Violation of Richardson's Criterion via Introduction of a Magnetic Field

TL;DR

This paper demonstrates that in magnetized plasmas, the classical Richardson criterion for shear flow instability can be violated, revealing new instability conditions influenced by magnetic fields and challenging traditional stability assessments.

Contribution

The study shows that magnetic fields can cause violations of the Richardson criterion in stratified shear flows, expanding understanding of plasma stability beyond classical criteria.

Findings

Certain profiles remain unstable despite stable stratification when magnetic fields are present.

Not all magnetic fields destabilize flows; some profiles adhere to the Richardson criterion.

Magnetic free energy is necessary for violating the Richardson criterion in magnetized plasmas.

Abstract

Shear flow instabilities can profoundly affect the diffusion of momentum in jets, stars, and disks. The Richardson criterion gives a sufficient condition for instability of a shear flow in a stratified medium. The velocity gradient $V'$ can only destabilize a stably stratified medium with squared Brunt-Vaisala frequency $N^2$ if $V'^2/4>N^2$. We find this is no longer true when the medium is a magnetized plasma. We investigate the effect of stable stratification on magnetic field and velocity profiles unstable to magneto-shear instabilities, i.e., instabilities which require the presence of both magnetic field and shear flow. We show that a family of profiles originally studied by Tatsuno & Dorland (2006) remain unstable even when $V'^2/4<N^2$, violating the Richardson criterion. However, not all magnetic fields can result in a violation of the Richardson criterion. We consider a class of flows originally considered by Kent (1968), which are destabilized by a constant magnetic field, and show that they become stable when $V'^2/4<N^2$, as predicted by the Richardson criterion. This suggests that magnetic free energy is required to violate the Richardson criterion. This work implies that the Richardson criterion cannot be used when evaluating the ideal stability of a sheared, stably stratified, and magnetized plasma. We briefly discuss the implications for astrophysical systems.