Trefoil knot timescales for reconnection and helicity

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 () plays several roles. First, during anti-parallel reconnection, sheets of oppositely signed helicity dissipation of equal magnitude on either side of the maximum of the enstrophy dissipation allow the global helicity to be preserved through the first reconnection, as suggested theoretically (Laing et al 2015 Sci. Rep. 5 9224) and observed experimentally (Scheeler et al 2014a Proc. Natl Acad. Sci. 111 15350–5). Next, to maintain the growth of the enstrophy and positive helicity within the trefoil while is preserved, forms in the outer parts of the trefoil so long as the periodic boundaries do not interfere. To prevent that, the domain size is increased as the viscosity . Combined, this allows two sets of trefoils to form a new scaling regime with linearly decreasing up to common ν-independent times tx 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 (2017 Nat. Phys. 9 253–8) when time is scaled by their respective nonlinear timescales tf. The timescales tf are based upon by the radii rf of the trefoils and their circulations Γ, so long as the strong camber of the experimental hydrofoil models is used to correct the published experimental circulations Γ that use only the flat-plate approximation.