Spectral diffusion in wave turbulence systems with nonlocal interaction.

Often in wave turbulence systems, the picture of the classical local energy cascade breaks down due to concentration of energy at the largest scales. This is a classical wave analogue of the Bose-Einstein condensation phenomenon which renders the Kolmogorov-Zakharov spectra irrelevant. In these situations, the dominant wave resonances involve motions with large space and time scale separations, which yields to reduced descriptions via the diffusion-type evolution in the k-space. I will review previous works applying these ideas for the three-wave internal waves and the Rossby waves, and report on our recent work on the four-wave systems, the gravity water waves and the MMT waves. I will empasize important differences between the concave up and concave down dispersion relations in terms of dominance of the angular or radia spectrall diffusion.