Translationally invariant cumulants in energy cascade models of turbulence

Abstract

In the context of random multiplicative energy cascade processes, we derive analytical expressions for translationally invariant one-and two-point cumulants in logarithmic field amplitudes. Such cumulants make it possible to distinguish between hitherto equally successful cascade generator models and hence supplement lowest-order multifractal scaling exponents and multiplier distributions.Although the underlying hydrodynamic equations are deterministic, the statistical description of fully developed turbulence has by now a long tradition [1]. Random multiplicative energy cascade models form a particularly simple and robust class of such statistical models. While different theoretical models can reproduce experimentally observed multifractal scaling exponents rather easily [2], observed multiplier distributions [3,4] eliminate many candidate cascade generators. Nevertheless, a number of competing generators remain, equally successful in reproducing both scaling exponents and multipliers. To make further progress in ferreting out the best cascade generator within this approximation, new observables are clearly called for.While most experiments have concentrated on measuring statistics in the energy dissipation density ε, we recently found a complete and analytical solution working in ln ε rather than ε itself [5]. In this letter, we show that cumulants in ln ε are analytically calculable even when restoring translational invariance to the solutions to emulate the spatial homogeneity of experimental turbulence statistics. Cumulants turn out to be powerful tools which for third and fourth

Analysis of velocity fluctuations and their intermittency properties in the surf zone using empirical mode decomposition

“…The cumulant approach has already been undertaken in the scaling turbulence framework in a few studies (see e.g. (Delour et al, 2001;Eggers et al, 2001;Chevillard et al, 2005)), where the cumulants of the cascade process (Eggers et al, 2001) or a polynomial development of the cumulant generating function (Delour et al, 2001;Chevillard et al, 2005) have been considered; see also Venugopal et al (2006) for an application to multifractal properties of rainfall.…”

Stochastic energy-cascade model for (1+1)-dimensional fully developed turbulence

“…This is to be contrasted with the geometrical RMCMs of (1) which, due to their hierarchical structure, are not translationally invariant in space. This non-invariance feeds through to all n-point observables and has to be removed at considerable cost through successive spatial sampling [17] before the latter can be compared to experimental counterparts.…”