Turbulent Prandtl Number in Two Spatial Dimensions: Two-Loop Renormalization Group Analysis

Abstract

Using the field theory renormalization group technique in the framework of the so-called double-expansion scheme, which takes additional divergences that appear in two dimensions into account, we calculate the turbulent Prandtl number in two spatial dimensions in the two-loop approximation in the model of a passive scalar field advected by the turbulent environment driven by the stochastic Navier–Stokes equation. We show that in contrast to the three-dimensional case, where the two-loop correction to the one-loop value of the turbulent Prandtl number is very small (less than 2% of the one-loop value), the two-loop value of the turbulent Prandtl number in two spatial dimensions, Pr_t = 0.27472, is considerably smaller than the corresponding value Pr_t⁽¹⁾ = 0.64039 obtained in the one-loop approximation, i.e., the two-loop correction to the turbulent Prandtl number in the two-dimensional case represents about 57% of its one-loop value and must be seriously taken into account. This result also means that there is a significant difference (at least quantitatively) between diffusion processes in two- and three-dimensional turbulent environments.