A Numerical Study of Dynamos in Spherical Shells with Conducting Boundaries

Summary

The problem of the generation of magnetic fields by convection in rotating spherical shells is considered in the case when the boundaries of the fluid shell exhibit a finite electrical conductivity. This problem is of geophysical interest because Lorentz forces acting in the boundaries provide a mechanical coupling that was not included in previous computations by Zhang & Busse (1988, 1989). The vanishing torques between fluid shell and boundaries determine the relative rotation between the three regions of the problem. But the finite conductivity does not seem to improve the numerical convergence for dynamo solutions.

INTRODUCTION

The mathematical difficulties in deriving solutions for growing magnetic fields in spherical geometries have long puzzled dynamo theoreticians. In contrast to the solutions of the kinematic dynamo problem found by Roberts (1970, 1972) and others in the case of periodic velocity fields in infinitely extended electrically conducting fluids, dynamo action often seems to disappear as soon as insulating boundaries are introduced. Motivated by this observation Bullard & Gubbins (1977) have investigated kinematic dynamos in a spherical domain of constant conductivity with insulating exterior for velocity fields with alternating signs as a function of radius. As expected the critical magnetic Reynolds number decreases significantly as the number of sign changes increases and the limit of a periodic velocity field is approached. When the radial velocity component does not change sign, a dynamo solution was not obtained unless an outer shellular region of finite conductivity was introduced.