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

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.

Evidence for enhancement of anisotropy persistence in kinematic magnetohydrodynamic turbulent systems with finite-time correlations

Using the field-theoretic renormalization group approach and the operator product expansion technique in the second order of the corresponding perturbative expansion, the influence of finite-time correlations of the turbulent velocity field on the scaling properties of the magnetic field correlation functions as well as on the anisotropy persistence deep inside the inertial range are investigated in the framework of the generalized Kazantsev-Kraichnan model of kinematic magnetohydrodynamic turbulence. Explicit two-loop expressions for the scaling exponents of the single-time two-point correlation functions of the magnetic field are derived and it is shown that the presence of the finite-time velocity correlations has a nontrivial impact on their inertial-range behavior and can lead, in general, to significantly more pronounced anomalous scaling of the magnetic field correlation functions in comparison to the rapid-change limit of the model, especially for the most interesting three-dimensional case. Moreover, by analyzing the asymptotic behavior of appropriate dimensionless ratios of the magnetic field correlation functions, it is also shown that the presence of finite-time correlations of the turbulent velocity field has a strong impact on the large-scale anisotropy persistence deep inside the inertial interval. Namely, it leads to a significant enhancement of the anisotropy persistence, again, especially in three spatial dimensions.

Anomalous Scaling in the Kinematic Magnetohydrodynamic Turbulence

The problem of the anomalous scaling in the kinematic magnetohydrodynamic turbulence is investigated using the field theoretic renormalization group method and the operator product expansion technique. The anomalous dimensions of all leading composite operators, which drive the anomalous scaling of the correlation functions of a weak passive magnetic field, are determined up to the second order of the perturbation theory (i.e., in the two-loop approximation in the field theoretic terminology) in the presence of a large scale anisotropy for physically the most interesting three-dimensional case. It is shown that the leading role in the anomalous scaling properties of the model is played by the anomalous dimensions of the composite operators near the isotropic shell, in accordance with the Kolmogorov’s local isotropy restoration hypothesis. The importance of the two-loop corrections to the anomalous dimensions of the leading composite operators is demonstrated.

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.

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.

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.

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. 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 a passive scalar field near two dimensions

The anomalous scaling of the single-time structure functions of a passive scalar field advected by the velocity field governed by the stochastic Navier-Stokes equation is investigated by the field theoretic renormalization group and the operator-product expansion with inclusion of additional ultraviolet divergences related to the spatial dimension d=2. Some modification of the standard minimal subtraction scheme is used to calculate the turbulent Prandtl number and the anomalous exponents within the one-loop approximation of a perturbation theory. It is shown that the taking into account of these additional divergences is in full agreement with results obtained in the framework of the usual analytical expansion technique, which are valid for d>2.

Scaling and statistical geometry in passive scalar turbulence

We show that the statistics of a turbulent passive scalar at scales larger than the pumping may exhibit multiscaling due to a weaker mechanism than the presence of statistical conservation laws. We develop a general formalism to give explicit predictions for the large scale scaling exponents in the case of the Kraichnan model and discuss their geometric origin at small and large scale.

Influence of anisotropy on anomalous scaling of a passive scalar advected by the Navier-Stokes velocity field

The influence of weak uniaxial small-scale anisotropy on the stability of the scaling regime and on the anomalous scaling of the single-time structure functions of a passive scalar advected by the velocity field governed by the stochastic Navier-Stokes equation is investigated by the field theoretic renormalization group and operator-product expansion within one-loop approximation of a perturbation theory. The explicit analytical expressions for coordinates of the corresponding fixed point of the renormalization-group equations as functions of anisotropy parameters are found, the stability of the three-dimensional Kolmogorov-like scaling regime is demonstrated, and the dependence of the borderline dimension d(c) is an element of (2,3] between stable and unstable scaling regimes is found as a function of the anisotropy parameters. The dependence of the turbulent Prandtl number on the anisotropy parameters is also briefly discussed. The influence of weak small-scale anisotropy on the anomalous scaling of the structure functions of a passive scalar field is studied by the operator-product expansion and their explicit dependence on the anisotropy parameters is present. It is shown that the anomalous dimensions of the structure functions, which are the same (universal) for the Kraichnan model, for the model with finite time correlations of the velocity field, and for the model with the advection by the velocity field driven by the stochastic Navier-Stokes equation in the isotropic case, can be distinguished by the assumption of the presence of the small-scale anisotropy in the systems even within one-loop approximation. The corresponding comparison of the anisotropic anomalous dimensions for the present model with that obtained within the Kraichnan rapid-change model is done.

Influence of helicity on the Kolmogorov regime in fully developed turbulence

The influence of helicity on the stability of the Kolmogorov scaling regime in fully developed turbulence in space dimension d=3 based on the stochastic Navier-Stokes equation with the self-similar Gaussian random stirring force delta -correlated in time and with the correlator proportional to k;{4-d-2epsilon} is investigated by the field-theoretic renormalization-group technique within two-loop approximation. The two-loop renormalization constant, the beta function, and the coordinate of the fixed point are found as functions of the helicity parameter. It is shown that the presence of helicity in the system does not destroy the stability of the Kolmogorov scaling regime.

Influence of helicity on anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time: two-loop approximation

The influence of helicity on the stability of scaling regimes, on the effective diffusivity, and on the anomalous scaling of structure functions of a passive scalar advected by a Gaussian solenoidal velocity field with finite correlation time is investigated by the field theoretic renormalization group and operator-product expansion within the two-loop approximation. The influence of helicity on the scaling regimes is discussed and shown in the plane of exponents epsilon-eta, where epsilon characterizes the energy spectrum of the velocity field in the inertial range E proportional to k(1-2epsilon), and eta is related to the correlation time at the wave number k, which is scaled as k(-2+eta). The restrictions given by nonzero helicity on the regions with stable fixed points that correspond to the scaling regimes are analyzed in detail. The dependence of the effective diffusivity on the helicity parameter is discussed. The anomalous exponents of the structure functions of the passive scalar field which define their anomalous scaling are calculated and it is shown that, although the separate composite operators which define them strongly depend on the helicity parameter, the resulting two-loop contributions to the critical dimensions of the structure functions are independent of helicity. Details of calculations are shown.