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The approximate equations of motion derived by Batchelor in 1953 are derived by a formal scale analysis, with the assumption that the percentage range in potential temperature is small and that the time scale is set by the Brunt-Väisälä frequency. Acoustic waves are then absent. If the vertical scale is small compared to the depth of an adiabatic atmosphere, the system reduces to the (non-viscous) Boussinesq equations. The computation of the saturation vapor pressure for deep convection is complicated by the important effect of the dynamic pressure on the temperature. For shallow convection this effect is not important, and a simple set of reversible equations is derived.
Ogura et al. (Thu,) studied this question.