Energy of linear quasi-neutral electrostatic drift waves

Certain kinds of nonlinear instabilities are related to the existence of negative-energy perturbations. In this paper an exact energy expression for linear quasi-neutral electrostatic perturbations is derived within the framework of dissipationless multi-fluid theory that is valid for any geometry. Taking the mass as a tensor with, in general, different masses parallel and perpendicular to an ambient magnetic field allows one to treat the full dynamics and also restrict consideration to parallel dynamics or to the completely adiabatic case. Application to slab configurations yields the result that the adiabatic approximation does not allow negative-energy perturbations, whereas inclusion of the parallel dynamics does. This is in agreement with a numerical study of drift-wave turbulence within the framework of collisional two-fluid theory by B. Scott. A dissipationless theory can be formulated in terms of a Lagrangian, from which the energy is immediately obtained. We start with the nonlinear theory, the linearized theory is written in terms of displacement vectors Ev(x, t) such that all constraints are taken into account, as in ideal MHD theory. The procedure used is the same as that developed in a forthcoming paper by Pfirsch and Sudan for ideal MHD theory, which also allows one to obtain the exact nonlinear energy with all the constraints built in. People working in this field have hitherto considered only (positive semi-definite) expressions for the nonlinear energy, which do not immediately allow one to investigate the existence of linear negative-energy perturbations relating to non-linear instability. (orig.)