Re-normalized quasi-linear approximation of plasma turbulence: Part 1. Modification of the Weinstock weak-coupling limit

A re-normalized expansion is presented for the solution of the fluctuating Vlasov equation for turbulent plasmas. Its first term can be identified with the Weinstock result in the weak-coupling approximation. The use of time-ordering operators allows one to solve the Weinstock U A evolution operator explicitly. Detailed results are given for the weak-coupling approximation, presented here as a simple re-normalization of the quasi-linear theory. Results of Weinstock and Birmingham & Bornatici, concerning the frequency shift, resonance broadening and Dupree damping appear to be deeply modified by similar corrective terms. These arise from a consistent handling of expressions involving non-commuting operators.