Stability of general plasma equilibria - I formal theory

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Abstract
A method is described for the detailed investigation of electrostatic instabilities in real experimental geometries. These have frequently been discussed in the plane slab model, and modifications of it, but the present work includes all geometrical effects from the outset. The starting point is the collisionless Boltzmann equation with the approximation that the scale length of the equilibrium is long compared to the ion gyro radius. The main interest is in perturbations of low frequency but of arbitrary wavelength, which may be comparable to the ion Larmor radius. Thus several instabilities such as drift wave, flute or trapped particle, come within the scope of the theory. Expressions are first obtained for the contribution to the charge density produced by an arbitrary electrostatic perturbation affecting particles whose unperturbed orbits are (i) trapped between magnetic mirrors; (ii) circulating around closed field lines; (iii) tracing out a magnetic surface. Together with Poisson's equation these expressions lead, via the appropriate Nyquist contours, to stability criteria valid for arbitrary equilibria. Finally it is shown how this method leads to a differential equation whose solution will determine the stability of an experimental configuration such as the multipole.