Time asymptotic perturbation theory in weak plasma turbulence II. Vlasov description

In electrostatic turbulence under appropriate conditions we show that we can have an equation for ϕk (time variation of electrostatic potential amplitude) which is essentially instantaneous in time. The coefficients in this equation are uniquely determined by the Poisson equation of the corresponding order. By applying time asymptotic methods of Bogoljubov and Mitropoiski to the corresponding hierarchy equations we obtain a kinetic equation for the spectrum 〈|ϕk|2〉. The resonant mode coupling terms agree with the usual ones. But the non-linear Landau damping terms, as given, e.g. by Frieman & Rutherford (1964), Kadomtsev (1965), Tsytovich (1968), and Rogister & Oberman (1969) are complemented by additional terms which are, at least formally, non-zero. For simplicity the calculations are only done for the weakly unstable (and weakly damped) modes.