General Theory of the Plasmoid Instability

TL;DR

This paper develops a comprehensive theory for plasmoid instability, deriving scaling relations for key parameters based on initial conditions and current sheet evolution, revealing a complex growth process with a quiescent phase followed by rapid development.

Contribution

It introduces a general theory using a least time principle, deriving non-power-law scaling relations for plasmoid instability parameters based on initial perturbations and current sheet dynamics.

Findings

Scaling relations depend on initial perturbation amplitude, Lundquist number, and evolution rate.

Instability involves a quiescent period followed by rapid growth.

Derived relations are not simple power laws, involving logarithmic factors.

Abstract

A general theory of the onset and development of the plasmoid instability is formulated by means of a principle of least time. The scaling relations for the final aspect ratio, transition time to rapid onset, growth rate, and number of plasmoids are derived, and shown to depend on the initial perturbation amplitude $\left({\hat w}_0\right)$, the characteristic rate of current sheet evolution $\left(1/\tau\right)$, and the Lundquist number $\left(S\right)$. They are not simple power laws, and are proportional to $S^{\alpha} \tau^{\beta} \left[\ln f(S,\tau,{\hat w}_0)\right]^\sigma$. The detailed dynamics of the instability is also elucidated, and shown to comprise of a period of quiescence followed by sudden growth over a short time scale.