Passive advection of a vector field: Anisotropy, finite correlation time, exact solution and logarithmic corrections to ordinary scaling

TL;DR

This paper analyzes the passive advection of a vector field in anisotropic turbulent flow with finite correlation time, providing exact results and revealing logarithmic corrections to scaling, contrasting with isotropic models.

Contribution

It extends previous models to finite correlation times, derives exact inertial-range behavior, and identifies logarithmic corrections to scaling in anisotropic turbulence.

Findings

Exact solutions due to vanishing multiloop diagrams

Two regimes corresponding to different fixed points

Logarithmic corrections replace anomalous scaling

Abstract

In this work we study the generalization of the problem, considered in [{\it Phys. Rev. E} {\bf 91}, 013002 (2015)], to the case of {\it finite} correlation time of the environment (velocity) field. The model describes a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow. Inertial-range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, with finite correlation time and preassigned pair correlation function. Due to the presence of distinguished direction ${\bf n}$, all the multiloop diagrams in this model are vanish, so that the results obtained are exact. The inertial-range behavior of the model is described by two regimes (the limits of vanishing or infinite correlation time) that correspond to the two nontrivial fixed points of the RG equations. Their stability depends on the relation between the exponents in the energy spectrum ${\cal E} \propto k_{\bot}^{1-\xi}$ and the dispersion law $\omega \propto k_{\bot}^{2-\eta}$. In contrast to the well known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale $L$.