Advection of a passive scalar field by turbulent compressible fluid: renormalization group analysis near $d=4$

Abstract

The field theoretic renormalization group (RG) and the operator product expansion (OPE) are applied to the model of a density field advected by a random turbulent velocity field. The latter is governed by the stochastic Navier-Stokes equation for a compressible fluid. The model is considered near the special space dimension $d=4$. It is shown that various correlation functions of the scalar field exhibit anomalous scaling behaviour in the inertial-convective range. The scaling properties in the RG+OPE approach are related to fixed points of the renormalization group equations. In comparison with physically interesting case $d=3$, at $d=4$ additional Green function has divergences which affect the existence and stability of fixed points. From calculations it follows that a new regime arises there and then by continuity moves into $d=3$. The corresponding anomalous exponents are identified with scaling dimensions of certain composite fields and can be systematically calculated as series in $y$ (the exponent, connected with random force) and $\epsilon =4-d$. All calculations are performed in the leading one-loop approximation.

arXiv comment11pages, 6 figures, LATEX2e. arXiv admin note: substantial text overlap with arXiv:1611.00327; text overlap with arXiv:1611.01302

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