Local Axisymmetric Diffusive Stability of Weakly-Magnetized, Differentially-Rotating, Stratified Fluids

TL;DR

This paper analyzes the local stability of stratified, differentially-rotating fluids with weak magnetic fields, revealing conditions for stability and the impact of multiple diffusion processes, with applications to the Sun's radiative zone.

Contribution

It generalizes the GSF double-diffusive analysis to include magnetic fields and resistivity, deriving new stability criteria for triple-diffusive systems.

Findings

Rotation must be constant on cylinders for stability.

Magnetic fields are crucial for stability conditions.

Adding a third diffusion process can stabilize otherwise unstable modes.

Abstract

We study the local stability of stratified, differentially-rotating fluids to axisymmetric perturbations in the presence of a weak magnetic field and of finite resistivity, viscosity and heat conductivity. This is a generalization of the Goldreich-Schubert-Fricke (GSF) double-diffusive analysis to the magnetized and resistive, triple-diffusive case. Our fifth-order dispersion relation admits a novel branch which describes a magnetized version of multi-diffusive modes. We derive necessary conditions for axisymmetric stability in the inviscid and perfect-conductor (double-diffusive) limits. In each case, rotation must be constant on cylinders and angular velocity must not decrease with distance from the rotation axis for stability, irrespective of the relative strength of viscous, resistive and heat diffusion. Therefore, in both double-diffusive limits, solid body rotation marginally satisfies our stability criteria. The role of weak magnetic fields is essential to reach these conclusions. The triple-diffusive situation is more complex, and its stability criteria are not easily stated. Numerical analysis of our general dispersion relation confirms our analytic double-diffusive criteria, but also shows that an unstable double-diffusive situation can be significantly stabilized by the addition of a third, ostensibly weaker, diffusion process. We describe a numerical application to the Sun's upper radiative zone and establish that it would be subject to unstable multi-diffusive modes if moderate or strong radial gradients of angular velocity were present.