Anomalous scaling in statistical models of passively advected vector fields

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.

12 pages, LaTeX source. Version accepted for publication in Theor. Math. Phys.: minor revisions. arXiv admin note: text overlap with arXiv:1212.3941, arXiv:1202.5992

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• Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field
• Anomalous scaling in magnetohydrodynamic turbulence: Effects of anisotropy and compressibility in the kinematic approximation
• Statistical symmetry restoration in fully developed turbulence: Renormalization group analysis of two models
• Turbulent compressible fluid: renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields
• Helical turbulent Prandtl number in the $A$ model of passive advection: Two loop approximation
• Advection by Compressible Turbulent Flows: Renormalization Group Study of Vector and Tracer Admixture