Magnetohydrodynamic stability at a separatrix. II. Determination by new conformal map technique

Abstract

It was shown in the first part to this paper how a simple magnetohydrodynamic model can be used to determine the stability of a tokamak plasma’s edge to peeling (external kink) modes. It was found that stability is determined by the value of Δ′, a normalized measure of the discontinuity in the radial derivative of the radial perturbation to the magnetic field at the plasma-vacuum interface. To avoid the possibility that numerical divergences near the X-point might lead to misleading conclusions about plasma stability, this paper calculates the value of Δ′ analytically. This is accomplished by showing that the method of conformal transformations can be applied to systems with a continuously varying nonzero boundary condition and using the technique to obtain analytical expressions for both the vacuum energy and Δ′. A conformal transformation is also used to obtain an equilibrium vacuum field surrounding a plasma with a separatrix and X-point. This allows the analytical expressions for the vacuum energy and Δ′ to be evaluated. The results here, combined with those in the first part of this paper, subsequently provide a quantitative description of the peeling mode’s growth rate as the plasma-vacuum boundary more closely approximates a separatrix with an X-point.