[Paper Review] Information theory demonstration of the Richardson cascade

Rory Cerbus, W. I. Goldburg|arXiv (Cornell University)|Feb 9, 2016

Fluid Dynamics and Turbulent Flows3 references3 citations

TL;DR

This paper uses information theory—specifically conditional entropy and mutual information—to demonstrate Richardson's eddy cascade hypothesis in (quasi-)2D turbulence without relying on Navier-Stokes equations or Kolmogorov's assumptions. It shows that information flows from large to small eddies, confirming the cascade direction and revealing that intermittency is a necessary feature of turbulent flows due to irreversible information transfer.

ABSTRACT

Turbulence theory is usually concerned with the statistical moments of the velocity or its fluctuations. One could also analyze the implicit probability distributions. This is the purview of information theory. Here we use information theory, specifically the conditional entropy, to analyze (quasi-)2D turbulence. We recast Richardson's "eddy hypothesis" that large eddies break up into small eddies in time in the language of information theory. In addition to confirming Richardson's idea, we find that self-similarity and turbulent length scales reappear naturally. Not surprisingly, we also find that the direction of information transfer is the same as the direction of the cascade itself. Consequently, intermittency may be considered a necessary companion to all turbulent flows.

Motivation & Objective

• To test Richardson’s eddy hypothesis—that large eddies break into smaller ones—using information theory instead of traditional fluid dynamics.
• To determine whether the direction of information transfer in turbulence aligns with the energy cascade direction.
• To investigate whether self-similarity and turbulent length scales emerge naturally from information-theoretic analysis of velocity fluctuations.
• To assess whether intermittency in turbulence is a necessary consequence of information transfer, independent of Kolmogorov scaling assumptions.

Proposed method

• The study uses conditional entropy H(Le|Sl) and H(Sl|Le) to quantify uncertainty in large eddy evolution given small eddy states and vice versa.
• A metric D(r) = H(Le|Sl) - H(Sl|Le) is defined to detect the direction of information transfer, where D(r) > 0 indicates a cascade from large to small scales.
• Experimental data from 2D soap film flows are analyzed using particle image velocimetry (PIV) to extract velocity differences δu(r) at various scales r.
• The probability distribution p(δu(r)) is used to compute information-theoretic quantities, avoiding reliance on statistical moments like Sₙ(r).
• Self-similarity is tested by normalizing D(r) with a characteristic length scale L, and collapse of curves across Reynolds numbers indicates scale invariance.
• Mutual information I(Sl;Le) is used to quantify shared information between large and small eddies, with I(Sl;Le) > I(Ll;Se) confirming net downscale information flow.

Experimental results

Research questions

• RQ1 Does information theory provide a framework to verify Richardson’s eddy cascade hypothesis without using Navier-Stokes equations or Kolmogorov assumptions?
• RQ2 Is the direction of information transfer in turbulence consistent with the physical cascade direction from large to small scales?
• RQ3 Do self-similarity and a turbulent length scale emerge naturally from information-theoretic analysis of velocity fluctuations?
• RQ4 Can mutual information between scales confirm that information flows irreversibly downscale, implying that 'forgetting' of large-scale forcing is impossible?
• RQ5 Is intermittency an inevitable feature of turbulent cascades due to irreversible information transfer?

Key findings

• The metric D(r) = H(Le|Sl) - H(Sl|Le) is positive for all r, confirming that large eddies statistically evolve into smaller eddies, validating Richardson’s cascade hypothesis.
• A region of approximately constant D(r) is observed, suggesting a constant flux of information analogous to the constant enstrophy flux in Kolmogorov theory.
• When normalized by a single length scale L, the D(r) curves collapse across different Reynolds numbers, indicating self-similarity and the emergence of a turbulent length scale.
• The ratio of L to the initial length scale L₀ decreases with increasing Reynolds number, suggesting a scaling dependence on flow intensity.
• Mutual information analysis confirms I(Sl;Le) > I(Ll;Se), proving net information transfer from large to small scales, which implies that large-scale forcing cannot be forgotten.
• The irreversible information transfer implies that intermittency is a necessary feature of all turbulent cascades, as complete statistical independence between scales is impossible.