On a class of non-axisymmetric flux rope solutions to the electromagnetic induction equation

Abstract

Abstract We display non-axisymmetric generalisations of a known exact axisymmetric flux rope solution to the induction equation. The assumed velocity field is a simple axisymmetric stagnation point flow. Solutions for magnetic fields with non-zero azimuthal wavenumber are given. These provide local models for fluxrope features seen in numerical computations of probable fast dynamos. Remarkably, they can have positive growth rates for any magnetic Reynolds number, and have finite energy (per unit length along the axis) provided only that the growth rate is less than one, in velocity-based units. Thus although not chaotic, this flow functions as a fast dynamo, albeit one with the unphysical feature that its velocity tends to infinity far from the origin. It is therefore desirable to exclude such a feature when choosing velocity fields as candidates for a realistic fast dynamo, to avoid finding one for a spurious reason.