Non-linear marginal convection in a rotating magnetic system

Abstract

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 bclow and cooling from above the system becomes unstable to small perturbations when the adverse density gradient as measured by the Rayleigh number, R, is suficiently 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,, based on U is small of order q and on exterior region in which R, is of order unity.

Dynamically consistent magnetic fields produced by differential rotation

“…The next stage is to take into account the mean Lorentz forces due to fluctuating fields. This remains a much more demanding task, and while Soward (1986) has made some progress recently in a plane layer model, there are to date no self-consistent solutions in a spherical geometry, nor have any mean magnetic fields been investigated that arise (as in FP87) from the dynamo process itself. Such an extension remains a challenging topic for future work.…”

Dynamo Theory

“…Two books have been written that are devoted entirely to the subject (Moffatt 1978, Krause & Radler 1980, and several others include substantial segments describing it (e.g. Ghil , Jacobs 1987, Parker 1979, Roberts & Soward 1978, Soward 1983. In addition, a few review articles have appeared, some of which we shall refer to below.…”

Magnetoconvection Patterns in Rotating Convection Zones

“…In defence of this step, now clearly seen to have been too optimistic, we should point to the studies of the linear stability by Eltayeb and Roberts (1970) and Eltayeb (1972Eltayeb ( , 1975, which gave results qualitatively insensitive to the relative orientations of g, fi and Bo, and which initially encouraged us to believe that geometrical considerations would not be crucial. The discovery of the anti-Taylor inequalities for the present situation, was already a foretaste of an impending disappointment, for it demonstrates that our model behaves at finite amplitude quite differently from the plane layer Boussinesq model in which rotation is parallel to gravity, a configuration extensively studied by Roberts andStewartson (1974, 1975), Soward (1980Soward ( , 1986 and Skinner and Soward (1989), and in which Taylor states occur.…”