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

A 2D linear theory of the instability of Sweet-Parker current sheets is developed. It is shown that the current sheet is unstable to two modes. Close to the center of the sheet the plasmoid instability is recovered: current sheets are unstable to the formation of a large wave number chain of plasmoids ($k_{max}L_{CS} \sim S^{3/8}$, where $k_{max}$ is the wave-number of fastest growing mode, $S=L_{CS} V_A/\eta$ is the Lundquist number, $L_{CS}$ 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 ($\gamma_{max}\tau_A\sim S^{1/4}$, where $\gamma_{max}$ is the maximum growth rate, and $\tau_A=Lsheet/V_A$). Away from the center of the sheet, it is found that the Kelvin-Helmholtz (KH) instability is triggered. The KH instability grows even faster than the plasmoid instability, $\gamma_{max} \tau_A \sim k_{max} L_{CS}\sim S^{1/2}$. The effect of viscosity ($\nu$) on the plasmoid instability is also addressed. In the limit of large Prandtl number, $Pm=\nu/\eta$, it is found that $\gamma_{max}\sim S^{1/4}Pm^{-5/8}$ and $k_{max} L_{CS}\sim S^{3/8}Pm^{-3/16}$; it is predicted that the critical Lundquist number for plasmoid instability in the $Pm\gg1$ regime is $S_c\sim 10^4 Pm^{1/2}$.