NONLINEAR MAGNETOCONVECTION IN A RAPIDLY ROTATING SPHERE AND TAYLORS CONSTRAINT

We investigate thermally driven convection in a rapidly rotating sphere in the presence of a prescribed azimuthal magnetic field B1(phi). Earlier work has looked at the linear problem. Here, we include the most important nonlinear effect; the geostrophic flow V-G(s)1(phi). This is determined through the standard condition that leads to Taylor's (1963) constraint in the limit of vanishing viscosity. The present work therefore follows on from earlier work on both kinematic alpha(2)- and alpha omega-dynamos and magnetoconvection. Examples of the latter have so far been restricted to plane-layer, duct and cylindrical geometries. The present work uses a spherical geometry and makes a further step towards physical realism in that the contributions from both the axisymmetric and non-axisymmetric components of the magnetic field to the Taylor integral are included. (The earlier magnetoconvection work only included the non-axisymmetric contributions while the kinematic dynamo calculations involved only the axisymmetric contributions). The problem is solved by integrating the governing partial differential equations forward in time. Ekman states [where the amplitude of the non-axisymmetric convection is controlled by the Ekman boundary layer and is O(E(1/4)), where E is the Ekman number] are found for values of the modified Rayleigh number (Ra) over tilde both above and (in at least one example) slightly below the critical value (Ra) over tilde(c) (in the absence of any differential rotation). The latter behaviour implies that the convective instability can be subcritical and this can be understood on the basis of the linear result that, for small amplitudes, a differential rotation can act to decrease (Ra) over tilde(c). (The reasons for this and the conditions under which this happens are not yet well understood, but are currently under investigation.) Two further main features have emerged from our calculations: the non-axisymmetric contribution to the Taylor integral typically dominates the axisymmetric contribution, and a complicated time-dependent behaviour emerges as the Rayleigh number is increased. The latter has so far prevented us from finding Taylor solutions.