# Quantification of non-stationarity across the scales of turbulence with information theory

Carlos Granero-Belinchon, Stéphane G. Roux, Nicolas B. Garnier

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Carlos Granero-Belinchon, Stéphane G. Roux, Nicolas B. Garnier. Quantification of non-stationarity across the scales of turbulence with information theory. STATPHYS 2025 - 29th International Conference On Statistical Physics, Jul 2025, Florence, Italy. Proceedings of STATPHYS29, 2025. hal-05125253

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Submitted on 23 Jun 2025

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# Quantification of non-stationarity across the scales of turbulence with information theory

Carlos Granero-Belinchon*,$^{a,b}$, Stéphane G. Roux$^c$, Nicolas B. Garnier$^c$

## ABSTRACT

We propose a multiscale statistical characterization of the stationarity of a stochastic process $X_t$ based on Shannon information theory. For this, we introduce an index $\Delta^{(\tau)} = H(\delta_{\tau}X_t) - h^{(\tau)}(X_t)$ that focus on the difference between: 1) the entropy of the increments of the stochastic process, $H(\delta_{\tau}X_t)$, that measures the information contained at the scale $\tau$ as defined by the increment $\delta_{\tau}X_t = X_t - X_{t-\tau}$, and 2) the entropy rate $h^{(\tau)}(X_t)$ of the process, that measures the extra information in the variable at time $t$ that is not present at time $t - \tau$. By varying the scale $\tau$, $\Delta^{(\tau)}$ quantifies the stationarity of $X_t$ at the scale of analysis. Consequently, this index is a multiscale quantity that is not restricted to specific statistical moments of the density distribution, nor to the covariance, but that probes the complete dependence structure of the stochastic process. We propose a characterization of the stationarity of the scales of turbulence based on this index. We study experimental turbulent velocity signals from channel flows at different Reynolds numbers. $\Delta^{(\tau)}$ shows the non-stationary nature of the small scales of turbulence and the transition to stationarity at large scales of the order of the integral scale. Very interestingly, $\Delta^{(\tau)}$ presents an universal behavior that is independent of both the experimental setup and the Reynolds number of the flow, see figure 1.

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Entropy rate $h^{(\tau)}$ (a) and index $\Delta^{(\tau)}$ (b), in function of the analysed scale $\tau$ for experimental turbulent velocity measures at different Reynolds numbers. The dotted straight lines in (a) have slopes 1 and $1/3$. The horizontal dashed lines in (b) correspond to the values 0 and $\ln(\sqrt{2})$.</figcaption></figure>

<footnote>$^a$Department of Mathematical and Electrical Engineering, IMT Atlantique, Lab-STICC, UMR CNRS 6285, 655 Av. du Technopôle, Plouzané, 29280, Bretagne, France.</footnote>

<footnote>$^b$Odyssey, Inria/IMT Atlantique, 263 Av. Général Leclerc, Rennes, 35042, Bretagne, France.</footnote>

<footnote>$^c$ENS de Lyon, CNRS, Laboratoire de Physique, 46, allée d'Italie, Lyon, F-69364, Rhône-Alpes, France</footnote>

<footnote>*Presenting author.</footnote>