Singular measures and information capacity of turbulent cascades

TL;DR

This paper uses information theory to analyze turbulence, revealing that even weak turbulence exhibits complex, singular measures with significant information transfer between scales, challenging traditional assumptions.

Contribution

It introduces an information-theoretic framework to distinguish equilibrium from turbulence and shows that turbulence involves singular measures with high information capacity.

Findings

Mutual information grows linearly with time in weak turbulence

Resonance surfaces can become singular measures

Resolved scales contain significant information about unresolved scales

Abstract

Is there really such a thing as weak turbulence? Here we analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence: the mutual information between modes is shown to be stationary and small in equilibrium but grows linearly with time in weak turbulence. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the statistics of any single amplitude, a stationary phase-space measure is far from Gaussian, as manifested by a large relative entropy. Though it might be upsetting to practitioners of weak turbulence approach, this is a rare piece of good news for turbulence modeling: the resolved scales carry significant information about the unresolved scales. The mutual information between large and small scales is the information capacity of turbulent cascade, setting the limit on the representation of subgrid scales in turbulence modeling.