A unified asymptotic theory of the anelastic approximation in geophysical gases and liquids

An asymptotic theory of the anelastic approximation is developed for fluids having arbitrary equations of state under two assumptions: weak compressibility and small Brunt-Väisälä frequency. We show that both Boussinesq approximation (BA) and anelastic approximation (AA) may be included in a unique quasi-incompressible approximation (QIA) already constructed by Durran for polytropic gases. The only difference between AA and BA is that, in the BA, the equations are with slowly varying coefficients, while in the AA the coefficients are fast varying. Applications are made to atmospheric air and to sea-water.