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.

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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.