We have established above that the one-point statistics is at the heart of the WT theory. All one-point statistical objects can be derived from the one-point amplitude generating function,
First-order PDE (80) can be easily solved by the method of characteristics. Its steady state solution is
At the tail of the PDF, , the solution can be represented as series in ,
Note that if the weakly nonlinearity assumption was valid uniformly to then we had to put to ensure positivity of and the convergence of its normalisation, . In this case which is a pure Rayleigh distribution corresponding to the Gaussian wave field. However, WT approach fails for for the amplitudes for which the nonlinear time is of the same order or less than the linear wave period and, therefore, we can expect a cut-off of at . Estimate for the value of can be obtained from the dynamical equation (14) by balancing the linear and nonlinear terms and assuming that if the wave amplitude at some happened to be of the critical value then it will also be of similar value for a range of 's of width (i.e. the -modes are strongly correlated when the amplitude is close to critical). This gives:
Depending on the position in the wavenumber space, the flux can be either positive or negative. As we discussed above, results in an enhanced probability of large wave amplitudes with respect to the Gaussian fields. Positive mean depleted probability and correspond to the wavebreaking value which is closer to the PDF core. When gets into the core, one reaches the wavenumbers at which the breakdown is strong, i.e. of the kind considered in . Consider for example the water surface gravity waves. Analysis of  predicts strong breakdown in the high- part of the energy cascade range. According to our picture, these high wavenumbers correspond to the highest positive values of and, therefore, the most depleted PDF tails with respect to the Gaussian distribution. When one moves away from this region toward lower 's, the value of gets smaller and, eventually, changes the sign leading to enhanced PDF tails at low 's. This picture is confirmed by the direct numerical simulations of the water surface equations reported in  the results of which are shown in figure 1.
This figure shows that at a high the PDF tail is depleted with respect to the Rayleigh distribution, whereas at a lower it is enhanced which corresponds to intermittency at this scale. Similar conclusion that the gravity wave turbulence is intermittent at low rather than high wavenumbers was reached on the basis of numerical simulations in . To understand the flux reversal leading to intermittency appears in the one-mode statistics, one has to consider fluxes in the multi-mode phase space which will be done in the next section.