Pressure, chaotic magnetic fields, and magnetohydrodynamic equilibria

Analyses of plasma behavior often begin with a description of the ideal magnetohydrodynamic equilibrium, this being the simplest model capable of approximating macroscopic force balance. Ideal force balance is when the pressure gradient is supported by the Lorentz force, ∇p=j×B. We discuss the implications of allowing for a chaotic magnetic field on the solutions to this equation. We argue that the solutions are pathological and not suitable for numerical calculations. If the pressure and magnetic field are continuous, the only nontrivial solutions have an uncountable infinity of discontinuities in the pressure gradient and current. The problems arise from the arbitrarily small length scales in the structure of the field, and the consequence of ideal force balance that the pressure is constant along the field-lines, B⋅∇p=0. A simple method to ameliorate the singularities is to include a small but finite perpendicular diffusion. A self-consistent set of equilibrium equations is described, and some algorithmic approaches aimed at solving these equations are discussed.