Field theoretical renormalization group methods are applied to the Obukhov-Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance $〈\mathbf{v}(t,\mathbf{x})\mathbf{v}{(t}^{\ensuremath{'}},\mathbf{x})〉\ensuremath{-}〈\mathbf{v}(t,\mathbf{x})\mathbf{v}{(t}^{\ensuremath{'}},{\mathbf{x}}^{\ensuremath{'}})〉\ensuremath{\propto}\ensuremath{\delta}(t\ensuremath{-}{t}^{\ensuremath{'}})|\mathbf{x}\ensuremath{-}{\mathbf{x}}^{\ensuremath{'}}{|}^{\ensuremath{\varepsilon}}.$ Inertial range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain essential or ``dangerous'' composite operators [powers of the local dissipation rate], whose negative critical dimensions determine anomalous exponents. The main technical result is the calculation of the anomalous exponents in the order ${\ensuremath{\varepsilon}}^{2}$ of the $\ensuremath{\varepsilon}$ expansion. Generalization of the results obtained to the case of a ``slow'' velocity field is also presented.

pdf/renormalization-group-operator-product-expansion-and-3mkf8pif4a.pdf

Renormalization group, operator product expansion, and anomalous scaling in a model of advected passive scalar

Renormalization group theory for fluid and plasma turbulence

The field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities (the density and the tracer fields) 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 ∝δ(t-t')k(4-d-y), where d is the dimension of space and y is an arbitrary exponent. The original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models; the corresponding renormalization group equations possess infrared attractive fixed points. It is shown that various correlation functions of the scalar field, its powers and gradients, 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. The validity of the one-loop approximation and comparison with Gaussian models are briefly discussed.

Anomalous scaling of passive scalar fields advected by the Navier-Stokes velocity ensemble: effects of strong compressibility and large-scale anisotropy.

Renormalization-group analysis of randomly stirred fluid with anisotropic distribution of random force is carried out at one-loop order. The axial anisotropy is introduced by free parameters of external forcing in the Navier-Stokes equation, but the anisotropy parameters are not assumed to be small. The region of stability of the Kolmogorov scaling regime in the space of anisotropy parameters has been determined for several space dimensionalities 2{lt}d{le}3. The Kolmogorov constant and the amplitudes of longitudinal and transverse projection operators with respect to the preferred direction in the energy spectrum have been calculated in situations where the competition between the anisotropy of the external forcing and the Navier-Stokes dynamics may affect the stability of the Kolmogorov regime. Extension to more complex magnetohydrodynamic systems is under investigation. {copyright} {ital 1997} {ital The American Physical Society}

Stability of Kolmogorov scaling in anisotropically forced turbulence

This Rapid Communication presents a simple closure for the two-point correlation transport equation in decaying isotropic turbulence. It relies essentially on an eddy viscosity ${\ensuremath{\nu}}_{t}$ which exhibits some remarkable universal facets over an impressively wide range of scales. This allows us to model the third-order structure functions in different decaying flows covering a large extent of Reynolds numbers. The model is numerically time integrated to predict the decay of second-order structure functions and compared to experiments in grid turbulence. Agreement between predictions and measurements is satisfactory.

/pdf/karman-howarth-closure-equation-on-the-basis-of-a-universal-4dz1eaquzq.pdf

Kármán-Howarth closure equation on the basis of a universal eddy viscosity.

The model of anisotropic magnetohydrodynamic turbulence is investigated by the renormalization group approach. It is demonstrated that the inclusion of small anisotropy into the model leads to increasing of the Lorentz force influence. This effect is especially important in the kinetic regime in which the Kolmogorov spectrum of pulsation energy takes place. It is quite different from the isotropic case where the Lorentz force has no influence on large scale properties of magnetohydrodynamic turbulence, even if the external injection of energy is very intensive. Therefore, the magnetic field behaves like a passive admixture. In the anisotropic MHD, nonlinear interactions generate modified "anisotropic Lorentz forces". These forces are relevant for certain values of dimensionless parameter a, which describes the spectrum of the magnetic noise, and the magnetic field ceases to be a passive admixture. In particular, this statement is true for the most realistic value of a = 1 when the random magnetic force amplitude has the same dimensionality as the energy injection rate.

Infrared properties of an anisotropically driven MHD turbulence

Calculation of spectra of turbulence in the energy-containing and inertial ranges

The phenomenon of magnetic field generation in an astrophysical plasma in the frame of developed magnetohydrodynamic (MHD) turbulence is considered. The functional quantum field renormalization group approach is applied to helical anisotropic MHD developed turbulence which is stabilized by the self-generated homogeneous magnetic field. The purpose of the study is to calculate the value as well as direction of the magnetic field in the stochastic dynamo model. The generated magnetic field is determined by ignoring divergent rotor part of Green function of the magnetic field. It is shown that the magnetic field direction is connected with unique existing vector n describing the anisotropic turbulence forcing.

/pdf/the-spontaneous-magnetic-field-direction-in-an-anisotropic-12up53pypa.pdf

The spontaneous magnetic field direction in an anisotropic MHD dynamo

A set of self-consistent equations in one-loop approximation in a statistical model of fully developed homogeneous isotropic turbulence, which is based on the principle of maximal randomness of the velocity field with a given energy spectral flux, is obtained These equations do not possess both infrared and ultraviolet divergences near the Kolmogorov values of indices The formal solution found of the system of equations yields Kolmogorov exponents, but this solution leads to negative Kolmogorov constant C k and negative viscosity It has been shown, that the turbulent fluid becomes stable if spontaneous parity violation is achieved Namely, the solution with the Kolmogorov indices and additional helical terms (which lead to positive both C k and effective viscosity) exists This solution predicts a large, closed to limit value, helical coefficient Θ in inertial range The relationship obtained between C k and e confirms this conclusion for the experimental value of C k 

Theory of Developed Turbulence: Principle of Maximal Randomness and Spontaneous Parity Violation

Influence of anisotropy on the scaling regimes in fully developed turbulence

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