On the toroidal-velocity anti-dynamo theorem under the presence of nonuniform electric conductivity

TL;DR

This paper demonstrates that non-uniform electric conductivity does not enable dynamo action in laminar Couette flows with stable quasi-Keplerian rotation, reaffirming the Elsasser toroidal-velocity anti-dynamo theorem.

Contribution

It shows that non-uniform conductivity distributions do not support dynamo instability in certain laminar flows, extending the anti-dynamo theorem to these cases.

Findings

No dynamo excitation in spherical geometry regardless of conductivity distribution.

Non-uniform conductivity does not enable dynamo action in cylindrical geometry.

Axial flows do not support dynamo mechanisms in these configurations.

Abstract

Laminar electrically conducting Couette flows with the hydrodynamically stable quasi-Keplerian rotation profile and non-uniform conductivity are probed for dynamo instability. In spherical geometry the equations for the poloidal and the toroidal field components completely decouple, resulting in free decay, regardless of the spatial distribution of the electric conductivity. In cylindrical geometry the poloidal and toroidal components do not decouple, but here also we do not find dynamo excitations for the cases that the electric conductivity only depends on the radius or -- much more complex -- that it only depends on the azimuthal or the axial coordinate. The transformation of the plane-flow dynamo model of Busse \& Wicht (1992) to cylindrical or spherical geometry therefore fails. It is also shown that even the inclusion of axial flows of both directions does {\em not} support the dynamo mechanism. The Elsasser toroidal-velocity antidynamo theorem, according to which dynamos without any radial velocity component cannot work, is thus not softened by non-uniform conductivity distributions.