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

N. V. Antonov

doi:10.1016/S0167-2789(00)00089-0

paper

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

arXiv:chao-dyn/9907018 · doi:10.1016/S0167-2789(00)00089-0

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 large-scale anisotropy. The energy spectrum of the velocity 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 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.

25 pages; REVTeX file with LATeX figures inside

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

Abstract

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