Direct and inverse cascades of energy, momentum and wave action in spectra of wind-driven waves
Abstract
The time-dependent, spatially uniform Hasselmann's kinetic equation for surface gravity waves in presence of wind forcing and white-capping dissipation is studied numerically. We use conventional parameterizations of wind wave input (Snyder et al. 1981; Plant 1982; Hsiao &Shemdin 1983; Donelan, Pierson 1987) that are consistent with weakly nonlinear scaling. We assume that strong dissipation due to white-capping is essential for short waves only (with frequencies above 1Hz) belonging to the spectral tail and can be neglected near the spectral peak. We compare our numerical results with the predictions of the theory of weak turbulence and found a very good coincidence.
It is shown that asymptotic behavior of wave spectra is in perfect agreement with stationary solutions of the Hasselmann equation -- Kolmogorov's solutions for direct (Zakharov & Filonenko 1966) and inverse (Zakharov &Zaslavskii 1982) cascades. This asymptotic behavior appears at rather early stages of wind wave evolution (physical time of order of few hours in our experiments); A strong tendency of solutions to self-similar behavior of duration limited solutions is found for rather wide range of initial conditions and external forcing; Good quantitative coincidence with recapitulative experimental data for duration limited wind wave growth (Young 1999, p.111) and for fetch-limited (JONSWAP) spectra parameterized by wave age C_p/Uwind is found. The findings here are quite robust and hopefully will be applied to the practical problems. Present wave prediction models are based on fairly crude parameterizations of the nonlinear energy transfers. In large part due to inaccuracies in these parameterizations, these models have had to rely on empirical fitting of general growth equation as a basis for constraining additional source-sink terms in the detailed balance equations. Results from this study could be used to reformulate a complete energy balance equation for wave generation, propagation and decay, which could lead to substantially improved predictions in the near future. The research was conducted under the U.S. Army Corps of Engineers, RDT&E program, grant DACA 42-00-C0044, ONR grant N00014-98-1-0070 and NSF grant NDMS0072803, INTAS grant 01-234 and Russian Foundation for Basic Research 01-05-64603, 01-05-64464, 02-05-65140. This support is gratefully acknowledged.- Publication:
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EGS - AGU - EUG Joint Assembly
- Pub Date:
- April 2003
- Bibcode:
- 2003EAEJA......287B