Equilibrium Spectra of Water Waves Forced by Intermittent Wind Turbulence

Abstract

With the help of fractal geometry used to model the intermittency of energy input from wind to wave components, the theoretical spectra of the equilibrium range in wind-generated gravity waves proposed by Phillips are refined. On account of the intermittency, it is proven that the classical frequency spectral exponent 4 must he replaced by 4 + (2 ? D), where D is the informational entropy dimension of the support subset, upon which the energy input from the wind to the gravity waves in the equilibrium range is concentrated. To a first approximation, it is found that D ≈ 1.88 and 4 + (2 ? D) ≈ 4.12. The variation of the Toba constant is found to be proportional to (u2*/gL0)(2?D)/2, where L0 is the wavelength of the longest wave component in the equilibrium range, that is, the lower limit wavenumber above which the processes of energy input from wind, spectral flux divergence, and loss by breaking are all significant and proportional. The refined wavenumber spectrum is less sensitive to wind strength than the original.