A Quasi-linear Eddy-Viscosity Model for the Flux of Energy and Momentum to Wind Waves Using Conservation-Law Equations in a Curvilinear Coordinate System

Abstract

The airflow above ocean waves is calculated using a quasi-linear model?one in which the effect of the waves on the mean flow is taken into account. The model uses curvilinear coordinates, in which one coordinate surface coincides with the instantaneous sea surface, and is consequently able to attain fine vertical resolution in the boundary layer just above the sea surface; the model equations are formulated in conservation-law form. The rates of energy and momentum input to the wave field are calculated from the oscillatory pressure and shear-stress components at the water surface. The equations are solved iteratively using a logarithmically spaced finite-difference mesh. The effect of air turbulence is modeled using a vertically varying shear-stress?dependent eddy viscosity, which acts on the wave-correlated oscillatory motions as well as on the mean flow field. For infinitesimal waves the model agrees with the results of Conte and Miles as the Newtonian viscosity and eddy viscosity that act on the oscillatory motions are reduced toward zero, and it converges slowly toward the results of Jacobs' analytical eddy viscosity model as the drag coefficient is reduced. In agreement with results from Janssen's simpler quasi-linear model, there is increased wave-induced drag for young wind seas with unidirectional JONSWAP spectra and Phillips constant proportional to the (?3/2) power of wave age. The present model gives similar values for wave drag and wave energy input to Janssen's, for the same values of roughness length and Phillips constant, and the spectral distribution of the rate of energy input to the waves is also in reasonable agreement. The variation of drag coefficient with wave age is quite close to the results obtained by Maat, Kraan, and Oost from analysis of HEXMAX field data.