Turbulent Prandtl number of a passively advected vector field in helical environment: two-loop renormalization group result

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.

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.