Quantifying reconnective activity in braided vector fields

We introduce a technique for evaluating the changing connectivity of a vector field whose integral curves (field lines) form tangled tubular bundles. Applications of such fields include magnetic flux ropes, relativistic plasma jets, stirred two-dimensional fluids, superfluid vortices, and polymer networks. The technique is based on maps of the field line winding—the average entanglement of a given field line with all other field lines. Previously this had been developed for divergence-free vector fields. By extending some previous theoretical results, we show how it can be applied to any vector field that forms a tubular bundle. We demonstrate the efficacy of this technique on data from laboratory plasma experiments with two interacting magnetic flux ropes. Performed in the UCLA Large Plasma Device, the plasma's magnetic field structure is too complex to identify a single dominant current sheet as an expected site of magnetic reconnection. Previously, this complex structure had restricted the ability to analyze the evolving magnetic connectivity, but this is no such restriction to our method. We demonstrate that the plasma establishes a periodically oscillating cycle of magnetic field structure variation which, while triggered by an ideal instability, is dominated by magnetic reconnection. This reconnection leads to periodically varying coherence of a merged central flux rope, a conclusion supported by analysis of the writhing structure of the magnetic field.

Figure 1

A braided vector field whose entangled field lines are contained within a tubular domain.

Figure 2

Geometrical interpretation of $\mathcal{L}({\mathbf{x}}_0,t)$, showing the definition of the angle $\mathrm{\Theta }$ between the curves $\gamma$ and $\tilde{\gamma }$ in each plane, relative to the $x_1$ direction (white arrows).

Figure 3

A braided field used in [15]. (a) The field is a superposition of a counter-rotating twisted field (black) and a straight background field (blue). (b) The distribution $\mathcal{L}({\mathbf{x}}_0,t)$ for this field.

Figure 4

The flux rope experiment, showing (a) the experimental setup (reproduced from [48]), (b) streamlines of the transverse velocity field $\mathbf{V}$ in the plane ${\mathcal{D}}_h$ at $t=0.64\mathrm{ms}$, (c) a small subset of the field lines of $\mathbf{B}$ at $t=1.28\mathrm{ms}$, (d) a color map of the magnetic connectivity from ${\mathcal{D}}_0$ to ${\mathcal{D}}_1$ at $t=0.64\mathrm{ms}$, (e) the field-line-integrated vertical current density at $t=0.64\mathrm{ms}$, and (f) the same at $t=1.28\mathrm{ms}$.

Figure 5

Writhe values for field lines typical of the experimental data. (a) Helices with $\mathcal{W}=0.0004$ (red), 0.0002 (blue), and 0.0001 (green). (b) and (c) show field lines taken from the actual data (for $t=0.7936\mathrm{ms}$), with the same $\mathcal{W}$ values as in (a). Curves are plotted with the aspect ratios (1,1,2.5) used in later figures.

Figure 6

Various global topological quantities charted in time. (a) $\mathcal{L}\left(t\right)$ is always positive and transitions to periodic behavior from uniform growth. (b) $\mathcal{R}\left(t\right)$ has some oscillatory behavior, but the mean value is consistently negative. (c) $\mathcal{R}\left(t\right)$ peaks in the middle of the field's evolution. (d) $\mathcal{W}\left(t\right)$ shows a significant rise between $t=0.5\mathrm{ms}$ and $t=0.64\mathrm{ms}$, then sets into an oscillatory cycle. (e) ${\mathcal{R}}_{\mathrm{abs}}\left(t\right)$ and ${\mathcal{I}}_{\mathrm{abs}}\left(t\right)$; the reconnective component is always significantly larger. (f) $\mathcal{R}\left(t\right)$ and $\mathcal{I}\left(t\right)$; generally both oscillate but during the first 400 steps the ideal contribution is net positive.

Figure 7

Scaled plots indicating the temporal relationship between various quantities associated with $\mathbf{B}$. (a) $\mathcal{T}\left(t\right)$ and $\mathcal{W}\left(t\right)$; local minima and maxima are temporally coincident (opposing). (b) $\mathcal{W}\left(t\right)$ and $\mathcal{R}\left(t\right)$; the oscillation period of the two quantities is roughly the same and local extrema in $\mathcal{W}$ always occur prior to those of $\mathcal{R}$. (c) $\mathcal{L}\left(t\right)$ and $\mathcal{R}\left(t\right)$; peaks in $\mathcal{R}\left(t\right)$ generally occur first.

Figure 8

Distributions of the quantities $\mathcal{L}({\mathbf{x}}_0,t),\mathcal{R}({\mathbf{x}}_0,t),\mathcal{W}({\mathbf{x}}_0,t)$. The three columns represent $\mathcal{L}$, $\mathcal{R}$, and $\mathcal{W}$, respectively. The vertical direction follows the snapshot order $t=0.7936\mathrm{ms},0.8448\mathrm{ms},0.896\mathrm{ms},0.9344\mathrm{ms},0.9728\mathrm{ms}$. The $\mathcal{L}$ distributions depict $\mathcal{L}{({\mathbf{x}}_0,t)}^2$ and the $\mathcal{R}$ figures depict $\pm \sqrt{\left|\mathcal{R}\right({\mathbf{x}}_0,t\left)\right|}$ so as to emphasize the features of the distributions.

Figure 9

Internal geometry of the central “flux rope” at $t=0.7936\mathrm{ms}$. (a) The set of field line start points; for black points $\mathcal{W}>0.0002$ and green $0.0001<\mathcal{W}<0.0002$. (b) and (c) show these field line sets separately (the green fieldlines are shown as light green), (d) together.

Figure 10

Internal geometry of the central “flux rope” at $t=0.896\mathrm{ms}$. (a) The set of field line start points; for blue points $\mathcal{W}>0.0002$ and green $0.0001<\mathcal{W}<0.0002$. (b) and (c) show these field line sets separately, (d) together.

Figure 11

Average mixing ratio of the high and low $\mathcal{W}$ field lines as a function of $z$ for the field line sets shown in Figs. 9 and 10. This is essentially the ratio of the number of low-$\mathcal{W}$ points to the number of high-$\mathcal{W}$ points in a local neighborhood of each high-$\mathcal{W}$ point, averaged over all such points at a given $z$.