Theory of the M = 1 Kink Mode in Toroidal Plasma

The energy principle of ideal magnetohydrodynamics (MHD) is used to study the ideal MHD stability of the m = 1 internal kink mode in a toroidal plasma. The equilibrium configurations that are considered allow for a broad region where the safety factor q is close to unity. This region may extend to the magnetic axis, or may be a singular layer. The minimization of the energy functional yields an implicit equation for the growth rate that can be solved by simple numerical means. The examples that are treated numerically retain the essential features of experimentally expected q profiles. The growth rate depends very sensitively on the q profile close to unity and increases with the width of the q almost-equal-to 1 layer. The highest values are of the order of the inverse aspect ratio epsilon divided by the poloidal Alfven time. Nonmonotonic profiles with q > 1 on axis are more unstable than monotonic q profiles with q < 1. In the latter case, the mode tends to be localized in the q almost-equal-to 1 region instead of in the center.