Axisymmetric Dynamos Sustained by a Modified Ohm's Law in a Toroidal Volume

TL;DR

This paper presents a new model demonstrating that incorporating an additional force into Ohm's law enables the sustained growth of axisymmetric magnetic fields in a toroidal volume, challenging traditional constraints like Cowling's theorem.

Contribution

The study introduces a modified Ohm's law model that allows for axisymmetric dynamo solutions, expanding understanding of magnetic field maintenance in toroidal geometries.

Findings

Magnetic energy can grow with a suitable toroidal flow.

The model produces a dipolar magnetic field.

Cowling's theorem is modified in this context.

Abstract

This work tackles a significant challenge in dynamo theory: the possibility of long-term amplification and maintenance of an axisymmetric magnetic field. We introduce a novel model that allows for non-trivial axially-symmetric steady-state solutions for the magnetic field, particularly when the dynamo operates primarily within a ``nearly-spherical'' toroidal volume inside a fluid shell surrounding a solid core. In this model, Ohm's law is generalized to include the dissipative force, arising from electron collisions, that tends to align the velocity of the shell with the rotational speed of the inner core and outer mantle. Our findings reveal that, in this context, Cowling's theorem and the neutral point argument are modified, leading to magnetic energy growth for a suitable choice of toroidal flow. The global equilibrium magnetic field that emerges from our model exhibits a dipolar character. The central insight of the model developed here is that if an additional force is incorporated into Ohm's law, symmetric dynamos become possible.