Binary correlation and phase space granulation in a turbulent plasma

Abstract

The time evolution of the average plasma distribution function is essentially determined by the coUective binary correlation ( 91 (x 1, v v t) 3f (x2, v2, t) ), where 91 is the fluctuating part of the distribution function. In the quasilinear theory, in the presence of a broad spectrum, the free stream part of the binary correlation is shown to phase mix to zero within a short interval of time (Davidson 1972) while in the resonance velocity domain the coherent part shows a strong positive correlation ,-,,O(~-2k), where ~k is the growth rate, but is devoid of any fine grained spatial structure. On the other hand a sufficiently narrow spectrum of a BGK wave shows a strong correlation as well as phase space structure. Dupree (1972) has pointed out an interesting departure from the above results when nonlinear interaction of particles with a wide spectrum of waves is taken into consideration. For a wide spectrum turbulence, the concept of trapping of a particle by the entire wave field is indeed vague. The particle 1 (x 1, v 0 is trapped only by the portion (k + 3k, ~o~+,k) ir'Ir 11 = oJJk and [ 91 I < I kl; with respect to the remaining portion of the spectrum it is untrapped. So the renormalized quasilinear (RQL) approximation (Misguich and Balescu 1975a) is not expected to yield an appreciably different result from that of the quasilinear theory so lar as the normal modes are eoneerned. But the situation changes when we attempt to calculate ( 91 t) 91 t)). The particle 2(x2, v2) is trapped by the free streaming mode (k, k'v 0 whenever v 2 .-. v 1 for all k belonging to the spectrum. The contribution to the correlation arising from the resonant interaction with the free streaming mode may therefore become important and on applying the RQL approximation theory, we find that i t i s indeed so. Two particle streams, each resonant with a wave of the same speed, modulate each other and give rise to phase space structure, although on a scale much smaller than those generated e.g., by trapped particles in a BGK wave. The purpose of the present paper is to obtain an expression for the collective binary correlation and determine the so caUed ' c lump' part in the RQL approximation theory. The generalized Dupree damping, in which is incorporated the effect of the turbulence mediated change in the slope of the distribution function, is found to