Subharmonic Dynamo Action in the Roberts Flow

Abstract

The paper deals with the dynamo action of the Roberts flow, that is, a flow depending periodically on two cartesian coordinates, X and Y , but being independent of the third one, Z . In particular the case is considered in which the magnetic fields, which are periodic in X, Y and Z , have period lengths in the XY -plane being integer multiples of that of the flow. Two approaches are used. Firstly, the equations governing the magnetic field are reduced to a matrix eigenvalue problem, which is solved numerically. Secondly, a mean magnetic field is defined by averaging over proper areas in the XY -plane, corresponding equations are derived, in which the induction effect of the flow occurs as an anisotropic f -effect, and analytic solutions are given. The results are of particular interest for the Karlsruhe dynamo experiment, which works with a Roberts type flow consisting of 52 cells inside a cylindrical volume. In order to check the reliability of predictions concerning self-excitation based on the mean-field approach, analogous predictions are derived for a rectangular box containing 50 cells, and are compared with results obtained with the help of direct solutions of the eigenvalue problem mentioned. It turns out that the simple mean-field approach in general underestimates the requirements for self-excitation. The corresponding results agree with those obtained in the subharmonic approach only if the side length L of the box, its height H and the edge length l of a spin generator satisfy $ L \\gg H \\gg l $ . In Appendix B, some comments on previous results concerning $\\cal {ABC}$ dynamos are made in the light of the subharmonic formalism used in the paper.