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

F. Plunian¹ - K.-H. Rädler²

¹ Laboratoires des Ecoulements Géophysiques et Industriels B.P. 53, 38041 Grenoble Cedex 9, France
² Astrophysical Institute Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany

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 α-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. Tables 2, Figs 5, Refs 20.