Trefoil knot structure during reconnection

Abstract

Three-dimensional images of evolving numerical trefoil vortex knots are used to study the growth and decay of the enstrophy and helicity. Negative helicity density ($h<0$) plays several roles. First, sheets of oppositely-signed helicity dissipation of equal magnitude on either side of the maximum of the enstrophy dissipation allow the global helicity ${\cal H}$ to be preserved through the first reconnection, as suggested theoretically (Laing et al 2015) and observed experimentally (Scheeler et al. 2014). Next, to maintain the growth of the enstrophy and positive helicity within the trefoil while ${\cal H}$ is preserved, $h<0$ forms in the outer parts of the trefoil so long as the periodic boundaries do not interfere. To prevent that, the domain size $\ell$ is increased as the viscosity $\nu\to0$. Combined, this allows two sets of trefoils to form a new scaling regime with linearly decreasing $(\sqrt{\nu}Z(t))^{-1/2}$ up to common $\nu$-independent times $t_x$ that the graphics show is when the first reconnection ends. During this phase there is good correspondence between the evolution of the simulated vortices and the reconnecting experimental trefoil of Kleckner and Irvine (2013) when time is scaled by their respective nonlinear timescales $t_f$. The timescales $t_f$ are based upon by the radii $r_f$ of the trefoils and their circulations $\Gamma$, so long as the strong camber of the experimental hydrofoil models is used to correct the published experimental circulations $\Gamma$ that use only the flat-plate approximation.