Mean-Field Magnetohydrodynamics Models as Scaling Limits of Stochastic Induction Equations

TL;DR

This paper rigorously derives a deterministic mean-field model from a stochastic induction equation for magnetic fields, revealing effects of turbulence anisotropy on magnetic decay and dynamo phenomena.

Contribution

It introduces a scaling limit approach to connect stochastic induction equations with deterministic mean-field magnetohydrodynamics models, highlighting turbulence anisotropy effects.

Findings

Isotropic turbulence leads to additional dissipation in the limit model.

Anisotropic turbulence can produce a dynamo effect.

The decay rate of magnetic fields is influenced by turbulence properties.

Abstract

We study the asymptotic properties of a stochastic model for the induction equations of the magnetic field in a three dimensional periodic domain. The turbulent velocity field driving the electromotive force on the magnetic field is modeled by a noise white in time. For this model we rigorously take a scaling limit leading to a deterministic model. While in case of isotropic turbulence this produces an additional dissipation in the limit model which influences also the decay rate of the Magnetic field in the stochastic model, the case of turbulence devoloped in a preferential direction allows us to find a dynamo effect.