Marginal stability diagrams for infinite-n ballooning modes in quasi-symmetric stellarators

Abstract

By varying the pressure-gradient and average shear at a selected surface in a given arbitrary stellarator equilibrium and by inducing a coordinate variation such that the perturbed state remains in equilibrium, a family of magnetohydrodynamic equilibria local to the surface is constructed. The equilibria are parameterized by the pressure-gradient and averaged magnetic shear. The geometry of the surface is not changed. The perturbed equilibria are analysed for infinite-n ballooning stability and marginal stability diagrams are constructed that are analogous to the (s, α) diagrams constructed for axisymmetric configurations.The method describes how pressure and rotational-transform gradients influence the local shear, which in turn influences the ballooning stability. Stability diagrams for the quasi-axially symmetric NCSX, a quasi-poloidally symmetric configuration and the quasi-helically symmetric HSX are compared. Regions of second-stability are observed in both NCSX and the quasi-poloidal configuration, whereas no second-stable region is observed for the quasi-helically symmetric device.To explain the different regions of stability, the curvature and local shear of the quasi-poloidal configuration are analysed. The results are seemingly consistent with the following simple explanation: ballooning instability results when the local shear is small in regions of bad curvature with sufficient pressure-gradient. Examples will be given that show that the structure and stability of the ballooning mode is determined by the structure of the potential function arising in the Schrödinger form of the ballooning equation.