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

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.