Compressible convection in a rotating spherical shell. I - Anelastic equations. II - A linear anelastic model. III - Analytic model for compressible vorticity waves

Abstract

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.