Effects of turbulent environment and random noise on self-organized critical behavior: Universality versus nonuniversality.

Self-organized criticality in the Hwa-Kardar model of a "running sandpile" [Phys. Rev. Lett.

Statistical symmetry restoration in fully developed turbulence: Renormalization group analysis of two models.

In this paper we consider the model of incompressible fluid described by the stochastic Navier-Stokes equation with finite correlation time of a random force. Inertial-range asymptotic behavior of fully developed turbulence is studied by means of the field theoretic renormalization group within the one-loop approximation. It is corroborated that regardless of the values of model parameters and initial data the inertial-range behavior of the model is described by the limiting case of vanishing correlation time.

Turbulent compressible fluid: Renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields.

We study a model of fully developed turbulence of a compressible fluid, based on the stochastic Navier-Stokes equation, by means of the field-theoretic renormalization group. In this approach, scaling properties are related to the fixed points of the renormalization group equations. Previous analysis of this model near the real-world space dimension 3 identified a scaling regime [N.

Directed percolation process in the presence of velocity fluctuations: Effect of compressibility and finite correlation time.

The direct bond percolation process (Gribov process) is studied in the presence of random velocity fluctuations generated by the Gaussian self-similar ensemble with finite correlation time. We employ the renormalization group in order to analyze a combined effect of the compressibility and finite correlation time on the long-time behavior of the phase transition between an active and an absorbing state. The renormalization procedure is performed to the one-loop order.

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

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 ∝ δ(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.

Passive advection of a vector field: Anisotropy, finite correlation time, exact solution, and logarithmic corrections to ordinary scaling.

In this work we study the generalization of the problem considered in [Phys. Rev. E 91, 013002 (2015)] to the case of finite correlation time of the environment (velocity) field.

Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field.

Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is studied by means of the field-theoretic renormalization group and the operator product expansion.

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

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.

Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: two-loop renormalization-group analysis of the Kazantsev-Kraichnan kinematic model.

The field theoretic renormalization group and operator product expansion are applied to the Kazantsev-Kraichnan kinematic model for the magnetohydrodynamic turbulence. The anomalous scaling emerges as a consequence of the existence of certain composite fields ("operators") with negative dimensions. The anomalous exponents for the correlation functions of arbitrary order are calculated in the two-loop approximation (second order of the renormalization-group expansion), including the anisotropic sectors.

Anomalous scaling of a passive scalar advected by the Navier-Stokes velocity field: two-loop approximation.

The field theoretic renormalization group and operator-product expansion are applied to the model of a passive scalar quantity advected by a non-Gaussian velocity field with finite correlation time. The velocity is governed by the Navier-Stokes equation, subject to an external random stirring force with the correlation function proportional to delta(t- t')k(4-d-2epsilon). It is shown that the scalar field is intermittent already for small epsilon, its structure functions display anomalous scaling behavior, and the corresponding exponents can be systematically calculated as series in epsilon.

Scaling behavior in a stochastic self-gravitating system.

A system of stochastic differential equations for the velocity and density of classical self-gravitating matter is investigated by means of the field theoretic renormalization group. The existence of two types of large-scale scaling behavior, associated with physically admissible fixed points of the renormalization-group equations, is established. Their regions of stability are identified and the corresponding scaling dimensions are calculated in the one-loop approximation (first order of the epsilon expansion).

Turbulence with pressure: anomalous scaling of a passive vector field.

The field theoretic renormalization group (RG) and the operator-product expansion are applied to the model of a transverse (divergence-free) vector quantity, passively advected by the "synthetic" turbulent flow with a finite (and not small) correlation time. The vector field is described by the stochastic advection-diffusion equation with the most general form of the inertial nonlinearity; it contains as special cases the kinematic dynamo model, linearized Navier-Stokes (NS) equation, the special model without the stretching term that possesses additional symmetries and has a close formal resemblance with the stochastic NS equation. The statistics of the advecting velocity field is Gaussian, with the energy spectrum E(k) proportional to k(1-epsilon) and the dispersion law omega proportional to k(-2+eta), k being the momentum (wave number).

Field-theoretic renormalization group for a nonlinear diffusion equation.

This paper is an attempt to relate two vast areas of the applicability of the renormalization group (RG): field-theoretic models and partial differential equations. It is shown that the Green function of a nonlinear diffusion equation can be viewed as a correlation function in a field-theoretic model with an ultralocal term, concentrated at a space-time point. This field theory is shown to be multiplicatively renormalizable, so that the RG equations can be derived in a standard fashion, and the RG functions (the beta function and anomalous dimensions) can be calculated within a controlled approximation.

Anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time: two-loop approximation.

The renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the Gaussian self-similar velocity field with finite, and not small, correlation time. The inertial-range energy spectrum of the velocity is chosen in the form E(k) proportional, variant k(1-2 epsilon ), and the correlation time at the wave number k scales as k(-2+eta). Inertial-range anomalous scaling for the structure functions and other correlation functions emerges as a consequence of the existence in the model of composite operators with negative scaling dimensions, identified with anomalous exponents.

Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field.

The field theoretic renormalization group (RG) is applied to the problem of a passive scalar advected by the Gaussian self-similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient. The energy spectrum in the inertial range has the form E(k) proportional to (1-epsilon), and the correlation time at the wave number k scales as k(-2+eta). It is shown that, depending on the values of the exponents epsilon and eta, the model in the inertial-convective range exhibits various types of scaling regimes associated with the infrared stable fixed points of the RG equations: diffusive-type regimes for which the advection can be treated within ordinary perturbation theory, and three nontrivial convection-type regimes for which the correlation functions exhibit anomalous scaling behavior.

Influence of compressibility on scaling regimes of strongly anisotropic fully developed turbulence.

A statistical model of strongly anisotropic fully developed turbulence of the weakly compressible fluid is considered by means of the field theoretic renormalization group. The corrections due to compressibility to the infrared form of the kinetic energy spectrum have been calculated in the leading order in the Mach number expansion. Furthermore, in this approximation the validity of the Kolmogorov hypothesis on the independence of the dissipation length of velocity correlation functions in the inertial range has been proved.

Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order epsilon(3).

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the Gaussian velocity field with zero mean and correlation function approximately equal to delta(t-t('))/k(d + epsilon). Inertial-range anomalous exponents, identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients, are calculated within the epsilon expansion to order epsilon(3) (three-loop approximation), including the exponents in anisotropic sectors. The main goal of the paper is to give the complete derivation of this third-order result, and to present and explain in detail the corresponding calculational techniques.

Anomalous scaling, nonlocality, and anisotropy in a model of the passively advected vector field.

A model of the passive vector quantity advected by the Gaussian velocity field with the covariance approximately delta(t-t('))|x-x(')|(epsilon) is studied; the effects of pressure and large-scale anisotropy are discussed. The inertial-range behavior of the pair correlation function is described by an infinite family of scaling exponents, which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques. The exponents are organized in a hierarchical order according to their degree of anisotropy, with the spectrum unbounded from above and the leading (minimal) exponent coming from the isotropic sector.

Anomalous scaling in two models of passive scalar advection: effects of anisotropy and compressibility.

The problem of the effects of compressibility and large-scale anisotropy on anomalous scaling behavior is considered for two models describing passive advection of scalar density and tracer fields. The advecting velocity field is Gaussian, delta correlated in time, and scales with a positive exponent epsilon. Explicit inertial-range expressions for the scalar correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal anomalous exponents (dependent only on epsilon and alpha, the compressibility parameter).

Anomalous exponents to order epsilon 3 in the rapid-change model of passive scalar advection.

Field-theoretic renormalization group is applied to the Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance -~delta(t-t('))|x-x(')|(epsilon). Inertial-range anomalous exponents, related to the scaling dimensions of tensor composite operators built of the scalar gradients, are calculated to the order epsilon(3) of the epsilon expansion. The nature and the convergence of the epsilon expansion in the models of turbulence are briefly discussed.

Anomalous scaling of a passive scalar in the presence of strong anisotropy.

Field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta(t,x), advected by the Gaussian strongly anisotropic velocity field with the covariance infinity delta(t-t('))/x-x(')/(epsilon). Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the structure functions S(n)(r) identical with<[straight theta(t,x+r)-straight theta(t,x)](n)> are obtained. They are represented by superpositions of power laws; the corresponding anomalous exponents, which depend explicitly on the anisotropy parameters, are calculated to the first order in epsilon in any space dimension d.

Manifestation of anisotropy persistence in the hierarchies of magnetohydrodynamical scaling exponents.

An example of a turbulent system where the failure of the hypothesis of small-scale isotropy restoration is detectable both in the "flattening" of the inertial-range scaling exponent hierarchy and in the behavior of odd-order dimensionless ratios, e.g., skewness and hyperskewness, is presented.

Persistence of small-scale anisotropies and anomalous scaling in a model of magnetohydrodynamics turbulence.

The problem of anomalous scaling in magnetohydrodynamics turbulence is considered within the framework of the kinematic approximation, in the presence of a large-scale background magnetic field. The velocity field is Gaussian, delta-correlated in time, and scales with a positive exponent xi. Explicit inertial-range expressions for the magnetic correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (independent of the anisotropy and forcing) anomalous exponents.