Fluctuation corrections of the turbulence spectrum by renormalization group methods

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 correction >0 in the exponent of the turbulence spectral function,E(k)k^–(5/3+). 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 correction is explained as being due to the fluctuations. is calculated by-expansion of the R.G.T.,=d–2. One gets²; extrapolating to=1 it is1/8.Herrn Prof. Dr. G. Ludwig zum 60. Geburtstag gewidmetThis 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