Does Kolmogorov mean field theory become exact for turbulence above some critical dimension?

Abstract

I give three different arguments for an upper critical dimension $d_{max}>3$ above which the 1941 Kolmogorov mean field theory becomes essentially exact, and anomalous scaling vanishes. The first argument concerns the number of degrees of freedom in a turbulent flow and indicates that $d_{max}=4$. The second argument is a naive estimate of dangerous fluctuations, and also suggests that $d_{max}=4$. The third argument is related to a known critical point of the GOY shell model when the amplitude of back energy transfer becomes small. This third argument does not give a numerical value for $d_{max}$. None of these arguments bears any known relationship to any of the others nor to the generally accepted qualitative physical picture of the dynamical origin of anomalous scaling in turbulence. Despite this, the three arguments together suggest that the suggestion of an upper critical dimension should be taken seriously.