Dynamo action in the ABC flows using symmetries

Abstract

This paper concerns kinematic dynamo action by the 1:1:1 ABC flow, in the highly conducting limit of large magnetic Reynolds number Rm. The flow possesses 24 symmetries, with a symmetry group isomorphic to the group O24 of orientation preserving transformations of a cube. This can be exploited to break up the linear eigenvalue problem into five distinct symmetry classes, or irreducible representations, which we label I–V. The paper discusses how to reduce the scale of the numerical problem to a subset of Fourier modes for a magnetic field in each representation, which then may be solved independently to obtain distinct branches of eigenvalues and magnetic field eigenfunctions.

Two numerical methods are employed: the first is to time step a magnetic field in a given symmetry class and obtain the growth rate and frequency by measuring the magnetic energy as a function of time. The second method involves a more direct determination of the eigenvalue using the eigenvalue solver ARPACK for sparse matrix systems, which employs an implicitly restarted Arnoldi method. The two methods are checked against each other, and compared for efficiency and reliability. Eigenvalue branches for each symmetry class are obtained for magnetic Reynolds numbers Rm up to 104 together with spectra and magnetic field visualisations. A sequence of branches emerges as Rm increases and the magnetic field structures in the different branches are discussed and compared. In a parallel development,results are presented for the corresponding fluid stability problem as a function of the Reynolds number Re.