The anomalous scaling exponents ζn​ of the longitudinal structure
functions Sn​ for homogeneous isotropic turbulence are derived from the
Navier-Stokes equations by using field theoretic methods to develop a low
energy approximation in which the Kolmogorov theory is shown to act effectively
as a mean field theory. The corrections to the Kolmogorov exponents are
expressed in terms of the anomalous dimensions of the composite operators which
occur in the definition of Sn​. These are calculated from the anomalous
scaling of the appropriate class of nonlinear Green's function, using an uv
fixed point of the renormalisation group, which thereby establishes the
connection with the dynamics of the turbulence. The main result is an algebraic
expression for ζn​, which contains no adjustable constants.
It is valid at orders n below , where g∗​ is the
fixed point coupling constant. This expression is used to calculate ζn​ for orders in the range to 10, and the results are shown to be in
good agreement with experimental data, key examples being ζ2​=0.7,
ζ3​=1 and .Comment: REVTeX, 59 pages, icludes 8 .eps file