Abstract
We apply the renormalization group idea to a stationary probability distribution which is supposed to represent a turbulent fluid. In contrast to the common procedure the R.G.T. is defined by eliminating successivelylow wave numbers instead of integrating from largek. This means that instead of starting from the short distance fluctuations, as near phase transitions, the procedure corresponds to the von Weizsäcker-Heisenberg averaging over nestedr-space volumes of decreasing size.
Ifd>2 we find a non-trivial fixed point of the R.G. equations. It is stable and attractive for every reasonable choice of the distribution function parameters. The only existing “critical exponent” is the field dimension. Its anomalous part gives rise to a correctionμ >0 in the exponent of the turbulence spectral function,E(k)∝k −(5/3+μ). The macroscopic part of the correlation function's scaling exponent, Kolmogoroff's 5/3, is determined by the scaling behaviour of the noise parameter which governs the probability distribution. The correctionμ is explained as being due to the fluctuations.μ is calculated byε-expansion of the R.G.T.,ε=d−2. One getsμ∝ε 2; extrapolating toε=1 it isμ≈1/8.
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References
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We take the opportunity here to correct an error which occurs in [4]: Instead of the free energy density the scaling procedure must be performed with the total free energy. Nevertheless all results of [4] are correct as this error is cancelled by the fact that the noise energyQ is assumed to be an intensive noise strength. ActuallyQ is the energy responsible for the turbulent noise and therefore it is extensive.
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Herrn Prof. Dr. G. Ludwig zum 60. Geburtstag gewidmet
This work has been done in part at the Max-Planck-Institut für Physik und Astrophysik, München. I would like to thank Prof. W. Zimmermann and Prof. W. Götze for their warm hospitality
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Grossmann, S., Schnedler, E. Fluctuation corrections of the turbulence spectrum by renormalization group methods. Z Physik B 26, 307–317 (1977). https://doi.org/10.1007/BF01570740
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DOI: https://doi.org/10.1007/BF01570740