# Topological analysis of vortices during reconnection

The diagnostics of the vorticity vector $\boldsymbol{\omega}$ in turbulent flows tends to focus upon the strain-induced growth of its maximum $\|\omega\|_\infty$ by stretching and the production of the enstrophy $Z$, the volume-integrated squared vorticity. This presentation will begin by showing how those grow during the first reconnection of trefoil vortex knots, anti-parallel pairs (JFM 839, R2, 2018) and nested vortex rings (JFM 854, R2, 2018). In all cases, before a viscosity $\nu$-independent time $t_x$ when the reconnection approximately ends, $Z\sim (T_c(\nu)-t)^{-2}$ for $t\leq t_x$. Those diagnostics are now being extended to include the topological numbers, twist, writhe and self-linking to provide supporting evidence for how, if the initial global helicity $\mathcal{H}(0)=\int_V h dV \neq 0$, $h(\mathbf{x})=\mathbf{u}(\mathbf{x})\cdot\mathbf{\omega}(\mathbf{x})$, that ${\mathcal H}(0)$ is preserved during reconnection despite significant changes in the $h(\mathbf{x})$ distribution. The simulations are also being extended to later times to see whether there is evidence for finite-time {\it dissipation anomalies} $\Delta E_\epsilon=\int_0^{t_\epsilon} \epsilon\,dt$, $\epsilon=\nu Z$. Meaning $\nu$-independent, temporally convergent and persistent energy dissipation rates $\epsilon$. This has now been demonstrated for one set of trefoil knots and one anti-parallel set. Allowing the size of the computational to increase as $\nu$ decreases, and required by the mathematics, is critical for maintaining the $B_{\nu}(t)=(\sqrt{\nu}Z)^{-1/2}$ scaling and finite $\Delta E_{\epsilon}$. For $t<t_x$, the fluctuations in the torsion $\tau$ increase, but twist does not, and there is viscous production of helicity density of opposite signs on the two sides of the reconnection (FDR 50, 011422, 2018), with helicity density of one sign supporting the growth of $\epsilon$ for $t_\epsilon> t >t_x$, and helicity density of the opposite sign being expelled into the outer regions (see cover of JFM 839).