A basing of the diffusion approximation derivation for the four-wave kinetic integral and properties of the approximation

International audienceA basing of the diffusion approximation derivation for the Hasselmann kinetic integral describing nonlinear interactions of gravity waves in deep water is discussed. It is shown that the diffusion approximation containing the second derivatives of a wave spectrum in a frequency and angle (or in wave vector components) is resulting from a step-by-step analytical integration of the sixfold Hasselmann integral without involving the quasi-locality hypothesis for nonlinear interactions among waves. A singularity analysis of the integrand expression gives evidence that the approximation mentioned above is the small scattering angle approximation, in fact, as it was shown for the first time by Hasselmann and Hasselmann (1981). But, in difference to their result, here it is shown that in the course of diffusion approximation derivation one may obtain the final result as a combination of terms with the first, second, and so on derivatives. Thus, the final kind of approximation can be limited by terms with the second derivatives only, as it was proposed in Zakharov and Pushkarev (1999). For this version of diffusion approximation, a numerical testing of the approximation properties was carried out. The testing results give a basis to use this approximation in a wave modelling practice

The choice of optimal Discrete Interaction Approximation to the kinetic integral for ocean waves

A lot of discrete configurations for the four-wave nonlinear interaction processes have been calculated and tested by the method proposed earlier in the frame of the concept of Fast Discrete Interaction Approximation to the Hasselmann's kinetic integral (Polnikov and Farina, 2002). It was found that there are several simple configurations, which are more efficient than the one proposed originally in Hasselmann et al. (1985). Finally, the optimal multiple Discrete Interaction Approximation (DIA) to the kinetic integral for deep-water waves was found. Wave spectrum features have been intercompared for a number of different configurations of DIA, applied to a long-time solution of kinetic equation. On the basis of this intercomparison the better efficiency of the configurations proposed was confirmed. Certain recommendations were given for implementation of new approximations to the wave forecast practice

Numerical wind wave model with a dynamic boundary layer

A modern version of a numerical wind wave model of the fourth generation is constructed for a case of deep water. The following specific terms of the model source function are used: (a) a new analytic parameterization of the nonlinear evolution term proposed recently in Zakharov and Pushkarev (1999); (b) a traditional input term added by the routine for an atmospheric boundary layer fitting to a wind wave state according to Makin and Kudryavtsev (1999); (c) a dissipative term of the second power in a wind wave spectrum according to Polnikov (1991). The direct fetch testing results showed an adequate description of the main empirical wave evolution effects. Besides, the model gives a correct description of the boundary layer parameters' evolution, depending on a wind wave stage of development. This permits one to give a physical treatment of the dependence mentioned. These performances of the model allow one to use it both for application and for investigation aims in the task of the joint description of wind and wave fields

Numerical Modelling of Wind Waves. Problems, Solutions, Verifications, and Applications

The time-space evolution of the field is described by the transport equation for the 2-dimensional wave energy spectrum density, S(x,t), spread in the space, x, and time, t. This equation has the forcing named the source function, F, depending on both the wave spectrum, S, and the external wave-making factors: local wind, W(x, t), and local current, U(x, t). The source function contains certain physical mechanisms responsible for a wave spectrum evolution. It is used to distinguish three terms in function F: the wind-wave energy exchange mechanism, In; the energy conservative mechanism of nonlinear wave-wave interactions, Nl; and the wave energy loss mechanism, Dis. Differences in mathematical representation of the source function terms determine general differences between wave models. The problem is to derive analytical representations for the source function terms said above from the fundamental wave equations. Basing on publications of numerous authors and on the last two decades studies of the author, the optimized versions of the all principal terms for the source function, F, have been constructed. Detailed description of these results is presented. The final version of the source function is tested in academic test tasks and verified by implementing it into numerical shells of the well known wind wave models: WAM and WAVEWATCH. Procedures of testing and verification are presented and described in details. The superiority of the proposed new source function in accuracy and speed of calculations is shown. Finally, the directions of future developments in this topic are proposed, and some possible applications of numerical wind wave models are shown, aimed to study both the wind wave physics and global wind-wave variability at the climate scale, including mechanical energy exchange between wind, waves, and upper water layer.Comment: 62 pages, 14 figures, 5 tables, 63 reference