Passive Advection of a Vector Field by Compressible Turbulent Flow: Renormalizations Group Analysis near d = 4

Nikolay Antonov,Tomáš Lučivjanský,Nikolay Gulitskiy,Maria Kostenko

doi:10.3390/universe5010037

Nikolay Antonov, Tomáš Lučivjanský + Show 2 more

Open Access

https://doi.org/10.3390/universe5010037

Copy DOI

Abstract

The renormalization group approach and the operator product expansion technique are applied to the model of a passively advected vector field by a turbulent velocity field. The latter is governed by the stochastic Navier-Stokes equation for a compressible fluid. The model is considered in the vicinity of space dimension d = 4 and the perturbation theory is constructed within a double expansion scheme in y and ε = 4 − d , where y describes scaling behaviour of the random force that enters the Navier-Stokes equation. The properties of the correlation functions are investigated, and anomalous scaling and multifractal behaviour are established. All calculations are performed in the leading order of y, ε expansion (one-loop approximation).

Highlights

Many phenomena in the nature are associated with hydrodynamic flows
Honkonen and Nalimov [48] showed that in the vicinity of space dimension d = 2 additional divergences appear in the model of the incompressible Navier-Stokes ensemble, and these divergences have to be properly taken into account
Its proper application requires a proof of a renormalizability of the model, i.e., a proof that only a finite number of divergent structures exists in a diagrammatic expansion [22]

Summary

Introduction

Many phenomena in the nature are associated with hydrodynamic flows. Ranging from microscopic up to macroscopic spatial scales, fluids can exist in very different states. In the theory of critical phenomena that parameter is ε = 4 − d, the deviation of the dimensionality of space d from its upper critical value, the term “epsilon expansion” Such expansions are still divergent, but they allow one to prove the existence of infrared (IR) scaling behavior (if such exists) and to systematically calculate the corresponding dimensions as series in ε. Honkonen and Nalimov [48] showed that in the vicinity of space dimension d = 2 additional divergences appear in the model of the incompressible Navier-Stokes ensemble, and these divergences have to be properly taken into account Their procedure results into improved perturbation expansion [49,50]. Where Cij (r/Lθ ) is a certain function finite at limit (r/Lθ ) → 0 and rapidly decaying for (r/Lθ ) → ∞

Quantum Field Theory Formulation

Renormalization Group Analysis

Scaling Regimes

Composite Fields

Operator Product Expansion

Conclusions