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

Abstract

The field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by the Gaussian nonsolenoidal velocity field with finite correlation time, in the presence of largescale anisotropy. The energy spectrum of the velocity in the inertial range has the form E(k) ∝ 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 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.

Compressible advection of a passive scalar: two-loop scaling regimes

“…The only condition which must be fulfilled by the function C(r/L) is that it must decrease rapidly for r ≡ |r| ≫ L, where L denotes an integral scale related to the stirring. The velocity field v(x) obeys a Gaussian distribution with zero mean and correlator [13] …”

“…The value α = 0 corresponds to the divergence-free (incompressible) advecting velocity field. The functionD v is chosen as follows [12,13] …”

“…Here γ * G is the fixed point value of the anomalous dimension γ G ≡ µ∂ µ ln Z G , where Z G is renormalization constant of multiplicatively renormalizable quantity G, i.e., G = Z G G R [13], and ∆ ω = 2−γ * ν is the critical dimension of frequency with γ ν = γ 1 which is defined further in the text (for more details see, e.g., [12,13]). …”

“…But the calculation of higher-order corrections is more difficult in the models with turbulent velocity field with finite correlation time than in the cases with δ-correlations in time. It is related to the fact that the diagrams for the finite correlated case involve two different dispersion laws, namely, ω ∝ k 2 for the scalar field and ω ∝ k 2−η for the velocity field which complicates situation even in the one-loop approximation [12,13]. But, as was discussed in [12,13,15], this difficulty can be avoided by the calculation of all renormalization constants in an arbitrary specific choice of the exponents ε and η that guarantees UV finiteness of the Feynman diagrams.…”

“…Afterwards, various generalized descendants of the Kraichnan model, namely, models with inclusion of large and small scale anisotropy [10], compressibility [11] and finite correlation time of the velocity field [12,13] were studied by the field theoretic approach. General conclusion is: the anomalous scaling, which is the most important feature of the Kraichnan rapid change model, remains valid for all generalized models.…”