Magnetohydrodynamics in a finite cylinder: Poloidal-toroidal decomposition

Abstract. The magnetohydrodynamic equations present two challenging algorithmic require-ments: 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 pe-riodic boundary conditions, complications appear in finite geometries which can, however, be circumvented. An implementation of the poloidal-toroidal decomposition for the magnetohydro-dynamic 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 represen-tation to an external field in a vacuum. 1 Poloidal-toroidal decomposition The requirement that velocity and magnetic elds be solenoidal, i.e. divergence-free, repre-sents one of the most challenging difculties in hydrodynamics and in magnetohydrodynamics. This condition is the approximation used in incompressible uid dynamics, and is the statement of the non-existence of magnetic monopoles in electromagnetism. Two main approaches exist for satisfying this requirement. The rst method is to conserv