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Zheng Shen and Liming Mei

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 (u 2 */gL 0)(2−D)/2, where L 0 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.

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Zheng Shen, Wei Wang, and Liming Mei

Abstract

One central problem in the study of wind-generated gravity waves is the energy balancing process in the equilibrium spectral subrange. In considering the predicted equilibrium spectral forms from physical models proposed by Kitaigorodskii, other investigators accepted that the statistical equilibrium state is effectively characterized by the wave action conservation law: δEt+C⃗g·∇E = 0, where E is the wave energy spectrum and C⃗g = ∇kω(k) is the group velocity. Here the continuous wavelet transform is used to analyze typical sets of wind-generated gravity wave data obtained both in the ocean and in a wind-wave channel. This “space scale” analysis is shown to provide the first visual evidence that when the fetch is not very short, the wave action conservation law mentioned above is not convenient to describe the dynamics of the wave components in the equilibrium range estimated from its energy spectrum.

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