4:45 PM - 5:00 PM
[MIS21-06] A Study on the Coupling between Atmospheric Lamb waves and Oceanic Tsunamis
Keywords:tsunamis, atmospheric Lamb waves, Tonga Volcano eruption
Introduction
Just after 1:00 p.m. JST on January 15, 2022, the volcano Hunga-Tonga-Hunga-Ha'apai in Tonga erupted, and pressure and sea level fluctuations were observed for several days over the entire globe, including Japan. Of these, the pressure fluctuation can be basically interpreted as an atmospheric Lamb wave. The sea level change, on the other hand, was different from that of a normal tsunami in that it started to be observed in many places before the arrival time of the tsunami, which is expected when a normal tsunami is triggered by an eruption, and was also observed on the other side of the continent, such as the Caribbean Sea. In this paper, we consider this phenomenon as a coupled system of atmospheric Lamb waves and oceanic tsunamis and discuss theoretically.
The system considered
For simplicity, we assume that the field is non-rotating and the system is two-dimensional. The ocean tsunami is assumed to have a sufficiently long wavelength to be described by the shallow water equation. On the other hand, for the atmosphere, the same equations as the shallow water equations can be obtained by linearizing the compressible system of equations and integrating vertically. This has a solution that propagates at the speed of sound, similar to Lamb waves. The modal solution is obtained by coupling the pressure and vertical velocity, assuming that they are continuous at the sea surface. It is also solved as an initial value problem if necessary.
Results: For realistic parameters
For typical values of temperature and sea depth (288K and 4000m, respectively), two coupled modes with phase speeds similar to those of normal atmospheric Lamb waves and tsunamis are obtained. The sign relation between the atmospheric pressure anomaly and the sea level anomaly is the same for the mode with the speed of atmospheric Lamb wave, and the opposite for the mode with the speed of tsunami. The former sign relation explains why the observed sea level anomaly is positive in the case of the Tonga eruption.
Results: In the case of the full resonance parameter
If the temperature and water depth are chosen so that the velocity of atmospheric Lamb waves and that of tsunamis are equal (\gamma RT = gD), the two are expected to resonate, but this case also yields two coupled modes with slightly different phase speeds. When the initial value problem is solved under the same conditions with atmospheric disturbance is given, two disturbances propagating at different speeds are obtained. The initial increase of the tsunami amplitude in the case of perfect resonance can be interpreted by the Proudman resonance.
Note
On the atmospheric side, there are not only Lamb waves but also internal gravity waves and sound waves, which are coupled with the tsunami. Therefore, the present results are limited to the case where the velocity of the tsunami-like mode is close to that of the Lamb wave (both the internal gravity wave and the sound wave become evanescent). The effect of this point will be discussed at the conference presentation.
Just after 1:00 p.m. JST on January 15, 2022, the volcano Hunga-Tonga-Hunga-Ha'apai in Tonga erupted, and pressure and sea level fluctuations were observed for several days over the entire globe, including Japan. Of these, the pressure fluctuation can be basically interpreted as an atmospheric Lamb wave. The sea level change, on the other hand, was different from that of a normal tsunami in that it started to be observed in many places before the arrival time of the tsunami, which is expected when a normal tsunami is triggered by an eruption, and was also observed on the other side of the continent, such as the Caribbean Sea. In this paper, we consider this phenomenon as a coupled system of atmospheric Lamb waves and oceanic tsunamis and discuss theoretically.
The system considered
For simplicity, we assume that the field is non-rotating and the system is two-dimensional. The ocean tsunami is assumed to have a sufficiently long wavelength to be described by the shallow water equation. On the other hand, for the atmosphere, the same equations as the shallow water equations can be obtained by linearizing the compressible system of equations and integrating vertically. This has a solution that propagates at the speed of sound, similar to Lamb waves. The modal solution is obtained by coupling the pressure and vertical velocity, assuming that they are continuous at the sea surface. It is also solved as an initial value problem if necessary.
Results: For realistic parameters
For typical values of temperature and sea depth (288K and 4000m, respectively), two coupled modes with phase speeds similar to those of normal atmospheric Lamb waves and tsunamis are obtained. The sign relation between the atmospheric pressure anomaly and the sea level anomaly is the same for the mode with the speed of atmospheric Lamb wave, and the opposite for the mode with the speed of tsunami. The former sign relation explains why the observed sea level anomaly is positive in the case of the Tonga eruption.
Results: In the case of the full resonance parameter
If the temperature and water depth are chosen so that the velocity of atmospheric Lamb waves and that of tsunamis are equal (\gamma RT = gD), the two are expected to resonate, but this case also yields two coupled modes with slightly different phase speeds. When the initial value problem is solved under the same conditions with atmospheric disturbance is given, two disturbances propagating at different speeds are obtained. The initial increase of the tsunami amplitude in the case of perfect resonance can be interpreted by the Proudman resonance.
Note
On the atmospheric side, there are not only Lamb waves but also internal gravity waves and sound waves, which are coupled with the tsunami. Therefore, the present results are limited to the case where the velocity of the tsunami-like mode is close to that of the Lamb wave (both the internal gravity wave and the sound wave become evanescent). The effect of this point will be discussed at the conference presentation.