Abstract.The results of theoretical and numerical study of the Hasselmann kinetic equation for deep water waves in presence of wind input and dissipation are presented. The guideline of the study: nonlinear transfer is the dominating mechanism of wind-wave evolution. In other words, the most important features of wind-driven sea could be understood in a framework of conservative Hasselmann equation while forcing and dissipation determine parameters of a solution of the conservative equation. The conservative Hasselmann equation has a rich family of self-similar solutions for duration-limited and fetch-limited wind-wave growth. These solutions are closely related to classic stationary and homogeneous weak-turbulent Kolmogorov spectra and can be considered as non-stationary and non-homogeneous generalizations of these spectra. It is shown that experimental parameterizations of wind-wave spectra (e.g. JONSWAP spectrum) that imply self-similarity give a solid basis for comparison with theoretical predictions. In particular, the selfsimilarity analysis predicts correctly the dependence of mean wave energy and mean frequency on wave age C p /U 10 . This comparison is detailed in the extensive numerical study of duration-limited growth of wind waves. The study is based on algorithm suggested by Webb (1978) that was first realized as an operating code by Perrie (1989, 1991). This code is now updated: the new version is up to one order faster than the previous one. The new stable and reliable code makes possible to perform massive numerical simulation of the Hasselmann equation with different models of wind input and dissipation. As a result, a strong tendency of numerical solutions to self-similar behavior is shown for rather wide range of wave generation and dissipation conditions. We found very good quantitative coincidence of these solutions with available results on duration-limited growth, as well as with experimental parametrization of fetch-limited spectra JONSWAP in terms of wind-wave age C p /U 10 .

The theory of weak turbulence developed for wind-driven waves in theoretical works and in recent extensive numerical studies concludes that non-dimensional features of self-similar wave growth (i.e. wave energy and characteristic frequency) have to be scaled by internal wave-field properties (fluxes of energy, momentum or wave action) rather than by external attributes (e.g. wind speed) which have been widely adopted since the 1960s. Based on the hypothesis of dominant nonlinear transfer, an asymptotic weakly turbulent relation for the total energy ϵ and a characteristic wave frequency ω* was derived The self-similarity parameter αss was found in the numerical duration-limited experiments and was shown to be naturally varying in a relatively narrow range, being dependent on the energy growth rate only.In this work, the analytical and numerical conclusions are further verified by means of known field dependencies for wave energy growth and peak frequency downshift. A comprehensive set of more than 20 such dependencies, obtained over almost 50 years of field observations, is analysed. The estimates give αss very close to the numerical values. They demonstrate that the weakly turbulent law has a general value and describes the wave evolution well, apart from the earliest and full wave development stages when nonlinear transfer competes with wave input and dissipation.

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A physical model for sea wave period from altimeter data is presented. Physical roots of the model are in recent advances of the theory of weak turbulence of wind-driven waves that predicts the link of instant wave energy to instant energy flux to/from waves. The model operates with wave height and its spatial derivative and does not refer to normalized radar cross-section r 0 measured by the altimeter. Thus, the resulting formula for wave period does not contain any empirical parameters and does not require features of particular satellite altimeter or any calibration for specific region of measurements. A single case study illustrates consistency of the new approach with previously proposed empirical models in terms of estimates of wave periods and their statistical distributions. The paper brings attention to the possible corruption of dynamical parameters such as wave steepness or energy fluxes to/from waves when using the empirical approaches. Applications of the new model to the studies of sea wave dynamics are discussed.