Sandpile behavior in discrete water - wave turbulence

I construct a sandpile model for evolution of the energy spectrum of the water waves in finite basins. This model takes into account loss of resonant wave interactions in discrete Fourier space and restoration of these interactions at larger nonlinearity levels. For weak forcing, the waveaction spectrum takes a critical ω shape where the nonlinear resonance broadening overcomes the effect of the Fourier grid spacing. The energy cascade in this case takes form of rare weak avalanches on the critical slope background. For larger forcing, this regime is replaced by a continuous cascade and Zakharov-Filonenko ω waveaction spectrum. For intermediate forcing levels, both scalings will be relevant, ω at small and ω at large frequencies, with a transitional region in between characterised by strong avalanches.