Negative dimensions of the turbulent dissipation field

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Abstract
Random multifractals that involve ensemble-averaged partition sums may give rise to negative dimensions. These models are highly relevant for interpreting fluctuations in fully developed hydrodynamic turbulence. From the experimental results that are obtained in a laboratory turbulent flow, it appears that self-similarity of the partition sums is only reached asymptotically. We demonstrate that this effect is due to correlations between subsequent refinements in a multifractal description. We give a way of correcting for the effects of correlations. We analyze an exactly solvable model that has the correlation depth as a parameter. This model has asymptotic self-similarity and displays a phase transition behavior in the limit of infinite deterministic refinements.