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

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

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

The Role of Wind Waves in Dynamics of the Air-Sea Interface

Wind waves are considered as an intermediate small-scale dynamic process at the air-sea interface,which modulates radically middle-scale dynamic processes of the boundary layers in water and air. It is shown that with the aim of a quantitative description of the impact said, one can use the numerical wind wave models which are added with the blocks of the dynamic atmosphere boundary layer (DABL) and the dynamic water upper layer (DWUL). A mathematical formalization for the problem of energy and momentum transfer from the wind to the upper ocean is given on the basis of the well known mathematical representations for mechanisms of a wind wave spectrum evolution. The problem is solved quantitatively by means of introducing special system parameters: the relative rate of the wave energy input, IRE, and the relative rate of the wave energy dissipation, DRE. For two simple wave-origin situations, the certain estimations for values of IRE and DRE are found, and the examples of calculating an impact of a wind sea on the characteristics of both the boundary layer of atmosphere and the water upper layer are given. The results obtained permit to state that the models of wind waves of the new (fifth) generation, which are added with the blocks of the DABL and the DWUL, could be an essential chain of the general model describing the ocean-atmosphere circulation.Comment: 11 pages, 4 figures, 1 tabl

Wave modelling - the state of the art

This paper is the product of the wave modelling community and it tries to make a picture of the present situation in this branch of science, exploring the previous and the most recent results and looking ahead towards the solution of the problems we presently face. Both theory and applications are considered. The many faces of the subject imply separate discussions. This is reflected into the single sections, seven of them, each dealing with a specific topic, the whole providing a broad and solid overview of the present state of the art. After an introduction framing the problem and the approach we followed, we deal in sequence with the following subjects: (Section) 2, generation by wind; 3, nonlinear interactions in deep water; 4, white-capping dissipation; 5, nonlinear interactions in shallow water; 6, dissipation at the sea bottom; 7, wave propagation; 8, numerics. The two final sections, 9 and 10, summarize the present situation from a general point of view and try to look at the future developments

On the problem of optimal approximation of the four-wave kinetic integral

International audienceThe problem of optimization of analytical and numerical approximations of Hasselmann's nonlinear kinetic integral is discussed in general form. Considering the general expression for the kinetic integral, a principle to obtain the optimal approximation is formulated. From this consideration it follows that the most well-accepted approximations, such as Discrete Interaction Approximation (DIA) (Hasselmann et al., 1985), Reduced Integration Approximation (RIA) (Lin and Perry, 1999), and the Diffusion Approximation proposed recently in Zakharov and Pushkarev (1999) (ZPA), have the same roots. The only difference among them is, essentially, the choice of the 4-wave configuration for the interacting waves. To evaluate a quality of any approximation for the 2-D nonlinear energy transfer, a mathematical measure of relative error is constructed and the meaning of approximation efficiency is postulated. By the use of these features it is shown that DIA has better accuracy and efficiency than ZPA. Following to the general idea of optimal approximation and by using the measures introduced, more efficient and faster versions of DIA are proposed

On the problem of optimal approximation of the four-wave kinetic integral

The problem of optimization of analytical and numerical approximations of Hasselmann's nonlinear kinetic integral is discussed in general form. Considering the general expression for the kinetic integral, a principle to obtain the optimal approximation is formulated. From this consideration it follows that the most well-accepted approximations, such as Discrete Interaction Approximation (DIA) (Hasselmann et al., 1985), Reduced Integration Approximation (RIA) (Lin and Perry, 1999), and the Diffusion Approximation proposed recently in Zakharov and Pushkarev (1999) (ZPA), have the same roots. The only difference among them is, essentially, the choice of the 4-wave configuration for the interacting waves. To evaluate a quality of any approximation for the 2-D nonlinear energy transfer, a mathematical measure of relative error is constructed and the meaning of approximation efficiency is postulated. By the use of these features it is shown that DIA has better accuracy and efficiency than ZPA. Following to the general idea of optimal approximation and by using the measures introduced, more efficient and faster versions of DIA are proposed