The axisymmetric antidynamo theorem revisited

TL;DR

This paper revisits the axisymmetric antidynamo theorem, proving exponential decay of magnetic fields in a general setting with variable conductivity and fluid flow, and explores conditions affecting decay rates.

Contribution

It extends the antidynamo theorem to more general, realistic conditions including compressible fluids and variable conductivity, providing new decay rate estimates.

Findings

Magnetic fields decay exponentially under broad conditions.

Decay rates can be very slow for certain flow fields and high magnetic Reynolds numbers.

Weak variations in density and conductivity lead to decay rates close to free decay.

Abstract

The axisymmetric kinematic dynamo problem is reconsidered and a number of open questions are answered. Apart from axisymmetry and smoothness of data and solution we deal with this problem under quite general conditions, i.e. we assume a compressible fluid of variable (in space and time) conductivity moving in an arbitrary (axisymmetric) domain. We prove unconditional, pointwise and exponential decay of magnetic field and electric current to zero. The decay rate of the external (meridional) magnetic field can become very small (compared to free decay) for special flow fields and large magnetic Reynolds numbers. We give an example of that. On the other hand, we show for fluids with weak variation of mass density and conductivity that the meridional and azimuthal decay rates do not drop significantly below those of free decay.