Visco-Resistive Plasmoid Instability

TL;DR

This paper analyzes the visco-resistive plasmoid instability in current sheets, deriving scaling laws for growth rates and widths, and discusses implications for fast magnetic reconnection in plasmas with viscosity effects.

Contribution

It provides new scaling laws for the linear and nonlinear growth of plasmoids in visco-resistive plasmas and explores their role in enabling fast magnetic reconnection.

Findings

Linear growth rate scales as S^{1/4}(1+Pm)^{-5/8}

Nonlinear growth time scales as S^{-3/16}(1+Pm)^{19/32}

Reconnection rate depends on Pm as (1+Pm)^{-1/2}

Abstract

The plasmoid instability in visco-resistive current sheets is analyzed in both the linear and nonlinear regimes. The linear growth rate and the wavenumber are found to scale as $S^{1/4} {\left( {1 + {P_m}} \right)}^{-5/8}$ and $S^{3/8} {\left( {1 + {P_m}} \right)}^{-3/16}$ with respect to the Lundquist number $S$ and the magnetic Prandtl number $P_m$. Furthermore, the linear layer width is shown to scale as $S^{-1/8} {(1+P_m)}^{1/16}$. The growth of the plasmoids slows down from an exponential growth to an algebraic growth when they enter into the nonlinear regime. In particular, the time-scale of the nonlinear growth of the plasmoids is found to be $\tau_{NL} \sim S^{-3/16} {(1 + P_m)^{19/32}}{\tau_{A,L}}$. The nonlinear growth of the plasmoids is radically different from the linear one and it is shown to be essential to understand the global current sheet disruption. It is also discussed how the plasmoid instability enables fast magnetic reconnection in visco-resistive plasmas. In particular, it is shown that the recursive plasmoid formation can trigger a collisionless reconnection regime if $S \gtrsim L_{cs} {(\epsilon_c l_k)^{-1}} {(1 + {P_m})^{1/2}}$, where $L_{cs}$ is the half-length of the global current sheet and $l_k$ is the relevant kinetic length scale. On the other hand, if the current sheet remains in the collisional regime, the global (time-averaged) reconnection rate is shown to be $\left\langle {{{\left. {d\psi /dt} \right|}_X}} \right\rangle \approx \epsilon_c v_{A,u} B_{u} {(1 + {P_m})^{-1/2}}$, where $\epsilon_c$ is the critical inverse aspect ratio of the current sheet, while $v_{A,u}$ and $B_{u}$ are the Alfv\'en speed and the magnetic field upstream of the global reconnection layer.