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

where and are the energy and the waveaction fluxes respectively and and are constants, and . 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 which follows from WKE (20). Then, the general steady state solution to (14) is

(27) |

where is the integral exponential function. At the tail we have

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

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].