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Helicity and enstrophy scaling for Navier-Stokes reconnection

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, 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 H 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). There is good correspondence between the evolution of the simulated vortices (Kerr 2018 Fluid Dyn. Res. 50 011422) and the reconnecting experimental trefoil of Kleckner and Irvine (2017 Nat. Phys. 9 253–8) and a new scaling regime forms with linearly decreasing 1/√(√νZ (t ), Z enstrophy. The regularity/singularity properties of the Navier-Stokes equations during this period are discussed in Kerr, J. Fluid Mech. (2018), vol. 839, R2. Finally, new numerical comparisons are made with their claims that they can measure the topological properties in their experiments with coiled ring (Scheeler et al 2017, Science 357, 487). This numerical work will include new, unpublished diagnostic methods that can, for the first time, both determine and validate the writhe Wr, twist Tw, self-linking LS, and centre-line helicity Hc in experiments and simulations when the trajectories of the vortex cores can be identified.