TY - JOUR

T1 - Renormalization group in turbulence theory

T2 - Exactly solvable Heisenberg model

AU - Adzhemyan, L. Ts

AU - Antonov, N. V.

N1 - Funding Information: for the pumping function in the random force correlator, and justified the extrapolation (exact, as well as approximate) of the RG analysis results from the domain of asymptotically small e to the actual value c = 2. Then, the perturbative approximate solution to the RG equations (the only accessible solution of a more complicated model, the stochastic Navier Stokes equation) already produces a qualitatively correct result for the IR-asymptotic behavior (powerlike behavior with an exactly known exponent) in the lowest order of the "improved perturbation theory," and the first few coefficients of the e expansion already give a good numerical estimate for the Kolmogorov constant in the energy spectrum of the turbulence. This work was supported by the Russian Foundation for Basic Research (Grant No. 96-02-17033) and the Competitive Center for Fundamental Natural Science, State Committee for Higher Education (Grant No. 95-0-5.1-30).

PY - 1998/5

Y1 - 1998/5

N2 - An exactly solvable Heisenberg model describing the spectral balance conditions for the energy of a turbulent liquid is investigated in the renormalization group (RG) framework. The model has RG symmetry with the exact RG functions (the β-function and the anomalous dimension γ) found in two different renormalization schemes. The solution to the RG equations coincides with the known exact solution of the Heisenberg model and is compared with the results from the ε expansion, which is the only tool for describing more complex models of developed turbulence (the formal small parameter ε of the RG expansion is introduced by replacing a δ-function-like pumping function in the random force correlator by a powerlike function). The results, which are valid for asymptotically small ε, can be extrapolated to the actual value ε = 2, and the few first terms of the ε expansion already yield a reasonable numerical estimate for the Kolmogorov constant in the turbulence energy spectrum.

AB - An exactly solvable Heisenberg model describing the spectral balance conditions for the energy of a turbulent liquid is investigated in the renormalization group (RG) framework. The model has RG symmetry with the exact RG functions (the β-function and the anomalous dimension γ) found in two different renormalization schemes. The solution to the RG equations coincides with the known exact solution of the Heisenberg model and is compared with the results from the ε expansion, which is the only tool for describing more complex models of developed turbulence (the formal small parameter ε of the RG expansion is introduced by replacing a δ-function-like pumping function in the random force correlator by a powerlike function). The results, which are valid for asymptotically small ε, can be extrapolated to the actual value ε = 2, and the few first terms of the ε expansion already yield a reasonable numerical estimate for the Kolmogorov constant in the turbulence energy spectrum.

UR - http://www.scopus.com/inward/record.url?scp=0032356851&partnerID=8YFLogxK

U2 - 10.1007/BF02575456

DO - 10.1007/BF02575456

M3 - Article

AN - SCOPUS:0032356851

VL - 115

SP - 562

EP - 574

JO - Theoretical and Mathematical Physics (Russian Federation)

JF - Theoretical and Mathematical Physics (Russian Federation)

SN - 0040-5779

IS - 2

ER -