An integral equation-based numerical solver for Taylor states in toroidal geometries

TL;DR

This paper presents a novel integral equation-based numerical method for calculating Taylor states, a type of Beltrami fields, in toroidal geometries, with applications to magnetohydrodynamics and potential for generalization to complex shapes.

Contribution

The authors introduce a well-conditioned integral equation approach using the Debye source representation for efficient computation of Taylor states in toroidal geometries.

Findings

The method accurately computes Taylor states in toroidal geometries.

The integral equation is well-conditioned except at discrete resonance values.

Numerical examples demonstrate the approach's effectiveness and generalizability.

Abstract

We develop an algorithm for the numerical calculation of Taylor states in toroidal and toroidal shell geometries using an analytical framework developed for the solution to the time-harmonic Maxwell equations. Taylor states are a special case of what are known as Beltrami fields, or linear force-free fields. The scheme of this work relies on the generalized Debye source representation of Maxwell fields and an integral representation of Beltrami fields which immediately yields a well-conditioned second-kind integral equation. This integral equation has a unique solution whenever the Beltrami parameter $\lambda$ is not a member of a discrete, countable set of resonances which physically correspond to spontaneous symmetry breaking. Several numerical examples relevant to magnetohydrodynamic equilibria calculations are provided. Lastly, our approach easily generalizes to arbitrary geometries, both bounded and unbounded, and of varying genus.