Instability of the magnetohydrodynamics system at small but finite Reynolds number

TL;DR

This paper investigates the stability and instability of magnetohydrodynamics solutions at small but finite Reynolds numbers, using the alpha-effect to construct solutions with complex oscillatory behavior and analyzing their stability properties.

Contribution

It provides new nonlinear stability and instability results for magnetohydrodynamics solutions at small Reynolds numbers using the alpha-effect method.

Findings

Proves nonlinear stability for certain initial conditions.

Establishes instability for a dense subset of initial velocities.

Analyzes oscillatory solutions on multiple scales.

Abstract

The aim of this paper is to give a result concerning the stability properties of the solutions of magnetohydrodynamics equations at small but finite Reynolds numbers. These solutions are found using the alpha-effect: this method gives us solutions which are highly oscillating spatially on the scale of the underlying flow but are growing on a larger scale depending on a parameter epsilon. We prove nonlinear stability and instability results for a dense subset of initial velocity field of the flow at given Reynolds number.