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The full, non4inear equations governing thermal convection in a compressible fluid have been re-examined in order to determine the conditions under which the Boussinesq approximation is applicable These conditions are (a) the vertical dimension of the fluid is much less than any scale height, and (b) the motion-induced fluctuations in density and pressure do not exceed, in order of magnitude, the total static variations of these quantities. Under these conditions the equations are formally equivalent to those for an incompressible system when the temperature gradient is replaced by its excess over the adiabatic and Cp replaces C .
Spiegel et al. (Tue,) studied this question.