MAGNETOCONVECTION IN RAPIDLY ROTATING BOUSSINESQ AND COMPRESSIBLE FLUIDS

Finite amplitude, convection is considered in a plane fluid layer rotating with angular velocity ω about a horizontal axis in the y-direction. The fluid is electrically conducting, and a uniform horizontal magnetic field of strength B0 is applied horizontally, in the x-direction, perpendicular to the rotation axis. The gravitational acceleration, g is parallel to Oz. The layer is infinite in x-extent but is bounded laterally in the y-direction by vertical walls, which may be either stress-free or rigid, i.e. prevent fluid motion. The model is intended to simulate crudely conditions near the equator of the solar convection zone. At the onset of convection in a stagnant layer, a Lorentz force, F, is created in the x-direction which, though weak, has a nonzero horizontal average that cannot be balanced by any other forces in the equation of motion. This force sets up a geostrophic flow, U, in the x-direction, i.e. a flow that depends on z alone. This flow affects F, and in a so-called ‘Taylor State” obliterates F. Such Taylor states are here shown to occur at small Elsasser numbers, [formula omitted], where ρ is density, σ is the electrical conductivity. At larger values of A, the amplitude of the geostrophic flow is determined by viscous friction, either at the vertical walls of the duct when these are rigid, or by viscous friction throughout the bulk of the fluid when the vertical walls are stress-free. Numerical solutions are presented for both these cases for several values of the Rayleigh number, R=gα2βd2/2ωk, where a is the coefficient of volume expansion, β is the applied adverse temperature gradient, and k is the thermal diffusivity. A symmetry-breaking bifurcation is located at large R in the case of stress-free side walls and sufficiently large A. For smaller R the solutions are steady and symmetric about the horizontal mid-level; for larger R they progress in opposite x-directions as horizontal waves hugging one horizontal boundary or the other. All these results were obtained for a Boussinesq fluid, but it is shown that a compressible layer exhibits analogous behavior. As demonstrated by Jones et al. (1990), convection is necessarily unsteady and unsymmetric in this case but, as R is increased at sufficiently large A, solutions develop that also progress in opposite x-directions as horizontal waves hugging one horizontal boundary or the other. The unfolding of the symmetry-breaking bifurcation of the Boussinesq model as the compressibility is increased from zero is elucidated. The final summary of the paper includes some astrophysical speculations about the possible bearing of this work on the regime of convection in the outer regions of the Sun. © 1990, Taylor & Francis Group, LLC. All rights reserved.