Ideal MHD stability of internal kinks in circular and shaped tokamaks

Stability limits for the internal kink mode are calculated for tokamaks with different current profiles and plasma cross-sections using ideal magnetohydrodynamics (MHD). The maximum stable poloidal beta at the q = 1 surface (beta(p)) is sensitive to the current profile, but for circular cross-sections it is typically between 0.1 and 0.2. Large aspect ratio theory gives similar predictions when the appropriate boundary conditions are applied at the plasma-vacuum surface. It is found that the internal kink is significantly destabilized by ellipticity. For JET geometry, the ideal MHD limit in beta(p) is typically between 0.03 and 0.1, and arbitrarily low limits can result if the shear is reduced at the q = 1 surface. A large aspect ratio expansion of the Mercier criterion retaining the effects of ellipticity and triangularity is used to illustrate the destabilizing influence of ellipticity.