Wavelet Analysis of Fractal Signals Application to Fully Developed Turbulence Data

Abstract

The recently developed multifractal formalism¹ has proven particularly fruitful in the characterization of singular measures arising in a variety of physical situations². Notable examples include the invariant probability distribution on a strange attractor, the dis tribution of voltage drops across a random resistor network, the distribution of growth probabilities on the boundary of a diffusion limited aggregate and the spatial distribution of dissipative regions in a turbulent flow. This formalism¹ accounts for the statistical scaling properties of these measures through the determination of their singularity spectrum f(α) or their generalized fractal dimensions D _q . The f(α) singularity spectrum provides a rather intuitive description of a multifractal measure in terms of interwoven sets of Hausdorff dimension f(α) corresponding to singularity strength α. This spectrum was shown to be intimately related, by means of a Legendre transform, to τ(q) = (q − 1)D _q . Actually, the concept of multifractal originated from a general class of multiplicative cascade models introduced by Mandelbrot^3a to account for the intermittent nature of the rate of turbulent energy dissipation. Along this line, measurements of the f(α) spectrum based on the local dissipation have been recently reported⁴ ; these experimental results bring conspicuous evidence for the multifractal nature of the dissipation field. An alternative description of the fine structures in fully developed turbulence consists in working exclusively with inertial range quantitities as proposed by Parisi and Frisch⁵. Basically their multifractal approach relies on the determination of the spectrum D(h) of Hölder exponents h of the velocity field from the inertial scaling properties of structure functions of variable order: S _p (ℓ) =< (δvℓ)^p >~ ℓ ^ζp , (p integer > 0), where δv _ℓ is a longitudinal velocity increment over a distance ℓ.