MHD Instabilities in Tokamaks

A general introduction to ideal magnetohydrodynamic (MHD) stability of tokamak plasmas is given, using linear perturbations of the ideal MHD equations. Subsequently the Energy Principle for ideal MHD instabilities is derived. The specific instabilities which are then discussed are loosely divided into two categories. Under the name "current driven instabilities", external and internal kink modes, which are modes with a large radial extent, are discussed. The internal m = 1 kink mode is responsible for sawtooth collapses and fishbone oscillations in tokamaks. Under the header "pressure driven instabilities", more localized modes are presented. These modes may limit the pressure gradient in the plasma without causing sizeable disruptions. The ballooning limit and the Mercier criterion are presented. The Troyon limit is mentioned as a synthesis of several of these stability boundaries.