Statistical geometry of a small surface patch in a developed sea

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Abstract
Based on short‐range asymptotics for the structure function, D(r) = L2μ(μ)r2−2μ − B(μ)r2, the fractal (μ>0) and the marginal fractal (μ = 0) regimes in surface geometry are studied. The analysis is self‐contained in that it provides a brief exposition of basic notions of fractal geometry (for a monofractal case) as applied to a small surface patch corresponding to the equilibrium wave number spectrum of the form αk−4+2μϒ(θ). Effects of the fractal codimension μ and of the inner and outer scales on the steepness of small‐scale wavelets are examined, and the cascade pattern in the geometry of an essentially fractal surface (i.e., when μ≥1/4) is shown to be solely responsible for the observed rms wave slope. On the contrary, the slope of a marginally fractal sea is influenced by the dominant wavelength. The inner scale of the equilibrium range, h, is evaluated for the case of the intermediate degrees of wave development (i.e., when the nondimensional fetch x∼104) by employing a geometrical‐statistical theory of breaking waves. The resulting theoretical predictions of whitecap and foam coverage are in good agreement with field observations at h≈0.5 m. Finally, a “fractal decomposition” for a surface patch is developed based on the Karhunen‐Loeve expansion. The resulting series formalizes the cascade process of constructing realizations of a Gaussian random patch. The influence of the sea maturity on surface geometry is highlighted by showing it as the main factor of L and μ. The work is aimed at improving the present understanding of microwave remote sensing signatures.

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