In decaying turbulence — the Taylor-Green vortex perturbed slightly and left to evolve — the curl-of-vorticity spectrum develops peaks at specific wavenumbers before the main dissipation burst arrives. These precursor peaks consistently lead the dissipation maximum, offering a spectral early warning of the energy cascade's climax. But small perturbations to initial conditions shift the timing unpredictably, demanding a statistical treatment of the lead time.
Tsuzuki runs an ensemble of 1,000 perturbed simulations to characterize this variability. The precursor typically leads, though rare lagging cases appear — strongly correlated with unusually high peak wavenumber values. Extreme-value theory applied to the tail distributions reveals bounded tails, enabling worst-case estimates. The maximum spectrum magnitude correlates tightly with maximum dissipation, and cross-correlations expose a reproducible phase relationship between high-curvature activity and dissipation events.
Turbulence appears chaotic, but its extreme events have structure. The precursor-dissipation relationship reveals that the cascade doesn't arrive unannounced — it builds through intermediate scales in a phase-locked sequence that small perturbations can shift but not destroy. Prediction in turbulence may not require resolving every eddy. It may require identifying the spectral signatures that obligately precede the events that matter — reading the orchestra's tuning before the crescendo arrives.
(arXiv:2603.01057)