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

TL;DR

This paper uses renormalization group and operator product expansion methods to analyze the anomalous scaling of a passive scalar field advected by a turbulent compressible fluid near four dimensions, revealing new regimes and fixed points.

Contribution

It introduces a RG+OPE framework for compressible turbulence near d=4, identifying new scaling regimes and calculating anomalous exponents systematically.

Findings

Correlation functions exhibit anomalous scaling in the inertial range.

Additional divergences at d=4 affect fixed point stability.

A new regime emerges at d=4, connecting to d=3 behavior.

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.