Numerical solutions of the kinematic dynamo problem

Abstract

The expansion method of Bullard & Gellman is used to find numerical solutions of the induction equation in a sphere of conducting fluid. Modifications are made to the numerical methods, and one change due to G. O. Roberts greatly increases the efficiency of the scheme. Calculations performed recently by Lilley are re-examined. His solutions, which appeared to be convergent, are shown to diverge when a higher level of truncation is used. Other similar dynamo models are investigated and it is found that these also do not provide satisfactory steady solutions for the magnetic field. Axially symmetric motions which depend on spherical harmonics of degree n are examined. Growing solutions, varying with longitude, 0, as e1^, are found for the magnetic field, and numerical convergence of the solutions is established. The field is predominantly an equatorial dipole with a toroidal field symmetric about the same axis. When n is large the problem lends itself to a two-scale analysis. Comparisons are made between the approximate results of the two-scale method and the numerical results. There is agreement whennis large. When n is small the efficiency of the dynamo is lowered. It is shown that the dominant effect of a large microscale magnetic Reynolds number is the expulsion of magnetic flux by eddies to give a rope-like structure for part of the field. Physical interpretations are given which explain the dynamo action of these motions, and of related flows which support rotating magnetic fields.