Approach and separation of quantum vortices with balanced cores

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 Schr\"odinger 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 Pad\'e 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 the $x_r$, the mid-point between the vortices, with extensions of the vortex lines formng 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. To define the structure further, lines are drawn that connect the four arms that extend from the reconnection plane, four angles $\theta_i$ between these arms are found, then summed, giving $\sum\theta_i>360^\circ$. This implies that the overall structure is convex or hyperbolic, as opposed to the acute angles of the anti-parallel pyramid.