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On weakly turbulent scaling of wind sea in simulations of fetch-limited growth

Published online by Cambridge University Press:  05 January 2011

ELODIE GAGNAIRE-RENOU ,
MICHEL BENOIT  and
SERGEI I. BADULIN
ELODIE GAGNAIRE-RENOU*
Affiliation:
Saint-Venant Laboratory for Hydraulics, Université Paris Est (joint research unit EDF R&D, CETMEF, Ecole des Ponts Paris Tech), c/o EDF R&D, 6 quai Watier, BP 49, 78401 Chatou, France Laboratoire National d'Hydraulique et Environnement (LNHE), EDF R&D, 6 quai Watier, BP 49, 78401 Chatou, France Laboratoire de Sondages Electromagnétiques de l'Environnement Terrestre (LSEET), Université du Sud Toulon-Var, BP 132, 83957 La Garde, France
MICHEL BENOIT
Affiliation:
Saint-Venant Laboratory for Hydraulics, Université Paris Est (joint research unit EDF R&D, CETMEF, Ecole des Ponts Paris Tech), c/o EDF R&D, 6 quai Watier, BP 49, 78401 Chatou, France Laboratoire National d'Hydraulique et Environnement (LNHE), EDF R&D, 6 quai Watier, BP 49, 78401 Chatou, France
SERGEI I. BADULIN
Affiliation:
P.P. Shirshov Institute of Oceanology of Russian Academy of Sciences, 36, Nakhimovskii pr., Moscow 117997, Russia Laboratory of Nonlinear Wave Processes, Novosibirsk State University, 630090, Novosibirsk, Russia
*
†Email address for correspondence: elodie.gagnaire-renou@edf.fr
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Abstract

Extensive numerical simulations of fetch-limited growth of wind-driven waves are analysed within two approaches: a ‘traditional’ wind-speed scaling first proposed by Kitaigorodskii (Bull. Acad. Sci. USSR, Geophys. Ser., Engl. Transl., vol. N1, 1962, p. 105) in the early 1960s and an alternative weakly turbulent scaling developed recently by Badulin et al. (J. Fluid Mech.591, 2007, 339–378). The latter one uses spectral fluxes of wave energy, momentum and action as physical scales of the problem and allows for advanced qualitative and quantitative analysis of wind-wave growth and features of air–sea interaction. In contrast, the traditional approach is shown to be descriptive rather than proactive. Numerical simulations are conducted on the basis of the Hasselmann kinetic equation for deep-water waves in a wide range of wind speeds from 5 to 30 m s −1 and for the ideal case of fetch-limited growth: permanent wind blowing perpendicularly to a straight coastline. Two different wave input functions, Sin, and two methods for calculating the nonlinear transfer term Snl (Gaussian quadrature method, or GQM, a quasi-exact method based on the use of Gaussian quadratures, and the discrete interaction approximation, or DIA) are used in the simulations. Comparison of the corresponding results firstly shows the relevance of the analysis of wind-wave growth in terms of the proposed weakly turbulent scaling, and secondly, allows us to highlight some critical points in the modelling of wind-generated waves. Three stages of wind-wave development corresponding to qualitatively different balance of the source terms, Sin, Sdiss and Snl, are identified: initial growth, growing sea and fully developed sea. Validity of the asymptotic weakly turbulent approach for the stage of growing wind sea is determined by the dominance of nonlinear transfers, which results in a rigid link between spectral fluxes and wave energy. This stage of self-similar growth is investigated in detail and presented as a consequence of three sub-stages of qualitatively different coupling of air flow and growing wind waves. The key self-similarity parameter of the asymptotic theory is estimated to be αss = 0.68 ± 0.1.

Further prospects of wind-wave modelling in the context of the presented weakly turbulent scaling are discussed.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows Waves/Free-surface Flows: Wind-wave interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 669 , 25 February 2011 , pp. 178 - 213
Copyright
Copyright © Cambridge University Press 2011

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References

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