Scaling of Navier-Stokes trefoil reconnection

The reconnection of a trefoil vortex knot is examined numerically to determine how its helicity and two of its vorticity norms behave. During an initial phase, the helicity is remarkably preserved, as reported in recent experiments (Scheeler et al. 2014a). In addition, the enstrophy (Z) has self-similar growth where all √ νZ(t) converge at a viscosity-independent time tx. This effect requires that the computational domain grow as the viscosity decreases, in accordance with known Sobolev space bounds for fixed domains. By rescaling time as δtν = (t − tx)/(Tc(ν) − tx), self-similar collapse onto a single curve for 0 . t 6 tx can be achieved for 1/ (√ νZ(t) )1/2 . Graphics show that tx is the end of first reconnection and by t ≈ 2tx, a viscosity independent dissipation rate = νZ appears. To address additional mathematical restrictions in Whole Space, very small viscosities at early times are considered. Over this period the Navier-Stokes ‖ω‖∞ are bounded by the Euler values and the velocity norm L3 barely changes until very late times. Despite this, the Navier-Stokes enstrophy can, for a brief period, grow faster than the Euler enstrophy. Taken together, these results could be a new template whereby smooth solutions without singularities or roughness can generate a ν → 0 dissipation anomaly (finite dissipation in a finite time), as observed in physical turbulent flows.