Passive advection of a vector field: effects of strong compressibility

TL;DR

This paper investigates how strong compressibility affects the passive advection of a vector field using advanced field theoretic methods, revealing anomalous scaling and calculating critical dimensions near four dimensions.

Contribution

It applies the renormalization group and operator product expansion to a compressible stochastic Navier-Stokes driven model, providing new insights into anomalous scaling behaviors.

Findings

Passive vector field exhibits anomalous scaling in the inertial range.

Critical dimensions of tensor operators are calculated in leading order.

Compressibility influences the scaling behavior of the advected field.

Abstract

The field theoretic renormalization group and the operator product expansion are applied to the stochastic model of a passively advected vector field. The advecting velocity field is generated by the stochastic Navier-Stokes equation with compressibility taken into account. 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 $\epsilon=4-d$, where $y$ describes scaling behaviour of the random force that enters a stochastic equation for the velocity field. We show that the correlation functions of the passive vector field in the inertial range exhibit anomalous scaling behaviour. The critical dimensions of tensor composite operators of passive vector field are calculated in the leading order of $y$,$\epsilon$ expansion.