Formulation of a Statistical Theory of Strong Plasma Turbulence

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Abstract
A new formulation of methods introduced by Dupree, and Orszag and Kraichnan, for solving the Vlasov equation for turbulent plasmas based on. algebraic use of an averaging operator is described. Formally exact sets of integrodifferential equations are thereby derived for determining the ensemble average of the one‐particle distribution function, and the electric field spectrum of turbulent plasmas. The formal difference between these equations and the approximate equations of Dupree, and Orszag and Kraichnan, is that the average “Vlasov” propagator U(t,t 0 ) of the latter is replaced by a new propagator U A (t,t 0 ) which involves the averaging operator. The potential usefulness of the present equations can be judged by the fact that, in a simple lowest‐order limit, they immediately reduce to the turbulence equation of Dupree. These equations are used to develop a modified perturbation theory which avoids certain time secularities. Cumulant expansions are introduced to evaluate average “Vlasov” propagators, and to rigorously treat the variation of the diffusion coefficient with velocity. It is explicitly shown that (diffusive) perturbed trajector corrections are equal to mean square deviations from the mean of “Vlasov” trajectories.