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...

Complex analytical structure of Stokes wave for two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave propagating with the constant velocity. Simulations with the quadruple (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with 2π/3 radians angle on the crest. A conformal map is used which maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase of the numerical precision and subsequent increase of the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one branch cut per horizontal spatial period λ of Stokes wave. Each branch cut extends strictly vertically above the corresponding crest of Stokes wave up to complex infinity. The lower end of branch cut is the square-root branch point located at the distance v c from the real line corresponding to the fluid surface in conformal variables. The increase of the scaled wave height H/λ from the linear limit H/λ = 0 to the critical value H max /λ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave (also called by the Stokes wave of the greatest height). Here H is the wave height from the crest to the trough in physical variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least 10 −26 . The tables use from several poles for near-linear Stokes wave up to about hundred poles to highly nonlinear Stokes wave with v c /λ ∼ 10 −6 .

We study the long-time evolution of surface gravity waves on deep water excited by a stochastic external force concentrated in moderately small wave numbers. We numerically implemented the primitive Euler equations for the potential flow of an ideal fluid with free surface written in Hamiltonian canonical variables, using the expansion of the Hamiltonian in powers of nonlinearity of up to terms of fourth order.We show that due to nonlinear interaction processes a stationary Fourier-spectrum of a surface elevation close to < |η k | 2 >∼ k −7/2 is formed. The observed spectrum can be interpreted as a weak-turbulent Kolmogorov spectrum for a direct cascade of energy. In all these cases the physical situations are similar. There is an ensemble of slowly decaying, weakly nonlinear waves in a medium with dispersion. Such systems have to be described statistically. However this is not traditional statistical mechanics, because the ensembles are very far from thermodynamic equilibrium. Nevertheless, one can develop a systematic approach for the statistical study of weakly nonlinear waves. This is the theory of weak (or wave) turbulence [4]. The main tools here are the kinetic equations for squared wave amplitudes. These equations describe the nonlinear resonant interaction processes taking place in the wave systems. Like in the turbulence in incompressible fluid, these processes lead to the cascades of some constants of motion (energy, wave action, momentum etc.) along the k−space. In isotropic systems it might be either a direct cascade of energy from small to large wave numbers or an inverse cascade of wave action to small wave numbers [5]. In an anisotropic system the situation could be much more complicated [6].The brilliant conjecture of Kolmogorov still is a hypothesis, supported by ample experimental evidence. On the contrary, the existence of powerlike Kolmogorov spectra, describing cascades in weak turbulence, is a rigorous mathematical fact. These spectra are the exact solutions of the stationary homogeneous kinetic equation, completely different from the thermodynamic RayleighJeans solutions.Nevertheless, the case is not closed. The weak turbulent theory itself is based on some assumptions, like phase stochasticity and the absence of coherent structures. This is the reason why justification of weak turbulent theory is an urgent and important problem.This justification can be done by a direct numerical solution of the primitive dynamic equation describing the wave ensemble. In pioneering works by Majda, McLaughlin and Tabak [7] it was done for the 1-D wave system. The results obtained by these authors are not easily interpreted. In some cases they demonstrate Kolmogorovtype spectra, in other cases -power spectra with essentially different exponents.In article [8] deviation from weak turbulent theory was explained by the role of coherent structures (solitons, quasi-solitons and collapses). If a 1-D system is free from coherent structures, weak-turbulent spectra are observed with a good deal of evidence [9,10,1...

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