INTRODUCTION

In this chapter I attempt a review of theories of convection in a spherical geometry in the presence of magnetic fields and rotation. The understanding of such motion is essential to a proper theory of the geodynamo. Even though, as discussed by Malkus and Braginsky (chapters 5 and 9), the nature of the driving mechanism for the convection is not certain, and is likely to be compositional in nature, we shall generally, following other authors, look only at thermal convection, which is the simplest to study. In addition, it will be assumed (incorrectly) that the core fluid has essentially constant viscosity, density, etc., allowing the Boussinesq approximation to be employed.

The excuse for these simplifications is readily to hand: the dynamical complexities induced by the interaction of Coriolis and Lorentz forces are still not fully resolved, and transcend the details of the forcing or of compressibility effects. The effects of this interaction on global fields are discussed by Fearn (chapter 7) but here we shall confine ourselves to a small part of the complete picture: the non-axisymmetric instabilities of an imposed (and prescribed) axisymmetric magnetic field and differential rotation in a rotating sphere. This task is the mirror-image of the ‘intermediate’ models of Braginsky (chapter 9) and the non-linear ‘macrodynamic’ dynamos driven by the a-effect, described by Fearn (chapter 7), in that these works parametrize the small, rather than the global fields.

In what follows, we shall begin by defining a geometry and non-dimensionalization for the system. We shall mainly be working in a spherical geometry, but use for illustration simplified (e.g., planar, cylindrical) geometry where appropriate.