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 root nu Z scaling, where v is viscosity, is identified before any epsilon = vZ dissipation scaling begins. Further resealing shows linearly decreasing B-nu(t) = (root nu Z)(-1)(/2) at configuration-dependent crossing times G. Gaps in the vortex structures identify the t(x) as when reconnection ends and collapse onto v-independent curves can be obtained using A(nu)(t) = (T-c(nu) - t(x))(B-nu(t) - B-nu(t(x))). The critical times T-c(nu) are identified empirically by extrapolating the linear B-nu(t) regimes to B-nu(similar to)(T-c) = 0, yielding an A(nu)(t) collapse that forms early as v varies by 256. These solutions are regular or non-singular, as shown by decreasing cubic velocity norms parallel to u parallel to(Ll3). For the anti-parallel vortices, first there is an exchange of circulation, from Gamma(y)(y = 0) to Gamma(z)(z = 0), mediated by the viscous circulation exchange integral epsilon(Gamma)(t), which is followed by a modified B-nu(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 l has to increase to maintain the collapse of A(nu)(t) and epsilon(Gamma) as nu decreases.