Non-Linear Marginal Convection in a Rotating Magnetic System

An inviscid, electrically conducting fluid is contained between two rigid horizontal planes and bounded laterally by two vertical walls. The fluid is permeated by a strong uniform horizontal magnetic field aligned with the side wall boundaries and the entire system rotates rapidly about a vertical axis. The ratio of the magnitudes of the Lorentz and Coriolis forces is characterized by the Elsasser number, A, and the ratio of the thermal and magnetic diffusivities, q. By heating the fluid from below and cooling from above the system becomes unstable to small perturbations when the adverse density gradient as measured by the Rayleigh number, R, is sufficiently large. With the viscosity ignored the geostrophic velocity, U, which is aligned with the applied magnetic field, is independent of the coordinate parallel to the rotation axis but is an arbitrary function of the horizontal cross-stream coordinate. At the onset of instability the value of U taken ensures that Taylor's condition is met. Specifically the Lorentz force, which results from marginal convection must not cause any acceleration of the geostrophic flow. It is found that the critical Rayleigh number characterising the onset of instability is generally close to the corresponding value for the usual linear problem, in which Taylor's condition is ignored and U is chosen to vanish. Significant differences can occur when q is small owing to a complicated flow structure. There is a central interior region in which the local magnetic Reynolds number, R(m), based on U is small of order q and on exterior region in which R(m) is of order unity.