On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

TL;DR

This paper investigates the mathematical properties of magneto-geostrophic equations, showing non-diffusive versions are ill-posed while diffusive ones are well-posed, with implications for magnetic field generation.

Contribution

It establishes the ill-posedness of non-diffusive magneto-geostrophic equations and demonstrates nonlinear instability in the diffusive case, revealing conditions for dynamo effects.

Findings

Non-diffusive equations are ill-posed in Sobolev spaces.

Critically diffusive equations are well-posed.

Existence of nonlinearly unstable steady states leading to dynamo effects.

Abstract

We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is well-posed. In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.