Plasmoid and Kelvin-Helmholtz instabilities in Sweet-Parker current sheets

TL;DR

This paper develops a 2D linear theory for instabilities in Sweet-Parker current sheets, revealing the dominance of Kelvin-Helmholtz instability over plasmoid instability and analyzing the effects of viscosity and magnetic Prandtl number.

Contribution

The paper introduces a new analytical 2D equilibrium solution and provides a comprehensive linear stability analysis including plasmoid and Kelvin-Helmholtz instabilities, with numerical verification.

Findings

Kelvin-Helmholtz instability grows faster than plasmoid instability.

Growth rate and wave-number increase along the outflow direction.

Critical Lundquist number for plasmoid instability scales with magnetic Prandtl number.

Abstract

A 2D linear theory of the instability of Sweet-Parker (SP) current sheets is developed in the framework of Reduced MHD. A local analysis is performed taking into account the dependence of a generic equilibrium profile on the outflow coordinate. The plasmoid instability [Loureiro et al, Phys. Plasmas {\bf 14}, 100703 (2007)] is recovered, i.e., current sheets are unstable to the formation of a large-wave-number chain of plasmoids ($k_{\rm max}\Lsheet \sim S^{3/8}$, where $k_{\rm max}$ is the wave-number of fastest growing mode, $S=\Lsheet V_A/\eta$ is the Lundquist number, $\Lsheet$ is the length of the sheet, $V_A$ is the Alfv\'en speed and $\eta$ is the plasma resistivity), which grows super-Alfv\'enically fast ($\gmax\tau_A\sim S^{1/4}$, where $\gmax$ is the maximum growth rate, and $\tau_A=\Lsheet/V_A$). For typical background profiles, the growth rate and the wave-number are found to {\it increase} in the outflow direction. This is due to the presence of another mode, the Kelvin-Helmholtz (KH) instability, which is triggered at the periphery of the layer, where the outflow velocity exceeds the Alfv\'en speed associated with the upstream magnetic field. The KH instability grows even faster than the plasmoid instability, $\gmax \tau_A \sim k_{\rm max} \Lsheet\sim S^{1/2}$. The effect of viscosity ($\nu$) on the plasmoid instability is also addressed. In the limit of large magnetic Prandtl numbers, $Pm=\nu/\eta$, it is found that $\gmax\sim S^{1/4}Pm^{-5/8}$ and $k_{\rm max} \Lsheet\sim S^{3/8}Pm^{-3/16}$, leading to the prediction that the critical Lundquist number for plasmoid instability in the $Pm\gg1$ regime is $\Scrit\sim 10^4Pm^{1/2}$. These results are verified via direct numerical simulation of the linearized equations, using a new, analytical 2D SP equilibrium solution.