Water-wave turbulence: statistics beyond the spectra

We perform numerical simulations of the dynamical equations for free water surface in presence of gravity in order to compare the numerically obtained statistics with the assumptions and predictions of the Wave Turbulence (WT) theory. Such a theory was derived under a weak nonlinearity assumption and in the infinite basin limit. Thus, its robustness is not obvious for larger nonlinearity levels which are necessary in numerics to activate the cascade in the discrete $k$-space and to reduce the computational time to an affordable level. As in several other recent numerical studies \cite{DKZ,onorato,naoto}, we confirm the Zakharov-Filonenko spectrum predicted by WT. We also go beyond finding the spectra and compute the one-mode probability density function (PDF) of the wave amplitudes. In agreement with recently developed theory \cite{clnp,cln}, this PDF is found to show an anomalously large, with respect to Gaussian fields, probability of strong waves. Also in agreement with the theory, we find that phases $\phi_k$ are correlated whereas factors $e^{i\phi_k}$ are de-correlated. However, at odds with the traditional WT picture, we find that at each $k$ there are two types of oscillations present: a dominant component at the linear frequency and a weaker one with a nonlinear frequency shift. We further observe that the energy cascade is very ``bursty'' in time and is somewhat similar to sporadic sandpile avalanches. We explain this as a cycle: a cascade arrest due to discrete $k$'s leads to accumulation of energy near the forcing scale which, in turn, leads to widening of the nonlinear resonance and, therefore, triggering of the cascade draining the turbulence levels and returning the system to the beginning of the cycle.