Time asymptotic perturbation theory in weak plasma turbulence I. Macroscopic description

The time asymptotic method of non-linear mechanics (Bogoljubov & Mitropoiski 1961) is used to solve the hierarchy equations of low-amplitude wave correlations. We start with a formal description of individual dispersive and undamped waves and derive the usual kinetic equation for the energy spectrum and the three-wave correlation in the lowest order (three-wave processes). We show that the equations are automatically closed (‘Quasi-Gaussian’) even in the case where the correlations have the same order of magnitude as the corresponding moments. This approach parallels the multiple-time formalism of Sandri and Frieman, but it represents an alternate systematic and unique method (without ‘extension’ in the sense of Sandri).