3.1. Estimation of air-pressure jump speed and direction
Analysis of the observed air-pressure signals demonstrated that the pressure jump propagated southward with a constant speed and direction while maintaining a similar magnitude (see Sect. 2.2). Here, we quantitatively infer the speed and direction of the pressure jump propagation based on the meteorological observation. Figure 6 shows the radar images (from KMA) of the rain rate at 23:00, 23:20, 23:40 and 24:00 of June 11th. These observations relate the pressure jump with the first rain precipitation, which is marked as a dashed line in Fig. 6. We can observe that a strong rain precipitation followed after the pressure jump, but this rain activity seems not to be directly linked to the pressure jump.
To estimate the speed and direction, we introduce some assumptions and perform numerical tests based on the air-pressure data. For the sake of simplicity, we assume that the pressure jump is aligned in a straight line and the moving direction is perpendicular to the alignment. Moreover, we assume that the speed and direction of pressure jump are constant, as demonstrated in Sect. 2.2. With these assumptions, the pressure jump directional angle, defined as the counter-clockwise angle with 0° toward east (see Fig. 1), is estimated between 275° and 295° based on the radar images of Fig. 6. On the other hand, the estimation of the propagation speed of pressure jump from radar images (Fig. 6) was not possible because of the low resolution and the uncertainty on the exact location of the jump in these images. To overcome this limitation, numerical tests allowing a more precise determination of the speed and direction are performed using the pressure records at KRB, ECD, MD and HSD observatories as input (see Fig. 1 for location). We interpolated the atmospheric pressure, and compared the observation at AH, OYD, KS and JH with the numerical simulations in Fig. 7. Based on these tests, we estimate the speed between 11 m/s and 13 m/s and the directional angle between 275° and 285°.
3.2. Resonance effects
From the previous section, the atmospheric pressure records support that the moving direction of the pressure jump was in the range of 275° and 285° and the moving speed was in the range of 11 m/s and 13 m/s. In the study area of the eastern Yellow Sea, the optimal moving speed of atmospheric disturbances leading to Proudman resonance and the occurrence of meteotsunami is estimated to 26–29 m/s (Choi et al. 2014). Therefore, the meteotsunami long-waves recorded on 11–12 June 2009 could not be associated with the Proudman resonance as our estimated moving speed of air-pressure jump (11–13 m/s) is much smaller than the optimal condition for such a resonance. During the previously known meteotsunami events in this region, the moving direction of atmospheric disturbances was toward the coast. However, on June 11–12, 2009, the atmospheric disturbances moved along the coast, i.e.,the moving direction was smaller than 290°, which is a favorable condition for edge waves to propagate and to be amplified by Greenspan resonance.
In order to explain the meteotsunami amplification, we turn to the Greenspan resonance which can intensify the propagating edge waves along the coast. Greenspan resonance states the relation of wave frequency, wave mode and the sea bed slope as follows,
$${C}_{edge}=gTtan\left[\left(2n+1\right)\beta \right]/2\pi$$.
When the speed of the atmospheric jump is close to the speed of one of the edge wave modes, the Greenspan resonance is expected. The sea bed from KRB to MD has a gentle slope (β) of approximated 0.0008 (Choi et al. 2014), and the resonant wave period (T) of 147–173 min corresponds to the speed (Cedge) 11–13 m/s of the fundamental mode (n = 0). Since both the trapped wave speed and the moving speed of air-pressure disturbance are in the same range, the condition for the Greenspan resonance is satisfied.
Furthermore, each period corresponds to the mode of the Greenspan resonance periods which are 160, 53, and 32 min for n = 0, 1 and 2, respectively, for the speed 12 m/s. Figure 8 shows the Fourier spectrum of sea-level anomaly data observed at BR, JH, KS and YG tide observation stations. The dominating periods, such as 144 − 120, 72 − 48, and 30 min, of the sea-level fluctuations (Figs. 2c and 8) are in good agreement with the analytically calculated periods of the Greenspan resonance.
3.3. Numerical simulations of June 2009 meteotsunami
The GeoClaw numerical code (LeVeque et al. 2011), a finite volume method solver of the nonlinear shallow water equations, is used to simulate the propagation of the tsunami-like waves in the Yellow Sea. To account for the air-pressure jump as a trigger and driver of meteotsunami propagation, the GeoClaw code was equipped with atmospheric pressure forcing terms following the governing equations presented in Kim and Omira (2021).
The atmospheric pressure records at KRB, ECD, MD and HSD observatories are used as the model's input to force the sea surface. In other coastal areas, with no available atmospheric data, these records were interpolated to estimate the air-pressure disturbances. Details on the interpolation process can be found in Kim and Omira (2021). The propagation of the meteotsunami waves is then simulated over a uniform bathymetric grid spacing of dx = dy = 20” (~ 500 m) (Fig. 1).
Numerical simulation results are presented for air-pressure disturbances propagating at a speed of 12 m/s and with a directional angle of 280°, both selected over the estimated range of 11–13 m/s and 275–285° (Sect. 3.1). Figure 9 depicts the snapshots of modelled water surface elevation and atmospheric pressure, and it is clear that the atmospheric disturbances (right panels in Fig. 9) travel southward, passing over Korean coastal areas of interest with almost unchangeable magnitude of the pressure jump. These results are in good agreement with the observed air-pressure signals at most Korean coast stations.
Numerical simulations show that the moving air-pressure disturbances guide the meteotsunami propagation, and waves reaching 0.2 m in height are estimated along the Korean coast. The results also allow observing the delayed arrival of the meteotsunami waves with respect to the atmospheric disturbances, as the propagation of the sea long-waves is highly affected by the shallower depths of the Yellow Sea bathymetry. As the waves reach near the continental shelf, edge waves are observed in Fig. 10 (d)-(g), and an effect of the nearshore seafloor topography concerns the alteration in the propagation direction of the incident meteotsunami waves.
The observed sea-level records at six tidal gauge stations are compared with numerical model results (Fig. 11). The numerical model results are in good agreement with the observation. Thus, this numerical simulation also has the capability to capture the Greenspan resonance in the eastern Yellow Sea.
At ECD, the observed and simulated water surface elevation is small because the island ECD is located far from the coastline of the Korean Peninsula (Fig. 1) and the resonance was not fully developed. Moreover, the tidal gauge is inside the harbor which is facing southward. Since the meteotsunami waves propagate from the northwest to southeast, the observed water surface elevation at ECD is relatively small for this event.
Sea-level observation stations at KS and JH face each other across a river estuary and the distance between the two gauges is relatively small (11.73 km). The sea-level oscillation results from numerical simulations fairly reproduce the observations at both tidal gauges. We notice that other large long-waves arrived three hours apart after the passage of the large jump in the atmospheric pressure over this area. The cause of the second and the third large waves, which are well reproduced in the numerical simulation, was partly explained by the Greenspan resonance, but it still needs to be explored in future study including the wave reflection by the nearby islands.