Plasmoid Instability in Forming Current Sheets

TL;DR

This paper derives new, complex scaling laws for plasmoid instability in evolving current sheets, revealing that previous models are inadequate for most astrophysical systems and enabling better understanding of rapid magnetic reconnection.

Contribution

The paper introduces a general set of scaling laws for plasmoid instability in time-evolving current sheets, applicable to astrophysical environments, based on a principle of least time.

Findings

Scaling laws depend on Lundquist number, Prandtl number, noise, and evolution rate.

Previous scalings are invalid for most astrophysical systems.

Predicted growth rates and aspect ratios are smaller than earlier models.

Abstract

The plasmoid instability has revolutionized our understanding of magnetic reconnection in astrophysical environments. By preventing the formation of highly elongated reconnection layers, it is crucial in enabling the rapid energy conversion rates that are characteristic of many astrophysical phenomena. Most of the previous studies have focused on Sweet-Parker current sheets, which, however, are unattainable in typical astrophysical systems. Here, we derive a general set of scaling laws for the plasmoid instability in resistive and visco-resistive current sheets that evolve over time. Our method relies on a principle of least time that enables us to determine the properties of the reconnecting current sheet (aspect ratio and elapsed time) and the plasmoid instability (growth rate, wavenumber, inner layer width) at the end of the linear phase. After this phase the reconnecting current sheet is disrupted and fast reconnection can occur. The scaling laws of the plasmoid instability are \emph{not} simple power laws, and depend on the Lundquist number ($S$), the magnetic Prandtl number ($P_m$), the noise of the system ($\psi_0$), the characteristic rate of current sheet evolution ($1/\tau$), as well as the thinning process. We also demonstrate that previous scalings are inapplicable to the vast majority of the astrophysical systems. We explore the implications of the new scaling relations in astrophysical systems such as the solar corona and the interstellar medium. In both these systems, we show that our scaling laws yield values for the growth rate, wavenumber, and aspect ratio that are much smaller than the Sweet-Parker based scalings.