Two-loop calculation of the turbulent Prandtl number

Evidence for equivalence of diffusion processes of passive scalar and magnetic fields in anisotropic Navier-Stokes turbulence

The influence of the uniaxial small-scale anisotropy on the kinematic magnetohydrodynamic turbulence is investigated by using the field theoretic renormalization group technique in the one-loop approximation of a perturbation theory. The infrared stable fixed point of the renormalization group equations, which drives the scaling properties of the model in the inertial range, is investigated as the function of the anisotropy parameters and it is shown that, at least at the one-loop level of approximation, the diffusion processes of the weak passive magnetic field in the anisotropically driven kinematic magnetohydrodynamic turbulence are completely equivalent to the corresponding diffusion processes of passively advected scalar fields in the anisotropic Navier-Stokes turbulent environments.

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.

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

Using the field-theoretic renormalization group technique in the two-loop approximation, the influence of helicity (spatial parity violation) on the turbulent vector Prandtl number is investigated in the model of a passive vector field advected by the turbulent helical environment driven by the stochastic Navier-Stokes equation. It is shown that the presence of helicity in the turbulent environment can significantly decrease the value of the turbulent vector Prandtl number by up to 15% of its nonhelical value. This result is compared to the corresponding results obtained recently for the turbulent Prandtl number of a passively advected scalar quantity as well as for the turbulent magnetic Prandtl number of a weak magnetic field in the framework of the kinematic magnetohydrodynamic turbulence. It is shown that the behavior of the turbulent vector Prandtl number as function of the helicity parameter is much closer to the corresponding behavior of the turbulent Prandtl number of the scalar quantity than to the behavior of the turbulent magnetic Prandtl number.

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

The turbulent Prandtl number in the model of a passive vector field advected by the turbulent environment driven by the stochastic Navier-Stokes equation is studied by using the field theoretic renormalization group technique in the two-loop approximation. It is shown that unlike the turbulent Prandtl number in the model of passively advected scalar field, as well as the turbulent magnetic Prandtl number of passively advected magnetic field in the framework of the kinematic magnetohydrodynamic turbulence, where the two-loop corrections to the corresponding Prandtl numbers are very small (less than 2% of their one-loop values), the two-loop correction to the turbulent Prandtl number of passively advected vector field is considerably larger; namely, it is 27% of its one-loop value. At the same time, the calculated two-loop value of the turbulent vector Prandtl number, Pr(v,t)=0.7307, is surprisingly very close to the two-loop value of the turbulent Prandtl number of passively advected scalar field, Pr(t)=0.7040.

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.

Turbulent magnetic Prandtl number in kinematic magnetohydrodynamic turbulence: two-loop approximation

The turbulent magnetic Prandtl number in the framework of the kinematic magnetohydrodynamic (MHD) turbulence, where the magnetic field behaves as a passive vector field advected by the stochastic Navier-Stokes equation, is calculated by the field theoretic renormalization group technique in the two-loop approximation. It is shown that the two-loop corrections to the turbulent magnetic Prandtl number in the kinematic MHD turbulence are less than 2% of its leading order value (the one-loop value) and, at the same time, the two-loop turbulent magnetic Prandtl number is the same as the two-loop turbulent Prandtl number obtained in the corresponding model of a passively advected scalar field. The dependence of the turbulent magnetic Prandtl number on the spatial dimension d is investigated and the source of the smallness of the two-loop corrections for spatial dimension d=3 is identified and analyzed.

Heisenberg approximation in passive scalar turbulence

We use Heisenberg's approximation to derive analytic expressions for eddy viscosity and eddy diffusivity from the transfer integrals of energy and mean-square scalar arising from the Navier-Stokes and passive scalar dynamics. In the same scheme, we evaluate the flux integrals for the transports of energy and mean-square scalar. These procedures allow for the evaluation of relevant amplitude ratios, from which we calculate the universal numbers, namely, Batchelor constant B, Kolmogorov constant C, and turbulent Prandtl number σ, under two different schemes (with and without ε expansion). Our results are comparable with existing theoretical, numerical, and experimental values. As a byproduct, we obtain a relation between C, B, and σ, namely, B=σ C. To compare our results with the experimental values, we calculate Batchelor constant in one dimension (B'). Within the same framework, we also see that with increasing values of space dimension d, the Prandtl number σ increases and approaches unity, while the Kolmogorov constant C and Batchelor constant B approach very close to each other. For large space dimensions, we find the asymptotic B=B(0)d(1/3), and evaluate B(0).

Comment on "Two-loop calculation of the turbulent Prandtl number"

We have revised the value of the turbulent Prandtl number obtained in the model of a passive scalar advected by the velocity field driven by the stochastic Navier-Stokes equation which was calculated by L. Ts. Adzhemyan [Phys. Rev. E 71, 056311 (2005)] by using the field-theoretic renormalization group approach within the two-loop approximation in the corresponding perturbative theory. It is shown that the correct two-loop contribution to the turbulent Prandtl number is essentially smaller than that calculated by Adzhemyan and, as a result, the final two-loop value of the turbulent Prandtl number is Pr(t)=0.7051 instead of Pr(t)=0.7693. The source of discrepancy between our result and that obtained by Adzhemyan is identified and discussed.

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.