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

Abstract

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.