Generalized Mercier stability criterion for stellarators

The Mercier criterion is a well-known stability criterion for tokamaks. It is derived from a 2 x 2 matrix eigenvalue problem arising from the expansion of resonant solutions about a singular surface where m -nq = 0, with m and n being the poloidal and toroidal mode numbers, respectively, and q being the safety factor. The stability criterion is that the eigenvalues must be real, otherwise, the solution oscillates, violating the Newcomb crossing criterion. Because of the non-axisymmetry of stellarators, different toroidal as well as poloidal harmonics couple to each other. It follows that each singular surface can have multiple resonant harmonics, with multiplicity M = 1. The corresponding matrix eigenvalue problem involves a 2M x 2M matrix, resulting in M pairs of positive and negative eigenvalues. The generalized stability criterion is that all eigenvalues must be real. While the original Mercier criterion can be expressed in terms of quadratures of equilibrium quantities over the singular surface, which can be evaluated anywhere, the generalized Mercier criterion can only be evaluated on rational q surfaces with a given set of resonant harmonics.