On the Nature of Potential Energy in Atmospheric Gravity Waves, or Why the Atmosphere Cannot Sustain a Tsunami

Motivated by the generation of exceptionally large gravito-elastic waves during the Hunga Tonga–Hunga Ha’apai explosion of 15 January 2022, we examine theoretically the nature of the main air wave branch \(GR_0,\) $G{R}_0,$ whose undispersed celerity, \(\sim 313\) $\sim 313$  m/s, suggests that it may represent a “tsunami” of the atmospheric column for an effective thickness \(H_{eff}\approx 10\) $H_{\mathit{eff}}\approx 10$  km. However, we find that its potential energy is about 90% elastic across a wide frequency band, thus negating the widely held perception that it constitutes an oscillation between kinetic and gravitational energy. Based on the systematic study of the effect of finite compressibility on the dispersion and potential energy of a classic oceanic tsunami, we confirm that this feature of the branch \(GR_0\) $G{R}_0$ stems from the similarity between its celerity and the average speed of sound in the atmosphere. We then show that this similarity is not fortuitous, but rather expected for a perfect gas, which, we conclude, cannot sustain “tsunamis”, i.e., oscillations between kinetic and gravitational energy.