Quasisymmetric magnetic fields in asymmetric toroidal domains

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. Subclasses of solutions are then constructed by specifying as input the form the flux function, which enforces boundary shape and nested flux surfaces. In particular, we exhibit smooth 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. These solutions are local because they lack periodicity in the toroidal angle. The problems of boundary shape and locality are also discussed. We find that magnetic fields with Euclidean isometries 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.