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

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 $ν=10^{-3}$. 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 $Δt_{\varepsilon,k}=t_\varepsilon-t_k$ 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 $X=-Δt_{\varepsilon,k}$ 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.

Figures (10)

• Figure 1: Example time series of the spectral peak wavenumber $k_{\mathrm{peak}}(t)$ (red) and viscous dissipation rate $\varepsilon(t)$ (green), illustrating typical, leading, and lagging realizations. Vertical lines indicate the timing markers $t_{k,\mathrm{first}}$, $t_{k,95}$, $t_{k,\mathrm{last}}$, and $t_\varepsilon$.
• Figure 2: (a) Conditional lag probability $\Pr(\Delta t_{\varepsilon,k}<-0.1)$ as a function of $K_{\max}$, comparing $t_{k,\mathrm{first}}$ and $t_{k,95}$. (b) Histogram of $K_{\max}$ across the $N_s=1000$ ensemble.
• Figure 3: (a) ECDF of $\Delta t_{\varepsilon,k}=t_\varepsilon-t_k$ for three $t_k$ definitions. (b) ECDF of the $K_{\max}$ plateau duration $T_{\mathrm{plat}}=t_{k,\mathrm{last}}-t_{k,\mathrm{first}}$. (c) Scatter plot of $M_{\max}$ versus $\varepsilon_{\max}$ showing a strong correlation across the ensemble.
• Figure 4: POT EVT analysis of upper tails. (a)--(c) Tail of the lag magnitude $X=-\Delta t_{\varepsilon,k}$ for different $t_k$ definitions, with GPD fit overlay. (d) Tail of $M_{\max}$ with GPD fit overlay. Dashed vertical lines indicate the threshold $u$ and dotted lines indicate the implied finite endpoint when $\hat{\xi}<0$.
• Figure 5: Time-resolved relationship between the curl-of-vorticity amplitude $M(t)$ and dissipation $\varepsilon(t)$. (a) Ensemble-mean cross-correlation $\mathrm{corr}[M(t),\varepsilon(t+\tau)]$. (b) Cross-correlation by $K_{\max}$ group (dominant groups $K_{\max}=34$ and $36$ shown). (c) ECDF of $\Delta t_{\varepsilon,M}=t_\varepsilon-t_M$ (negative indicates $M$ peaks after $\varepsilon$). (d)--(f) Distribution and ECDFs of the optimal lag $\tau^\ast=\arg\max_\tau \,\mathrm{corr}[M(t),\varepsilon(t+\tau)]$.