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

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.

Passive scalar transport by a non-Gaussian turbulent flow in the Batchelor regime

We analyze passive scalar advection by a turbulent flow in the Batchelor regime. No restrictions on the velocity statistics of the flow are assumed. The properties of the scalar are derived from the statistical properties of velocity; analytic expressions for the moments of scalar density are obtained. We show that the scalar statistics can differ significantly from that obtained in the frames of the Kraichnan model.

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.

Passive scalars: Mixing, diffusion, and intermittency in helical and nonhelical rotating turbulence

We use direct numerical simulations to compute structure functions, scaling exponents, probability density functions, and effective transport coefficients of passive scalars in turbulent rotating helical and nonhelical flows. We show that helicity affects the inertial range scaling of the velocity and of the passive scalar when rotation is present, with a spectral law consistent with ∼k_{⊥}^{-1.4} for the passive scalar variance spectrum. This scaling law is consistent with a phenomenological argument [P. Rodriguez Imazio and P. D. Mininni, Phys. Rev. E 83, 066309 (2011)PLEEE81539-375510.1103/PhysRevE.83.066309] for rotating nonhelical flows, which follows directly from Kolmogorov-Obukhov scaling and states that if energy follows a E(k)∼k^{-n} law, then the passive scalar variance follows a law V(k)∼k^{-n_{θ}} with n_{θ}=(5-n)/2. With the second-order scaling exponent obtained from this law, and using the Kraichnan model, we obtain anomalous scaling exponents for the passive scalar that are in good agreement with the numerical results. Multifractal intermittency models are also considered. Intermittency of the passive scalar is stronger than in the nonhelical rotating case, a result that is also confirmed by stronger non-Gaussian tails in the probability density functions of field increments. Finally, Fick's law is used to compute the effective diffusion coefficients in the directions parallel and perpendicular to rotation. Calculations indicate that horizontal diffusion decreases in the presence of helicity in rotating flows, while vertical diffusion increases. A simple mean field argument explains this behavior in terms of the amplitude of velocity fluctuations.

Anomalous scaling of the magnetic field in the helical Kazantsev-Kraichnan model

The field-theoretic renormalization group and the operator product expansion are used to investigate the influence of spatial parity violation of the conductive turbulent environment on the anomalous scaling behavior of correlation functions of a weak magnetic field in the framework of the Kazantsev-Kraichnan rapid change model. Two-loop expressions for the critical dimensions of the leading composite operators, which drive the anomalous scaling of the two-point single-time correlation functions of the magnetic field in the presence of large-scale anisotropy, are found to be functions of the helicity parameter. It is shown that the presence of helicity in the system leads to a significantly stronger manifestation of anomalous scaling than in the nonhelical case. At the same time, it is also shown that helicity does not destroy the standard hierarchy of the anisotropic anomalous exponents in the framework of which the leading contribution to anomalous scaling is given by the isotropic shell.

Turbulent magnetic Prandtl number in helical kinematic magnetohydrodynamic turbulence: two-loop renormalization group result

Using the field theoretic renormalization group technique, the influence of helicity (spatial parity violation) on the turbulent magnetic Prandtl number in the kinematic magnetohydrodynamic turbulence is investigated in the two-loop approximation. It is shown that the presence of helicity decreases the value of the turbulent magnetic Prandtl number and, at the same time, the two-loop helical contribution to the turbulent magnetic Prandtl number is at most 4.2% (in the case with the maximal helicity) of its nonhelical value. These results demonstrate, on one hand, the potential importance of the presence of asymmetries in processes in turbulent environments and, on the other hand, the rather strong stability of the properties of diffusion processes of the magnetic field in the conductive turbulent environment with the spatial parity violation in comparison to the corresponding systems without the spatial parity violation. In addition, obtained results are compared to the corresponding results found for the two-loop turbulent Prandtl number in the model of passively advected scalar field. It is shown that the turbulent Prandtl number and the turbulent magnetic Prandtl number, which are the same in fully symmetric isotropic turbulent systems, are essentially different when one considers the spatial parity violation. It means that the properties of the diffusion processes in the turbulent systems with a given symmetry breaking can considerably depend on the internal tensor structure of advected quantities.

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.

Four-point correlation function of a passive scalar field in rapidly fluctuating turbulence: Numerical analysis of an exact closure equation

A numerical analysis is made on the four-point correlation function in a similarity range of a model of two-dimensional passive scalar field ψ advected by a turbulent velocity field with infinitely small correlation time. The model yields an exact closure equation for the four-point correlation Ψ{4} of ψ, which may be casted into the form of an eigenvalue problem in the similarity range. The analysis of the eigenvalue problem gives not only the scale dependence of Ψ{4} , but also the dependence on the configuration of the four points. The numerical analysis gives S4(R)∝R{ζ{4}} in the similarity range in which S2(R)∝R{ζ{2}} , where S_{N} is the structure function defined by S{N}(R)≡⟨[ψ(x+R)-ψ(x)]{N} and ζ{4}≠2ζ{2} . The estimate of ζ_{4} by the numerical analysis of the eigenvalue problem is in good agreement with numerical simulations so far reported. The agreement supports the idea of universality of the exponent ζ{4} in the sense that ζ_{4} is insensitive to conditions of ψ outside the similarity range. The numerical analysis also shows that the correlation C(R,r)≡[ψ(x+R)-ψ(x)]{2}[ψ(x+r)-ψ(x)]{2}> is stronger than that given by the joint-normal approximation, and scales like C(R,r)∝(r/R){χ} for r/R<<1 with R and r in the similarity range, where χ is a constant depending on the angle between R and r .

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.

Anomalous scaling of a passive scalar advected by a turbulent velocity field with finite correlation time and uniaxial small-scale anisotropy

The influence of uniaxial small-scale anisotropy on the stability of the scaling regimes and on the anomalous scaling of the 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 one-loop approximation. Possible scaling regimes are found and classified in the plane of exponents epsilon-eta , where epsilon characterizes the energy spectrum of the velocity field in the inertial range E proportional, variantk;{1-2epsilon} , and eta is related to the correlation time of the velocity field at the wave number k which is scaled as k;{-2+eta} . It is shown that the presence of anisotropy does not disturb the stability of the infrared fixed points of the renormalization group equations, which are directly related to the corresponding scaling regimes. The influence of anisotropy on the anomalous scaling of the structure functions of the passive scalar field is studied as a function of the fixed point value of the parameter u , which represents the ratio of turnover time of scalar field and velocity correlation time. It is shown that the corresponding one-loop anomalous dimensions, which are the same (universal) for all particular models with a concrete value of u in the isotropic case, are different (nonuniversal) in the case with the presence of small-scale anisotropy and they are continuous functions of the anisotropy parameters, as well as the parameter u . The dependence of the anomalous dimensions on the anisotropy parameters of two special limits of the general model, namely, the rapid-change model and the frozen velocity field model, are found when u-->infinity and u-->0 , respectively.