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.
First, the generation and propagation of long ocean waves due to the atmospheric-pressure variation have been simulated using the numerical model based on the nonlinear shallow water equations, where the atmospheric-pressure waves of various pressure-profile patterns travel eastward over East China Sea. Before the oscillation attenuation in Urauchi Bay, Japan, the incidence of long waves can continue owing to an oscillation system generated between the main island of Kyushu and Okinawa Trough. Second, the simple estimate equations are proposed to predict both the wave height and wavelength of long waves caused by an atmospheric-pressure wave, using atmospheric-pressure data above the ocean. Third, numerical simulation has been generated for the oscillation in the harbors of C-, I-, L-, and T-type shapes, as well as Urauchi Bay with two bay heads like a T-type harbor. Finally, we discuss disaster measures, including the real-time prediction of meteotsunami generation, as well as both the structural and the nonstructural preparations.
- long wave
- atmospheric pressure
- harbor oscillation
- secondary undulation
- submarine trough
- East China Sea
- real-time prediction
At Urauchi Bay of Kamikoshiki Island, situated in the western offing of Kyushu Island, Japan, as shown in Figure 1, heavy harbor oscillations occurred during February 24–26, 2009, where the maximum total amplitude of water level reached 3.0 m , resulting in that eight fishing boats were capsized and several houses were flooded, as shown in Figures 2–4. In terms of time, Japan Standard Time (JST) is used in this chapter. According to the Grid-Point-Value (GPV) pressure data, published by Japan Meteorological Agency (JMA), atmospheric-pressure waves propagated almost eastward over East China Sea, during this term.
Such atmospheric-pressure waves propagating over the sea surface have often generated significant long ocean waves, through an amplification mechanism, that is, the Proudman resonance , especially when the phase velocity of the atmospheric-pressure wave is close to that of the long ocean waves, as examined by, for example, Hibiya and Kajiura  and Vilibic et al. , where they numerically reproduced the large harbor oscillation in Nagasaki Bay, Kyushu, Japan, and that in Ciudadella Harbor, Balearic Islands, Spain, respectively. Once long ocean waves are generated by meteorological disturbance due to the instability of a wintry weather system, as well as a storm, and reach a nearshore zone, the wave height of the secondary undulation increases owing to the decrease of water depth, like a tsunami caused by a submarine earthquake (e.g., ), a land slide (e.g., ), etc., such that the long waves are called “meteotsunamis.” Meteotsunamis amplified depending on the conditions of atmospheric-pressure waves  can become external forces to create huge oscillation, severe inundation, etc. to coastal areas. Long ocean waves supposed to be meteotsunamis have been discussed based on observed data for many coastal zones, considering local characteristics concerning both geographic features and meteorological phenomena (e.g. [8, 9]); Bailey et al.  reported meteotsunamis caused by storms, which attacked the east coasts of the United States, facing the continental shelf; recent meteotsunami cases around the world were summarized by Tanaka and Ito . In nearshore zones, meteotsunamis are amplified through not only shoaling but also harbor oscillation in ports, harbors, and bays. Harbor oscillation, also called seiche, with the harbor paradox , depends on incident-wave period, harbor shape, and water depth. The oscillation in harbors of various horizontal shapes has been studied using linear theories , hydraulic experiments , nonlinear numerical models , etc.
In this chapter, first, we numerically simulate long ocean waves due to atmospheric-pressure waves with different pressure-profile patterns, including the atmospheric-pressure waves that caused the large harbor oscillation in Urauchi Bay on February 25, 2009. Second, simple estimate equations concerning both the wave height and wavelength of long waves generated by atmospheric-pressure variation are proposed using atmospheric-pressure data above the ocean, for easy prediction methods are required for disaster prevention by, for example, fisheries cooperatives and local authorities, although the numerical computation is necessary to research both the mechanisms and characteristics of meteotsunamis. Third, we apply a numerical model based on the nonlinear shallow water equations, to study oscillation in harbors of various shapes, including the types of “L,” “I” with a narrow region, “I” with a seabed crest or trough, “C,” and “T,” as well as Urauchi Bay, which has two heads like a T-type harbor. Finally, we discuss disaster measures against meteotsunamis, generated to propagate toward the west coasts of Kyushu. Several methods for the real-time prediction of meteotsunami generation are proposed, using an inverse analysis, as well as the proposed simple prediction equations, after which both the structural and the nonstructural preparations for meteotsunamis are summarized.
2. Numerical model and calculation conditions
A set of nonlinear shallow water equations, in consideration of atmospheric-pressure gradient at the sea surface, is solved in the horizontal two dimensions by applying a finite difference method. The fundamental equations are
3. Long-wave generation due to atmospheric-pressure waves
3.1. The relationship between the parameters of atmospheric-pressure waves and long-wave generation
In the large area along the west coasts of Kyushu, as well as Yamaguchi Prefecture nearby Kyushu, secondary undulation, supposed to be caused by atmospheric-pressure disturbance above East China Sea, often increases from February to April, sometimes leading to disasters as mentioned above. In this section, we discuss the relationship between the parameters of atmospheric-pressure waves and long-wave generation in the ocean. The computational domain is part of East China Sea, where the longitude is from 123.0 to 131.0°E, and the latitude is from 30.0 to 32.5°N, with the actual seabed configuration. The still water depth in East China Sea near the main island of Kyushu is shown in Figure 5, where it is around 800 m at the deepest site in Okinawa Trough. The grid widths ∆
In the computation, it is assumed that the atmospheric pressure is uniform from north to south, and atmospheric-pressure waves travel eastward at a constant phase velocity over East China Sea. The distribution of atmospheric pressure along the latitude lines is classified into four patterns shown in Figure 6, based on the GPV pressure data, where the atmospheric pressure
An atmospheric-pressure wave of pattern (a), for example, has three parameters, that is, wavelength
where the initial position of the pressure peak,
Figure 7 shows the numerical calculation results of water surface displacements at Point indicated in Figure 5, owing to an assumed atmospheric-pressure wave of pattern (a), where
Shown in Figure 8 is the wave height and period of the long wave with the maximum wave height at Point indicated in Figure 5, for various values of
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 as
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,
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
The numerical result for the water surface displacement at Point 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 , 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 (
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.
4. Oscillation system between the main island of Kyushu and Okinawa Trough
The amplification of harbor oscillation requires continuous wave energy incidence into the harbor. Figure 14 shows the water surface displacements at Point , off the mouth of Urauchi Bay, owing to the atmospheric-pressure wave of pattern (a), where
In order to examine the generation of an oscillation system between Okinawa Trough and the main island of Kyushu, we perform numerical experiments for a hypothetical seabed configuration. Figure 15(a) shows the actual seabed configuration along the latitude of 31.8°E, where Urauchi Bay is located as shown in Figure 5, while Figure 15(b) shows the hypothetical seabed configuration, where the trough length is extended to make the distance between wave reflection points larger. In both cases, the perfect reflection boundary condition is adopted at the west coasts of the main island of Kyushu.
In the one-dimensional computation for long waves, the nonlinear surface wave equations based on a variational principle  is applied to consider both the strong nonlinearity and dispersion of long waves over the shallower areas, as well as the deeper trough, where the velocity potential is assumed to show a linear distribution in the vertical direction. The water surface profile is given by
5. Simple method to estimate long waves due to an atmospheric-pressure wave
5.1. Estimate equations for the wave height and wavelength of generated long waves
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
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 −
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 atmospheric-pressure wave,
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
Through the recovery of low pressure after
Conversely, if we observe the time variation of atmospheric pressure at several offshore sites, to obtain the recovery rate of 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
Figure 21 shows the numerical calculation results of water surface displacements at
Shown in Table 1 are the numerical results of wave amplitude
|Values of pressure parameters||Numerical results obtained using Eqs. (1)–(3)||Estimated values from Eqs. (8) and (9)|
6. Numerical calculation for harbor oscillation in harbors of various shapes
6.1. Numerical calculation conditions
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
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 T-type, while the actual bay is Urauchi Bay. We examine numerical calculation results for the amplification factor of wave height due to oscillation in these harbors.
6.2. Amplification in the L-type harbors
Figure 23 shows L-type harbors, as well as an I-type harbor, where the harbor-axis length is 2000 m, while the bending position of the L-type harbors is different. The still water depth
Shown in Figure 24 is the amplification factor
6.3. Amplification in the I-type harbors with a narrowed area
Figure 25 shows I-type harbors with a narrowed area, where the position, or the width, of the narrowed area is different. The still water depth
Shown in Figure 26 is the amplification factor
6.4. Amplification in the C-type harbor
Depicted in Figure 27 is a C-type harbor, where two I-type harbors are connected with a rectangular-section channel, such that the C-type harbor has two mouths. The still water depth
Figure 28 shows the amplification factor
6.5. Amplification in the I-type harbors with a seabed crest or trough
Shown in Figure 29 are the seabed configurations of I-type harbors with a seabed crest or a seabed trough, where the still water depth is 10.5 or 29.5 m at the longitudinal center, respectively; except at the longitudinal center, the seabed is uniformly sloping inside the harbors. The still water depth is 20.0 m at both the head and mouth of the harbors, as well as outside the harbors in the computational domains. The length and width of the harbors are 2000 and 400 m, respectively.
Figure 30 shows the amplification factor
6.6. Amplification in the T-type harbors
A T-type harbor has two heads, as shown in Figure 31, where an I-type and L-type harbors are also depicted for comparison. The harbor width is 600 m, and the still water depth
Figure 32 shows the amplification factor
6.7. Harbor oscillation in Urauchi Bay
6.7.1. Amplification in Urauchi Bay
Urauchi Bay has two bay heads, as shown in Figure 1, such that the bay has a shape similar to that of a T-type harbor. Figure 33 shows the amplification factor
6.7.2. Water surface displacements at the ports of Urauchi Bay
The time variations of the water surface displacements at Oshima and Kuwanoura Fishing Ports are shown in Figure 34, where those for
6.7.3. The damping processes of oscillations in the T-type harbor and Urauchi Bay
In order to study the damping process of oscillation in the T-type harbor shown in Figure 31, we continuously give incident waves to obtain a quasi-steady state of harbor oscillation, after which the incidence of waves is stopped when
Conversely, Figure 36 shows the time variations of the maximum water level at Oshima and Kuwanoura Fishing Ports facing Urauchi Bay shown in Figure 1. The wave period of the incident waves,
7. Countermeasures against meteotsunamis
7.1. The real-time prediction of meteotsunami generation
7.1.1. The application of an inverse analysis
We discuss disaster measures against meteotsunamis, generated to propagate toward the west coasts of Kyushu. In order to predict the generation and propagation of meteotsunamis in real time, it is necessary to obtain atmospheric-pressure variation far from Kyushu. If we know the sites, concerning the generation of meteotsunamis through atmospheric-pressure variation, the valuable information on atmospheric pressure is restricted, such that the following inverse analysis is available:
We give some atmospheric-pressure variation at a site, to generate numerical simulation for atmosphere in a huge area including the Asian Continent, the Indian Ocean, and East China Sea, with a typical atmospheric condition for each season.
If an atmospheric-pressure wave appears over East China Sea, we give the atmospheric-pressure wave at the sea surface as an external force, to obtain the amplitude distribution of long waves along the west coasts, as well as the islands, of Kyushu, by applying the numerical model based on Eqs. (1)–(3).
We repeat the abovementioned calculation process for various conditions on atmospheric pressure, with atmospheric-pressure variation at different sites.
Using the results, we analyze inverse problems, where we give the distributions of long-wave amplitude, observed by, for example, the nationwide ocean wave information network for ports and harbors (NOWPHAS) conducted by the Ministry of Land, Infrastructure, Transport and Tourism, at the coasts of Kyushu, with the corresponding atmospheric-pressure conditions, to identify the sites in the Asian Continent and the Indian Ocean, which concerns the generation of meteotsunamis in East China Sea, through atmospheric-pressure variation.
According to the real-time variation in atmospheric pressure at the important sites, we can pick up bays and ports, which involve the risk of meteotsunami attack, to make adequate preparations for the meteotsunamis over a few days.
Conversely, we can also utilize a pattern recognition system for atmospheric-pressure distributions, instead of the inverse analysis, to exemplify dangerous atmospheric-pressure patterns.
7.1.2. Prediction for the amplitude of long waves using atmospheric pressure above East China Sea
We can predict approximate values for meteotsunami parameters, including long-wave amplitude, based on real-time variation in atmospheric pressure at several sites in East China Sea. If we obtain atmospheric-pressure data from barometers at plural islands, such as Danjyo Islands and Uji islands, shown in Figure 5, far from the west coasts of Kyushu, we can imagine the propagation direction of atmospheric-pressure waves and predict the possible largest long-wave amplitude
If fishing cooperatives and town offices obtain GPV atmospheric-pressure data, presented by JMA, to find out an atmospheric-pressure wave traveling east, they can predict the propagation direction, as well as the possible largest amplitude, of long-waves in East China Sea, where the accuracy of the predicted parameters is improved using Eqs. (8) and (9). It is important to catch every occurrence of large secondary undulation easily, even though both predictive accuracy and hitting ratio are relatively low. The fishing cooperatives and town offices, where the simple derivation process of Eqs. (8)–(10) is preferably understood, should be aware of the importance of the daily monitoring for variation in atmospheric pressure as a routine work.
7.2. Structural measures
The following structural measures against meteotsunamis are useful, depending on conditions including bay shape and water depth distribution:
Breakwaters are raised for ports with experience of large harbor oscillation, where several dozen centimeters may be enough. In case high breakwaters work against the loading of fishes and cargos, lockages of less than 1 m in height are suitable.
The bay width is narrowed with jetties, to protect ports and towns at bay heads, without inconvenience for daily steerage. It should be noted that the flow velocity, due to not only meteotsunamis but also tides, between the jetties may be larger, resulting in seabed scour, and that wave energy may be trapped behind jetties, leading to water surface oscillation prolonged in the bay. Furthermore, some device is required to advance seawater exchange, for part of the bay is occlusive. If the district to be protected is a narrow area, a water gate between two jetties is effective.
Permeable breakwaters with impounding reservoirs are constructed for coasts at high risk of overflow.
Fishery facilities are built, or moved, to adequate places, for corves etc., located near a node of harbor oscillation, may be flown away owing to flow of large velocity. The fish that got away is always big. The right places should be determined considering both water level and flow velocity, based on the characteristics of harbor oscillation in each bay or port.
Both drainage pipes and street gutters are designed to prevent inundation due to the intrusion of seawater into the residential area through the pipes and gutters. Although the walls of the castle, which is a world heritage, in Galle, Sri Lanka, rejected the tsunamis caused by the 2004 Indian Ocean earthquake, the seawater entered the inside of the walls through drainage pipes, leading to the flood.
River banks are constructed in consideration of meteotsunamis ascending rivers, as well as downflows due to heavy rain. The wave height of nonlinear tsunamis due to a submarine earthquake increases, when they travel upstream along a river with relatively narrow width, depending on the mouth shape of the river, according to the numerical results from the three-dimensional calculation .
7.3. Nonstructural measures
The coastal structures are permitted to be built considering cost effectiveness, nearshore environment, etc., such that nonstructural measures, including evacuation and preparation against meteotsunamis, are necessary as follows:
When a meteotsunami is predicted to be generated in the ocean, fishing boats and vessels are put offshore, if it is not stormy; if it is possible, they are put up on the land. Otherwise, mooring ropes should be tied firmly to prevent the flowage of fishing boats. If the ropes are too short, boats and vessels may be damaged when they collide with seawalls, or go on shore, owing to water level rise and onshore currents. Conversely, when the water level lowers, boats are hung by mooring ropes, as sketched in Figure 37, and they become upside-down, after which they are waterlogged as the water level rises. Note, however, that if mooring ropes are too long, boats are damaged owing to their collisions, such that bumpers should be attached to both boats and seawalls, unless the mooring positions are not moved to calm spots in harbor oscillation. Mooring facilities should be developed for temporary mooring at calm positions against meteotsunamis.
Waterproof tools, such as waterproof walls and sandbags, should be prepared for inundation of architectures including houses, shops, fishery facilities, factories, etc. Figure 38 shows the examples of waterproof walls, equipped at Kinki Area Seaside Disaster Prevention Center in Osaka Prefecture, Japan.
It is most important to notify inhabitants immediately that meteotsunamis are predicted to approach the coasts, using a community wireless system and speakers, or door-to-door visits. The prediction of disasters including meteotsunamis is probabilistic, commonly without high accuracy in their parameters, such that education to increase public awareness about disaster prevention is essential. It is crisis management that covers all the cases, whether the boy who cries wolf is right or not.
First, the generation and propagation of long ocean waves due to the atmospheric-pressure variation were simulated using the numerical model based on the nonlinear shallow water equations, where the atmospheric-pressure waves of four pressure-profile patterns traveled eastward over East China Sea, as well as the atmospheric-pressure waves that caused the large harbor oscillation in Urauchi Bay on February 25, 2009. The wave height of the long waves increased as the moving velocity of the pressure-recovery point was close to that of the long ocean waves. Before the oscillation attenuation in Urauchi Bay, the incidence of long waves can continue owing to an oscillation system generated between the main island of Kyushu and Okinawa Trough.
Second, the simple estimate equations were proposed to predict both the wave height and wavelength of severe meteotsunamis, using observed or GPV atmospheric-pressure data concerning the pressure profile of atmospheric-pressure waves or the recovery rate of atmospheric pressure in the ocean, without complicated calculation. The estimated values for both the wave height and wavelength of the long ocean waves showed good agreement with the corresponding computational data.
Third, numerical simulation was generated for the oscillation in the harbors of various shapes. The amplification factor at the head of the L-type harbor for the second mode increased, as its bending position was nearer to the harbor mouth. As the narrowed area of the I-type harbor was located nearer to the harbor mouth, the amplification factor at the head for the first mode decreased, while that for the second mode increased. The C-type harbor showed the amplification depending on the position with the phase difference between the waves coming through two mouths. When the I-type harbor has the seabed crest, the amplification factor at the head for the first mode was larger than that for the second mode, while the reverse was true, when the I-type harbor has the seabed trough. Although the oscillation in Urauchi Bay had the second mode specific to T-type harbors, where antinodes appeared at their two harbor heads, future work is required to make clear the reason why the damping processes were different between Urauchi Bay and the T-type harbor.
Finally, the disaster measures were discussed against meteotsunamis, generated to propagate toward the west coasts of Kyushu. The methods of real-time prediction for meteotsunami generation were proposed using the inverse analysis, as well as the simple prediction equations, after which both the structural and the nonstructural measures against meteotsunamis were summarized.
Sincere gratitude is extended to Satsumasendai City Office for providing the photographs shown as Figures 2 and 3. I also express my gratitude to Mr. Kosuke Fukita, Mr. Soichiro Hidaka, Mr. Taisuke Inoue, Mr. Hiroyuki Matsukawa, Mr. Shota Minami, Mr. Yuki Suetsugu, Mr. Taishi Toyofuku, and Dr. Kei Yamashita, who contributed to the numerical simulation, etc., when they were student members of our laboratory. I am grateful to the reviewer for suggestive comments on countermeasures against meteotsunamis. This work was supported by JSPS Grant-in-Aid for Scientific Research (C) Grant Number JP17K06585.