Renormalization group in turbulence theory: Exactly solvable Heisenberg model

L. Ts. Adzhemyan ; N. V. Antonov

Teoretičeskaâ i matematičeskaâ fizika, Tome 115 (1998) no. 2, pp. 245-262

Résumé

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 $\beta$-function and the anomalous dimension $\gamma$) 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 $\varepsilon$ expansion, which is the only tool for describing more complex models of developed turbulence (the formal small parameter $\varepsilon$ of the RG expansion is introduced by replacing a $\delta$-function-like pumping function in the random force correlator by a powerlike function). The results, which are valid for asymptotically small $\varepsilon$, can be extrapolated to the actual value $\varepsilon=2$, and the few first terms of the $\varepsilon$ expansion already yield a reasonable numerical estimate for the Kolmogorov constant in the turbulence energy spectrum.