Magnetohydrodynamics in a finite cylinder: Poloidal-toroidal decomposition

The magnetohydrodynamic equations present two challenging algorithmic requirements: that both fields be solenoidal and that the magnetic field match an unknown external field. The poloidal-toroidal decomposition represents a three-dimensional solenoidal vector field via two scalar potentials. Widely used in Cartesian and spherical geometries with periodic boundary conditions, complications appear in finite geometries which can, however, be circumvented. An implementation of the poloidal-toroidal decomposition for the magnetohydrodynamic equations in a finite cylinder is described, which uses a spectral spatial discretisation. A Green's function method is proposed for matching the magnetic field in a spectral representation to an external field in a vacuum