Transition from a coherent three wave system to turbulence with application to the fluid closure

We start from a Mattor-Parker system and its generalization to include diffusion and derive the Random Phase equations. It is shown that the same type of fluid closure holds in the coherent and turbulent regimes. This is due to the fact that the Random Phase levels (1/I ₁ = 1/I ₂ + 1/I ₃), where I_j is the intensity of wave packet `j', are attractors for the wave dynamics both in the coherent and incoherent cases. Focus here is on the wave dynamics with phase velocities varying due to nonlinear frequency shifts. Thus a Maxwellian distribution function is kept in all cases.