Linearized analysis of inhomogeneous plasma equilibria: General theory

A generalized framework is presented for analyzing the linearized equations for perturbations of inhomogeneous plasma equilibria in which there is a collisionless species, some properties of the solutions of the linearized equations are described, and a basis is provided for numerical computations of the linearized properties of such equilibria. It is useful to expland the perturbation potentials in eigenfunctions of the field operator which appears in the linearized equations, and to define a dispersion matrix whose analytical properties determine the nature of the solutions of the initial-value problem. It is also useful to introduce auxiliary functions to replace the usual perturbation distribution functions, and to expand the auxiliary functions in eigenfunctions of the equilibrium Liouville operators. By introducing the auxiliary functions, great freedom is achieved in the choice of the field operator which appears in the linearized equations. This freedom can be useful in some problems to define expansion functions for the potentials that are particularly suitable for studying specific normal modes.