Sandpile behaviour in discrete water-wave turbulence

Abstract

We construct a sandpile model for evolution of the energy spectrum of the water surface waves in finite basins. This model take 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 $\omega^{-10}$ shape where the nonlinear resonance broadening is equal to 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 $\omega^{-8}$ waveaction spectrum. For intermediate forcing levels, both scalings will be seen: $\omega^{-10}$ at small and $\omega^{-8}$ at large frequencies. The interval between these two scaling regions is characterised by strong avalanches.