Dynamos driven by modified Beltrami flows

A. A. Bachtiar and R. W. James

Abstract

Bachtiar, Ivers and James (2006, BIJ), and Bachtiar & James (2010), showed that a planar fluid velocity $\mathbf{v}$ can support dynamo action in a conducting sphere. BIJ also tried to convert some historical flows to planar flows, via a process they termed 'planarizing'. In particular BIJ studied one of many flows considered by Pekeris, Accad and Shkoller (1973, PAS). The PAS flow was chosen because it produced a dynamo at low truncation levels and critical magnetic Reynolds number Rc, this possibly influenced by it being a helical Beltrami flow (i.e. $\mathrm{\nabla }\times \mathbf{v}\parallel \mathbf{v}$). BIJ were able to planarize the poloidal part of the PAS flow, but not the toroidal part. The present paper considers more partly planarized PAS flows; and two modifications of PAS, labelled biPAS and quasiPAS, that can be fully planarized. BiPAS flows are just a sum of two PAS flows, and quasiPAS are PAS-like with a modified radial cell structure for the toroidal flow. We have studied 128 models using these flows, found 96 dynamos including 84 new dynamos, but found no dynamos with fully planar flows. The partly planarized PAS, the biPAS and quasiPAS flows are not Beltrami. But 20 (normalized using $\mathrm{rms}(\mathbf{v})$, or 22 using $max|\mathbf{v}|$) $R_c$ of the associated dynamos are lower than the $R_c$ of the Beltrami dynamos from which they were derived, showing that the Beltrami property is not essential for low $R_c$.

Keywords: magnetohydrodynamics, dynamo theory, Beltrami flows, planar dynamos.

AMS Subject Classification: Primary 76W05; secondary 85A30, 86A25.