Connections between adiabatic and low-resistivity magnetohydrodynamics

A first step in the hierarchy of increasingly complex magnetohydrodynamic (MHD) models goes from the nondissipative Grad--Shafranov equilibrium model to the slow-evolution resistive Grad--Hogan model. In principle, as the resistivity tends to zero, the limit of solutions of a resistive problem might be a solution of the adiabatic equations, only in a generalized sense. Specifically, the adiabatic limit could exhibit discontinuities across singular surfaces where dissipation can occur, while the body of the fluid becomes truly adiabatic. For a suitable test case, a two-dimensional analog of an oscillating Doublet, behavior typical of such an approach to a singular adiabatic limit is presented. A boundary layer or sheath appears around the separatrix of this configuration. The sheath contains the bulk of the dissipative effects and its thickness decreases rapidly with resistivity. For example, flux transport is mainly confined to the sheath and current is nearly invariant outside it. The resistive equations are solved numerically by the alternating dimension (AD) technique devised by Grad. With decreasing resistivity, the run time of the algorithm grows rapidly to durations infeasible, even for a supercomputer. In contrast, the AD method for adiabatic codes has short run times and can be executed easily on a microcomputer. Thus, it is useful to approximate solutions of low-resistivity problems by solving appropriate singular adiabatic problems. It is shown how to obtain the given data of a suitable adiabatic problem from resistive data. The crucial connecting link between the two MHD models is the flux function μ(ψ), which appears as part of the given data in the adiabatic formulation. This is given directly by resistive information outside the sheath and, inside the sheath region, indirectly by extrapolation of the observed behavior of low-resistivity solutions of the Grad--Hogan equations