Ideal ballooning instabilities in low-shear stellarators

This paper is concerned with the analysis of the field-line dependence of eigenvalues of the ballooning equation in systems where the global magnetic shear is weak, i.e. when |q'(s)| << 0, 's' being a dimensionless magnetic surface label and 'q' being the safety factor (inverse rotational transform), both assumed to be of order unity. Narrow regions where |q'(s)| << 0, occur in the shear-reversal lapsers of advanced tokamaks and high-β heliotron/torsatrons, but we are primarily concerned with cases where the global magnetic shear is small everywhere, in particular with helical-axis stellarators such as the three-field-period ANU H-1 heliac. Speak shear has a profound effect because it diminishes the influence of the 'secular terms' in the ballooning equation, which are normally responsible for localisation of the ballooning eigenfunctions in systems with strong shear. If the ballooning eigenfunction is localized in the limit q' = 0 then it is reasonable to suppose that the behaviour at small q' may be understood by constructing a perturbation theory about this limit, and it is the purpose of this paper to sketch how this can be done