Sandpile behaviour in discrete water-wave turbulence

I construct a sand pile model for the evolution of the energy spectrum of 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 wave action spectrum takes a critical omega(-10) shape where the nonlinear resonance broadening overcomes the effect of the Fourier grid spacing. The energy cascade in this case takes the form of rare weak avalanches on the critical slope background. For larger forcing, this regime is replaced by a continuous cascade and a Zakharov-Filonenko omega(-8) wave action spectrum. For intermediate forcing levels, both scalings will be relevant omega(-10) at small frequencies and omega(-8) at large frequencies, with a transitional region in between, characterized by strong avalanches.