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Self-similarity of wind-driven seas

Copernicus Publications on behalf of European Geosciences Union
Nonlinear Processes in Geophysics
Authors:
Sergei Badulin at Institute of Oceanology. PP Shirshov Russian Academy of Sciences
Sergei Badulin
A. N. Pushkarev at Novosibirsk State University
A. N. Pushkarev
Donald T. Resio at University of North Florida
Donald T. Resio
V. E. Zakharov at Russian Academy of Sciences
V. E. Zakharov
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Abstract and Figures

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 self-similarity analysis predicts correctly the dependence of mean wave energy and mean frequency on wave age Cp / U10 . 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 Resio and 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 Cp / U10 .
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... Zaslavskii (2000) was the first to present the self-similar solutions for swell assuming the angular narrowness of the swell spectra and stated explicit analytical results. In fact, more general consideration, in the spirit of Badulin et al. (2002Badulin et al. ( , 2005a, leads to important findings and raises questions independently of the assumption of angular narrowness. ...
... The right-hand term S nl describes the effect of wave-wave resonant interactions and can be written in an explicit form (see the Appendices in Badulin et al., 2005a, for a collection of the formulas). The cumbersome term S nl causes many problems for wave modeling whenever Eq. (1) is extensively used. ...
... 4-6) were derived in particularly sophisticated and cumbersome ways. Later on, simpler and more physically transparent approaches were presented (Zakharov and Pushkarev, 1999;Balk, 2000;Pushkarev et al., 2003Pushkarev et al., , 2004Badulin et al., 2005a;Zakharov, 2010). These more general approaches allow us to find higher-order terms of the anisotropic Kolmogorov-Zakharov solutions (Eq. ...
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  • Jul 2016
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... which leads via Eq. (19) to, ...
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  • Aug 2017
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Article
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Full-text available
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