Anomalous scaling of a passive scalar advected by the synthetic compressible flow

The field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by a Gaussian nonsolenoidal velocity field with finite correlation rime in the presence of large-scale anisotropy. The energy spectrum of the velocity in the inertial range has the form E(k) proportional to k(1-epsilon), and the correlation time at the wave number k scales as k(-2+eta). It is shown that depending on the values of the exponents epsilon and eta, the model exhibits various types of inertial-range scaling regimes with nontrivial anomalous exponents. Explicit asymptotic expressions for the structure functions and other correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (independent of the anisotropy) anomalous exponents, calculated to the first-order in epsilon and eta in any space dimension. These anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. The anomalous exponents depend explicitly on the degree of compressibility. (C) 2000 Elsevier Science B.V. All rights reserved.