Spectrum of the Curl of Vorticity as a Precursor to Dissipation in 3D Taylor--Green Turbulence

Predicting when a three-dimensional turbulent flow reaches its dissipation peak is essential for both theory and adaptive algorithms in simulations and experiments. Using direct numerical simulations (DNSs) of the Taylor–Green vortex (TGV) at resolutions of ${256}^3$–${1024}^3$, we introduce and test a small-scale weighted diagnostic: the spectrum of $\mathrm{\mid }\mathrm{\nabla }\times \mathit{\omega }{\mathrm{\mid }}^2$ (with $\mathit{\omega }=\mathrm{\nabla }\times \mathbf{u}$), which, for incompressible flow, is equivalent to a $k^4$-weighted energy spectrum. We show that the peak wavenumber of this spectrum, $k_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}[\mathrm{\mid }\mathrm{\nabla }\times \mathit{\omega }{\mathrm{\mid }}^2]$, advances rapidly to intermediate-small scales and then levels off before the dissipation rate $\epsilon (t)={\sum }_{k}2\nu k^{2}E(k)$ reaches its maximum. Across all resolutions, we observe robust temporal ordering $t_k<t_{\epsilon }<t_{\mathrm{\Pi }}$, where $t_k$ marks the onset of the rapid rise of $k_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}[\mathrm{\mid }\mathrm{\nabla }\times \mathit{\omega }{\mathrm{\mid }}^2]$, $t_{\epsilon }$ is the time of the maximal $\epsilon (t)$, and $t_{\mathrm{\Pi }}$ is when the cumulative flux $\mathrm{\mid }\mathrm{\Pi }(K)\mathrm{\mid }$ attains its largest peak scale. This early-warning signal correlates with the morphological transition to filament-dominated structures visible in $Q$-criterion isosurfaces and is consistent with integral-scale trends ($L_{\mathrm{i}\mathrm{n}\mathrm{t}},\lambda ,\eta$). The diagnostic is simple to compute from standard DNS data and highlights the incipient formation of high-curvature structures, where viscosity acts most strongly.