Anomalous scaling in magnetohydrodynamic turbulence: Effects of anisotropy and compressibility in the kinematic approximation

TL;DR

This paper uses field theoretic methods to analyze anomalous scaling in magnetohydrodynamic turbulence within the kinematic approximation, accounting for anisotropy and compressibility effects, and calculates critical exponents systematically.

Contribution

It introduces a renormalizable field theoretic model for passive magnetic fields in compressible turbulence and computes anomalous exponents including anisotropic effects in a systematic way.

Findings

Magnetic field correlation functions show anomalous scaling in the inertial range.

Anomalous exponents can be calculated as series in the parameter y.

The model is Galilean covariant, unlike Gaussian ensemble models.

Abstract

The field theoretic renormalization group and the operator product expansion are applied to the model of passive vector (magnetic) field advected by a random turbulent velocity field. The latter is governed by the Navier--Stokes equation for compressible fluid, subject to external random force with the covariance $\propto \delta(t-t') k^{4-d-y}$, where $d$ is the dimension of space and $y$ is an arbitrary exponent. From physics viewpoints, the model describes magnetohydrodynamic turbulence in the so-called kinematic approximation, where the effects of the magnetic field on the dynamics of the fluid are neglected. The original stochastic problem is reformulated as a multiplicatively renormalizable field theoretic model; the corresponding renormalization group equations possess an infrared attractive fixed point. It is shown that various correlation functions of the magnetic field and its powers demonstrate anomalous scaling behavior in the inertial-convective range already for small values of~$y$. The corresponding anomalous exponents, identified with scaling (critical) dimensions of certain composite fields ("operators" in the quantum-field terminology), can be systematically calculated as series in $y$. The practical calculation is performed in the leading one-loop approximation, including exponents in anisotropic contributions. It should be emphasized that, in contrast to Gaussian ensembles with finite correlation time, the model and the perturbation theory presented here are manifestly Galilean covariant.