Multifractal Scaling of Probability Density Function : a Tool for Turbulent Data Analysis

(Received 26 September 1995, revised 22 February 1996, accepted 4 March 1996)

Abstract
The probabilistic reformulation of the multifractal model [1] is obtained directly from the structure functions written as integrals over cumulative distribution functions (c.d.f.) by the steepest descent method. The saddle point being a function of scale, we perform a change of variable to obtain expressions that are asymptotically valid in the inertial range. Starting directly from the inertial range behavior of the c.d.f., our algorithm yields values for the scaling exponents and codimension that are identical to those obtained from structure functions. Furthermore, a simple interpretation of multifractality in terms of global c.d.f. scaling is shown to collapse the inertial range c.d.f. into a single curve, directly related to the codimension. Our method determines a new length scale, larger than the integral scale, that gives a quantitative measure of the degree of multifractality of the data. Finally, some possible future applications are mentioned.