A new derivation of quasilinear theory

Abstract

A new derivation of quasilinear theory for plasma oscillations is given, subject to the neglect of wave-wave interactions. The theory, which applies equally to damping and growing waves, requires that the individual wave resonances be narrow and the electric field spectrum sufficiently broad.

Comment on revised quasilinear equations dealing with wave-number spectra

“…The first term will be positive for a large range of v if y fc is negative over a small range. A proof that D(v, t) is always positive has been suggested as necessary by Montgomery & Bodner (1971), but is not at present available and appears to be impossible without further approximations.…”

“…We can regard the second form of D (v, t) as a function of a complex variable where k is now a complex variable also. The integral is nearly of the Cauchy form except that the complex frequency (o k + iy k is a function of the integration variable k. However, the analytic continuation of D for y k < 0 just involves the residue at the pole k'v -w k > -iy k > = 0 and we can expand the dominator about k x k' and recover the Cauchy integral for the resonant electrons if the various approximations used by Klozenberg & Bernstein (1970) are valid.…”

Reformulation of quasi-linear theory

“…, Vahala & Montgomery (1970), Montgomery & Bodner (1971), Bernstein & Klozenberg (1971), Abraham-Shrauner (1971a, b) and Bodner (1971) attack the difficulty that standard plasma quasi-linear theory (for a survey see Sagdeev & Galeev 1969) encounters in dealing with damped waves. Yet the quantum derivation (Harris 1970) appears to have no such difficulty, and to be applicable equally to growing and damped waves.…”