Multifractals, Multiscaling and the Energy Cascade of Turbulence

Abstract

The multifractal formalism [1,2], which originated in the context of turbulence and chaotic systems [3], has become a standard tool to analyze phenomena observed in fractal aggregates, semiconductors, disordered systems and so on [4]. To describe a multifractal set one usually divides it into small boxes (of size ℓ say) and introduces a partition sum of order q (or a generalized correlation function), C _q(ℓ), defined as the sum over the box-probabilities, each raised to a power q. From the scaling behavior C _q(ℓ) ~ ℓ^τ(q), the multifractal f(α) spectrum then follows from a Legendre transform of τ(q) [1,2]. To obtain the entire f(α) spectrum the moment q must vary from -∞ to +∞. Sometimes this demands very cumbersome calculations and it can be hard to obtain a good convergence of the scaling exponents. Instead by using the concept of multiscaling it is possible to obtain a part of the f(α) spectrum by calculating only one moment of the partition sum (or one correlation function). The idea of multiscaling was introduced by Kadanoff and co-workers to account for the scaling behavior of power spectra of temperature signals obtained in convective turbulence [5] and to describe distribution functions computed in simple models for sandpiles [6]. It was found that standard finite size scaling given by a few exponents did not describe the data satisfactory. Instead, by the multiscaling ansatz, the data could be nicely fitted using an infinity of scaling exponents. Recently, it was shown that it is possible to relate the concepts of multiscaling and multifractality to each other in a very natural way, namely by introducing a lower cut-off in the calculation of the correlation functions [7]. As this cut-off varies, a cross-over in the scaling of the correlation function C _q(ℓ) appears at a certain length scale. Below this length scale the correlation function stops being a power law, and this cross-over point moves as the cut-off varies. Using the multiscaling idea one then rescales the various correlation curves and collapse then onto a single scaling function (apart a small correction) of which a section is given by the f(α) spectrum. Again, this means that we obtain the multifractal spectrum from calculating only one moment in the partition sum. In turbulence, this type of behavior appears very naturally, if a multifractal model is assumed [8]. Here one can relate various values of the scaling exponents to various dissipative cut-offs [4]. In order words, there will be different viscous cut-offs (for each value of the scaling exponent) occurring at different length scales as one moves below the Kolmogorov inner scale. This defines a kind of intermediate viscous regime, which again by a multiscaling transformation can be rescaled onto one curve, part of which is the singularity spectrum for turbulence [8].