Momentum of vortex tangles by weighted area information

Here we show how to apply a recently introduced method based on the geometric interpretation of linear momentum of vortex lines to determine dynamical properties of a network of knots and links. To show how the method works and to prove its feasibility, we consider the evolution of quantum vortices governed by the Gross-Pitaevskii equation. Accurate estimates of the momentum of interacting and reconnecting vortex rings, links, and knots are determined. The method is of general validity and it proves particularly useful in practical situations where no analytical information is available. It can be easily adapted to situations where morphological information can be extracted from experimental or computational data, thus providing a powerful tool for real-time diagnostics of vortex filaments or other networks of filamentary structures.

Figure 1

(a) Two defects of same circulation $\mathrm{\Gamma }=2\pi$ forming a Hopf link in space (arrows denote vorticity direction). (b) Graph ${\vec{\mathrm{\Lambda }}}_i$ obtained by the planar projection of the Hopf link along the direction ${\widehat{\mathbf{e}}}_i$: each region, $R_1, R_2$, and $R_3$, is pierced by the streamlines due to the circulation of all vortex strands. (c) The index ${\mathcal{I}}_j={\mathcal{I}}_j\left(R_j\right)$ ($j=1,2,3$) is given by the algebraic sum of the signed intersections $\widehat{\mathbf{\rho }}\cap \vec{\partial }{R}_j$. (d) ${\mathcal{I}}_1\left(R_1\right)=+1, {\mathcal{I}}_2\left(R_2\right)=+1, {\mathcal{I}}_3\left(R_3\right)=+2$.

Figure 2

Application of the geometric method to the evolution of Hopf link defects, just before ($t=35$) and after ($t=37$) a reconnection event. Top row: (left) 3D configuration of defects in space; (right) numerical values of standard area and index of the individual graph regions identified in the projection planes $z=0, y=0, x=0$. Bottom row: graph regions color-coded (online) according to the values of ${\mathcal{I}}_j=0,\pm 1$ [magenta (dark gray): $\mathcal{I}=+1$; white: $\mathcal{I}=0$; cyan (light gray): $\mathcal{I}=-1$].

Figure 3

Top row: plots of the momentum components $P_x, P_y$, and $P_z$ against time $t$. Bottom row: direction and intensity of the linear momentum [big arrow (blue online)] just before and after the reconnection event.

Figure 4

Top row: interaction and reconnection of two planar rings (from [9]). Bottom row: direction and intensity of the linear momentum [big arrow (blue online)] just before and after the reconnection event.

Figure 5

Top row: initial configuration of (a) trefoil knot ${\mathcal{T}}_{2,3}$ and (b) cinquefoil ${\mathcal{T}}_{2,5}$. Bottom row: direction and intensity of linear momentum shown by the big arrow (blue online) immediately after simultaneous reconnections.