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 time in the presence of large-scale anisotropy. The energy spectrum of the velocity in the inertial range has the form E(k)∝k1−ε, and the correlation time at the wave number k scales as k−2+η. It is shown that depending on the values of the exponents ε and η, 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 ε and η 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.