The non-modal onset of the tearing instability

TL;DR

This paper explores the non-modal, transient growth phenomena in the onset of the classical MHD tearing instability, revealing that transient effects can dominate the initial dynamics, especially at high Lundquist numbers.

Contribution

It demonstrates that non-normal operators lead to significant transient growth in tearing instability, with growth scaling as S^{1/4} and identifies optimal initial perturbations.

Findings

Transient growth scales as O(S^{1/4}) for high Lundquist numbers.

Maximum transient growth occurs on time scales of O(S^{1/4}).

Optimal initial conditions are wave packets concentrated at the current sheet.

Abstract

We investigate the onset of the classical magnetohydrodynamic (MHD) tearing instability (TI) and focus on non-modal (transient) growth rather than the tearing mode. With the help of pseudospectral theory, the operators of the linear equations are shown to be highly non-normal, resulting in the possibility of significant transient growth at the onset of the TI. This possibility increases as the Lundquist number $S$ increases. In particular, we find evidence, numerically, that the maximum possible transient growth, measured in the $L_2$-norm, for the classical setup of current sheets unstable to the TI, scales as $O(S^{1/4})$ on time scales of $O(S^{1/4})$ for $S\gg 1$. This behaviour is much faster than the time scale $O(S^{1/2})$ when the solution behaviour is dominated by the tearing mode. The size of transient growth obtained is dependent on the form of the initial perturbation. Optimal initial conditions for the maximum possible transient growth are determined, which take the form of wave packets and can be thought of as noise concentrated at the current sheet. We also examine how the structure of the eigenvalue spectrum relates to physical quantities.