Resistive magnetohydrodynamic simulations of the ideal tearing mode

TL;DR

This paper uses resistive magnetohydrodynamic simulations to demonstrate that the tearing instability can trigger rapid reconnection in thin current sheets at high Lundquist numbers, explaining explosive plasma phenomena.

Contribution

It provides the first numerical confirmation that the ideal tearing mode leads to fast reconnection at high Lundquist numbers, with implications for solar and astrophysical flares.

Findings

Tearing mode growth rate can reach approximately 0.6 times the inverse Alfvén time.

Linear eigenmodes and dispersion relations are confirmed at high Lundquist numbers.

Secondary reconnection events obey the same critical scaling, leading to faster reconnection.

Abstract

We study the linear and nonlinear evolution of the tearing instability on thin current sheets by means of two-dimensional numerical simulations, within the framework of compressible, resistive magnetohydrodynamics. In particular we analyze the behavior of current sheets whose inverse aspect ratio scales with the Lundquist number $S$ as $S^{-1/3}$. This scaling has been recently recognized to yield the threshold separating fast, ideal reconnection, with an evolution and growth which are independent of $S$ provided this is high enough, as it should be natural having the ideal case as a limit for $S\to\infty$. Our simulations confirm that the tearing instability growth rate can be as fast as $\gamma\approx 0.6\,{\tau_A}^{-1}$, where $\tau_A$ is the ideal Alfv\'enic time set by the macroscopic scales, for our least diffusive case with $S=10^7$. The expected instability dispersion relation and eigenmodes are also retrieved in the linear regime, for the values of $S$ explored here. Moreover, in the nonlinear stage of the simulations we observe secondary events obeying the same critical scaling with $S$, here calculated on the \emph{local}, much smaller lengths, leading to increasingly faster reconnection. These findings strongly support the idea that in a fully dynamic regime, as soon as current sheets develop, thin and reach this critical threshold in their aspect ratio, the tearing mode is able to trigger plasmoid formation and reconnection on the local (ideal) Alfv\'enic timescales, as required to explain the explosive flaring activity often observed in solar and astrophysical plasmas.