The Mercier Criterion

Abstract

We have presented a method to test for instabilities with low mode numbers m and n, such as kink modes. Both theoretical analysis and calculations of axially symmetric equilibria suggest that modes with high m and n, including interchange modes, may be equally critical in the search for stable high pressure configurations [34]. They can be assessed via the Mercier criterion, which is concerned with modes localized about some rational surface [39]. The assumption is made that the plasma is covered by a family of nested flux surfaces, which we have seen to be rather tenuous when the geometry is truly three-dimensional [28]. Specialized variations δW of the energy E_P are introduced that are restricted to the neighborhood of a rational surface or a closed magnetic line. Then a necessary condition for stability is derived by careful optimization of δW. A preliminary version of this condition involves integrals over the relevant closed line, but considerations about the ergodicity of magnetic lines on a surface with irrational rotational transform lead to a better version involving surface integrals that occur naturally in our model of magnetohydrodynamic equilibrium. For details we refer to the book by Mercier and Luc [39], noting only that their derivation is somewhat at odds with the resonances that occur at rational surfaces in three dimensions. More specifically, the analysis suggests that in the neighborhood of a rational surface, lower energy levels can be achieved by equilibrium solutions that have island structure.