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. 62, 1813 (1989)10.1103/PhysRevLett.62.1813; Phys. Rev. A 45, 7002 (1992)10.1103/PhysRevA.45.7002] with a turbulent motion of the environment taken into account is studied with the field theoretic renormalization group (RG). The turbulent flow is modeled by the synthetic d-dimensional generalization of the anisotropic Gaussian velocity ensemble with finite correlation time, introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990)10.1007/BF02161420; Commun. Math. Phys. 146, 139 (1992)10.1007/BF02099212]. The Hwa-Kardar model with time-independent (spatially quenched) random noise is considered alongside the original model with white noise. The aim of the present paper is to explore fixed points of the RG equations which determine the possible types of universality classes (regimes of critical behavior of the system) and critical dimensions of the measurable quantities. Our calculations demonstrate that influence of the type of random noise is extremely large: in contrast to the case of white noise where the system possesses three fixed points, the case of spatially quenched noise involves four fixed points with overlapping stability regions. This means that in the latter case the critical behavior of the system depends not only on the global parameters of the system, which is the usual case, but also on the initial values of the charges (coupling constants) of the system. These initial conditions determine the specific fixed point which will be reached by the RG flow. Since now the critical properties of the system are not defined strictly by its parameters, the situation may be interpreted as a universality violation. Such systems are not forbidden, but they are rather rare. It is especially interesting that the same model without turbulent motion of the environment does not predict this nonuniversal behavior and demonstrates the usual one with prescribed universality classes instead [J. Stat. Phys. 178, 392 (2020)10.1007/s10955-019-02436-8].

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. This indicates that the Galilean symmetry of the model violated by the "colored" random force is restored in the inertial range. This regime corresponds to the only nontrivial fixed point of the renormalization group equation. The stability of this point depends on the relation between the exponents in the energy spectrum E∝k^{1-y} and the dispersion law ω∝k^{2-η}. The second analyzed problem is the passive advection of a scalar field by this velocity ensemble. Correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial-convective range. We demonstrate that in accordance with Kolmogorov's hypothesis of the local symmetry restoration the main contribution to the operator product expansion is given by the isotropic operator, while anisotropic terms should be considered only as corrections.

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. V. Antonov et al., Theor. Math. Phys. 110, 305 (1997)TMPHAH0040-577910.1007/BF02630456]. The aim of the present paper is to explore the existence of additional regimes, which could not be found using the direct perturbative approach of the previous work, and to analyze the crossover between different regimes. It seems possible to determine them near the special value of space dimension 4 in the framework of double y and ɛ expansion, where y is the exponent associated with the random force and ɛ=4-d is the deviation from the space dimension 4. Our calculations show that there exists an additional fixed point that governs scaling behavior. Turbulent advection of a passive scalar (density) field by this velocity ensemble is considered as well. We demonstrate that various correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial-convective range. The corresponding anomalous exponents, identified as scaling dimensions of certain composite fields, can be systematically calculated as a series in y and ɛ. All calculations are performed in the leading one-loop approximation.

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. The model describes a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow. Inertial-range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, with finite correlation time and preassigned pair correlation function. Due to the presence of distinguished direction n, all the multiloop diagrams in this model vanish, so that the results obtained are exact. The inertial-range behavior of the model is described by two regimes (the limits of vanishing or infinite correlation time) that correspond to the two nontrivial fixed points of the RG equations. Their stability depends on the relation between the exponents in the energy spectrum E∝k(⊥)(1-ξ) and the dispersion law ω∝k(⊥)(2-η). In contrast to the well-known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale L.

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. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form ∝δ(t-t')/k(⊥)(d-1+ξ), where k(⊥)=|k(⊥)| and k(⊥) is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow")--the d-dimensional generalization of the ensemble introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990)]. The stochastic advection-diffusion equation for the transverse (divergence-free) vector field includes, as special cases, the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized Navier-Stokes equation. In contrast to the well-known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale L has a logarithmic behavior: Instead of powerlike corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of L. The key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized. The detailed proof of this fact is given for the correlation functions of arbitrary order.

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. The anomalous scaling and the hierarchy of anisotropic contributions become stronger due to those second-order contributions.