Resistive instabilities in sinusoidal shear flows with a streamwise magnetic field

TL;DR

This paper explores how viscosity and resistivity introduce new instabilities in sinusoidal shear flows with a magnetic field, revealing a resistively-unstable Alfvén wave active for any nonzero magnetic field when magnetic Prandtl number is less than one.

Contribution

It identifies and analyzes a new resistively-unstable Alfvén wave mode in MHD shear flows, extending understanding of flow stability with finite resistivity and viscosity.

Findings

Discovery of a resistively-unstable Alfvén wave mode for Pm<1.

Demonstration that this mode saturates into a quasi-stationary state.

Reduced model explaining the excitation mechanism of the instability.

Abstract

We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin-Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we call a resistively-unstable Alfv\'en wave due to its connection to shear Alfv\'en waves, exists for any nonzero magnetic field strength as long as the magnetic Prandtl number $Pm < 1$. We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch (1991). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons.