Optimization of the magnetic dynamo

TL;DR

This paper uses an optimization approach to identify the velocity fields that maximize magnetic energy growth in dynamo theory, revealing the critical magnetic Reynolds number needed for sustained magnetic fields.

Contribution

It introduces an optimization framework for dynamo theory, identifying the minimal flow conditions for magnetic field amplification in a periodic box.

Findings

Optimal velocity field exists for magnetic energy growth.

Critical magnetic Reynolds number is 15% above the threshold for transient growth.

Flow measurement by dissipation yields a single optimal at the dynamo threshold.

Abstract

In stars and planets, magnetic fields are believed to originate from the motion of electrically conducting fluids in their interior, through a process known as the dynamo mechanism. In this Letter, an optimization procedure is used to simultaneously address two fundamental questions of dynamo theory: "Which velocity field leads to the most magnetic energy growth?" and "How large does the velocity need to be relative to magnetic diffusion?" In general, this requires optimization over the full space of continuous solenoidal velocity fields possible within the geometry. Here the case of a periodic box is considered. Measuring the strength of the flow with the root-mean-square amplitude, an optimal velocity field is shown to exist, but without limitation on the strain rate, optimization is prone to divergence. Measuring the flow in terms of its associated dissipation leads to the identification of a single optimal at the critical magnetic Reynolds number necessary for a dynamo. This magnetic Reynolds number is found to be only 15% higher than that necessary for transient growth of the magnetic field.