Helicity conservation under quantum reconnection of vortex rings

Here we show that under quantum reconnection, simulated by using the three-dimensional Gross-Pitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that the total length of the vortex system reaches a maximum at the reconnection time, while both writhe helicity and twist helicity remain separately unchanged throughout the process. Self-helicity is computed by two independent methods, and topological information is based on the extraction and analysis of geometric quantities such as writhe, total torsion, and intrinsic twist of the reconnecting vortex rings.

Figure 1

Time-evolution of interaction and reconnection of two quantum vortex rings; isosurfaces of $\rho =0.06$. (a) $t=0$, (b) $t=10$, (c) reconnection time $t=t^{*}=11.81$, and (d) $t=14$.

Figure 2

Normalized total length $L/\xi$, plotted against time $t$. The peak in $L/\xi$ is taken as signature of the reconnection time at $t=t^{*}=11.81$.

Figure 3

(a) Normalized total energy (Hamiltonian) $(K+I)/E$, normalized kinetic energy $K/E$, and normalized interaction energy $I/E$ plotted against time $t$. (b) Kinetic helicity $H$ plotted against time $t$. The vertical (red) line denotes the reconnection time $t=t^{*}=11.81$.

Figure 4

A closeup view of the vortex centerlines (red) and reference ribbons (green and blue, pale gray, and darker otherwise) (a) immediately before reconnection at $t=11.81$ and (b) immediately after reconnection at $t=11.88$. The transition is shown at maximum numerical resolution: (c) vortex centerlines just before ($t<t^{*}$, red) and after ($t>t^{*}$, black) reconnection; the state in-between (at $t=t^{*}$) shows that reconnection is numerically triggered by a jump at just two nodal points (circles).

Figure 5

(a) Writhe (Wr), (b) normalized total torsion $\left(T\right)$, (c) normalized intrinsic twist $\left(N\right)$ (with $\overline{\theta }\approx {50}^{\circ }$), and (d) self-linking number (SL) plotted against time $t$. The vertical line (red) denotes the reconnection time $t=t^{*}=11.81$.