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 MUCH LESS-THANgkh , 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 epsilon=omega/gkh . Further, we show that, in this same low-frequency limit, several forms of the anelastic approximation, including the Lantz-Braginsky-Roberts 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. However, in a stable stratification, the differences between the two approximations are stark and only the pseudo-incompressible approximation remains valid.