**7. Conclusions**

The paper is devoted to the wind wave simulations based on the initial equations of potential motion of fluid with a free surface. The system of equations includes the evolutionary kinematic and dynamic surface conditions and Laplace equation for the velocity potential. In this paper, a case of the double-periodic domain of infinite depth is considered. The construction of the exact numerical scheme for a longterm integration of these equations in the Cartesian coordinate system is impossible, since the surface moves between the grid knots. Instead, the system of the curvilinear coordinates (1) fitted with the surface is introduced. The main advantage of this coordinate system is that the surface coincides with a coordinate line *ζ* ¼ 0*:* The penalty follows immediately after turning the simple Cartesian coordinates into the curvilinear, nonstationary, and nonorthogonal coordinate system. Fortunately, the evolutionary Eqs. (4) and (5) become just slightly complicated, while Laplace equation transforms into the full elliptic equation. At each time step, these equations can be represented as Poisson equation with the right-hand side depending on the solution itself as well as on the metric coefficient. Since the norm of the right sight of the equation is usually small, the solution of Poisson equation can be found with the three-diagonal matrix algorithm and with iterations over the right-hand side. This procedure being formulated in the Fourier space is greatly simplified by the assumption of periodicity since in this case the derivatives over the horizontal coordinates are represented by the absolute value of wave number j j *k* in the diagonal terms. When constructing a numerical scheme, we noticed that the significant simplification of the problem can be achieved by separation of the velocity potential into the linear and nonlinear components (see Ref. [7]). It is assumed that the linear component satisfies Laplace equations with the known solution. The equation for the nonlinear component can be obtained by extracting Laplace equation from the initial Poisson equation. Such procedure has a lot of advantages since the nonlinear component is on the average less by 1–2 decimal orders than the linear one. It means that for solution of the reduced Poisson equation the lesser number of levels in vertical, the lesser number of iterations and a smaller accuracy criterion can be used. The use of two components in the evolutionary equation does not seem to provide noticeable advantages; however, this way deserves further consideration.

The adiabatic version of the model was validated by simulation of a running Stokes wave with the steepness *AK* ¼ 0*:*40 in Ref. [7]. It was shown that the amplitudes of Stokes modes remain practically constant up to the accuracy of 10�<sup>7</sup> . The current version of the model after some technical improvements of the numerical scheme provides accuracy up to 10�12. Then, the adiabatic version of the model was used for reproduction of a quasi-stationary regime for investigation of the statistical properties of sea waves [1, 7, 8].

For calculations of development of wave field under the action of wind, it was necessary to include the algorithms for calculations of input and dissipation of energy. The scheme for calculation of the energy input was developed by Chalikov and Rainchik [3] on the basis of coupling the one-dimensional phase-resolving model and the two-dimensional boundary layer model with the second-order turbulence closure scheme. The parameterization suggested is still quasi-linear (similar to Miles'scheme [30]), but in our opinion, it is the only scheme confirmed by the extended results of the numerical simulations. The theoretical and observational data on *β*function are dramatically scattered (see Ref. [4], **Figure 1**).

For stabilization of the solution, the algorithm of high-frequency dumping in the Fourier space suggested by Chalikov and Sheinin [28] was used. The numerous attempts were made to improve that scheme (for example by reduction of the spectral interval of dumping) but without much success.

The most complicated problem is the parameterization of dissipation due to wave breaking. Such algorithm should not describe a process of breaking as it is, which within the frame of such model is impossible, but it should prevent the numerical instability that interrupts a run (see discussion in Ref. [26]). Currently, the algorithm used is very simple. It is based on the diffusion operator with a highly selective coefficient of 'viscosity.' It works satisfactorily, but we are far from thinking that it cannot be substantially improved or completely replaced by another one.

The results described in this paper show that the wave field development under the action of wind is reproduced quite realistically. The area of application of such models is very wide. Such modeling should be used for improvement of the algorithms of the energy input and dissipation. A model with the periodic boundary conditions can be used for the local interpretation of the spectral forecast in terms of real waves. The finite-difference version of the model can be used for simulation of wave regimes in the basins with real shapes and bathymetry (see, e.g., Ref. [5]).
