Statistical symmetry restoration in fully developed turbulence: Renormalization group analysis of two models

TL;DR

This paper uses renormalization group analysis to study turbulence models, showing that Galilean symmetry is restored in the inertial range and analyzing scalar advection with anomalous scaling behaviors.

Contribution

It demonstrates symmetry restoration in turbulence models with colored noise and analyzes scalar advection, providing new insights into inertial-range dynamics.

Findings

Galilean symmetry is restored in the inertial range.

Inertial-range behavior is governed by a nontrivial fixed point.

Scalar field exhibits anomalous scaling consistent with Kolmogorov's hypothesis.

Abstract

In this paper we consider the model of incompressible fluid described by the stochastic Navier-Stokes equation with finite correlation time of a random force. Inertial-range asymptotic behavior of fully developed turbulence is studied by means of the field theoretic renormalization group within the one-loop approximation. It is corroborated that regardless of the values of model parameters and initial data, the inertial-range behavior of the model is described by limiting case of vanishing correlation time. It indicates that the Galilean symmetry of the model violated by the "colored" random force is restored in the inertial range. This regime corresponds to the only nontrivial fixed point of the renormalization group equation. The stability of this point depends on the relation between the exponents in the energy spectrum ${\cal E} \propto k^{1-y}$ and the dispersion law $\omega \propto k^{2-\eta}$. The second analyzed problem is the passive advection of a scalar field by this velocity ensemble. Correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial-convective range. We demonstrate that in accordance with Kolmogorov's hypothesis of the local symmetry restoration, the main contribution to the operator product expansion is given by the isotropic operator, while anisotropic terms should be considered only as corrections.