Using two innovations, smooth, but distinctly different, scaling laws for the numerical reconnection of pairs of initially orthogonal and anti-parallel quantum vortices are obtained using the three-dimensional Gross-Pitaevskii equations, the simplest mean-field non-linear Schrödinger equation for a quantum fluid. The first innovation suppresses temporal fluctuations by using an initial density profile that is slightly below the usual two-dimensional steady-state Padé approximate profiles. The second innovation is to find the trajectories of the quantum vortices from a pseudo-vorticity constructed on the three-dimensional grid from the gradients of the wave function. These trajectories then allow one to calculate the Frenet-Serret frames and the curvature of the vortex lines. For the anti-parallel case, the scaling laws just before and after reconnection obey the dimensional $\delta \sim |t_r - t|^{1/2}$ prediction with temporal symmetry about the reconnection time $t_r$ and physical space symmetry about $x_r$, the mid-point between the vortices, with extensions of the vortex lines forming the edges of an equilateral pyramid. For all of the orthogonal cases, before reconnection $\delta_{in} \sim (t - t_r)^{1/3}$ and after reconnection $\delta_{out} \sim (t - t_r)^{2/3}$, which are respectively slower and faster than the dimensional prediction. In these cases, the reconnection takes place in a plane defined by the directions of the curvature and vorticity.

Co-authors: C. Rorai (NORDITA), J. Skipper (University of Warwick), K. R. Sreenivasan (NYU)