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, non-linear interactions in deep water; 4, white-capping dissipation; 5, non-linear 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. Keywords

Abstract. 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.

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On the basis of the author's earlier results, a new source function for a numerical wind-wave model optimized by the criterion of accuracy and speed of calculation is substantiated. The proposed source function includes (a) an optimized version of the discrete interaction approximation for parametrization of the nonlinear evolution mechanism, (b) a generalized empirical form of the input term modified by adding a special block of the dynamic boundary layer of the atmosphere, and (c) a dissipation term quadratic in the wave spectrum. Particular attention is given to a theoretical substantiation of the least investigated dissipation term. The advantages of the proposed source function are discussed by its comparison to the analogues used in the widespread models of the third generation WAM and WAVEWATCH. At the initial stage of assessing the merits of the proposed model, the results of its testing by the system of academic tests are presented. In the course of testing, some principals of this procedure are formulated. The possibility of using the testing results to study the physics of evolution processes in wind waves is shown. It is noted that the specially added block of the dynamic boundary layer of the atmosphere makes it possible to give an exhaustive description of the air-sea-interface's characteristics, which may be used to improve wave forecasting. This new modeling quality allows us to make a statement about the construction of a model of the next (fourth) generation.