TY - JOUR

T1 - Renormalization group, operator product expansion and anomalous scaling in models of turbulent advection

AU - Antonov, N. V.

PY - 2006/6/23

Y1 - 2006/6/23

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33745082451&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/39/25/S04

DO - 10.1088/0305-4470/39/25/S04

M3 - Article

AN - SCOPUS:33745082451

VL - 39

SP - 7825

EP - 7865

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 25

M1 - S04

ER -