On the growth of a fully non-linear Stokes wave by turbulent shear flow .1. Eddy-viscosity model

The transfer of energy from turbulent wind to fully non-linear Stokes wave, with a sharp crest and the interior angle of 120 degrees, through the turbulent Reynolds stress in the immediate neighbourhood of the surface is explored. Resonance between the Tollmien-Schlichting waves for a given wind profile and the free-surface waves is shown to be a possibility. The results, for growth of Stokes wave by a wind are compared with the Miles (1962) critical-layer mechanism. In the present model the wind mean velocity profile is assumed to be logarithmic, and effects of turbulence on the interaction between wind and water are nor neglected. It is shown that the present asymptotic theory predicts the fivefold increase in the growth rate of water waves compared with the critical-layer mechanism. This finding is in more agreement with the observation made at the sea. The present theory is also applied to longer waves, and it is argued that the contributing agencies, for this significant increase in the growth of water waves, are the shape of the surface wave and the effects of turbulence. The latter agrees with the recent findings of Belcher and Hunt (1993) for slow moving water waves, and clearly confirms that the critical layer is not the major contributing agency for growth of surface waves. It is further shown that the present theory is capable of predicting growth rate for slow waves (c/U* < 10) and fast waves (c/U* > 20), however, the theory fails to give finite growth rate in the intermediate region (10 < c/U* < 20).