Compressible convection in the presence of rotation and a magnetic field

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Abstract
The onset of convection in a polytropic atmosphere with rotation and magnetic field is considered in a geometry such that rotation, magnetic field and gravity are mutually perpendicular. The anelastic approximation is used together with the low diffusion magnetogeostrophic approximation. The factors determining the pattern of convection are discussed, with particular reference to the question of whether convection rolls are primarily aligned with the rotation axis or the magnetic field. As in the Boussinesq problem, the global Elsasser number ℬ2/2μρΩη plays a key role. It is shown that the onset of convection in this system cannot be steady, but must occur in the form of travelling waves. In compressible convection the pressure fluctuations contribute to the buoyancy force as well as the temperature fluctuations: in rotating systems these pressure fluctuations are out-of-phase with the temperature fluctuations, and this phase difference gives rise to the travelling waves. Growth rates and frequencies have been computed for a range of models. The relevance of these results to the pattern of large scale convection in the Sun is considered, in the light of recent observations suggesting that toroidal rolls may be an important element in the dynamics of the convection zone. The question of whether a “banana cell” motion, selected when the Proudman—Taylor theorem applies, dominates over axisymmetric toroidal rolls is shown to involve the effective value of the turbulent diffusivity.