Approach and separation of quantised vortices with balanced cores

Abstract

The scaling laws for the reconnection of isolated pairs of quantised vortices are characterised by numerically integrating the three-dimensional Gross–Pitaevskii equations, the simplest mean-field equations for a quantum fluid. The primary result is the identification of distinctly different temporal power laws for the pre- and post-reconnection separation distances $\unicode[STIX]{x1D6FF}(t)$ for two configurations. For the initially anti-parallel case, the scaling laws before and after the reconnection time $t_{r}$ obey the dimensional $\unicode[STIX]{x1D6FF}\sim |t_{r}-t|^{1/2}$ prediction with temporal symmetry about $t_{r}$ and physical space symmetry about the mid-point between the vortices $x_{r}$. The extensions of the vortex lines close to reconnection form the edges of an equilateral pyramid. For all of the initially orthogonal cases, $\unicode[STIX]{x1D6FF}\sim |t_{r}-t|^{1/3}$ before reconnection and $\unicode[STIX]{x1D6FF}\sim |t-t_{r}|^{2/3}$ after reconnection are respectively slower and faster than the dimensional prediction. For both configurations, smooth scaling laws are generated due to two innovations. The first innovation is to use an initial low-energy vortex-core density profile that suppresses unwanted density fluctuations as the vortices evolve in time. The other innovation is the accurate identification of the position of the vortex cores from a pseudo-vorticity constructed on the three-dimensional grid from the gradients of the wave function. These trajectories allow us to calculate the Frenet–Serret frames and the curvature of the vortex lines, secondary results that might hold clues for the origin of the differences between the scaling laws of the two configurations. Reconnection takes place in a reconnection plane defined by the average tangents $\boldsymbol{T}_{av}$ and curvature normal $\boldsymbol{N}_{av}$ directions of the pseudo-vorticity curves at the points of closest approach, at time $t\approx t_{r}$. To characterise the structure further, lines are drawn that connect the four arms that extend from the reconnection plane, from which four angles $\unicode[STIX]{x1D703}_{i}$ between the lines are defined. Their sum is convex or hyperbolic, that is $\sum _{i=1,4}\unicode[STIX]{x1D703}_{i}>360^{\circ }$, for the orthogonal cases, as opposed to the acute angles of the pyramid found for the anti-parallel initial conditions.