Scaling laws of resistive magnetohydrodynamic reconnection in the high-Lundquist-number, plasmoid-unstable regime

The Sweet–Parker layer in a system that exceeds a critical value of the Lundquist number $(S)$ is unstable to the plasmoid instability. In this paper, a numerical scaling study has been done with an island coalescing system driven by a low level of random noise. In the early stage, a primary Sweet–Parker layer forms between the two coalescing islands. The primary Sweet–Parker layer breaks into multiple plasmoids and even thinner current sheets through multiple levels of cascading if the Lundquist number is greater than a critical value $Sc≃4×104$⁠. As a result of the plasmoid instability, the system realizes a fast nonlinear reconnection rate that is nearly independent of $S$⁠, and is only weakly dependent on the level of noise. The number of plasmoids in the linear regime is found to scales as $S3/8$⁠, as predicted by an earlier asymptotic analysis [N. F. Loureiro et al, Phys. Plasmas 14, 100703 (2007)]. In the nonlinear regime, the number of plasmoids follows a steeper scaling, and is proportional to $S$⁠. The thickness and length of current sheets are found to scale as $S−1$⁠, and the local current densities of current sheets scale as $S−1$⁠. Heuristic arguments are given in support of theses scaling relations.