Scaling theory of hydrodynamic turbulence

A phenomenological scaling theory for incompressible fluid turbulence in the limit of infinite Reynolds number is proposed. The local vorticity and local dissipation are taken as scaling variables with scaling dimensions $\frac{2}{3}-\frac{\zeta }{2}$ and $\frac{\mu }{2}$, respectively. The 1941 Kolmogorov theory corresponds to $\mu =\zeta =0\mathrm{}$. Experimentally, $\zeta$ is small and $\mu \approx \frac{1}{2}$. This choice of scaling variables gives immediate and simple predictions about measured or readily measurable scaling exponents. An additional dimensionality-dependent scaling relation, $\mu =d-\frac{8}{3}+2\mathrm{}\zeta$, is proposed and supported by a physically plausible argument. This relation, which is consistent with experiment, suggests that the 1941 Kolmogorov theory is exact for $2<d<\mathrm{}\frac{8}{3}$ and has small corrections for $d=3\mathrm{}$. Dynamical reasons for this behavior are suggested. The relation of scaling behavior to intermittency of the dissipation is briefly discussed.