Information Content of Turbulence

TL;DR

This paper explores the informational content of turbulence by applying Shannon entropy concepts, revealing how turbulence complexity varies with Reynolds number and drawing parallels with chaotic maps.

Contribution

It introduces an entropy-based framework for analyzing turbulence, linking turbulence dynamics to information theory and demonstrating the relationship between entropy and Reynolds number.

Findings

Entropy density decreases as Reynolds number increases in turbulent soap films.

In the logistic map, entropy increases with the control parameter, illustrating chaos.

Correlations significantly influence the entropy in modified logistic maps.

Abstract

We treat a turbulent velocity field as a message in the same way as a book or a picture. All messages can be described by their entropy per symbol $h$, defined as in Shannon's theory of communication. In a turbulent flow, as the Reynolds number $Re$ increases, more correlated degrees of freedom are excited and participate in the turbulent cascade. Experiments in a turbulent soap film suggest that the spatial entropy density $h$ is a decreasing function of $Re$, namely $h \propto -\log Re$ + const. In the logistic map, also analyzed here, increasing the control parameter $r$ increases $h$. A modified logistic map with additional coupling to past iterations suggests the significance of correlations.