Relationship between unpredictability and intermittency in shell models of turbulence and experiments

TL;DR

The paper tackles how predictability of turbulent velocity signals degrades across scales and how intermittency contributes to forecast uncertainty. It combines probabilistic analog forecasting with a GOY shell-model, a GOY-derived multiscale pseudo-velocity, and experimental data from a high-Reynolds-number turbulent flow to perform scale-resolved predictions. The results show that extreme, localized small-scale increments are strongly associated with incorrect mean forecasts and larger uncertainty, with predictability declining from large to small scales; the GOY model enables explicit scale-by-scale assessment, while experiments reveal richer dynamics and occasional deviations from white-noise innovations. Overall, the work links intermittency to forecast errors, demonstrates the utility of analog-based probabilistic forecasting for turbulence, and suggests precursors to extreme events in the velocity field, contributing to a deeper understanding of spontaneous stochasticity in high-Reynolds turbulent systems.

Abstract

We study the predictability of turbulent velocity signals using probabilistic analog-forecasting. Here, predictability is defined by the accuracy of forecasts and the associated uncertainties. We study the Gledzer--Ohkitani--Yamada (GOY) shell model of turbulence as well as experimental measurements from a fully developed turbulent flow. In both cases, we identify the extreme values of velocity at small scales as localized unpredictable events that lead to a loss of predictability: worse mean predictions and increase of their uncertainties. The GOY model, with its explicit scale separation, allows to evaluate the prediction performance at individual scales, and so to better relate the intensity of extreme events and the loss of forecast performance. Results show that predictability decreases systematically from large to small scales. These findings establish a statistical connection between predictability loss across scales and intermittency in turbulent flows.

Figures (12)

• Figure 1: Normalized time series of the real part $u_i$ of the GOY complex variables a) near integral $u_6$, b) mid inertial $u_{10}$, and c) near dissipative $u_{14}$.
• Figure 2: Normalized time series of a) the GOY pseudo-velocity $\mathsf{v}_{\text{GOY}}(t)$, b) the experimental velocity $\mathsf{v}_{\text{Exp}}(t)$ from Modane wind tunnel.
• Figure 3: Power spectral density (PSD) of individual $u_n$ (colored lines) and of the pseudo-velocity (gray line). The PSD is plotted against a surrogate wavenumber axis $k$, normalized by the forcing scale $k_\text{forcing}$. The theoretical $k^{-5/3}$ inertial range scaling is shown for reference (dashed black line).
• Figure 4: Logarithm of the MSE of the forecast $\frac{1}{N} \sum_{t=1}^{N} \left( x_t - \hat{x}_t^{(p)} \right)^2$ as a function of GOY shell index. Variables are normalized prior to any computation as $x_t \mapsto \frac{x_t - \mu}{\sigma}$, where $\mu$ and $\sigma$ are the mean and standard deviation, respectively. The color scale encodes the index for consistency with other figures.
• Figure 5: Time series of the increments $\delta_{dt} u_i = u_i(t+dt) - u_i(t)$ (blue), innovation $\hat{\epsilon}_t^{(p)}$ (green), and prediction variance $\hat{\sigma}_t^{2\,(p)}$ (black) for the real parts of GOY shells $6$, $10$, and $14$.