J. Phys. I France
Volume 7, Numéro 6, June 1997
Page(s) 759 - 765
DOI: 10.1051/jp1:1997190
J. Phys. I France 7 (1997) 759-765

A Renormalization Group Study of Three-Dimensional Turbulence

Ph. Brax

DAMTP, University of Cambridge, Silver Street, Cambridge CB39EW, UK

(Received 24 January 1997, revised 12 March 1997, accepted 4 April 1997)

We study the three-dimensional Navier-Stokes equation with a random Gaussian force acting on large wavelengths. Our work has been inspired by Polyakov's analysis of steady states of two-dimensional turbulence. We investigate the time evolution of the probability law of the velocity potential. Assuming that this probability law is initially defined by a statistical field theory in the basin of attraction of a renormalization fixed point, we show that its time evolution is obtained by averaging over small scale features of the velocity potential. The probability law of the velocity potential converges to the fixed point in the long time regime. At the fixed point, the scaling dimension of the velocity potential is determined to be -4/3. We give conditions for the existence of such a fixed point of the renormalization group describing the long time behaviour of the velocity potential. At this fixed point, the energy spectrum of three-dimensional turbulence coincides with a Kolmogorov spectrum.

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