Fully developed turbulence and statistical mechanics

TLDR

This review surveys recent advances in the statistical theory of fully developed turbulence, emphasizing analogies and differences with Hamiltonian statistical mechanics, especially critical phenomena. The authors discuss spectral equations as a mean‑field–like framework, reformulating Kolmogorov 1941 theory to yield quantitative insights into turbulence energetics (power‑law spectra, cascades, dissipation) and to illuminate Navier‑Stokes/Euler properties, while also exploring quantum‑field‑theory inspired methods for higher‑order statistics. The reformulation provides quantitative predictions for turbulence energetics and reveals Navier‑Stokes/Euler properties, yet experimental evidence of intermittency indicates that Kolmogorov 1941 theory is only approximate.

Abstract

This paper gives a self contained review of some recent progress of the statistical theory of fully developed turbulence. The emphasis is on both analogies and differences with Hamiltonian statistical mechanics, in particular critical phenomena. The method of spectral equations, which plays to a certain extent the role of a mean field theory, is discussed in detail. It is here viewed as a reformulation of the Kolmogorov 1941 theory leading to quantitative insight into the energetics of turbulence (power-law spectra, direct and inverse energy cascades, energy dissipation in the limit of zero viscosity, etc.). In addition, it sheds light on the proven and conjectured properties of the Navier-Stokes and Euler equations which are reviewed in terms more accessible than those of the mathematical literature. There are strong experimental indications (intermittency) that the Kolmogorov 1941 theory is only approximate. Some of the current efforts to handle higher than second order statistics by formal methods inspired from quantum field theory or critical phenomena are also discussed.