Multi-mode correlations and the entropy of turbulence

TL;DR

This paper introduces a new approach focusing on multi-mode correlations in turbulence, revealing hidden structures and proposing a way to classify turbulence states through entropy measures.

Contribution

It develops a framework to analyze multi-mode correlations in shell models of turbulence, linking cumulant growth to entropy and universality classes.

Findings

Higher multi-mode cumulants grow with order

Sum of squared cumulants relates to relative entropy

Relative entropy may grow logarithmically with modes

Abstract

We suggest a new focus for turbulence studies -- multi-mode correlations -- which reveal the hitherto hidden nature of turbulent state. We apply this approach to shell models describing basic properties of turbulence. The family of such models allows one to study turbulence close to thermal equilibrium, which happens when the interaction time weakly depends on the mode number. As the number of modes increases, the one-mode statistics approaches Gaussian (like in weak turbulence), the occupation numbers grow, while the three-mode cumulant describing the energy flux stays constant. Yet we find that higher multi-mode cumulants grow with the order. We derive analytically and confirm numerically the scaling law of such growth. The sum of all squared dimensionless cumulants is equal to the relative entropy between the full multi-mode distribution and the Gaussian approximation of independent modes; we argue that the relative entropy could grow as the logarithm of the number of modes, similar to the %mutual information and entanglement entropy in critical phenomena. Therefore, the multi-mode correlations give a new way to characterize turbulence states and possibly divide them into universality classes.