Enstrophy and circulation scaling for Navier–Stokes reconnection

As reconnection begins and the enstrophy $Z$ grows for two configurations, helical trefoil knots and anti-parallel vortices, two regimes of self-similar collapse are observed. First, during trefoil reconnection a new $\sqrt{\unicode[STIX]{x1D708}}Z$ scaling, where $\unicode[STIX]{x1D708}$ is viscosity, is identified before any $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D708}Z$ dissipation scaling begins. Further rescaling shows linearly decreasing $B_{\unicode[STIX]{x1D708}}(t)=(\sqrt{\unicode[STIX]{x1D708}}Z)^{-1/2}$ at configuration-dependent crossing times $t_{x}$. Gaps in the vortex structures identify the $t_{x}$ as when reconnection ends and collapse onto $\unicode[STIX]{x1D708}$-independent curves can be obtained using $A_{\unicode[STIX]{x1D708}}(t)=(T_{c}(\unicode[STIX]{x1D708})-t_{x})(B_{\unicode[STIX]{x1D708}}(t)-B_{\unicode[STIX]{x1D708}}(t_{x}))$. The critical times $T_{c}(\unicode[STIX]{x1D708})$ are identified empirically by extrapolating the linear $B_{\unicode[STIX]{x1D708}}(t)$ regimes to $B_{\unicode[STIX]{x1D708}}^{{\sim}}(T_{c})=0$, yielding an $A_{\unicode[STIX]{x1D708}}(t)$ collapse that forms early as $\unicode[STIX]{x1D708}$ varies by 256. These solutions are regular or non-singular, as shown by decreasing cubic velocity norms $\Vert u\Vert _{L_{\ell }^{3}}^{}$. For the anti-parallel vortices, first there is an exchange of circulation, from $\unicode[STIX]{x1D6E4}_{y}(y=0)$ to $\unicode[STIX]{x1D6E4}_{z}(z=0)$, mediated by the viscous circulation exchange integral $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6E4}}(t)$, which is followed by a modified $B_{\unicode[STIX]{x1D708}}(t)$ collapse until the reconnection ends at $t_{x}$. Singular Leray scaling and mathematical bounds for higher-order Sobolev norms are used to help explain the origins of the new scaling and why the domain size $\ell$ has to increase to maintain the collapse of $A_{\unicode[STIX]{x1D708}}(t)$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6E4}}$ as $\unicode[STIX]{x1D708}$ decreases.