Quasisymmetric magnetic fields in asymmetric toroidal domains

TL;DR

This paper investigates the existence and construction of quasisymmetric magnetic fields in asymmetric toroidal domains, revealing mathematical challenges and conditions for boundary and topological constraints.

Contribution

It derives a novel system of PDEs for quasisymmetric fields in asymmetric domains and demonstrates local solutions satisfying boundary conditions.

Findings

Symmetric magnetic fields can be embedded in asymmetric domains.

Global extension of quasisymmetric fields faces topological obstructions.

Local solutions can satisfy boundary conditions on parts of the domain.

Abstract

We explore the existence of quasisymmetric magnetic fields in asymmetric toroidal domains. These vector fields can be identified with a class of magnetohydrodynamic equilibria in the presence of pressure anisotropy. First, using Clebsch potentials, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric magnetic fields in bounded domains. In regions where flux surfaces and surfaces of constant field strength are not tangential, this system can be further reduced to a single degenerate nonlinear second order partial differential equation with externally assigned initial data. Then, we exhibit regular quasisymmetric vector fields which correspond to local solutions of anisotropic magnetohydrodynamics in asymmetric toroidal domains such that tangential boundary conditions are fulfilled on a portion of the bounding surface. The problems of boundary shape and locality are also discussed. We find that symmetric magnetic fields can be fitted into asymmetric domains, and that the mathematical difficulty encountered in the derivation of global quasisymmetric magnetic fields lies in the topological obstruction toward global extension affecting local solutions of the governing nonlinear first order partial differential equations.