Velocity difference statistics in turbulence

Abstract

We unify two approaches that have been taken to explain the non-Gaussian probability distribution functions (PDFs) obtained in measurements of longitudinal velocity differences in turbulence, and we apply our approach to Couette-Taylor turbulence data. The first approach we consider was developed by Castaing and coworkers, who obtained the non-Gaussian velocity difference PDF from a superposition of Gaussian distributions for subsystems that have a particular energy dissipation rate at a fixed length scale [Castaing et al., Physica D 46, 177 (1990)]. Another approach was proposed by Beck and Cohen, who showed that the observed PDFs can be obtained from a superposition of Gaussian velocity difference PDFs in subsystems conditioned on the value of an intensive variable (inverse "effective temperature") in each subsystem [Beck and Cohen, Physica A 322, 267 (2003)]. The intensive variable was defined for subsystems assuming local thermodynamic equilibrium, but no method was proposed for determining the size of a subsystem. We show that the Castaing and Beck-Cohen methods are related, and we present a way to determine subsystem size in the Beck-Cohen method. The application of our approach to Couette-Taylor turbulence (Reynolds number 540 000) yields a log-normal distribution of the intensive parameter, and the resultant velocity difference PDF agrees well the observed non-Gaussian velocity difference PDFs.

From time series to superstatistics

“…A relatively fast dynamics is given by the velocity of the Brownian particle, and a slow dynamics is given, e.g., by temperature changes of the environment. The two effects are associated with two well separated time scales, which result in a superposition of two statistics, or in a short, a superstatistics (SS) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. The stationary distributions of superstatistical systems typically exhibit non-Gaussian behavior with fat tails, which can decay with a power law, or as a stretched exponential, or in an even more complicated way.…”

Stretched exponentials from superstatistics

“…The corresponding superstatistics is relevant for turbulent systems [28,29,30]. The integration over β cannot be done analytically in this case.…”

“…Recent applications of the superstatistics concept include a variety of physical systems. Examples are Lagrangian [24,25,26,27] and Eulerian turbulence [28,29,30], defect turbulence [31], atmospheric turbulence [32,33], cosmic ray statistics [34], solar flares [35], solar wind statistics [36], networks [37,38], random matrix theory [39], and mathematical finance [40,41,42].…”

A probabilistic approach to heterogeneity in space plasmas: the case of magnetic field intensity in solar wind

“…These different approaches move from different physical and statistical assumptions: the composition law (Castaing et al, 1990;Beck and Cohen, 2003;G. Consolini et al: Heterogeneity in space plasmas Jung and Swinney, 2005), the Tsallis' statistics (Leubner and Vörös, 2005a, b) and the Extreme Deviation Theory (Frisch and Sornette, 1997). For example, the widely used model (Castaing et al, 1990) is derived from the assumption that the PDF of the velocity flucluations δv r at the separation scale r can be written as a convolution of a Gaussian distribution P G (δv r | σ ) with a weight function G λ (σ ) representing the statistical weight of the Gaussian PDF characterized by the standard deviation σ , i.e.…”