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Renormalization group in turbulence theory: Exactly solvable Heisenberg model
L. Ts. Adzhemyan,
N. V. Antonov Saint-Petersburg State University
Abstract:
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.
Received: 24.11.1997
DOI:
10.4213/tmf870
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