Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field

Abstract

The field theoretic renormalization group (RG) is applied to the problem of a passive scalar advected by the Gaussian self-similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient. The energy spectrum in the inertial range has the form $E(k)\propto k^{1-ε}$, and the correlation time at the wavenumber k scales as $k^{-2+η}$. It is shown that, depending on the values of the exponents $ε$ and $η$, the model in the inertial range reveals various types of scaling regimes associated with the infrared stable fixed points of the RG equations: diffusive-type regimes for which the advection can be treated within ordinary perturbation theory, and three nontrivial convection-type regimes for which the correlation functions exhibit anomalous scaling behavior. 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. For the first nontrivial regime the anomalous exponents are the same as in the rapid-change version of the model; for the second they are the same as in the model with time-independent (frozen) velocity field. In these regimes, the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator. For the last regime the exponents are nonuniversal (they can depend also on the amplitudes); however, the nonuniversality can reveal itself only in the second order of the RG expansion. Comments: Extended version accepted to Phys. Rev. E. 35 pages; REVTeX source with LATeX figures inside.