Oblate Spheroidal Kinematic Dynamos

D. J. Ivers

Abstract

The kinematic dynamo problem is solved for a uniformly electrically-conducting fluid filling an oblate spheroidal volume with insulating exterior. The solution method uses a class of oblate spheroidal toroidal-poloidal fields in a non-orthogonal coordinate system. Scaling of coordinates and fields to a spherical geometry leads to a modified form of the kinematic dynamo problem with a geometric anisotropic diffusion and a modified current-free condition in the exterior, which must be solved explicitly. The scaling allows the use of well-developed spherical harmonic techniques in angle and large aspect ratios. Dynamo solutions are found for three axisymmetric flows in oblate spheroids with semi-axis ratios \(1 \leqslant a/c \leqslant 25\). For larger aspect ratios strong magnetic fields may occur in any region of the spheroid, depending on the flow, but the external fields for all three flows are weak and concentrated near the axis or periphery of the spheroid.

Keywords: magnetohydrodynamics, dynamo theory, oblate spheroid, axisymmetric flow.

AMS Subject Classification: Primary 85A30,76W05,86A25,65Z05.