Hierarchy of extreme-event predictability in turbulence revealed by machine learning

Abstract

Extreme-event predictability in turbulence is strongly state dependent, yet event-by-event predictability horizons are difficult to quantify without access to governing equations or costly perturbation ensembles. Here we train an autoregressive conditional diffusion model on direct numerical simulations of the two-dimensional Kolmogorov flow and use a CRPS-based skill score to define an event-wise predictability horizon. Enstrophy extremes exhibit a pronounced hierarchy: forecast skill persists from $\approx 1$ to $> 4$ Lyapunov times across events. Spectral filtering shows that these horizons are controlled predominantly by large-scale structures. Extremes are preceded by intense strain cores organizing quadrupolar vortex packets, whose lifetime sharply separates long- from short-horizon events. These results identify coherent-structure persistence as a governing mechanism for the predictability of turbulence extremes and provide a data-driven route to diagnose predictability limits from observations.

Figures (3)

• Figure 1: Predictability evolution of extreme events prior to occurrence. The heatmap shows ${S}_e(T)$ against $t-t_e$ (bottom axis) and normalized time $(t-t_e)/T_{\lambda}$(top axis), with darker shadings indicating higher predictability. Vertical ticks to the left of each shading mark the individual predictability limit (horizons) $T^{*}_e$ for each corresponding extreme event. The box plot above the heatmap summarizes the distribution of the event-wise predictability score ($S_{e}(T)$) for all events. The right axis indicates the predictability regime of each event (high predictability, medium predictability, or low predictability) based on its corresponding $T^*_e$.
• Figure 2: Influence of spatial filtering scale $r_c$ on predictability limit $T^*_e$. (a) Scatter plots of filtered versus baseline $T^*_e$. (b) Evolution of $T^*_e$ for high (orange) and low (blue) predictability limits events. The vertical dashed area ($\lambda_c/L = 0.3$) marks the region of predictability breakdown. (c) Visualization of structural degradation in vorticity fields, with the baseline presented on the right and the breakdown scale ($\lambda_c/L=0.35$) on the left.
• Figure 3: (a) Ensemble-averaged evolution of $\Omega$ (green, left axis) and $|Q_{min}|$ (purple, right axis), with shaded regions denoting one standard deviation. The insets display representative snapshots of the $Q$-criterion and vorticity at the peak of the extreme event. (b) Temporal evolution of coherent-structure statistics within the analysis window centered on the strain core, defined by the maximum of $|Q_{min}|$. (c) Cumulative distribution functions (CDFs) of structural lifetimes $\tau$ for high-predictability (orange) and low-predictability (blue) events. The inset shows the bootstrap distribution of the mean lifetime difference $\Delta\tau = \langle\tau\rangle_{\mathrm{high}} - \langle\tau\rangle_{\mathrm{low}}$, yielding $p = 0.017$.