Magnetohydrodynamic Stability at a Separatrix: Part I

TL;DR

This paper re-examines the stability of the Peeling mode at a separatrix in tokamaks, generalizing previous cylindrical models to realistic geometries, and finds that stability depends on a single parameter with near-separatrix modes potentially having very slow growth rates.

Contribution

It extends the analysis of Peeling mode stability from cylindrical to arbitrary tokamak geometries, providing a more accurate criterion based on a single parameter $ riangle'$.

Findings

Stability is governed by the parameter $ riangle'$ related to the magnetic field jump.

Near a separatrix, modes can have negative energy but very slow growth rates.

The generalized equations describe the Peeling mode independently of coupling to ballooning modes.

Abstract

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-Ballooning 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 field's normal component. We also find that near a separatrix it is possible for the energy principle's $\delta W$ 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.