Turbulent Prandtl number in two spatial dimensions: Two-loop renormalization group analysis

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, $\mathrm{Pr_t}=0.27472$, is considerably smaller than the corresponding value $\mathrm{Pr_t}^{(1)}=0.64039$ 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.