The pressure waves of the Tonga volcanic eruption reached Japan, 8,000 km away, approximately 7 hours after the eruption. Its speed was estimated as 310 m/s, which is interpreted as the average speed of sound for the lower atmosphere (Fig. 2). The pressure variation continued for several hours. The frequency analyses of pressure waves concerning the arrival time of each wave packet showed that it contained four-wave categories: one long period Lamb wave, two short period acoustic waves, and one long period gravity wave (Fig. 3), which have different wave celerity, a kind of apparent speed as a function of distance from the sound source to the sensor.
The earliest arriving component seen in Fig. 3 is the Lamb wave11. This pressure wave is trapped at approximately 20 km altitude from the ground surface and propagates at the sound speed averaged over the wave trapping region. The component has a wide frequency range that was extended below the detection limit of the infrasound sensors. The datasets of barometers showed that the peak pressure change was approximately 200 Pa, and the period of passing the half-wavelength was approximately 1000 s. It is known that such low-frequency Lamb waves barely attenuate and can easily travel a path of several thousand kilometers. Even the Lamb wave signals that propagate from the other side of the Earth, or those that encircle the Earth more than once, are later observed. The observed presence of the Lamb wave component strongly implies that the eruption mainly generated Lamb waves with periods as long as or even longer than 200 seconds, suggesting the vast magnitude of the explosion event.
The second and third arrivals seen in Fig. 3 are acoustic waves in 0.003–0.05 Hz. These waves arrived approximately 2,000 and 10,000 seconds after the first Lamb wave arrival. These acoustic waves are higher in frequency than the Lamb waves. They can propagate with a vertical velocity component under the vertical profile of sound speed, which depends on the air temperature at each local region along the path. Each second and third wave has a wave celerity of 290 and 230 m/s, respectively, calculated from the horizontal distances from the sound source to the sensor and arrival times of observations. If the wave originated from the same sound source, the third wave would have an exceptionally high attenuation in the high-frequency band above 1 Hz.
The fourth arrival seen in Fig. 3 is the gravity wave. Steady oscillations of approximately 10 minutes were observed following the sound waves at many sites (Fig. 1). The lower curve in Fig. 3b, the lowermost-frequency component of the differential pressure gauge, clearly shows the arrival of the gravity wave. According to the accompanying barometers, raw amplitudes of the oscillations exceed 20 Pa at many stations. Although the waveform of the changes slightly differs from station to station, the observation time began earlier in the south and later in the north, which is consistent with the idea that a wave packet propagated as a kind of internal gravity wave at approximately 200–220 m/s from the Tonga volcano.
Gravity waves excited Tsunami over the Pacific Ocean.
The gravity wave observed by KUT sensors could be the origin of the tsunami generated by the Tonga volcanic eruption through Proudman resonance12. It should be emphasized that this propagation speed is close to the typical tsunami speed in the Pacific Ocean. Both gravity and tsunami waves presumably travelled with almost the same speed of 200–220 m/s (Fig. 4). Moreover, the reported time (11:00 UTC) of sea-level fluctuations along the Japanese coastline was well correlated to the arrivals of the wave packet of the gravity wave. To understand the possible behaviour of gravity waves and tsunamis in the interaction process, we implemented illustrative two-dimensional numerical modelling based on the hydrodynamic equations describing the dynamic coupling of a compressible atmosphere with oceanic tsunamis below it (see Methods 2). The result shows that the explosive release of heat in the stratosphere excites all candidates of observed waves, i.e., Lamb waves that are trapped near the ground surface, acoustic gravity waves that propagate with “bouncing” between the ground and mesopause (~ 90 km), and gravity waves that have a vertically modal structure between the ground surface and mesopause and dispersedly propagate with a wide range of horizontal speeds, including those of tsunamis. Some acoustic waves bear large amplitudes into the thermosphere and possibly disturb the ionosphere13. Tsunamis are indeed excited resonantly by the gravity waves, despite that the amplitude of the corresponding gravity waves is much smaller than that of Lamb or acoustic waves. Notably, the gravity wave-tsunami resonance occurs over a relatively wide range of characteristic velocities. Close examination reveals that tsunamis are excited by gravity waves whose phase speed is comparable to tsunamis (~ 200 m/s). These waves propagate collectively as a “wave packet” with a group velocity that is slower than the phase speed of the tsunami. It is also notable in the time evolution (Supplementary Movie). The tsunamis amplify gradually within a region located a few hundred km from the volcano, and the amplitude scarcely changes afterward. This implies that the high tsunamis that struck the islands of Tonga within several kilometers require excitation mechanisms other than the gravity waves, such as pyroclastic flow and submarine landslides, whose effects should be considered elsewhere.
It has been observed that the sea level fluctuation increased along the coast of Japan around the beginning of the detected gravity wave packet.
It also implies that the observed coincidence of the arrivals of gravity waves and tsunamis should be interpreted not as the continuous resonance between the gravity waves and tsunamis but as their propagations in tandem after resonant amplification in the region relatively close to the volcano. This point remains verified by careful comparison between barometric and tsunami observations over the Pacific.
Two packets of acoustic wave
We have identified two kinds of ray paths for acoustic waves. One is refracted from the stratosphere; the other is refracted from the lower thermosphere. According to the three-dimensional ray-tracing calculations of infrasound emitted by a volcano eruption10, the celerity of some of the waves turning from the stratosphere is approximately 300 m/s, and those from the lower thermosphere are approximately 238 m/s. The parabolic equation method14 was applied to estimate the two-dimensional transmission loss of acoustic energy in the azimuth direction toward Japan (see Methods 1). Moreover, the sound waves turning from the lower thermosphere have a lower frequency than those from the stratosphere. These numerical results are consistent with the observation (Fig. 3b). In this calculation, the third wave is coming through the lower thermospheric path for the longer propagating distance. In these calculations, several bounces on the sea surface are needed to reach Japan, suggesting that the acoustic waves travelled back and forth between the atmosphere and the ocean surface multiple times.
What Tonga atmospheric waves taught us
We identified the wave propagation characteristics that are excited by global-scale volcanic eruptions. Four-wave packets are observed to be almost similar/coherent among the 25 sites in Japan categorized into Lamb waves, two paths of acoustic waves, and atmospheric gravity waves. Lamb wave and our simple equation give us the yield and lower estimation of the explosive energy of the Tonga eruption. The total energy is 46 ± 14 megatons (see Methods 3), one of the most significant energy we have ever encountered. Acoustic waves with several bounces on the sea surface are needed to reach Japan, suggesting that the infrasound travelled back and forth between the atmosphere and the ocean surface multiple times. These significant eruptions have allowed us to see the global vertical energy exchange in the atmosphere. Both waves give us great geoscientific suggestions. Lamb waves and acoustic waves are issues in the geoscience field, but the last gravity wave contains a more prominent theme. The gravity wave excited an unexpected tsunami in the Pacific Ocean. The velocity of the gravity waves strikingly well matched the velocity of tsunamis, resulting in the coupling of the gravity waves with the tsunami and the generation of an unpredictable tsunami. The tsunami was also excited even in the Caribbean Sea, separated by American continent. In other words, this tsunami could occur anywhere on earth. Thus, gravity waves are of interest not only for geoscience but also for disaster mitigation.