} \end{array} \right. $ 

$$\begin{aligned} \left( \mathbf{d} \right): \ P(\mathbf{x}, t\_0) &= \mathbf{0}.05e^{\mathbf{x}} P\_0 \left\{ \mathbf{1} + \cos \left[ 6\pi (\mathbf{x} - \mathbf{x}\_c) / L \right] \right\} / 4 \\ &+ P\_0 \left\{ \mathbf{1} - \sin \left[ 3\pi (\mathbf{x} - \mathbf{x}\_c - \mathbf{x}\_d) / L \right] \right\} / 2 \end{aligned} \tag{7}$$

respectively, while P(x, t0) = P0(x < xc � L/2) and P(x, t0) = 0.0 (x > xc + L/2). In Eq. (7), xd is the initial position of the second pressure peak, and the power κ is 0.02x.

The parameters of each pattern are evaluated based on the GPV pressure data on the days when large harbor oscillation occurred in Urauchi Bay. For example, the time variation of GPV pressure distribution on February 25, 2009, when the largest harbor oscillation was observed in Urauchi Bay from 2009 to 2018, is shown in Figure 9.

Figure 10 shows the pressure profiles along three latitudes of 30.0, 30.5, and 31.0°N, at 3:00 on February 25, 2009, according to the GPV pressure data shown in Figure 9. An atmospheric-pressure wave, where the pressure gap was 4–5 hPa, and the total wavelength was 80–120 km, traveled almost eastward over East China Sea, at the phase velocity of around 140 km/h from 3:00 to 4:00, 120 km/h from 4:00 to 5:00, and 150 km/h from 5:00 to 6:00, such that the wave profile of the atmospheric pressure on the day is described with pattern (d), where the mean values of the parameters, that is, L, Pmax, and Cp, are 90.0 km, 4.0 hPa, and 38.6 m/s, respectively.

#### Figure 9.

The time variation of the GPV pressure distribution on February 25, 2009, when the huge harbor oscillation of 3.0 m in total amplitude was observed in Urauchi Bay. The GPV pressure data were published by Japan Meteorological Agency.

Figure 7 shows the numerical calculation results of water surface displacements at Point ① indicated in Figure 5, owing to an assumed atmospheric-pressure wave of pattern (a), where L = 10.0 km, Cp = 20.0 m/s, and Pmax = 1.0, 2.0, or 3.0 hPa. Point ① is located off the mouth of Urauchi Bay, where the huge harbor oscillation of 3.0 m in total amplitude was observed, as mentioned above. The wave height of the generated long waves at Point ① is almost in proportion to Pmax, which has been also confirmed at the other monitoring points near Danjyo Islands or Uji Islands. According to Figure 7, many long waves propagate through Point ①, owing to the

The water surface displacements at Point ① indicated in Figure 5, for various values of Pmax. The wave profile of atmospheric pressure is pattern (a), where L = 10.0 km, Cp = 20.0 m/s, and Pmax = 1.0, 2.0, or 3.0 hPa. The

Shown in Figure 8 is the wave height and period of the long wave with the maximum wave height at Point ① indicated in Figure 5, for various values of Cp, where the wave profile of atmospheric pressure is pattern (a); L = 30.0 km and Pmax = 1.0 hPa; the waves are defined using the zero-up-cross method. Inside the area from 125.5 to 127.0°E, and from 30.0 to 32.5°N, the average value of still water depth is 80 m, or 100 m, over the continental shelf, such that the Proudman resonance for long ocean waves can occur when the phase velocity of an

atmospheric-pressure wave, Cp, is around the phase velocity of linear shallow water

The wave height and period of the long wave with the maximum wave height at point ① indicated in Figure 5, for various values of Cp. The wave profile of atmospheric pressure is pattern (a), where L = 30.0 km and

gh p ≈30 m=s. According to Figure 8, the wave height of the long

travel of one atmospheric-pressure wave.

still water depth is about 22.0 m at Point ①.

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ocean waves increases as Cp is close to 32.0 m/s.

waves, that is, ffiffiffiffiffi

Figure 8.

86

Pmax = 1.0 hPa.

Figure 7.

Figure 10.

The GPV pressure distributions along three latitudes of 30.0, 30.5, and 31.0°N, at 3:00 on February 25, 2009. The solid, dotted, and chain double-dashed lines show the pressure along the latitudes of 30.0, 30.5, and 31.0° N, respectively.

Figure 11.

The numerical result for the time variation of water level distribution on February 25, 2009. The wave profile of atmospheric pressure is pattern (d), where L = 90.0 km, Pmax = 4.0 hPa, and Cp = 38.6 m/s.

Depicted in Figure 11 is the numerical result for the time variation of water level distribution due to the atmospheric-pressure waves, where the pressure profile is pattern (d), and its parameters L, Pmax, and Cp are 90.0 km, 4.0 hPa, and 38.6 m/s, respectively. The waves show refraction over Okinawa Trough, for the phase velocity of the generated long waves decreases over the deep trough, after which they propagate to the northeast, as pointed out by Katayama et al. [16].

with the abovementioned mean values of their parameters. The wave height of the long waves due to the atmospheric-pressure wave of pattern (b) is lower than that in the other cases, for the atmospheric pressure does not decrease after its increase. If the sea surface, which has been pressed down, is relieved owing to attenuation in atmospheric pressure, the balance between the atmospheric pressure and the water surface gradient is not maintained, resulting in the production and propagation of free-surface waves, and the Proudman resonance appears when the moving velocity of the recovery point of atmospheric pressure matches the phase velocity of long ocean waves. Although the reason why the harbor oscillation in Urauchi Bay was rather large on March 3, 2010, is thought to be linked to the instability in atmo-

The numerical results for the water surface displacements at Point ① indicated in Figure 5. The wave profiles of atmospheric pressure are patterns (b), (c), and (d), where the parameters (L, Pmax, and Cp) are (100.0 km, 3.0 hPa, 20.0 m/s), (100.0 km, 4.0 hPa, 33.0 m/s), and (90.0 km, 4.0 hPa, 25.0 m/s), respectively.

The numerical result for the water surface displacement at point ① indicated in Figure 5, on February 25, 2009. The wave profile of atmospheric pressure is pattern (d), where L = 90.0 km, Pmax = 4.0 hPa, and

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

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

Conversely, the long waves generated by the atmospheric-pressure wave of pattern (d) show remarkable wave height of 1.1 m, where the atmospheric pressure decreases after its increase. The wave period of the first wave is about 1300 s, while that of the second and the third waves is about 1250 and 900 s, respectively. These values of wave period, as well as the numbers of exited long waves, concern the amplification of harbor oscillation, as discussed in the following sections. The long waves due to the atmospheric-pressure wave of pattern (c) also show the maximum

spheric pressure before the day, future work is required.

Figure 12.

Figure 13.

89

Cp = 38.6 m/s.

The numerical result for the water surface displacement at Point ① indicated in Figure 5 is shown in Figure 12, where the wave height of the first three waves is over 1 m, and the wave period of the first to the fifth waves is about 1000, 750, 700, 760, and 660 s, respectively.

According to the observed data [9], large harbor oscillation also occurred in Urauchi Bay on March 3, 5, and 6, 2010, where the wave profiles of atmospheric pressure are described by patterns (b), (c), and (d), respectively, based on the corresponding GPV pressure data, and the mean values of the parameters (L, Pmax, and Cp) are (100.0 km, 3.0 hPa, 20.0 m/s), (100.0 km, 4.0 hPa, 33.0 m/s), and (90.0 km, 4.0 hPa, 25.0 m/s), respectively. Shown in Figure 13 are the numerical calculation results for the water surface displacements at Point ① indicated in Figure 5, originating from the atmospheric-pressure waves of patterns (b), (c), and (d), Long-Wave Generation due to Atmospheric-Pressure Variation and Harbor Oscillation… DOI: http://dx.doi.org/10.5772/intechopen.85483

#### Figure 12.

The numerical result for the water surface displacement at point ① indicated in Figure 5, on February 25, 2009. The wave profile of atmospheric pressure is pattern (d), where L = 90.0 km, Pmax = 4.0 hPa, and Cp = 38.6 m/s.

#### Figure 13.

Depicted in Figure 11 is the numerical result for the time variation of water level distribution due to the atmospheric-pressure waves, where the pressure profile is pattern (d), and its parameters L, Pmax, and Cp are 90.0 km, 4.0 hPa, and 38.6 m/s, respectively. The waves show refraction over Okinawa Trough, for the phase velocity of the generated long waves decreases over the deep trough, after which

The numerical result for the time variation of water level distribution on February 25, 2009. The wave profile

The GPV pressure distributions along three latitudes of 30.0, 30.5, and 31.0°N, at 3:00 on February 25, 2009. The solid, dotted, and chain double-dashed lines show the pressure along the latitudes of 30.0, 30.5, and 31.0°

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The numerical result for the water surface displacement at Point ① indicated in Figure 5 is shown in Figure 12, where the wave height of the first three waves is over 1 m, and the wave period of the first to the fifth waves is about 1000, 750, 700,

According to the observed data [9], large harbor oscillation also occurred in Urauchi Bay on March 3, 5, and 6, 2010, where the wave profiles of atmospheric pressure are described by patterns (b), (c), and (d), respectively, based on the corresponding GPV pressure data, and the mean values of the parameters (L, Pmax, and Cp) are (100.0 km, 3.0 hPa, 20.0 m/s), (100.0 km, 4.0 hPa, 33.0 m/s), and (90.0 km, 4.0 hPa, 25.0 m/s), respectively. Shown in Figure 13 are the numerical calculation results for the water surface displacements at Point ① indicated in Figure 5, originating from the atmospheric-pressure waves of patterns (b), (c), and (d),

they propagate to the northeast, as pointed out by Katayama et al. [16].

of atmospheric pressure is pattern (d), where L = 90.0 km, Pmax = 4.0 hPa, and Cp = 38.6 m/s.

760, and 660 s, respectively.

Figure 10.

Figure 11.

88

N, respectively.

The numerical results for the water surface displacements at Point ① indicated in Figure 5. The wave profiles of atmospheric pressure are patterns (b), (c), and (d), where the parameters (L, Pmax, and Cp) are (100.0 km, 3.0 hPa, 20.0 m/s), (100.0 km, 4.0 hPa, 33.0 m/s), and (90.0 km, 4.0 hPa, 25.0 m/s), respectively.

with the abovementioned mean values of their parameters. The wave height of the long waves due to the atmospheric-pressure wave of pattern (b) is lower than that in the other cases, for the atmospheric pressure does not decrease after its increase. If the sea surface, which has been pressed down, is relieved owing to attenuation in atmospheric pressure, the balance between the atmospheric pressure and the water surface gradient is not maintained, resulting in the production and propagation of free-surface waves, and the Proudman resonance appears when the moving velocity of the recovery point of atmospheric pressure matches the phase velocity of long ocean waves. Although the reason why the harbor oscillation in Urauchi Bay was rather large on March 3, 2010, is thought to be linked to the instability in atmospheric pressure before the day, future work is required.

Conversely, the long waves generated by the atmospheric-pressure wave of pattern (d) show remarkable wave height of 1.1 m, where the atmospheric pressure decreases after its increase. The wave period of the first wave is about 1300 s, while that of the second and the third waves is about 1250 and 900 s, respectively. These values of wave period, as well as the numbers of exited long waves, concern the amplification of harbor oscillation, as discussed in the following sections. The long waves due to the atmospheric-pressure wave of pattern (c) also show the maximum wave height of about 0.3 m, and the wave period of the long wave with the maximum wave height is around 2600 s.
