Turbulent shear flow over fast-moving waves

We divide the interaction between wind and ocean surface waves into three parameter regimes, namely slow, intermediate and fast waves, that are distinguished by the ratio c/u* (c is the wave phase speed and u*, is the friction velocity in the wind). We develop here an analytical model for linear changes to the turbulent air flow caused by waves of small slope that is applicable to slow and to fast waves. The wave-induced turbulent shear stress is parameterized here with a damped mixing-length model, which tends to the mixing-length model in an inner region that lies close to the surface, and is then damped exponentially to zero in an outer region that lies above the inner region. An adjustable parameter in the damped mixing-length model controls the rate of decay of the wave-induced stress above the inner region, and shows how the results vary from a model with no damping, which corresponds to using the mixing-length model throughout the flow, to a model with full damping, which, following previous suggestions, correctly represents rapid distortion of the wave-induced turbulence in the outer region. Solutions for air flow over fast waves are obtained by analysing the displacement of streamlines over the wave; they show that fast waves are damped, thereby giving their energy up to the wind. There is a contribution to this damping from a counterpart of the non-separated sheltering mechanism that gives rise to growth of slow waves (Belcher & Hunt 1993). This sheltering contribution is smaller than a contribution from the wave-induced surface stress working against the orbital motions in the water. Solutions from the analysis for both slow and fast waves are in excellent agreement with values computed by Mastenbroek (1996) from the nonlinear equations of motion with a full second-order closure model for the turbulent stresses. Comparisons with data for slow and intermediate waves show that the results agree well with laboratory measurements over wind-ruffled paddle-generated waves, but give results that are a factor of about two smaller than measurements of purely wind-generated waves. We know of no data for fast waves with which to compare the model. The damping rates we find for fast waves lead to e-folding times for the decay of the waves that are a day or longer. Although this wind-induced damping of fast waves is small, we suggest that it might control low-frequency waves in a fully-developed sea.