Low-Frequency Internal Gravity Waves are Pseudo-incompressible

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.