Marginal Stability Diagrams for Infinite-n Ballooning Modes in Quasi-symmetric Stellarators

By perturbing the pressure and rotational-transform profiles at a selected surface in a given equilibrium, and by inducing a coordinate variation such that the perturbed state is in equilibrium, a family of magnetohydrodynamic equilibria local to the surface and parameterized by the pressure gradient and shear is constructed for arbitrary stellarator geometry. The geometry of the surface is not changed. The perturbed equilibria are analyzed for infinite-n ballooning stability and marginal stability diagrams are constructed that are analogous to the (s; alpha) diagrams constructed for axi-symmetric 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 (National Compact Stellarator Experiment), a quasi-poloidally-symmetric configuration and the quasi-helically-symmetric HSX (Helically Symmetric Experiment) are presented. 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 analyzed. The results are seemingly consistent with the simple explanation: ballooning instability results when the local shear is small in regions of bad curvature. 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 Schroedinger form of the ballooning equation