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

TL;DR

This study investigates the scaling behavior of resistive magnetohydrodynamic reconnection in high-Lundquist-number plasmoid-unstable regimes, revealing nearly constant reconnection rates and specific scaling laws for plasmoid number and current sheet dimensions.

Contribution

The paper provides numerical evidence for scaling laws of plasmoid formation and reconnection rates in high-Lundquist-number regimes, extending previous asymptotic predictions.

Findings

Reconnection rate becomes nearly independent of Lundquist number S.

Number of plasmoids scales as S^{3/8} in linear regime and proportional to S in nonlinear regime.

Current sheet thickness and length scale as S^{-1}.

Abstract

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 $S_{c}\simeq4\times10^{4}$. 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 $S^{3/8}$, as predicted by an earlier asymptotic analysis (Loureiro \emph{et al.}, Phys. Plasmas \textbf{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.