Hydromagnetic convective instability of a rotating, self-gravitating fluid sphere containing a uniform distribution of heat sources

Abstract

The desire to understand better the magneto-hydrodynamics of the Earth's and planetary interiors has recently motivated a number of studies on convective motions in hydromagnetic rotating systems. These studies have, however, been restricted to planar geometry, the convective layer being confined between two horizontal planes in externally applied uniform gravitational and magnetic fields. This paper takes a step further to the geophysical and astrophysical contexts by restoring curvature effects. The linear stability of a uniformly rotating, self-gravitating fluid sphere in the presence of a co-rotating zonal magnetic field is studied when buoyancy is provided by a uniform distribution of heat sources. The analysis is limited to the case where the Chandrasekhar number, Q , and the Taylor number, λ 2 , are both large. (These are, respectively, dimension-less measures of Lorentz and Coriolis forces relative to the viscous forces.) It is shown that for all values of λ and Q the motions appearing at marginal convection are necessarily time-dependent and associated with a temperature fluctuation which is always symmetric with respect to the equatorial plane. The critical Rayleigh number R c ( λ, Q ), which is a dimensionless measure of the temperature contrast necessary for the onset of convection, is found to be qualitatively the same as for the planar model only when λ ≥ O(Q) , although even in this case certain characteristic curvature effects arise. The motions prevalent at marginal stability, when O ( Q 3/2 ) ≥ λ ≫ Q , occur in the form of a thin cylindrical shell of thickness O (( Q/λ ) 2/3 ) and whose distance from the axis of rotation varies between 0.4 and 0.6 spherical radii depending on the value of q , which is the ratio of the thermal to the magnetic diffusivities. The waves will drift westward or eastward according to whether q ≶ 2.5. (The cause of disagreement in this result with Busse (1975 b ) is explained in an appendix). For λ = O(Q) convection occurs in the whole volume of the sphere and the waves drift westward for all values of q . When λ ≪ Q , not only is R c incorrectly given by that for the plane layer model but also modal degeneracies of convection in the plane layer are removed by the curvature and boundedness of the system. For this range of λ and Q convection again fills the whole sphere but all forms of diffusion are concentrated in multiple boundary layers on the surface of sphere. The waves drift westward. The results are compared with parellel studies, including Braginsky's MAC waves (i. e. Hide's slow magnetohydrodynamic waves) and Busse's recent dynamo model. In particular, it is argued that the last of these may not be representative of planetary magnetism because of a convective growth of field (not considered by Busse) associated with convection patterns occuring in the whole sphere rather than in a cylindrical shell.