Extreme-value statistics of curl-of-vorticity precursor peaks in perturbed Taylor-Green vortex turbulence

TL;DR

This study statistically analyzes the timing and extreme behavior of precursor peaks in turbulence, revealing their typical lead over dissipation peaks and their correlation with high-curvature activity.

Contribution

It introduces a statistical framework using extreme-value theory to characterize precursor peak timing and their relation to dissipation in turbulent flows.

Findings

Precursor peaks generally lead dissipation peaks in turbulence.

Extreme-value analysis provides bounds on worst-case lag times.

Strong correlation between maximum curl-of-vorticity peaks and dissipation bursts.

Abstract

Precursor peaks in the wavenumber $k_{\mathrm{peak}}(t)$ maximizing the curl-of-vorticity spectrum have been observed to precede the dissipation peak in decaying turbulence. Because small perturbations in the initial condition can shift peak times, the associated lead time should be characterized statistically. We perform a pseudospectral DNS ensemble of $N_s=1000$ perturbed Taylor--Green vortex realizations at $N=256^3$ and $\nu=10^{-3}$. For each run we extract $k_{\mathrm{peak}}(t)$, several definitions of the precursor time $t_k$, the dissipation-peak time $t_\varepsilon$, and run-wise extrema including $K_{\max}=\max_t k_{\mathrm{peak}}(t)$ and $M_{\max}=\max_t\max_k \mathcal{C}(k,t)$, where $\mathcal{C}(k,t)$ is the isotropic curl-of-vorticity spectrum. The distribution of $\Delta t_{\varepsilon,k}=t_\varepsilon-t_k$ shows that the precursor typically leads, while rare lagging realizations occur and are strongly conditioned on $K_{\max}$. Using peaks-over-threshold extreme-value theory, we fit generalized Pareto models to the right tails of $X=-\Delta t_{\varepsilon,k}$ and $M_{\max}$; negative shape parameters indicate bounded tails and enable worst-case lag and endpoint estimates. Finally, $M_{\max}$ correlates strongly with $\varepsilon_{\max}$ and ensemble cross-correlations reveal a reproducible phase offset, supporting a dynamical coupling between high-curvature activity and dissipation bursts.