Renormalization group in turbulence theory: Exactly solvable Heisenberg model

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 epsilon expansion, which is the only tool for describing more complex models of developed turbulence (the formal small parameter epsilon 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 epsilon, can be extrapolated to the actual value epsilon = 2, and the few first terms of the epsilon expansion already yield a reasonable numerical estimate for the Kolmogorov constant in the turbulence energy spectrum.