Anomalous scaling in statistical models of passively advected vector fields

TL;DR

This paper investigates the anomalous scaling behavior of passively advected vector fields in turbulence, revealing how anisotropy affects critical exponents through a field-theoretic renormalization group approach.

Contribution

It introduces a comprehensive analysis of anomalous scaling in passive vector fields with general nonlinear terms, including anisotropic effects, using the renormalization group and operator product expansion.

Findings

Anomalous scaling in passive vector fields is confirmed.

Critical exponents depend on tensor operator rank and anisotropy.

Isotropic case results align with Kolmogorov's hypothesis.

Abstract

The field theoretic renormalization group and the operator product expansion are applied to the stochastic model of passively advected vector field with the most general form of the nonlinear term allowed by the Galilean symmetry. The advecting turbulent velocity field is governed by the stochastic Navier--Stokes equation. It is shown that the correlation functions of the passive vector field in the inertial range exhibit anomalous scaling behaviour. The corresponding anomalous exponents are determined by the critical dimensions of tensor composite fields (operators) built solely of the passive vector field. They are calculated (including the anisotropic sectors) in the leading order of the expansion in $y$, the exponent entering the correlator of the stirring force in the Navier--Stokes equation (one-loop approximation of the renormalization group). The anomalous exponents exhibit an hierarchy related to the degree of anisotropy: the less is the rank of the tensor operator, the less is its dimension. Thus the leading terms, determined by scalar operators, are the same as in the isotropic case, in agreement with the Kolmogorov's hypothesis of the local isotropy restoration.