Anomalous scaling of a passive vector field in $d$ dimensions: Higher-order structure functions

Abstract

The problem of anomalous scaling in the model of a transverse vector field ${\theta }_i(t,x)$ passively advected by the non-Gaussian, correlated in time turbulent velocity field governed by the Navier--Stokes equation, is studied by means of the field-theoretic renormalization group and operator product expansion. The anomalous exponents of the $2n$-th order structure function $S_{2n}(r)=<[\theta (t,x)-\theta (t,x+r){]}^{2n}>$, where $\theta$ is the component of the vector field parallel to the separation $r$, are determined by the critical dimensions of the family of composite fields (operators) of the form $(\partial \theta \partial \theta {)}^{2n}$, which mix heavily in renormalization. The daunting task of the calculation of the matrices of their critical dimensions (whose eigenvalues determine the anomalous exponents) simplifies drastically in the limit of high spatial dimension, $d\to \infty$. This allowed us to find the leading and correction anomalous exponents for the structure functions up to the order $S_{56}$. They reveal intriguing regularities, which suggest for the anomalous exponents simple "empiric" formulae that become practically exact for $n$ large enough. Along with the explicit results for modest $n$, they provide the full description of the anomalous scaling in the model. Key words: passive vector field, turbulent advection, anomalous scaling, renormalization group, operator product expansion.

arXiv comment20 pages, LaTeX source with one EPS figure. Published version; minor revisions

References in corpus (27)

• Particles and fields in fluid turbulence
• ANOMALOUS SCALING OF THE PASSIVE SCALAR
• Normal and Anomalous Scaling of the Fourth-Order Correlation Function of a Randomly Advected Passive Scalar
• Integrability of three-particle evolution equations in QCD
• Anomalous Scaling Exponents of a White-Advected Passive Scalar
• Renormalization Group, Operator Product Expansion, and Anomalous Scaling in a Model of Advected Passive Scalar
• Influence of compressibility on scaling regimes of strongly anisotropic fully developed turbulence
• Anomalous Scaling in the N-Point Functions of Passive Scalar
• Persistence of small-scale anisotropies and anomalous scaling in a model of magnetohydrodynamics turbulence
• Renormalization-group approach to the stochastic Navier--Stokes equation: Two-loop approximation
• Turbulence with Pressure: Anomalous Scaling of a Passive Vector Field
• Quantum integrability in (super) Yang-Mills theory on the light-cone
• Nonperturbative Spectrum of Anomalous Scaling Exponents in the Anisotropic Sectors of Passively Advected Magnetic Fields
• Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order ${\epsilon }^3$
• Anisotropic non-perturbative zero modes for passively advected magnetic fields
• Anomalous exponents in the rapid-change model of the passive scalar advection in the order ${\epsilon }^3$
• Anomalous scaling of a passive scalar advected by the Navier--Stokes velocity field: Two-loop approximation
• Anomalous scaling, nonlocality and anisotropy in a model of the passively advected vector field
• Pressure and intermittency in passive vector turbulence
• Anomalous scaling of passively advected magnetic field in the presence of strong anisotropy
• Spectrum of Anisotropic Exponents in Hydrodynamic Systems with Pressure
• Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev--Kraichnan kinematic model
• Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection
• Anomalous scaling and anisotropy in models of passively advected vector fields
• Two-Loop Calculation of the Anomalous Exponents in the Kazantsev--Kraichnan Model of Magnetic Hydrodynamics
• Anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow
• Towards a geometrical classification of statistical conservation laws in turbulent advection

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