Growth rate degeneracies in kinematic dynamos

TL;DR

This paper demonstrates that in steady flows, the dynamo growth rate remains the same under two different magnetic boundary conditions due to the adjointness of the induction operator, provided the flow is reversible.

Contribution

It reveals a surprising equivalence of dynamo growth rates under different boundary conditions in steady flows with reversible velocity fields.

Findings

Growth rates are identical for normal and tangent boundary conditions.

Magnetic eigenmodes differ significantly despite identical growth rates.

Reversibility of the flow is key to this degeneracy.

Abstract

We consider the classical problem of kinematic dynamo action in simple steady flows. Due to the adjointness of the induction operator, we show that the growth rate of the dynamo will be exactly the same for two types of magnetic boundary conditions: the magnetic field can be normal (infinite magnetic permeability, also called pseudo-vacuum) or tangent (perfect electrical conductor) to the boundaries of the domain. These boundary conditions correspond to well-defined physical limits often used in numerical models and relevant to laboratory experiments. The only constraint is for the velocity field u to be reversible, meaning there exists a transformation changing u into -u. We illustrate this surprising property using S2T2 type of flows in spherical geometry inspired by Dudley and James (1989). Using both types of boundary conditions, it is shown that the growth rates of the dynamos are identical, although the corresponding magnetic eigenmodes are drastically different.