An Energy Integral and Variational Principle for Plasma Kinetic Modes

An energy principle is extended to include kinetic processes, finite gyroradius effects and wave-particle interactions, in electromagnetic instabilities in a toroidal plasma. The present arguments are based on the energy integrals which are derived from the eigenmode equations for electromagnetic perturbations. First, the finite gyroradius effects are included in the energy integral and it is shown that this integral can be used for determining the eigenmode by minimizing the excess energy of the perturbed state. The generalized stability conditions for kink-like mode and ballooning mode are obtained by use of this improved energy principle. For the case where wave-particle interaction is important, the energy integral becomes a rate equation of energy exchange, and a variational form can independently be constructed for the mode analysis.