Low-Frequency Internal Gravity Waves are Pseudo-incompressible

TL;DR

This paper demonstrates that low-frequency internal gravity waves are inherently pseudo-incompressible and shows that common approximations like the anelastic formulation fail to accurately describe these waves in stable stratification.

Contribution

The study clarifies the conditions under which the pseudo-incompressible approximation accurately models internal gravity waves, highlighting its superiority over the anelastic approximation in low-frequency regimes.

Findings

Internal gravity waves are naturally pseudo-incompressible at low frequencies.

The anelastic approximation poorly reproduces the behavior of these waves in stable stratification.

Pseudo-incompressible approximation remains valid in stable stratification, unlike the anelastic approximation.

Abstract

Starting from the fully compressible fluid equations in a plane-parallel atmosphere, we demonstrate that linear internal gravity waves are naturally pseudo-incompressible in the limit that the wave frequency $\omega$ is much less than that of surface gravity waves, i.e., $\omega \ll \sqrt{g k_h}$ where $g$ is the gravitational acceleration and $k_h$ is the horizontal wavenumber. We accomplish this by performing a formal expansion of the wave functions and the local dispersion relation in terms of a dimensionless frequency $\varepsilon = \omega / \sqrt{g k_h}$. Further, we show that in this same low-frequency limit, several forms of the anelastic approximation, including the Lantz-Braginsky-Roberts (LBR) formulation, poorly reproduce the correct behavior of internal gravity waves. The pseudo-incompressible approximation is achieved by assuming that Eulerian fluctuations of the pressure are small in the continuity equation. Whereas, in the anelastic approximation Eulerian density fluctuations are ignored. In an adiabatic stratification, such as occurs in a convection zone, the two approximations become identical. But, in a stable stratification, the differences between the two approximations are stark and only the pseudo-incompressible approximation remains valid.