The Dissipation Range of Wind-Wave Spectra Observed on a Lake

Abstract

Based on an interpretation of a field experiment it is argued that, due to breaking of wind waves in deep water, the dissipation of energy is restricted to a range of frequencies ω > ωg, much higher than the frequency ωm of the dominant waves. In this dissipation range the spectrum has the form S(ω) = βg2ω−5 where g is the acceleration due to gravity and β = 0.025. For spectral wave components at ω ≤ ωg, only a local balance between energy input from the wind and the weak, third-order, nonlinear interaction is important. Asymptotically as ω ≫ ωm the wind input becomes unimportant, and the wave spectrum has the Kitaigorodskii form of a Kolgomorov analog S(ω) = 2aε0⅓ g4/3 ω−4 where ε0 is a constant flow of mean energy per unit surface area through the spectrum dissipated at high frequencies (when multiplied by g and water density ρw). From a method of M. S. Longuet-Higgins we estimate the magnitude of the dissipation (due to wave breaking) and find the Kolmogorov constant to be a ≈ 0.6. When a model, explained by Phillips, for wind energy input to the wave spectrum is applied to a simplified spectral model prescribing the scales of dissipation and growth of spectral wave components, good agreement is found with measurements by Donelan et al. of the coefficient 2aε0⅓ and its dependence on the frequency ωm of the dominant waves at the spectral peak.