Elodie Gagnaire-Renou

La Houille Blanche - Revue internationale de l eau, est une publication de la Société Hydrotechnique de France

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, S_in, and two methods for calculating the nonlinear transfer term S_nl (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, S_in, S_diss and S_nl, 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 α_ss = 0.68 ± 0.1.Further prospects of wind-wave modelling in the context of the presented weakly turbulent scaling are discussed.

The estimation of nonlinear wave‐wave interactions is one of the central problems in the development of operational and research models for ocean wave prediction. In this paper, we present results obtained with a numerical model based on a quasi‐exact computation of the nonlinear wave‐wave interactions called the Gaussian quadrature method (GQM) that gives both precise and computationally efficient calculations of the four‐wave interactions. Two situations are presented: a purely nonlinear evolution of the spectrum and a duration‐limited case. Properties of the directional wave spectrum obtained using GQM and the Discrete Interaction Approximation Method (DIM) are compared. Different expressions for the wind input and dissipation terms are considered. Our results are consistent with theoretical predictions. In particular, they reproduce the self‐similar evolution of the spectrum. The bimodality of the directional distribution of the spectrum at frequencies lower and greater than the peak frequency is shown to be a strong feature of the sea states, which is consistent with high‐resolution field measurements. Results show that nonlinear interactions constitute the key mechanism responsible for bimodality, but forcing terms also have a quantitative effect on the directional distribution of the spectrum. The influence of wind and dissipation parameterizations on the high‐frequency shape of the spectrum is also highlighted. The imposition of a parametric high‐frequency tail has a significant effect not only on the high‐frequency shape of the spectrum but also on the energy level and peak period and on the global directional distribution.