Renormalization group in the~theory of turbulence: Three-loop approximation as $d\to\infty$
Teoretičeskaâ i matematičeskaâ fizika, Tome 158 (2009) no. 3, pp. 460-477
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We use the renormalization group method to study the stochastic Navier–Stokes equation with a random force correlator of the form $k^{4-d-2\varepsilon}$ in a $d$-dimensional space in connection with the problem of constructing a $1/d$-expansion and going beyond the framework of the standard $\varepsilon$-expansion in the theory of fully developed hydrodynamic turbulence. We find a sharp decrease in the number of diagrams of the perturbation theory for the Green's function in the large-$d$ limit and develop a technique for calculating the diagrams analytically. We calculate the basic ingredients of the renormalization group approach (renormalization constant, $\beta$-function, fixed-point coordinates, and ultraviolet correction index $\omega$) up to the order $\varepsilon^3$ (three-loop approximation). We use the obtained results to propose hypothetical exact expressions (i.e., not in the form of $\varepsilon$-expansions) for the fixed-point coordinate and the index $\omega$.
Keywords:
renormalization group, fully developed turbulence.
@article{TMF_2009_158_3_a10,
author = {L. Ts. Adzhemyan and N. V. Antonov and P. B. Goldin and T. L. Kim and M. V. Kompaniets},
title = {Renormalization group in the~theory of turbulence: {Three-loop} approximation as $d\to\infty$},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {460--477},
publisher = {mathdoc},
volume = {158},
number = {3},
year = {2009},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2009_158_3_a10/}
}
TY - JOUR AU - L. Ts. Adzhemyan AU - N. V. Antonov AU - P. B. Goldin AU - T. L. Kim AU - M. V. Kompaniets TI - Renormalization group in the~theory of turbulence: Three-loop approximation as $d\to\infty$ JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2009 SP - 460 EP - 477 VL - 158 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2009_158_3_a10/ LA - ru ID - TMF_2009_158_3_a10 ER -
%0 Journal Article %A L. Ts. Adzhemyan %A N. V. Antonov %A P. B. Goldin %A T. L. Kim %A M. V. Kompaniets %T Renormalization group in the~theory of turbulence: Three-loop approximation as $d\to\infty$ %J Teoretičeskaâ i matematičeskaâ fizika %D 2009 %P 460-477 %V 158 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2009_158_3_a10/ %G ru %F TMF_2009_158_3_a10
L. Ts. Adzhemyan; N. V. Antonov; P. B. Goldin; T. L. Kim; M. V. Kompaniets. Renormalization group in the~theory of turbulence: Three-loop approximation as $d\to\infty$. Teoretičeskaâ i matematičeskaâ fizika, Tome 158 (2009) no. 3, pp. 460-477. http://geodesic.mathdoc.fr/item/TMF_2009_158_3_a10/