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

2001/03/25 by Mark Nelkin, Nelkin, Mark
Economics, Econometrics and Finance · Engineering · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Chaotic Dynamics (nlin.CD) #Complex Systems and Time Series Analysis #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Soft Condensed Matter (cond-mat.soft) #cond-mat.soft #nlin.CD

paper · pdf · doi:10.48550/arxiv.nlin/0103046

7 pages LaTeX, no figures

arxiv created 2001/03/25 · openalex publication_date 2001/03/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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