Turbulent air flow over sea waves; simplified model for applications

A simplified model of the air flow over surface water waves propagating at arbitrary phase velocity (as compared to the wind speed, e.g. at 10 m) and direction (relative to the wind direction) is presented. The air flow is divided into an outer (OR) and an inner (IR) regions. In the OR the wave-induced motions experience inviscid undulation, while in the IR they are strongly affected by turbulent shear stresses. Introduction of the OR and the IR allows a considerable simplification in the description of the air flow above waves.

The critical height (the height where the wind speed and the wave phase velocity are equal) is for most cases located inside the IR. Hence, its singular behaviour is strongly suppressed by turbulent. stresses. This fact allows a simple description of the wind velocities in the OR, which is based on the approximate solution of the Rayleigh equation suggested by Miles (1957).

The description of the IR is based on the solution of the vorticity equation accounting for the turbulent, diffusion. The turbulent shear stress is parameterised via the eddy viscosity coefficient adopting the mixing length closure scheme. Exponential damping of the shear stress variations with height towards the OR is introduced. which leads to further simplification. This damping describes phenomenologically the basic feature of the wave boundary laver: rapid distortion of turbulence in the OR.

The model is reduced to a set of explicit analytical formulas which describe the wind velocities and shear stresses in the boundary layer above the waves. Results of the simplified model (velocities and shear stress profiles, the growth rate) are in reasonable agreement with those obtained by a two-dimensional numerical model based on the second-order closure scheme.
This fact is encouraging, and has an
important consequence: the description of the air flow dynamics over waves is not. sensitive to the details of the closure schemes. It is only important to provide the vertical damping of variations in turbulence characteristics on a scale comparable with the IR height.

The simplified model is compared with data of the laboratory experiment of Hsu and Hsu (1983), and reasonable agreement in velocity and shear stress distributions is found.