THREE-DIMENSIONAL RECONNECTION INVOLVING MAGNETIC FLUX ROPES

We first examine the spatial structure of the magnetic flux ropes at t = 0.270 ms (2.7τ_A, where τ_A is the Alfvén transit time). This is relatively early in the discharge and is before the oscillations seen in Figure 3 have fully developed. From the 3D magnetic field data set, we can compute the field lines to gain more insight into the magnetic structure. In Figure 4(a), we have drawn two sets of representative field lines; one for each cathode. The field lines are seeded at a radial distance of 0.5 cm from the center of each flux rope at z = 64 cm. We can identify the two types of twisting of the field lines. One is the "writhe," which is the kinking up of the entire flux rope. The other is the "twist," which is the curling of a field line about the axis of the flux rope. Inspection of the field lines shows that writhe is approximately 180°, and the twist is 180°–270°. We can check this value against what we expect from the measured magnetic field. The perpendicular magnetic field in the upper rope peaks at r = 1.9 cm and B_⊥(1.9 cm) = 5.2 G. The total number of turns (twist and writhe) the field line will go through is turns = LB_θ/2πrB_z. For our values, we find that the field line should go through 1.4 turns. This is consistent with the twist and writhe found above. We can also compute the current density using Ampére's law, $\bm J =({c}/{{4\pi }})\nabla \times \bm B$. Cuts of the parallel current density, J_z are shown in Figure 4(b). Near the cathodes at z = 64 cm, we can see two well-defined current channels. As they progress toward the anode, the currents writhe about each other and move closer together in the perpendicular direction. Furthermore, the profile of each current channel becomes elongated. At z = 830 cm, the two current channels are nearly indistinguishable and form a single-merged configuration. Note that for our experimental setup, J_zB_0z < 0 for the flux ropes.

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One method of visualizing the time dependence of the flux rope configuration is to track magnetic field lines. As a first cut, we calculate the field lines that start from the center of each of the current channels in the z = 64 cm plane and track their motion in time. The results are generally the same for field lines 0.4 cm in any direction from the center, so we consider this field line to be representative of the zeroth-order motion of the whole rope.

Hodograms of the two central field lines at early times in the discharge are shown in the line plots of Figure 5(a). The curves begin at t = 0.200 ms (indicated by an arrow) and continue for 0.190 ms (f = 5.3 kHz). We can see that the flux ropes make fairly regular rotations. The area swept out by the flux rope increases at z positions closer to the mesh anode. The edge of the anode (diameter is 16.5 cm) extends well beyond the area of rotation. In the 3D plot, we show the surface swept out by each field line.

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The field lines are not line-tied at the anode end of the volume. That is, they are not constrained to terminate at a fixed location. In previous experiments (Van Compernolle et al. 2011; Sun et al. 2008), these boundary conditions have been observed. The sheath resistance of the anode is likely responsible for breaking line tying, as well as the fact that it is a semitransparent mesh and therefore not a perfect conducting wall. At later times, the field line motion is still periodic, but becomes irregular, as shown in Figure 5(b). In the middle of the flux rope, the area swept by the field line has become elongated to the point of being nearly linear. At the far end, the area swept is much smaller than in early times, and the path is much more erratic.

To try to understand this motion, we plot the separation Δs between the two field lines as a function of time and z position in Figure 6. The dashed line shows the separation at z = 507 cm. It does not change appreciably with time, but this is not too surprising given the small displacements made by the central field lines in this plane. The solid colored lines show the separation of the flux ropes in increasingly farther planes. In these planes, there are two times where Δs decreases rapidly, indicating that the two flux ropes are colliding. The dot and square correspond to the same symbols on the hodogram in Figure 5. The first collision appears to start in the z = 830 cm plane, and then propagate back down the field lines, while the propagation direction is reversed for the second.

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We can try to understand these collisions by looking at how the cross sections of the flux ropes behave in time. In Figure 7, we show the cross sections at four times during the first collision. The cross section is computed by seeding field lines on a circle in the z = 64 cm plane and computing their endpoints in the z = 830 cm plane. The radius of the circle is 0.5 cm, which is approximately at the half-maximum of the current profile. The t = 1.630 ms panel shows the pre-collision state of the flux ropes. At t = 1.650 ms, a 2 cm finger has stretched out from the bottom flux rope, but its center has not yet moved appreciably. At t = 1.670 ms, the main body of the flux rope has begun to catch up with the finger. Finally, at t = 1.690 ms, we can see that the centers of both flux ropes are substantially closer than in their original positions. So, while the Δs plot shows the zeroth-order motion, it is apparent that the flux ropes do not move as rigid bodies, but can instead stretch out and locally collide.

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When flux ropes with the same magnetic field handedness collide, the perpendicular components of the magnetic field at the point of collision will be antiparallel. This can create 2D field nulls at which magnetic reconnection can occur. We see evidence for reconnection in the form of current layers shown in Figure 8. At the anode end of the flux ropes, we see current layers (shown in bright yellow) directed opposite to the injected current as would be expected during reconnection. At the cathode end of the flux ropes, the current profile has not changed appreciably. These current layers are not present at early time in the discharge when the flux ropes do not seem to be interacting (see Figure 4(b)).

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In three dimensions, reconnection can occur even in the absence of a true magnetic null (Hesse & Schindler 1988; Schindler et al. 1988). One can identify regions in the magnetic field configuration known as QSLs (Priest & Démoulin 1995), where the spatial field line connectivity is changing dramatically yet continuously. To quantitatively define field line connectivity, consider two boundary surfaces that intersect a set of field lines. For example, one might consider the photospheric endpoints on either end of a coronal loop. For relevance to this experiment, take two transverse planes at z = z₀ and z = z₁. Let each field line that intersects the z = z₀ plane at coordinates (x, y) have corresponding points (X(x, y), Y(x, y)), where it intersects the z = z₁ plane. One measure of field connectivity is then the squashing degree (Titov et al. 2002)

$$\begin{equation} Q = \frac{{\left({\frac{{\partial X}}{{\partial x}}} \right)^2 + \left({\frac{{\partial X}}{{\partial y}}} \right)^2 + \left({\frac{{\partial Y}}{{\partial x}}} \right)^2 + \left({\frac{{\partial Y}}{{\partial y}}} \right)^2 }}{{\left| {\frac{{B_z \left({z_0 } \right)}}{{B_z \left({z_1 } \right)}}} \right|}}. \end{equation} \tag{ 1 }$$

The numerator of this expression indicates by how much the endpoints in the z₁ plane change relative to small movements in the z₀ plane. The denominator is a factor that scales away changes in connectivity simply due to magnetic mirroring. It also ensures that the value of Q is invariant under reversal of the boundary planes, so in effect we can assign a value of Q to each field line. Then, regions in the magnetic field where Q ≫ 2 (its minimum possible value) define QSLs. For the calculations presented here, we take the planes at the two ends of the measurement volume as our boundaries: z = 64 cm and z = 830 cm. The general differential equation for a field line is

$$\begin{equation} \frac{{d\bm r}}{{ds}} = \frac{{\bm B\left({\bm r} \right)}}{{\left| {\bm B\left({\bm r} \right)} \right|}}, \end{equation} \tag{ 2 }$$

where r(s) = (x(s), y(s), z(s)) is the vector pointing to points along the field line parameterized by s. Because of the strong guide field, we can re-parameterize the field lines by z-coordinates. The set of equations is reduced by one and simplified

$$\begin{equation} \begin{array}{*{20}c} {\dsty\frac{{dx}}{{dz}} = \dsty\frac{{B_x \left({x,y,z} \right)}}{{B_z \left({x,y,z} \right)}}} \hfill \\\\[-6pt] {\dsty\frac{{dy}}{{dz}} = \dsty\frac{{B_y \left({x,y,z} \right)}}{{B_z \left({x,y,z} \right)}}} \hfill \\ \end{array}. \end{equation} \tag{ 3 }$$

For each timestep, we seed field lines on a grid of points on the z = 64 cm plane, and using a Runge–Kutta differential equation solver, we calculate their end points in the z = 830 cm plane. The squashing degree is then computed by numerically differentiating the end points. Because of the strong guide field, the ratio of B_z at each plane is nearly unity and is therefore neglected.

In order to accurately compute field lines, the magnetic field data set must be divergenceless to a high degree of precision and preferably described analytically by a set of splines. Nonzero divergence can be introduced from measurement errors such as small misalignment of the probe head or even by interpolation between grid points. We follow the procedure described in detail by Mackey et al. (2006) to "divergence clean" the data set. Briefly, $\bm B$ is integrated in Fourier space to find the magnetic vector potential A, which is then fitted with tricubic splines. The splines are then differentiated analytically to get the divergenceless magnetic field. This allows us to compute field lines to high degree of precision.

The gradients of the endpoint functions are particularly sharp, so to resolve them we use a numerical grid much finer than the measurement grid. To quantify how well resolved the distribution is, one metric is

$$\begin{equation*} \det \left[ {\begin{array}{@{}cc@{}} {\dsty\frac{{\partial X}}{{\partial x}}} & {\dsty\frac{{\partial X}}{{\partial y}}} \\ \\[-6pt] {\dsty\frac{{\partial Y}}{{\partial x}}} & {\dsty\frac{{\partial Y}}{{\partial y}}} \\ \end{array}} \right]. \end{equation*}$$

This matrix can be thought of as the Jacobian matrix of the mapping along field lines from one boundary to another. The value of the determinant tells us by how much the area of an infinitesimal surface shrinks or expands under this mapping. From flux conservation, the determinant must equal (B_z(z₀)/B_z(z₁)). Since this is nearly unity for our configuration, we can use this as a check for the resolution. Even a small error in resolution will lead to a fairly large determinant since it is a difference of two large numbers, so it is a particularly sensitive metric. From this test, we find that to see the overall structure of the Q distribution (peaks up to Q ∼ 1000 are completely resolved), we need 15 points per measurement point, but if we want to completely resolve specific peaks, as many as 300 points may be necessary. Typically the peak value of Q is not as interesting as the peak width, so to save computational time we do not often make such detailed calculations.

It is fair to ask if it is meaningful to speak of a QSL whose width is much narrower than the measurement grid. The answer to this lies in the fact that Q does not depend on local magnetic field gradients, but instead on the global structure of the magnetic field. As long as the magnetic field is sufficiently resolved to allow field lines to be calculated accurately, local gradients will not have a significant effect on Q. In the appendices of two papers, Démoulin et al. considered this problem in detail from an analytical standpoint (Démoulin et al. 1996) and through numerical experiment (Démoulin et al. 1997). In our own error analysis, we did not see significant changes in the basic structure of the Q distribution when we simulated two sources of error: adding random spatial noise to the B measurements, and randomly misaligning each plane by one grid spacing in the perpendicular direction. However, some tests with a spatial low-pass filter of the magnetic field did reduce some of the finer structure in the Q distribution. Nonetheless, we present results using the unfiltered data set because we will be drawing most of our conclusions from the coarse structure, and because it may be of interest for future researchers to try to determine if the fine structure is due to a physical effect.

To get a sense of what the Q distribution looks like in our experiment, we show it in the z = 64 cm plane at t = 1.688 ms in Figure 9. This is during the point of closest approach in the first collision shown in Figure 6. At this time we find the QSL to be the most pronounced in magnitude of Q and in width. Two isocontours of parallel current density are overplotted in white to show the approximate locations of the two flux ropes in this plane. The primary structure is an S-shaped QSL that wraps around each flux rope. A cut across this QSL is shown in the lineplot below the QSL distribution. The width of the QSL at Q = 200 is 0.054 cm, and splits off into two peaks with bases at Q ∼ 2000. Such QSL splitting has been observed previously (Titov et al. 2008) but as discussed earlier it is difficult to say if we are truly resolving the QSL peaks.

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In a previous work (Milano et al. 1999), a resistive MHD simulation was performed of two merging flux ropes, and the Q distribution was calculated from the results. We see good qualitative agreement in the shape and location of the QSL.

The 3D structure of the QSLs can be visualized by seeding field lines along a contour of a particular value of Q, and drawing the flux tube bounded by the field lines on the QSL. In Figure 10(a), the blue semitransparent layer is the Q = 200 surface. The red and yellow field lines are representative field lines of the flux ropes. We can see that the QSL threads in between the two flux ropes, which is consistent with the idea that reconnection is occurring between them. The midsection of the layer has a smaller aspect ratio than its wide edges. The data resolve the electron skin depth and ion gyroradius. The electron gyroradius is far too small; we need probes less than 100–200 μm in size. For reference, the Debye length is less than 10 μm.

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To examine the field line structure for the QSL, in Figure 10(b) we plot field lines on the Q = 2000 surface inside the Q = 200 surface shown in the last figure. (The Q = 200 surface has similar structure, but it is complicated by the higher Q peaks inside it.) Pairs of field lines are seeded at z = 64 on either side of the QSL. The red set of field lines turn to one side of the QSL and collect in one corner at the far end of the QSL. The yellow field lines follow a similar but oppositely directed path.

The initial separation of the field lines is ∼0.01 cm and is 1 cm at the far end. In previous analytical models (Sato & Hayashi 1979) and simulations (Galsgaard et al. 2003) of interacting coronal loops, a similar structure described as a hyperbolic flux tube (HFT) has been observed. When HFTs become twisted, strong current layers have been shown to develop. Comparison of our structure with these results shows that our QSL has the structure of an HFT that has been twisted by 180°. While the source of twist is from field line bending rather than boundary flows, the structure is the same. This shows that the HFT has some relevance beyond its original application. It is expected that during reconnection, field lines will rapidly move across the QSL in a process known as magnetic flipping (Priest & Forbes 1992). To test this, we can calculate the velocity of the field line endpoints in a boundary plane. We assume line-tying at the z = 64 cm plane. At a timestep t_i, a grid of points in the z = 830 cm plane is mapped back to the z = 64 cm plane. We then map these points back to the z = 830 cm plane at t_{(i − 1)} and t_{(i + 1)}. Taking the time derivative of the endpoints in the z = 830 cm plane gives us the (perpendicular) velocity of the field line endpoint in this plane. We show the spatial velocity distribution at one timestep (Figure 11(a)). Blue arrows show the direction of the field line velocity at representative locations. The magnetic field is frozen into a fluid only in the limit σ → ∞. The resistivity of the background plasma is determined by Coulomb collisions. The mean free path for a background electron under these conditions is 10 cm and therefore comparing field lines at two times is an approximation. In Figure 11(b), we show the Q distribution mapped into the z = 830 cm plane. This can be thought of as a 2D cut of Figure 10 if we had drawn all the Q surfaces.

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The two distributions have a remarkably similar structure. We find that the fastest moving field lines are situated along the QSLs. The motion of the field lines is such that they are flipping along the long dimension of the QSL. We note that the field line velocity in these plots is on the order of 2–8 × 10⁵ cm s⁻¹, and that the perpendicular QSL size is about 1.5 cm, so the field line endpoints in this plane change on a timescale of 3 μs. This is much faster than the 100 μs Alfvén transit time along a field line. In MHD simulations of two interacting coronal loops, Aulanier et al. (2006) find that field lines slip across the QSLs at velocities on the order of the Alfvén speeds. They categorize reconnection that occurs at these speeds as "slip-running reconnection." This is in contrast to slow magnetic field diffusion and fast abrupt reconnection across a true separatrix. They conjecture that such slipping is responsible for hard X-ray sources that move along chromospheric ribbons during a solar flare. We have observed similar field line motion, but no heating of the bulk plasma. However, the injected power density is 10–20 times larger than the Ohmic heating rate from expected from the current layers, so we may simply not be able to resolve it. Nonetheless, further comparisons to this model may be a possible avenue of further research.

We have to date no detailed measurement of plasma flow but theory and simulations indicate that in 3D reconnection flows are required. In a case simpler than this one, studied by Linton & Priest, the flux tubes were initially untwisted (Linton & Priest 2003) but strong flows (a significant fraction of the Alfvén speed) were generated. Reconnection gives rise to flux tubes that become twisted and are pulled toward each other. In the simple 2D picture, oppositely directed field lines are entrained in a flow toward an X point or neutral line and particles jet are outward boosted by the energy released from the magnetic field, which is destroyed. In the realistic 3D case the flows will be far more complex. In these experiments, the flux tubes flattened out and current sheets were formed (see Figure 8). These narrow sheets were subject to the tearing instability. Counter rotating flows above and below a finite reconnection region were observed in another simulation study by Horning & Priest (2003). Reconnection occurs somewhere within the QSL and one expects to find oppositely directed flows on either side of this. There is a hint of this in Figure 12 if one accepts that the field line motion can be associated with a flow. The only way to resolve this is to directly measure the flow field and it will be done sometime in the future.

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The main dynamics of the three flux ropes can be captured by a series of snapshots of the parallel current density as displayed in Figure 15. From top to bottom, the distance of the data plane to the emitting cathode increases, i.e., δz = 64 cm, δz = 383 cm, and δz = 831 cm. The guide magnetic field comes out of the plane. Close to the cathode the individual flux ropes can easily be identified. The flux ropes are line tied to the emitting cathode, and their center-to-center spacing is still about 3.81 cm, which is set by the center-to-center spacing of the holes in the carbon mask in front of the cathode. Farther away, at δz = 383 cm, the flux ropes have approached one another and are starting to merge. Individual ropes can still be made out. The ropes have rotated by about 90° compared to the data at δz = 64 cm. This writhe is clockwise in the picture, corresponding to a left-handed writhe. Far away, at δz = 831 cm, the three flux ropes have merged and reconnected. A single extended flux rope is observed.

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In Figure 15, time advances from left to right by δt = 46 μs. At the far away data plane, δz = 831 cm, it is obvious that the merged flux ropes are rotating about the guide field with a rotation radius on the order of 5 cm. The rotation is counter-clockwise or right-handed, i.e., rotation in the electron diamagnetic direction. At δz = 64 cm, the rotation is apparent in the strength of the parallel current density in the flux ropes. At a fixed time, one of the ropes has the strongest current density. The next flux rope to have the strongest current associated with it is the adjacent one in the counterclockwise direction. The rate at which the strength of the current channel cycles is the same as the rate of rotation of the flux ropes. At intermediate distance from the cathode, δz = 383 cm, the rotation of the three flux ropes as a whole can be made out. At the same time, flux ropes can be seen to approach each other and later on distance from one another. At one instant one set of two flux ropes will be closest to each other, while at another instant a set of two other flux ropes is closest. Because of this complex dynamics, the QSL associated with the flux ropes will appear between different pairs of flux ropes at different times.

Figure 16 shows the rotation of the flux ropes in more detail. Magnetic field lines were started at the center of each flux rope at δz = 64 cm. In one rotation period, each field line traces out a surface. These surfaces are depicted as blue, red, and green transparent surfaces. The emitting cathode is on the far side of the figure. The insets show two slices of the hodograms. The colors indicate the time evolution. The ropes rotate in the counterclockwise direction, i.e., right-handed rotation in the electron diamagnetic direction. Close to the cathode the rotation of the flux ropes is very minimal, since the flux ropes are line-tied to the cathode. The hodograms of the three flux ropes are spatially separated there. Far away from the cathode, the hodograms overlap once the rotation radius becomes larger than the rope-to-rope distance.

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Figure 17 shows the center-to-center distance between two of the flux ropes versus time. Again, field lines were started at δz = 64 cm at the center of each flux rope and followed throughout space and time. From this the inter-rope spacing was computed. Figure 17 shows the rope separation at four axial locations. Close to the cathode, the ropes are line-tied and the rope separation is basically given by the separation of the holes in the carbon mask in front of the cathode (δz = 3.81 cm). Farther away the ropes can be seen to approach each other and then separate again. At large axial distances from the cathode the rope centers approach to within 1 cm. Note however that although we can trace the field line and obtain inter-rope separation measurements, the flux ropes themselves have in fact merged to one extended flux rope, as was apparent from the bottom panels of Figure 15.

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The QSL can be calculated as described for the two flux ropes. Starting from the divergence cleaned magnetic field data, the slip squash factor Q is calculated. Field lines are then seeded on contours of constant Q, and the surfaces that those field lines define are the quasi-separatrix layers. A snapshot of the magnetic field lines and quasi-separatrix layer at t = 4 ms is shown in Figure 18. This is the time at which the distance between the red and green flux ropes is minimized, as shown in Figure 17. The close proximity of the two flux ropes enhances the possibility of magnetic reconnection and results in a quasi-separatrix layer popping up in between the two ropes.

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Another snapshot of the QSLs and magnetic field lines at a later time is shown in Figure 19. At this instant in time, there are two QSLs, shown as transparent light blue and pink surfaces. The light blue QSL exists between the red and green flux ropes; the pink QSL exists between the red and blue flux ropes. The fact that multiple QSLs can be observed at the same time is one of the main differences between the three and two flux rope experiments. QSLs can be found between any pair of flux ropes. In the experiments there are times when only one QSL is present, as in Figure 18. At other times two or more QSLs (Figures 19 and 20) are present. It is interesting to note however that the peak slip squash factor Q in the three flux rope experiment was on the order of 50. This is far less than Q ∼ 1000 which was observed in the two flux rope experiment. Return currents as displayed in Figure 8 were not observed for the three flux rope case. The reason for this fundamental difference is not understood. Both the two flux rope and three flux rope experiments had a similar guide field, similar current densities in the flux ropes, and similar distance between ropes. In the two flux rope experiment evidence of reconnection in the form of return currents and high Q values were found, whereas in the three flux ropes case no clear evidence of reconnection was found unless one closely inspects the perpendicular (B_x–B_y) magnetic field lines.

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The appearance of the QSLs is a dynamic situation. When the ropes are not moving toward one another and there is no reconnection there are no QSLs and at other times there are one or more. This situation is illustrated in Figure 20 which is a red/blue anaglyph which appears three dimensional when viewed with glasses which can be purchased at a great number of stores as 3D has become popular once again. The field line structure within the QSLs of the figures, such as Figures 18 and 19, is displayed in Figure 20.

The field line structure within the QSLs of Figure 19 is displayed in Figure 21. Red and yellow field lines were started on either side of the QSLs. They are evenly spaced at the right edge of the QSL, but end up at opposite sides of the QSL at the left of the picture. Pairs of yellow and red field lines starting out close together at the right end up separated on the left edge. This result is qualitatively identical to Figure 10(b) for the two flux rope experiment and is reminiscent of hyperbolic fluxtubes.

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The 3D field line structure is so complex that reconnection sites are not easily identifiable, although the existence of a QSL indicates they are there. By tedious inspection of axial locations one can find reconnection events in the transverse plane by studying the morphology of field lines as a function of time. This is shown in Figure 22, where lines of B_x–B_y are drawn for four different timesteps, δt = 0.64 μs apart. The field lines approach each other, reconnect at t = t₀ + 44.80 μs and then move apart. In three dimensions the large component of B_z obscures this. Since the flux tubes move about in 3D, any two of them could cause a reconnection site at any position in the volume and at any given time reconnection could be happening at several locations.

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