Discussions

In this paper, we used direct numerical simulations of the free water surface in order to examine the statistical properties of the water-wave field beyond the energy spectrum. Our first aim was to check recent predictions of the WT theory about the PDF and intermittency, about the character of correlations of the wave amplitudes and phases. We particularly focused on the question how the effects of discreteness and finite nonlinearity change statistics with respect to the WT closure developed for weak nonlinearities and for a continuous wavenumber space.

Firstly, following [1,2,3] we see formation of a quasi-steady spectrum consistent with the Zakharov-Filonenko spectrum predicted by WT. Secondly, we measured PDF for the wave amplitudes and observe an anomalously large, with respect to Gaussian fields, probability of strong waves. This result is in agreement with recent theoretical predictions of [4,5]. Thirdly, we measure correlations for the amplitudes, phases and phase factors and we observe agreement with predictions of [4,5]. Namely, the amplitude and the phase correlations behave as statistically independent variables, whereas the phases develop strong auto-correlations over the nonlinear time. Note that these properties are fundamental for the WT closure to work, so in a way we provide a numerical validation for the WT approach.

We also find that at each $k$ there are two sharp frequency peaks: a dominant one at the linear frequency and a weaker one with a frequency shift arising due to the $k/2$-mode. Somewhat related to this two-peak frequency structure is the observed time behavior of the phase. We observe calm periods during which the phase oscillates within $2 \pi$-wide margins intermittent with sudden phase runs during which it experiences a monotonic change significantly greater than $2 \pi$.

Finally, we observe that the energy cascade is ``bursty'' in time and is somewhat similar to sporadic sandpile avalanches. We give a plausible explanation of this behavior as an interplay of effects of discreteness and nonlinearity. Because in between of the avalanche discharges the resonances are absent then, at least qualitatively, one can refer to the KAM theory and say that the evolution should remain close to the corresponding integrable case, - the linear system in our case. This picture is supported by a simple analysis of quasi-resonances given in this paper which indicates that there exists a single threshold value of turbulence intensity at the forcing scale separating the no-cascade and unlimited (in $k$) cascade regimes.

A further numerical study of the avalanche effect is desirable, particularly using a different wavenumber grid and using a more direct method of measuring the turbulent flux and its correlations for different inertial interval points.