An integral equation-based numerical solver for Taylor states in toroidal geometries
For researchers in plasma physics and magnetohydrodynamics, this provides a robust numerical method for computing Taylor states in complex geometries, though it is an incremental improvement over existing integral equation techniques.
The paper develops a numerical solver for Taylor states (Beltrami fields) in toroidal geometries using an integral equation method, achieving well-conditioned solutions except at discrete resonances. Numerical examples for magnetohydrodynamic equilibria are provided.
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 $λ$ 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.