Direct determination of the f(α) singularity spectrum and its application to fully developed turbulence

Abstract

This paper focuses on a method proposed for determining f(\ensuremath{\alpha}) (the singularity spectrum of a multifractal) directly from experimental data, without first calculating the ``generalized dimensions'' ${D}_{q}$ and applying the usual Legendre transforms. We describe the method, which is based on theorems by Shannon, Eggelston, and Billingsley, and apply it to one-dimensional cuts of the dissipation field of fully developed turbulence in laboratory and atmospheric flows. The accuracy of this method can be understood in terms of computation of conventional thermodynamic quantities in microcanonical and canonical ensembles. Comparisons with other direct and indirect methods of computing f(\ensuremath{\alpha}) are made.