The Wind-Induced Wave Growth Rate and the Spectrum of the Gravity-Capillary Waves

Abstract

A form of the spectrum of gravity-capillary waves is suggested, based on a balance of the wind input, the spectral flux divergence, the viscous dissipation, and the modulation from the wave-drift interaction. The spectrum in the alongwind direction is expressed as where k is the wavenumber, m and α are constants, u* is the wind friction velocity, δ is the threshold wind friction velocity, c is the short-wave phase velocity, and mo is the zeroth moment of nondirectional frequency spectrum of long-wave orbital velocity. For shorter capillary waves, the addition of dissipation due to the eddy viscosity is needed. A kinematic wave breaking criterion is applied in the calculation of the short-wave dissipation. It is found that this short-wave dissipation, due to wave-drift interactions, has the effect of suppressing the spectrum at higher wind speed and yields a good agreement with the measurements of both the wave spectrum and the wave directional spreading rate. It is also found that the fluctuation of average wind stress and dissipations due to molecular viscosity and eddy viscosity have an important influence on the wave growth rate and the gravity-capillary wave spectrum. The molecular viscosity plays a significant role at lower wind conditions; the eddy-viscous dissipation dominates at higher wavenumbers. The threshold wind friction velocity is determined by a balance between the wind input with fluctuation and the molecular-viscous dissipation. Its value in the fully developed stage is different from that when the wind starts to blow. The fluctuation of the average wind stress arising from the wind turbulence has been considered in the calculation of the short-wave growth rate, and its effect of increasing short-wave growth rate at low winds has been shown. ERS-1 scatterometer models and radar backwater measurements in the open ocean are used to confirm the proposed spectrum model, especially the directional spreading function. The results show a good agreement between our model and the field measurements.