Superstatistical turbulence models

Recently there has been some progress in modeling the statistical properties of turbulent flows using simple superstatistical models. Here we briefly review the concept of superstatistics in turbulence. In particular, we discuss a superstatistical extension of the Sawford model and compare with experimental data. Turbulence is a spatio-temporal chaotic dynamics generated by the Navier-Stokes equation ~̇v = −(~v∇)~v + ν∆~v + ~ F . (1) In the past 5 years there has been some experimental progress in Lagrangian turbulence measurements, i.e. tracking single tracer particles in the turbulent flow. Due to the measurements of the Bodenschatz [1, 2, 3] and Pinton groups [4, 5] we now have a better view of what the statistics of a single test particle in a turbulent flow looks like. The recent measurements have shown that the acceleration ~a as well as velocity difference ~u = ~v(t+ τ)− ~v(t) on short time scales τ exhibits strongly non-Gaussian behavior. This is true for both, single components as well as the absolute value of ~a and ~u. Moreover, there are correlations between the various components of ~a, as well as between velocity and acceleration. The corresponding joint probabilities do not factorize. Finally, the correlation functions of the absolute value |~a| and |~u| decay rather slowly. How can we understand all this by simple stochastic models? There is a recent class of models that are pretty successful in explaining all these statistical properties of Lagrangian turbulence (as well as of other turbulent systems, such as Eulerian turbulence [6, 7, 8], atmospheric turbulence [9, 10, 11] and defect turbulence [12]). These are turbulence models based on superstatistics [13]. Superstatistics is a concept from nonequilibrium statistical mechanics, in short it means a ‘statistics of statistics’, one given by ordinary Boltzmann factors and another one given by fluctuations of an intensive parameter, e.g. the inverse temperature, or the energy dissipation, or a local variance. While the idea of fluctuating intensive parameters is certainly not new, it is the application to spatio-temporally chaotic systems such as turbulent flow that makes the concept interesting. The first turbulence model of this kind was introduced in [14], in the meantime the idea has been further refined and extended [15, 16, 3, 8]. The basic idea is to generate a superposition of two statistics, in short a ‘superstatistics’, by stochastic differential equations whose parameters fluctuate on a relatively large spatiotemporal scale. In Lagrangian turbulence, this large time scale can be understood by the fact that the particle is trapped in vortex tubes for quite a while [3]. Superstatistical turbulence models reproduce all the experimental data quite well. An example is shown in Fig. 1. The