Vortex line topology during vortex tube reconnection

This paper addresses reconnection of vortex tubes, with particular focus on the topology of the vortex lines (field lines of the vorticity). This analysis of vortex line topology reveals key features of the reconnection process, such as the generation of many small flux rings, formed when reconnection occurs in multiple locations in the vortex sheet between the tubes. Consideration of three-dimensional reconnection principles leads to a robust measurement of the reconnection rate, even once instabilities break the symmetry. It also allows us to identify internal reconnection of vortex lines within the individual vortex tubes. Finally, the introduction of a third vortex tube is shown to render the vortex reconnection process fully three-dimensional, leading to a fundamental change in the topological structure of the process. An additional interesting feature is the generation of vorticity null points.

Figure 1

Selected vortex lines during the interaction of antiparallel vortex tubes (red, threads; cyan, reconnected bridges). Shading on the end planes and in the volume shows $\left|\mathbf{\omega }\right|$. Taken from the simulation described in Sec. 2 at (a) $t=30$, (b) $t=45$.

Figure 2

Vorticity field lines (a) before and (b) after 3D reconnection. The shaded sphere in (a) represents the nonideal region, within which $(\mathbf{\nabla }\times \mathbf{\omega })·\mathbf{\omega }\ne 0$.

Figure 3

Vortex lines in the vicinity of (a) 3D null point and (b) a separator. After Ref. [32].

Figure 4

Vorticity field lines plotted from $30\%\left|\mathbf{\omega }\right|$ contours at $x=0$ (red) and $y=0$ (blue) at (a) $t=0$, (b) $t=30$, (c) $t=60$, (d) $t=90$, (e) $t=120$, and (f) $t=150$. Inset: $\left|\mathbf{\omega }\right|$ isosurface of $30\%$ maximum $\left|\mathbf{\omega }\right|$ at the $x=-3$ boundary, plotted over the full numerical domain.

Figure 5

Contour plots of ${\omega }_x$ (shaded, filled contours) and ${(\mathbf{\nabla }\times \mathbf{\omega })}_z$ (unfilled contours; solid positive, dashed negative) in the $x=0$ plane, at (a) $t=35$, (b) $t=51$, (c) $t=75$, (d) $t=120$.

Figure 6

(a) Vorticity flux measured at $x=0$ (solid), $y=0$ (dashed), and total of both (dotted) as a function of time. (b) Reconnection rate measured from rate of change of vorticity flux through the symmetry and dividing planes (solid) and by integrating ${(\mathbf{\nabla }\times \mathbf{\omega })}_z$ along the central axis (dashed).

Figure 7

${(\mathbf{\nabla }\times \mathbf{\omega })}_z$ contour plot in the dividing plane $\left(y=0\right)$ at $t=45$ and $t=90$.

Figure 8

Threads (red) plotted from $30\%$ maximum ${\omega }_x$ contours at $x=0$, bridges (blue) plotted from $30\%$ maximum ${\omega }_y$ contours at $y=0$, additional vortex ring field lines (green) plotted from $30\%$ maximum ${\omega }_y$ (of opposite sign to bridges) contours at $y=0,\text{Re}=4000$ at (a) $t=54$, (b) $t=63$, (c) $t=111$, and (d) $t=138$.

Figure 9

Vorticity flux in the dividing plane due to additional vortex rings as a function of time. For $\text{Re}=4000$ (purple), 2000 (red) and 800 (blue).

Figure 10

Isosurfaces of $|\mathbf{v}·\mathbf{\omega }|$ (cyan), $|\mathbf{\omega }·(\mathbf{\nabla }\times \mathbf{\omega }\left)\right|$ (red), and $\left|\mathbf{\omega }\right|$ (inset, green), in each case at 10% of the domain maximum, for (a) $t=45$ and (b) $t=75$.

Figure 11

Contour plots, from left to right, of vorticity magnitude (red, threads; blue, bridges) field line helicity $h$, and field line “slipping” rate $\mathrm{\Psi }$, in the plane $x=-3$. Threads and bridges are distinguished in the left frame by tracing vortex lines from a grid of points and determining whether they connect to $x=3$ or back to $x=-3$.

Figure 12

Estimates of the cumulative flux reconnected internally within the tubes. Solid curve is the maximum measure of total flux reconnected [see Eq. (21) of Ref. 46]; dashed curve is the minimal measure or “net” flux reconnected [see Eq. (19) of Ref. 46].

Figure 13

$30\%$ of maximum vorticity field lines at $x=-3,3$ boundaries (red and blue) and $z=-12,0$ boundaries (green $\left({\mathcal{P}}_s\right)$ and yellow $\left({\mathcal{P}}_c\right))$ at (a) $t=0$, (b) $t=15$, (c) $t=30$, (d) $t=45$, (e) $t=60$, and (f) $t=90$ for $\text{Re}=2000$.

Figure 14

(a) Vorticity isosurface at $30\%$ of the maximum vorticity at the $z=0$ boundary at $t=30$. (b) Sample field lines at $t=45$, demonstrating that field lines are not antiparallel prior to reconnection; boundary shading shows $\left|\mathbf{\omega }\right|$. (c) Contours of absolute vorticity in the symmetry plane $x=0$ at $t=24$ and (d) the same plot in the dividing plane. The red square marks the location of the identified vortex nulls.

Figure 15

$30\%$ of maximum vorticity field lines at $x=0$ (green) and $y=0$ (blue) with fans (red) and spines (black) plotted from null points for (a) $\text{Re}=800,t=48$, and (b) $\text{Re}=2000,t=93$.

Figure 16

(a) Estimated change in flux between ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ by integrating $\nu {(\mathbf{\nabla }\times \mathbf{\omega })}_z$ along the central axis (dashed) and direct flux counting (solid line). (b) Change in flux between ${\mathcal{A}}_1$ and ${\mathcal{P}}_c$ (solid line) and between ${\mathcal{A}}_1$ and the ${\mathcal{P}}_s$ (dashed).