Dynamo Onset as a First-Order Transition: Lessons from a Shell Model for Magnetohydrodynamics

TL;DR

This study models the dynamo transition in magnetohydrodynamic turbulence as a first-order phase transition, revealing a complex phase diagram with hysteresis and fractal boundary behavior.

Contribution

It introduces a shell model approach to analyze dynamo onset, framing it as a nonequilibrium first-order transition with a detailed phase diagram and hysteresis phenomena.

Findings

Dynamo onset behaves as a first-order phase transition.

The phase diagram exhibits a fractal boundary between dynamo and no-dynamo states.

Hysteresis and nucleation-like phenomena are observed at the transition.

Abstract

We carry out systematic and high-resolution studies of dynamo action in a shell model for magnetohydrodynamic (MHD) turbulence over wide ranges of the magnetic Prandtl number $Pr_{\rm M}$ and the magnetic Reynolds number $Re_{\rm M}$. Our study suggests that it is natural to think of dynamo onset as a nonequilibrium, first-order phase transition between two different turbulent, but statistically steady, states. The ratio of the magnetic and kinetic energies is a convenient order parameter for this transition. By using this order parameter, we obtain the stability diagram (or nonequilibrium phase diagram) for dynamo formation in our MHD shell model in the $(Pr^{-1}_{\rm M}, Re_{\rm M})$ plane. The dynamo boundary, which separates dynamo and no-dynamo regions, appears to have a fractal character. We obtain hysteretic behavior of the order parameter across this boundary and suggestions of nucleation-type phenomena.