Superstatistics and isotropic turbulence

In this work, we analyze the capacity of the superstatistics construction to provide modeling of the velocity field probability density functions (PDFs) of isotropic turbulence. Generalizing along the lines of the kappa distribution, superstatistics is understood here as a PDF for the statistical temperature that depends on a single dimensionful parameter θ² and a dimensionless parameter κ₀, which both depend on the size of the fluid eddies and the Reynolds number, and possibly on auxiliary dimensionless constants that depend only on the Reynolds number. We show that such superstatistics –in some sense, the simplest class of models– cannot provide PDFs for scales outside the dissipation subrange for the currently accessible Reynolds numbers in Direct Numerical Simulations (DNS). The obstruction results from realizability constraints and an associated bound, and is related to the flatness factor of the velocity derivative distribution. Greater values of the flatness extend the applicability of superstatistics to larger scales. We argue that phenomenologically effective superstatistics models will require a value of flatness F∼25 or larger in order to cover the inertial subrange scales. The argument is assisted by constructing and analyzing a family of models which derive from modifying the gamma distribution in the regime of large statistical temperatures and nearly realize the realizability bound.