Renormalization group, operator product expansion and anomalous scaling in models of turbulent advection

Abstract

Recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field-theoretic renormalization group (RG) and operator product expansion (OPE). In that approach, the anomalous scaling is established as a consequence of the existence in the corresponding field-theoretic models of an infinite number of 'dangerous' composite fields (operators) with negative critical dimensions, which are identified with the anomalous exponents. This allows one to calculate the exponents in a systematic perturbation expansion, similar to the ε expansion in the theory of critical phenomena. The RG and OPE approach is presented in a self-contained way for the example of a passive scalar field (temperature, concentration of an impurity, etc) advected by a self-similar Gaussian velocity ensemble with vanishing correlation time, the so-called Kraichnan's rapid-change model, where the anomalous exponents are known up to order O(ε3). Effects of anisotropy, compressibility and the correlation time of the velocity field are discussed. Passive advection by non-Gaussian velocity field governed by the stochastic Navier–Stokes equation and passively advected vector (e.g. magnetic) fields are considered.