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LIT-1243: Scaling of information in turbulence

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Properties5
StatusActive — read 2026-10-20 (NOTE-970) in full; worth reading for a clean demonstration that a scale-dependent information measure of a near-Gaussian turbulent signal reduces to the second-order structure function (THEORY-750), and as a worked example of why entropy rates cannot carry the direction of the cascade (THEORY-748). Read it knowing that its "information" is a lag-dependent conditional entropy of the velocity, not a quantity distributed over or transferred between scales; that its claim to "prove the existence of an energy cascade" with information theory does not follow, since one-point skewness alone gives unequal sign-conditioned entropy rates; and that no result ties an information quantity to the energy balance.
Version2
Tagsnatural-sciences, information-theory
Cited as “paper” byNOTE-970 — Granero-Belinchón, Roux and Garnier — Scaling of information in turbulence
Cited as “source” byTHEORY-748 — Block entropies, and so every lag-τ entropy rate of a stationary signal, are unchanged by the reflection x → −x and by time reversal, so they cannot detect the odd-order asymmetry the 4/5 law ties to energy flux; Granero-Belinchón, Roux and Garnier's entropy rate conditioned on the sign of the increment breaks the reflection invariance, but h⁺ ≠ h⁻ shows only that the two-point distribution lacks central symmetry, which a skewed one-point distribution produces with exactly symmetric increments, so the test does not by itself detect skewed increments or an energy cascade, THEORY-750 — For a stationary Gaussian signal the lag-τ entropy rate with one-point embedding is h_τ = H(X) + ½ log(1 − c(τ)²), so where the normalized second-order structure function 1 − c(τ) is small and scales as τ^ζ₂, h_τ = (ζ₂/2) log τ + constant; the "scaling of information" that Granero-Belinchón, Roux and Garnier measure in near-Gaussian wind-tunnel turbulence, slope 1/3 in the inertial range and 1 in the dissipative range, is the scaling of the second-order structure function written in nats

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Granero-Belinchón, C., Roux, S. G. & Garnier, N. B. (2016). Scaling of information in turbulence. EPL 115(5): 58003. doi:10.1209/0295-5075/115/58003. Preprint: arXiv:1607.05511 (19 July 2016).

Read in full in NOTE-970. The record holds two results from it: THEORY-750 (the entropy-rate scaling is the second-order structure function in nats) and THEORY-748 (entropy rates are blind to the sign of the cascade, and the sign-conditioned test detects one-point skewness as readily as skewed increments).

Why this is here

A3 recommended it with a caution the record keeps: energy transfer and information transfer should not be identified because both can be given a direction across scales. Bears on QUESTION-219. The reading supports the caution and goes further: the paper defines no information transfer at all.

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