Geodesic
Turbulent Prandtl number in two spatial dimensions: Two-loop renormalization group analysis
Teoretičeskaâ i matematičeskaâ fizika, Tome 200 (2019) no. 2, pp. 250-258

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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.
Keywords: developed turbulence, renormalization group.
Mots-clés : passive advection 
@article{TMF_2019_200_2_a5,
     author = {E. Jur\v{c}i\v{s}inov\'a and M. Jur\v{c}i\v{s}in and R. Remecky},
     title = {Turbulent {Prandtl} number in two spatial dimensions: {Two-loop} renormalization group analysis},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {250--258},
     publisher = {mathdoc},
     volume = {200},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_200_2_a5/}
}
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E. Jurčišinová; M. Jurčišin; R. Remecky. Turbulent Prandtl number in two spatial dimensions: Two-loop renormalization group analysis. Teoretičeskaâ i matematičeskaâ fizika, Tome 200 (2019) no. 2, pp. 250-258. http://geodesic.mathdoc.fr/item/TMF_2019_200_2_a5/