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

Amplification of the anomalous scaling in the Kazantsev-Kraichnan model with finite-time correlations and spatial parity violation

By using the field theoretic renormalization group technique together with the operator product expansion, simultaneous influence of the spatial parity violation and finite-time correlations of an electrically conductive turbulent environment on the inertial-range scaling behavior of correlation functions of a passively advected weak magnetic field is investigated within the corresponding generalized Kazantsev-Kraichnan model in the second order of the perturbation theory (in the two-loop approximation). The explicit dependence of the anomalous dimensions of the leading composite operators on the fixed point value of the parameter that controls the presence of finite-time correlations of the turbulent field as well as on the parameter that drives the amount of the spatial parity violation (helicity) in the system is found even in the case with the presence of the large-scale anisotropy. In accordance with the Kolmogorov's local isotropy restoration hypothesis, it is shown that, regardless of the amount of the spatial parity violation, the scaling properties of the model are always driven by the anomalous dimensions of the composite operators near the isotropic shell. The asymptotic (inertial-range) scaling form of all single-time two-point correlation functions of arbitrary order of the passively advected magnetic field is found. The explicit dependence of the corresponding scaling exponents on the helicity parameter as well as on the parameter that controls the finite-time velocity correlations is determined. It is shown that, regardless of the amount of the finite-time correlations of the given Gaussian turbulent environment, the presence of the spatial parity violation always leads to more negative values of the scaling exponents, i.e., to the more pronounced anomalous scaling of the magnetic correlation functions. At the same time, it is shown that the stronger the violation of spatial parity, the larger the anomalous behavior of magnetic correlations.

Anomalous scaling in kinematic magnetohydrodynamic turbulence: Two-loop anomalous dimensions of leading composite operators

Using the field theoretic formulation of the kinematic magnetohydrodynamic turbulence, the explicit expressions for the anomalous dimensions of leading composite operators, which govern the inertial-range scaling properties of correlation functions of the weak magnetic field passively advected by the electrically conductive turbulent environment driven by the Navier-Stokes velocity field, are derived and analyzed in the second order of the corresponding perturbation expansion (in the two-loop approximation). Their properties are compared to the properties of the same anomalous dimensions obtained in the framework of the Kazantsev-Kraichnan model of the kinematic magnetohydrodynamics with the Gaussian statistics of the turbulent velocity field as well as to the analogous anomalous dimensions of the leading composite operators in the problem of the passive scalar advection by the Gaussian (the Kraichnan model) and non-Gaussian (driven by the Navier-Stokes equation) turbulent velocity field. It is shown that, regardless of the Gaussian or non-Gaussian statistics of the turbulent velocity field, the two-loop corrections to the leading anomalous dimensions are much more important in the case of the problem of the passive advection of the vector (magnetic) field than in the case of the problem of the passive advection of scalar fields. At the same time, it is also shown that, in phenomenologically the most interesting case with three spatial dimensions, higher velocity correlations of the turbulent environment given by the Navier-Stokes velocity field play a rather limited role in the anomalous scaling of passive scalar as well as passive vector quantities, i.e., that the two-loop corrections to the corresponding leading anomalous dimensions are rather close to those obtained in the framework of the Gaussian models, especially as for the problem of scalar field advection. On the other hand, the role of the non-Gaussian statistics of the turbulent velocity field becomes dominant for higher spatial dimensions in the case of the kinematic magnetohydrodynamic turbulence but remains negligible in the problem of the passive scalar advection.

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].

Passive Advection of a Vector Field by Compressible Turbulent Flow: Renormalizations Group Analysis near d = 4

The renormalization group approach and the operator product expansion technique are applied to the model of a passively advected vector field by a turbulent velocity field. The latter is governed by the stochastic Navier-Stokes equation for a compressible fluid. The model is considered in the vicinity of space dimension d = 4 and the perturbation theory is constructed within a double expansion scheme in y and ε = 4 − d , where y describes scaling behaviour of the random force that enters the Navier-Stokes equation. The properties of the correlation functions are investigated, and anomalous scaling and multifractal behaviour are established. All calculations are performed in the leading order of y, ε expansion (one-loop approximation).

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.

Simultaneous influence of helicity and compressibility on anomalous scaling of the magnetic field in the Kazantsev-Kraichnan model

Using the field theoretic renormalization group technique and the operator product expansion, the systematic investigation of the influence of the spatial parity violation on the anomalous scaling behavior of correlation functions of the weak passive magnetic field in the framework of the compressible Kazantsev-Kraichnan model with the presence of a large-scale anisotropy is performed up to the second order of the perturbation theory (two-loop approximation). The renormalization group analysis of the model is done and the two-loop explicit expressions for the anomalous and critical dimensions of the leading composite operators are found as functions of the helicity and compressibility parameters and their anisotropic hierarchies are discussed. It is shown that for arbitrary values of the helicity parameter and for physically acceptable (small enough) values of the compressibility parameter, the main role is played by the composite operators near the isotropic shell in accordance with the Kolmogorov's local isotropy restoration hypothesis. The anomalous dimensions of the relevant composite operators are then compared with the anomalous dimensions of the corresponding leading composite operators in the Kraichnan model of passively advected scalar field. The significant difference between these two sets of anomalous dimensions is discussed. The two-loop inertial-range scaling exponents of the single-time two-point correlation functions of the magnetic field are found and their dependence on the helicity and compressibility parameters is studied in detail. It is shown that while the presence of the helicity leads to more pronounced anomalous scaling for correlation functions of arbitrary order, the compressibility, in general, makes the anomalous scaling more pronounced in comparison to the incompressible case only for low-order correlation functions. The persistence of the anisotropy deep inside the inertial interval is investigated using the appropriate odd ratios of the correlation functions. It is shown that, in general, the persistence of the anisotropy is much more pronounced in the helical systems, while in the compressible turbulent environments this is true only for low-order odd ratios of the correlation functions.

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.

Helical turbulent Prandtl number in the A model of passive vector advection

Using the field theoretic renormalization group technique in the two-loop approximation, turbulent Prandtl numbers are obtained in the general A model of passive vector advected by fully developed turbulent velocity field with violation of spatial parity introduced via the continuous parameter ρ ranging from ρ=0 (no violation of spatial parity) to |ρ|=1 (maximum violation of spatial parity). Values of A represent a continuously adjustable parameter which governs the interaction structure of the model. In nonhelical environments, we demonstrate that A is restricted to the interval -1.723≤A≤2.800 (rounded to 3 decimal places) in the two-loop order of the field theoretic model. However, when ρ>0.749 (rounded to 3 decimal places), the restrictions may be removed, which means that presence of helicity exerts a stabilizing effect onto the possible stationary regimes of the system. Furthermore, three physically important cases A∈{-1,0,1} are shown to lie deep within the allowed interval of A for all values of ρ. For the model of the linearized Navier-Stokes equations (A=-1) up to date unknown helical values of the turbulent Prandtl number have been shown to equal 1 regardless of parity violation. Furthermore, we have shown that interaction parameter A exerts strong influence on advection-diffusion processes in turbulent environments with broken spatial parity. By varying A continuously, we explain high stability of the kinematic MHD model (A=1) against helical effects as a result of its proximity to the A=0.912 (rounded to 3 decimal places) case where helical effects are completely suppressed. Contrary, for the physically important A=0 model, we show that it lies deep within the interval of models where helical effects cause the turbulent Prandtl number to decrease with |ρ|. We thus identify internal structure of interactions given by the parameter A, and not the vector character of the admixture itself being the dominant factor influencing diffusion-advection processes in the helical A model.

Diffusion in anisotropic fully developed turbulence: Turbulent Prandtl number

Using the field theoretic renormalization group technique in the leading order of approximation of a perturbation theory the influence of the uniaxial small-scale anisotropy on the turbulent Prandtl number in the framework of the model of a passively advected scalar field by the turbulent velocity field driven by the Navier-Stokes equation is investigated for spatial dimensions d>2. The influence of the presence of the uniaxial small-scale anisotropy in the model on the stability of the Kolmogorov scaling regime is briefly discussed. It is shown that with increasing of the value of the spatial dimension the region of stability of the scaling regime also increases. The regions of stability of the scaling regime are studied as functions of the anisotropy parameters for spatial dimensions d=3,4, and 5. The dependence of the turbulent Prandtl number on the anisotropy parameters is studied in detail for the most interesting three-dimensional case. It is shown that the anisotropy of turbulent systems can have a rather significant impact on the value of the turbulent Prandtl number, i.e., on the rate of the corresponding diffusion processes. In addition, the relevance of the so-called weak anisotropy limit results are briefly discussed, and it is shown that there exists a relatively large region of small absolute values of the anisotropy parameters where the results obtained in the framework of the weak anisotropy approximation are in very good agreement with results obtained in the framework of the model without any approximation. The dependence of the turbulent Prandtl number on the anisotropy parameters is also briefly investigated for spatial dimensions d=4 and 5. It is shown that the dependence of the turbulent Prandtl number on the anisotropy parameters is very similar for all studied cases (d=3,4, and 5), although the numerical values of the corresponding turbulent Prandtl numbers are different.

Turbulent Prandtl number in the A model of passive vector admixture

Using the field theoretic renormalization group technique in the second-order (two-loop) approximation the explicit expression for the turbulent vector Prandtl number in the framework of the general A model of passively advected vector field by the turbulent velocity field driven by the stochastic Navier-Stokes equation is found as the function of the spatial dimension d>2. The behavior of the turbulent vector Prandtl number as the function of the spatial dimension d is investigated in detail especially for three physically important special cases, namely, for the passive advection of the magnetic field in a conductive turbulent environment in the framework of the kinematic MHD turbulence (A=1), for the passive admixture of a vector impurity by the Navier-Stokes turbulent flow (A=0), and for the model of linearized Navier-Stokes equation (A=-1). It is shown that the turbulent vector Prandtl number in the framework of the A=-1 model is exactly determined already in the one-loop approximation, i.e., that all higher-loop corrections vanish. At the same time, it is shown that it does not depend on spatial dimension d and is equal to 1. On the other hand, it is shown that the turbulent magnetic Prandtl number (A=1) and the turbulent vector Prandtl number in the model of a vector impurity (A=0), which are essentially different at the one-loop level of approximation, become very close to each other when the two-loop corrections are taken into account. It is shown that their relative difference is less than 5% for all integer values of the spatial dimension d≥3. Obtained results demonstrate strong universality of diffusion processes of passively advected scalar and vector quantities in fully symmetric incompressible turbulent environments.

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. The original stochastic problem is reformulated as a multiplicatively renormalizable field-theoretic model; the corresponding renormalization group equations possess an infrared attractive fixed point. It is shown that various correlation functions of the magnetic field and its powers 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.

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.