The evolution of wave correlations in uniformly turbulent, weakly non-linear systems

A formalism is developed describing the time evolution of wave correlations in a uniformly turbulent ensemble of weakly non-linear systems. The statistics are built into the formalism a priori. With closure (of the hierarchy of equations for wave correlations) appropriate to the inclusion of resonant three-wave interactions, a (non-linear) kinetic equation for the two-wave correlations is derived and various properties of this equation are discussed. The effects of both bilinear and trilinear non-linearities are considered. Modifications of the formalism in situations where there is a weak (linear) instability γ_к, (γ_к/ω_к 1) are also considered.