Steady state solutions.

Steady power-law solutions of WKE which correspond to a direct cascade of energy and an inverse waveaction cascade are,

$$n(k) = C_{1}P^{1/3}k^{-4}$$ (25)

$$n(k) = C_{2}Q^{1/3}k^{-23/6},$$ (26)

where $P$ and $Q$ are the energy and the waveaction fluxes respectively and $C_1$ and $C_2$ are constants, and $k = \vert{\bf k}\vert$. The first of these solutions is the famous ZF spectrum [8] and it has a great relevance to the small-scale part of the sea surface turbulence. It has been confirmed in a number of recent numerical works [1,2,3], but we will also confirm it in our simulation.

Now, let us consider the steady state solutions for the one-mode PDF. Note that in the steady state $\gamma /\eta = n$ which follows from WKE (20). Then, the general steady state solution to (14) is

$$P=\hbox{const} \, \exp{(-s/n)} -({F}/{\eta}) Ei({s}/{n}) \exp{(-s/n)},$$ (27)

where $Ei(x)$ is the integral exponential function. At the tail $s \gg n_k$ we have

$$P \to - \frac{F}{s\gamma}$$ (28)

if $F \ne 0$. The $1/s$ tail decays much slower than the exponential (Rayleigh) part and, therefore, it describes strong intermittency. On the other hand, $1/s$ tail cannot be infinitely long because otherwise the PDF would not be normalizable. As it was argued in [5], the $1/s$ tail should with a cutoff because the WT description breaks down at large amplitudes $s$. This cutoff can be viewed as a wavebreaking process which does not allow wave amplitudes to exceed their critical value, $P(s) =0$ for $s > s_{nl}$.

Relation between intermittency and a finite flux in the amplitude space was observed numerically also for the Majda-Mc-Laughlin-Tabak model by Rumpf and Biven [22].