Power law wave number spectra of fractal particle distributions advected by flowing fluids

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Abstract
Recently, it has been shown that an initial cloud of particles advected by a fluid may, under common circumstances (e.g., when the particles float on the fluid surface), eventually becomes distributed on a fractal set in space. This paper considers the characterization of such fractal spatial patterns by wave number spectra. If a scaling range exists in which the spectrum has an observable power law dependence, k−ρ, then the exponent ρ is given by ρ=D2+1−M, where D2 is the correlation dimension of the fractal attractor and M is the dimension of the relevant space. We find that observability of the power law may be obscured by fluctuations in the k‐spectrum, but that averaging can be employed to compensate for this. Theoretical results and supporting numerical computations utilizing a random map are presented. In the companion paper by Sommerer [Phys. Fluids 8, 2441 (1996)], an experimental example utilizing particles floating on the surface of a flowing fluid is given. (More generally we note that our result for the k‐spectrum power law exponent ρ should apply to fractal patterns in other physical contexts.)