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

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 $\propto \delta (t-t^{\prime })k^{4-d-y}$, 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.

14 pages, LaTeX source, two EPS figures. arXiv admin note: text overlap with arXiv:1410.1262

References in corpus (8)

• Inertial-Range Kinetic Turbulence in Pressure-Anisotropic Astrophysical Plasmas
• Passive advection of a vector field: Anisotropy, finite correlation time, exact solution and logarithmic corrections to ordinary scaling
• Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev--Kraichnan kinematic model
• Inertial-Range Reconnection in Magnetohydrodynamic Turbulence and in the Solar Wind
• Anomalous scaling of passive scalar fields advected by the Navier-Stokes velocity ensemble: Effects of strong compressibility and large-scale anisotropy
• Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field
• Compressible advection of a passive scalar: Two-loop scaling regimes
• Sign singularity and flares in solar active region NOAA 11158

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• Helical turbulent Prandtl number in the $A$ model of passive advection: Two loop approximation
• Advection of a passive scalar field by turbulent compressible fluid: renormalization group analysis near $d=4$
• Stochastic Navier-Stokes equation and advection of a tracer field: One-loop renormalization near $d=4$
• Two-Loop Turbulent Helical Magnetohydrodynamics: Large-Scale Dynamo and Energy Spectrum
• Advection by Compressible Turbulent Flows: Renormalization Group Study of Vector and Tracer Admixture