Toroidal visco-resistive magnetohydrodynamic steady states contain vortices

Abstract

Poloidal velocity fields seem to be a fundamental feature of resistive toroidal magnetohydrodynamic (MHD) steady states. They are a consequence of force balance in toroidal geometry, do not require any kind of instability, and disappear in the “straight cylinder” (infinite aspect ratio) limit. If a current density j results from an axisymmetric toroidal electric field that is irrotational inside a torus, it leads to a magnetic field B such that ∇×(j×B) is nonvanishing, so that the Lorentz force cannot be balanced by the gradient of any scalar pressure in the equation of motion. In a steady state, finite poloidal velocity fields and toroidal vorticity must exist. Their calculation is difficult, but explicit solutions can be found in the limit of low Reynolds number. Here, existing calculations are generalized to the more realistic case of no-slip boundary conditions on the velocity field and a circular toroidal cross section. The results of this paper strongly suggest that discussions of confined steady states in toroidal MHD must include flows from the outset.