In order to develop equations whose solution will clarify the role played by large solar convection zone density variations in differential rotation transports, anelastic equations for convection of a compressible fluid in a deep, rotating spherical shell are derived in the first part of the study. The model equations represent a generalization of a Boussinesq system that has been studied extensively with the solar differential rotation problem in mind, and are expected to apply best in the deep part of a convection zone where departures of the fluid from an adiabatic atmosphere are smallest. The second part of the study focuses on the onset of convection for a compressible fluid in a rotating spherical shell via linear inelastic fluid equations for a depth of 40% of the radius, constant kinematic viscosity and thermometric diffusivity, Taylor numbers up to 100,000, and density stratifications up to seven e-folds across the zone. The perturbations are expanded in spherical harmonics, and the radially dependent equations are solved with a Newton-Raphson relaxation method.
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Gilman et al. (1981) studied this question.