Harmonic and subharmonic solutions of the Roberts dynamo problem. Application to the Karlsruhe experiment

TL;DR

This paper investigates the Roberts dynamo problem using numerical eigenvalue analysis and analytical mean-field theory, comparing results to experimental data from the Karlsruhe dynamo experiment and revealing limitations of the mean-field approach.

Contribution

It introduces two approaches—numerical eigenvalue solutions for harmonic/subharmonic fields and analytical mean-field theory—for analyzing the Roberts dynamo, and compares these with experimental data.

Findings

Eigenvalue solutions provide detailed dynamo behavior.

Mean-field theory has limitations in accurately predicting the experiment.

Comparison reveals shortcomings of simplified mean-field models.

Abstract

Two different approaches to the Roberts dynamo problem are considered. Firstly, the equations governing the magnetic field are specified to both harmonic and subharmonic solutions and reduced to matrix eigenvalue problems, which are solved numerically. Secondly, a mean magnetic field is defined by averaging over proper areas, corresponding equations are derived, in which the induction effect of the flow occurs essentially as an anisotropic alpha-effect, and they are solved analytically. In order to check the reliability of the statements on the Karlsruhe experiment which have been made on the basis of a mean-field theory, analogous statements are derived for a rectangular dynamo box containing 50 Roberts cells, and they are compared with the direct solutions of the eigenvalue problem mentioned. Some shortcomings of the simple mean-field theory are revealed.