4:30 PM - 4:45 PM
[MTT45-05] Theoretical considerations on the atmospheric and oceanic waves excited by the Tonga volcano
★Invited Papers
Keywords:Tonga Volcano, Lamb waves, Tsunamis, meteotsunamis, internal gravity waves, acoustic gravity waves
Introduction
Just after 1:00 p.m. (JST) on January 15, 2022, Hunga-Tonga-Hunga-Ha'apai volcano in Tonga erupted, and pressure and sea-level changes were observed for several days around the globe, including Japan. In particular, the sea level change started to be observed in many areas before the arrival time of the tsunami, which is expected in the case of a normal eruption, and was also observed on the other side of the continent such as the Caribbean Sea.
In this study, we survey a wide range of parameters using a linear model to theorize how this atmospheric and oceanic disturbance could be excited by the eruption.
Basic equations
We consider the linear response of the compressible atmosphere and the ocean tsunami coupled with it to the forcing by the volcanic eruption. For simplicity, the system is assumed to be two-dimensional (x, t) horizontally and vertically, and the rotation of the earth and spherical effects are neglected. We assume that the tsunami can be described by the shallow water equation. The atmospheric equation and the tsunami equation are solved numerically at the sea level (z=0) as an initial value problem, coupled by giving the vertical direct current as the boundary condition from the ocean to the lower end of the atmosphere and the pressure disturbance as the forcing from the lower end of the atmosphere to the ocean. The computational domain is 3200 km or 12800 km horizontally and 480 km vertically (with the exception of the sponge layer above 400 km).
Initial results
The results of localized thermal forcing at various spatio-temporal scales are insensitive to parameters if the forcing is short enough and the spatial scale is small enough, and can be summarized as follows.
Excitation characteristics of atmospheric waves
Localized thermal forcing with a wide range of parameters excites Lamb waves and propagates them far away. When the fundamental field is not isothermal, Lamb waves are dispersive up to a pulse width of about 10 km, after which they behave almost non-dispersively. The altitude dependence of the amplitude and pulse width is also weak. On the other hand, the characteristics of internal gravity waves (vertical wavelength, period, etc.) are very sensitive to the height of thermal forcing. In particular, when the forcing height is higher than about 20 km, internal gravity waves with long vertical wavelengths and gravity sound waves are excited with large amplitudes. The former may resonate with tsunamis (see below), and the latter may excite the oscillations of the solid earth.
Tsunami excitation
In a wide range of forcing parameters, tsunamis propagating with Lamb waves are excited. Barometric pressure anomaly and sea level displacement have the same sign, which is opposite to the static response such as storm surge. On the other hand, the internal gravity wave, which is generated when the forcing height is high, resonantly excites tsunamis because it has a wide range of phase velocities covering the typical tsunami velocity (about 200 m/s at 4000 m depth). As a result, the ratio of the tsunami amplitude to the surface pressure amplitude is much larger than that of the tsunami excited by the Lamb wave. Bearing in mind the dispersive nature of internal gravity waves, the sea level change propagated as a "normal tsunami" after the Lamb wave may have been excited near the volcano by internal gravity waves.
Resonance between tsunami and Lamb wave
Assuming that the atmospheric basic field is isothermal (temperature T) and the water depth D is γRT/g, the Lamb wave and the tsunami resonate perfectly. However, when the actual calculation is carried out under this condition, two modes with different propagation velocities appear, and it is found that the excitation amplitude of the tsunami has an upper limit. This characteristic can be reproduced by vertically integrating the equation on the atmospheric side, constructing the equation in the same phase as the shallow water equation, and then combining it with the equation on the ocean side for modal analysis. The upper limit of the excitation amplitude can be estimated approximately even when the water depth does not satisfy the resonance condition with Lamb waves, including realistic cases.
Future development
Combining the results of this study with various aspects of observed waves, we may be able to estimate the eruptive behavior (e.g., distribution of heat from magma to atmospheric heating and seawater evaporation). Details including the response to momentum and mass forcing will be presented at the comference.
Just after 1:00 p.m. (JST) on January 15, 2022, Hunga-Tonga-Hunga-Ha'apai volcano in Tonga erupted, and pressure and sea-level changes were observed for several days around the globe, including Japan. In particular, the sea level change started to be observed in many areas before the arrival time of the tsunami, which is expected in the case of a normal eruption, and was also observed on the other side of the continent such as the Caribbean Sea.
In this study, we survey a wide range of parameters using a linear model to theorize how this atmospheric and oceanic disturbance could be excited by the eruption.
Basic equations
We consider the linear response of the compressible atmosphere and the ocean tsunami coupled with it to the forcing by the volcanic eruption. For simplicity, the system is assumed to be two-dimensional (x, t) horizontally and vertically, and the rotation of the earth and spherical effects are neglected. We assume that the tsunami can be described by the shallow water equation. The atmospheric equation and the tsunami equation are solved numerically at the sea level (z=0) as an initial value problem, coupled by giving the vertical direct current as the boundary condition from the ocean to the lower end of the atmosphere and the pressure disturbance as the forcing from the lower end of the atmosphere to the ocean. The computational domain is 3200 km or 12800 km horizontally and 480 km vertically (with the exception of the sponge layer above 400 km).
Initial results
The results of localized thermal forcing at various spatio-temporal scales are insensitive to parameters if the forcing is short enough and the spatial scale is small enough, and can be summarized as follows.
Excitation characteristics of atmospheric waves
Localized thermal forcing with a wide range of parameters excites Lamb waves and propagates them far away. When the fundamental field is not isothermal, Lamb waves are dispersive up to a pulse width of about 10 km, after which they behave almost non-dispersively. The altitude dependence of the amplitude and pulse width is also weak. On the other hand, the characteristics of internal gravity waves (vertical wavelength, period, etc.) are very sensitive to the height of thermal forcing. In particular, when the forcing height is higher than about 20 km, internal gravity waves with long vertical wavelengths and gravity sound waves are excited with large amplitudes. The former may resonate with tsunamis (see below), and the latter may excite the oscillations of the solid earth.
Tsunami excitation
In a wide range of forcing parameters, tsunamis propagating with Lamb waves are excited. Barometric pressure anomaly and sea level displacement have the same sign, which is opposite to the static response such as storm surge. On the other hand, the internal gravity wave, which is generated when the forcing height is high, resonantly excites tsunamis because it has a wide range of phase velocities covering the typical tsunami velocity (about 200 m/s at 4000 m depth). As a result, the ratio of the tsunami amplitude to the surface pressure amplitude is much larger than that of the tsunami excited by the Lamb wave. Bearing in mind the dispersive nature of internal gravity waves, the sea level change propagated as a "normal tsunami" after the Lamb wave may have been excited near the volcano by internal gravity waves.
Resonance between tsunami and Lamb wave
Assuming that the atmospheric basic field is isothermal (temperature T) and the water depth D is γRT/g, the Lamb wave and the tsunami resonate perfectly. However, when the actual calculation is carried out under this condition, two modes with different propagation velocities appear, and it is found that the excitation amplitude of the tsunami has an upper limit. This characteristic can be reproduced by vertically integrating the equation on the atmospheric side, constructing the equation in the same phase as the shallow water equation, and then combining it with the equation on the ocean side for modal analysis. The upper limit of the excitation amplitude can be estimated approximately even when the water depth does not satisfy the resonance condition with Lamb waves, including realistic cases.
Future development
Combining the results of this study with various aspects of observed waves, we may be able to estimate the eruptive behavior (e.g., distribution of heat from magma to atmospheric heating and seawater evaporation). Details including the response to momentum and mass forcing will be presented at the comference.