In the first part to this papercite{part1} it was shown how a simple Magnetohydrodynamic model could be used to determine the stability of a Tokamak plasmas edge to a Peeling (External Kink) mode. Stability was found to be determined by the value of
$Delta$, a normalised measure of the discontinuity in the radial derivative of the radial perturbation to the magnetic field at the plasma-vacuum interface. Here we calculate $Delta$, but in a way that avoids the numerical divergences that can arise near a separatrices X-point. This is accomplished by showing how the method of conformal transformations may be generalised to allow their application to systems with a non-zero boundary condition, and using the technique to obtain analytic expressions for both the vacuum energy and $Delta$. A conformal transformation is used again to obtain an equilibrium vacuum field surrounding a plasma with a separatrix. This allows the subsequent evaluation of the vacuum energy and $Delta$. For a plasma-vacuum boundary that approximates a separatrix, the growth rate $gamma$ normalised by the Aflven frequency $gamma_A$ is then found to have $ln(gamma/gamma_A)=-{1/2} ln (q/q)$. Consequences for Peeling mode stability are discussed.
The rapid deposition of energy by Edge Localised Modes (ELMs) onto plasma facing components, is a potentially serious issue for large Tokamaks such as ITER and DEMO. The trigger for ELMs is believed to be the ideal Magnetohydrodynamic Peeling-Balloon
ing instability, but recent numerical calculations have suggested that a plasma equilibrium with an X-point - as is found in all ITER-like Tokamaks, is stable to the Peeling mode. This contrasts with analytical calculations (G. Laval, R. Pellat, J. S. Soule, Phys Fluids, {bf 17}, 835, (1974)), that found the Peeling mode to be unstable in cylindrical plasmas with arbitrary cross-sectional shape. However the analytical calculation only applies to a Tokamak plasma in a cylindrical approximation. Here, we re-examine the assumptions made in cylindrical geometry calculations, and generalise the calculation to an arbitrary Tokamak geometry at marginal stability. The resulting equations solely describe the Peeling mode, and are not complicated by coupling to the ballooning mode, for example. We find that stability is determined by the value of a single parameter $Delta$ that is the poloidal average of the normalised jump in the radial derivative of the perturbed magnetic fields normal component. We also find that near a separatrix it is possible for the energy principles $delta W$ to be negative (that is usually taken to indicate that the mode is unstable, as in the cylindrical theory), but the growth rate to be arbitrarily small.
