Numerical study of $\delta$-function current sheets arising from resonant magnetic perturbations

TL;DR

This study numerically investigates the formation of singular current sheets in ideal MHD equilibria under resonant perturbations, using two different computational approaches to validate the results.

Contribution

It introduces a comparative numerical analysis of the Hahm-Kulsrud-Taylor problem using GS and SPEC codes, advancing understanding of current sheet formation in ideal MHD.

Findings

Excellent agreement between GS and SPEC solutions.

Convergence of MRxMHD solutions to ideal MHD with increasing regions.

Validation of numerical methods for singular current sheet formation.

Abstract

General three-dimensional toroidal ideal magnetohydrodynamic equilibria with a continuum of nested flux surfaces are susceptible to forming singular current sheets when resonant perturbations are applied. The presence of singular current sheets indicates that, in the presence of non-zero resistivity, magnetic reconnection will ensue, leading to the formation of magnetic islands and potentially regions of stochastic field lines when islands overlap. Numerically resolving singular current sheets in the ideal MHD limit has been a significant challenge. This work presents numerical solutions of the Hahm-Kulsrud-Taylor (HKT) problem, which is a prototype for resonant singular current sheet formation. The HKT problem is solved by two codes: a Grad-Shafranov (GS) solver and the SPEC code. The GS solver has built-in nested flux surfaces with prescribed magnetic fluxes. The SPEC code implements multi-region relaxed magnetohydrodynamics (MRxMHD), where the solution relaxes to a Taylor state in each region while maintaining force balance across the interfaces between regions. As the number of regions increases, the MRxMHD solution approaches the ideal MHD solution assuming a continuum of nested flux surfaces. We demonstrate excellent agreement between the numerical solutions obtained from the two codes through a thorough convergence study.