Resistive tearing: numerical exploration of nonmodal effects

TL;DR

This paper develops non-modal stability tools for resistive MHD, demonstrating how transient growth can influence plasma instabilities and providing insights into their nonlinear evolution.

Contribution

It introduces a detailed derivation of non-modal stability analysis for resistive MHD and explores the role of transient growth in plasma tearing modes.

Findings

Transient growth is norm-dependent and varies by an order of magnitude.

Linear transient growth can be reproduced in nonlinear simulations.

Transient growth alone does not trigger full plasmoid instability.

Abstract

The fluid dynamics community has found success in explaining both the onset and coherent structure formation in wall-bounded turbulence through examining transient growth and pseudoresonance. Whether similar effects are important in plasmas well-described by magnetohydrodynamics is an open question. In nuclear fusion experiments, the onset of turbulence often enhances undesirable heat transport between the core and the edge, and a vast number of studies have attempted to understand these phenomena via linear and nonlinear numerical codes. If there are plasma experiments where instabilities and turbulence onset are dominated by nonmodal effects, the tools of non-modal stability and resolvent analysis could prove to be essential to practitioners. Towards that goal, in this work we provide a detailed derivation of non-modal stability tools and the resolvent operator for incompressible, resistive magnetohydrodynamics (MHD). Concretely, optimal initial conditions that maximize transient growth are computed, along with bounds on linear transient growth, and are numerically verified with a nonlinear solver to show that these linear effects can be reproduced in nonlinear simulations. The potential of such initial conditions to initiate the tearing mode in a spectrally stable system is explored. We found the approach common in nonmodal hydrodynamic stability of using pseudospectral methods to be infeasible for many parameter combinations due to numerical ill-conditioning. Further, transient growth in the Harris current sheet proved to be highly norm-dependent, with the observed differences being of an order of magnitude. Nonlinear simulations showed that transient growth occurs to some extent in all norms, both in the infinitesimal as well as finite limit. No case was observed of transient growth being strong enough to trigger the full plasmoid instability.