Fractal dimensions of lines in chaotic advection

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Abstract
The difference between locally self-similar fractals, called K fractals, and H fractals, which are globally self-similar fractals, is recalled . The self-similar cascade of energy [and enstrophy in the two-dimensional (2-D) case], which has been the original incentive for introducing fractals in the study of turbulence, does not seem to generate H-fractal interfaces in 2-D turbulent flows, according to the numerical evidence currently available. It only produces spiral singularities that are an example of K-fractal structures, and presumably a consequence of the structure of the turbulence that is neglected in pictures of the turbulence, based on a self-similar cascade of energy. It may be that the experimental evidence for fractal interfaces in 3-D turbulence is, in fact, evidence that these interfaces are K fractal. On the other hand, it could be that only very unsteady flows can produce H-fractal interfaces through a folding process that the unsteadiness adds to the stretching of the flow [see J. M. Ottino, The Kinematics of Mixing: Stretchings, Chaos and Transport (Cambridge U. P., Cambridge, 1989)]. That idea is investigated by releasing lines in a blinking vortex [J. Fluid Mech. 143, 1 (1984)] known to advect fluid elements ‘‘chaotically’’ in a certain range of parameters. The numerical evidence obtained supports the claim that interfaces in chaotic advection are H fractal. The length of a line in a blinking vortex grows exponentially, whereas the length of a line in a single steady vortex only grows linearly. The two ‘‘fractal’’ dimensions of the line investigated here increase with time, but they do not both asymptote to 2. The asymptotic value of the Kolmogorov capacity decreases as the time spent by the vortex in each location is decreased.

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