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We report results of sumulation of wave turbulence. Both inverse and direct cascades are observed. The definition of "mesoscopic turbulence" is given. This is a regime when the number of modes in a system involved in turbulence is high enough to qualitatively simulate most of the processes but significantly smaller then the threshold, which gives us quantitative agreement with the statistical description, such as kinetic equation. Such a regime takes place in numerical simulation, in essentially finite systems, etc. : 02.60Cb, 47.11.+j, 47.35.+i, 47.27.Eq The theory of wave turbulence is developed for infinitely large system. In weakly nonlinear dispersive media, the turbulence is described by a kinetic equation for squared wave amplitudes (weak turbulence). However, all real systems are finite. Computer simulation of wave turbulence can also be perfomed only in finite system (typically in a box with periodic boundary conditions). It is important to know how strong discreteness of a system impacts the physical picture of wave turbulence. PACSLet a turbulence be realized in a Q-dimensional cube with side L. Then, wave vectors form a cubic lattice with the lattice constant ∆k = 2π/L. Suppose that four-wave resonant conditions are dominating. Exact resonances satisfy the equationsIn infinite medium, Eqs. (1) and (2) In a finite system, (1) and (2) are Diophantine equations which might have or have no exact solutions. The Diophantine equation for four-wave resonant processes are not studied yet. For three-wave resonant processes, they are studied for Rossby waves on the β-plane [1]. However, not only exact resonances are important. Individual harmonics in the wave ensemble fluctuate with inverse time Γ k , dependent on their wavenumbers. Suppose that all Γ ki for waves, composing a resonant quartet, are of the same order of magnitude Γ ki ∼ Γ. Then resonant equation (2) has to be satisfied up to 1) e-mail: kao@landau.ac.ru accuracy ∆ ∼ Γ, and the resonant surface is blurred into the layer of thickness δk/k ≃ Γ k /ω k . This thickness should be compared with the lattice constant ∆k. Three different cases are possible 1. δk ≫ ∆k. In this case the resonant layer is thick enough to hold many approximate resonant quartets on a unit of resonant surface square. These resonances are dense, and the theory is close to the classical weak turbulent theory in infinite media. The weak turbulent theory offers recipes for calculation of Γ k . The weak-turbulent Γ k are the smallest among all given by theoretical models. To be sure that the case is realized, one has to use weak-turbulent formulae for Γ k .2. δk < ∆k. This is the opposite case. Resonances are rarefied, and the system consists of a discrete set of weakly interacting oscillators. A typical regime in this situation is the "frozen tur-, which is actually a system of KAM tori, accomplished with a weak Arnold's diffusion.3. The intermediate case δk ≃ ∆k can be called "mesoscopic turbulence". The density of approximate resonances is high enough to provide the energy tran...

Abstract.The results of theoretical and numerical study of the Hasselmann kinetic equation for deep water waves in presence of wind input and dissipation are presented. The guideline of the study: nonlinear transfer is the dominating mechanism of wind-wave evolution. In other words, the most important features of wind-driven sea could be understood in a framework of conservative Hasselmann equation while forcing and dissipation determine parameters of a solution of the conservative equation. The conservative Hasselmann equation has a rich family of self-similar solutions for duration-limited and fetch-limited wind-wave growth. These solutions are closely related to classic stationary and homogeneous weak-turbulent Kolmogorov spectra and can be considered as non-stationary and non-homogeneous generalizations of these spectra. It is shown that experimental parameterizations of wind-wave spectra (e.g. JONSWAP spectrum) that imply self-similarity give a solid basis for comparison with theoretical predictions. In particular, the selfsimilarity analysis predicts correctly the dependence of mean wave energy and mean frequency on wave age C p /U 10 . This comparison is detailed in the extensive numerical study of duration-limited growth of wind waves. The study is based on algorithm suggested by Webb (1978) that was first realized as an operating code by Perrie (1989, 1991). This code is now updated: the new version is up to one order faster than the previous one. The new stable and reliable code makes possible to perform massive numerical simulation of the Hasselmann equation with different models of wind input and dissipation. As a result, a strong tendency of numerical solutions to self-similar behavior is shown for rather wide range of wave generation and dissipation conditions. We found very good quantitative coincidence of these solutions with available results on duration-limited growth, as well as with experimental parametrization of fetch-limited spectra JONSWAP in terms of wind-wave age C p /U 10 .

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