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

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.