An analytic model of the generation of surface gravity waves by turbulent air flow

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Abstract
Turbulent air flow over a surface gravity wave of small amplitude is studied on the basis of a family of first-order closure models, of which the eddy viscosity model and Prandtl's mixing-length model are members. Results are obtained by the method of matched asymptotic expansions in three layers. The problem is modelled by taking into account the combined effects of turbulence and molecular viscosity, which accommodates a proper imposition of the boundary conditions at the wave surface. The detailed structure of the various wave-induced field variables throughout the flow is then investigated. In addition, it is found that the growth rate of the waves by wind depends on the turbulence model. In particular, the more sensitively the mixing length depends on the shear in the mean air flow, the higher the growth rate. The validity of the results we obtain is restricted to small drag coefficient and small phase speed. Comparisons are made with other theoretical studies and with recent laboratory and field observations.