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,kminus;rgr;, then the exponent rgr; is given by rgr;=D2+1minus;M, whereD2is the correlation dimension of the fractal attractor andMis the dimension of the relevant space. We find that observability of the power law may be obscured by fluctuations in thekhyphen;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 lsqb;Phys. Fluids8, 2441 (1996)rsqb;, an experimental example utilizing particles floating on the surface of a flowing fluid is given. (More generally we note that our result for thekhyphen;spectrum power law exponent rgr; should apply to fractal patterns in other physical contexts.) copy;1996 American Institute of Physics.
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