Critical behavior of vorticity in two-dimensional turbulence

Abstract

We point out some similitudes between the statistics of high Reynolds number turbulence and critical phenomena. An analogy is developed for two-dimensional decaying flows, in particular by studying the scaling properties of the two-point vorticity correlation function within a simple phenomenological framework. The inverse of the Reynolds number is the analogue of the small parameter that separates the system from criticality. It is possible to introduce a set of three critical exponents; for the correlation length, the autocorrelation function and a so-called susceptibility, respectively. The exponents corresponding to the well-known enstrophy cascade theory of Kraichnan and Batchelor are, remarkably, the same as the Gaussian approximation exponents for spin models. The limitations of the analogy, in particular the lack of universal scaling functions, are also discussed.