Velocities of Flare Kernels and the Mapping Norm of Field Line Connectivity

Before its eruption, the filament was observed for more than one Carrington rotation and underwent several partial eruptions described in Srivastava et al. (2014). Its eruption on 2012 August 31 was accompanied by a C8.4-class solar flare. The eruption is analyzed in Williams et al. (2014) using spectroscopic data from Hinode/EIS. The 3D reconstruction of the ejection in the interplanetary space was presented in Wood et al. (2016).

The GOES X-ray emission (Figure 1(a)) of the accompanying flare started to rise at about 19:30 UT and peaked at about 20:47 UT. The gradient of the GOES X-ray emission has peaked at 19:44 UT, corresponding to the impulsive phase of the flare. We also investigated the hard X-ray emission sources related to this event observed by the Reuven-Ramaty High Energy Solar Spectroscopic Imager (RHESSI; Lin et al. 2002) and were only able to reconstruct the footpoint sources at ≈19:47 UT (Figure 1(c)). X-ray emissions at earlier and later times were difficult to reconstruct due to the presence of other events elsewhere on the Sun.

The rising of the filament can be first recognized in running difference images of the 171 Å bandpass of AIA at ≈19:10 UT (not shown). During the early stage of the eruption, a pair of flare ribbons appeared. The negative-polarity ribbon (hereafter, "NR") formed a J-shaped structure (Figures 1(b)–(d)). We denote the hook of NR as NRH. The straight part of NR as well as the outer part of NRH were anchored in regions of weak negative polarity, with NR extending into stronger field of AR 11562 (Figure 1(b)). The conjugate ribbon, located to the southwest, is spatially coincident with regions of the positive polarity; we denote it PR. After ≈20:15 UT, as the filament kept rising, both ribbons rapidly elongated and PR formed an extended hook PRH to the west outside of the region shown in Figure 1(b). Because PRH is large and its structure is complex, spanning many flux concentrations, we do not analyze it further.

In addition to the ribbon elongation, the ribbons themselves showed the classical separation, i.e., motion in direction perpendicular to the PIL. Velocity of the separation was measured using time–distance diagrams in straight portions of the ribbons and was found to be typically ≤15 km s⁻¹. We also found discontinuities in the separation of ribbons. For example, the straight portion of NR moved quickly ("jumped") through a supergranule located at ≈[–680'', –350''] at around 20:15 UT. The conjugate ribbon PR jumped through several supergranules located at ≈[–620'', –470''] between 19:45 UT—20:30 UT (see the animation accompanying Figure 1(d)). In general, the observed velocities of ribbon separation are rather small compared to velocities of elongation and kernels along ribbons (see Sections 2.3 and 2.4). Thus, here we further neglect the contributions of the separation to the overall dynamics of kernels.

The NR and PR were connected by an arcade of flare loops (see the animation accompanying Figure 1(c)). The first flare loops appeared at about 19:45 UT in the 94 and 131 Å filter channels and were sheared with respect to the PIL. We note that flare loops observed progressively later became less sheared. Arcade of flare loops thus likely underwent the strong-to-weak shear transition (Su et al. 2007). The formation of arcade of flare loops was accompanied by ribbon elongation and an apparent slipping motion of the flare loops, which are analyzed in Sections 2.2–2.4.

In the period between ≈19:41–19:52, NRH underwent a complex evolution. While propagating westward, it slightly expanded and its tip elongated toward the SW. The elongation was accompanied by the apparent slipping motion of flare loops and motion of flare kernels (see Figure 4 and Section 2.3.1). Slipping flare loops were studied using the observations of ≈11 MK plasma carried out in the 131 Å filter channel. Flare ribbons and kernels were studied using the 1600 Å filter data. In Figure 2, observations from both filter channels are combined in such a way that 131 Å filtergrams are overplotted with 1600 Å contours corresponding to 500 DN s⁻¹ px⁻¹ (animated version of Figure 2 is available online).

The apparent slipping motion of the flare loops was observed shortly after the onset of the eruption at ≈19:41 UT. Two patches of slipping loops were observed at the tip of NRH. Their footpoints are marked as "1a" and "1b" in Figure 2(a). One patch of slipping loops is anchored in the lower kernel 1b at the tip of the ribbon (Figures 2(a)–(h)). Another patch of slipping loops is anchored in the upper kernel 1a, which is broader and propagates in the opposite direction with respect to the elongation (Figures 2(d)–(m)).

In order to derive the velocity of apparent slipping motion of the observed flare loops, a time–distance diagram was constructed along the broken arrow shown in Figure 2(f). The position of the cut was chosen to be parallel to NRH, as well as to avoid one of the legs of the filament (denoted as "B" in Figure 2(a)). The time–distance diagram of the 131 Å filter channel data is shown in Figure 3. In panel (b), different line styles were used to distinguish between the apparent slipping motion of flare loops and ribbon emission visible in Figure 3(a). Dotted lines in Figure 3(b) denote apparent slipping motion of flare loops. The velocity of the apparent slipping motion of the flare loops was found to be between ≈24 and 85 km s⁻¹. Apparent slipping velocities of this order are typical (e.g., Dudík et al. 2014, 2016; Li & Zhang 2014, 2015; Jing et al. 2017). Solid lines fit the apparent motion of flare loops tracked in proximity to their foot-points. Velocities were found to range between 24 and 66 km s⁻¹, again on the same order of magnitude. The bright feature in the bottom-right corner fitted with dashed lines is NRH, which after ≈19:48 UT crossed the cut as it entered its contracting phase.

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Panel (c) of this figure shows the time–distance diagram constructed using the 304 Å filter channel data along the same cut as in the case of the 131 Å filter channel data. This diagram was constructed to help identify the moving features described before. The structures visible here are loop foot-points (solid lines in Figure 3(b)) and the propagating ribbon (dashed lines in Figure 3(b)), but not the slipping flare loops (dotted lines in Figure 3(b)), which are only visible in the 131 Å channel. On the contrary, because the features fitted with solid lines are present in time–distance diagrams from both filter channels, foot-points of flare loops are strongly multithermal.

Finally, no slipping loops were identified along other portions of the NRH. At position "A," (Figure 2), no slipping flare loops can be clearly distinguished due to obscuration by a cool, bright material, visible in many AIA filters, indicating that it likely originates at transition-region temperatures. It is possible that this is a portion of the erupting material returning back to the Sun. Furthermore, no slipping loops are visible between positions "C" and "D," corresponding to the relatively dim tip of the elongating NRH (Figure 2(a)).

We now analyze observations of flare kernels and flare ribbons carried out with the 1600 Å filter channel of the instrument (Figure 1(d); the animation is available online). At a 24 s cadence, the 1600 Å channel images the upper photosphere as well as the transition region, with a strong contribution from the C iv ion (Lemen et al. 2012; Simões et al. 2019). We note that we reviewed the 304 Å observations as well, which have higher cadence of 12 s. However, the outer edge of the hook NRH is in this filter, as well as other EUV filters of AIA obscured by the erupting filament, which can be optically thick at EUV wavelengths (Anzer & Heinzel 2005; Heinzel et al. 2008). Therefore, at least for this flare, motions of fast kernels and ribbons are best studied in the AIA 1600 Å filter channel, where the filament itself is invisible. In order to examine motion of kernels, we also analyzed running-difference images using data from this channel.

Figure 4 shows running-difference images of the same field of view as in Figure 2 imaged in the 1600 Å filter channel of AIA. In NRH, a pair of flare kernels appeared at about 19:39 UT. These were found to be the kernels 1a and 1b located at footpoints of slipping loops reported on in Section 2.2. Three minutes later, the kernels moved along NRH in opposite directions for about four minutes (Figures 4(c)–(k)). In the remainder of this paper, we will refer to the motion of these kernels as Case 1. Another apparently bright kernel, henceforth referred to as Case 2, moved along the straight part of NRH. There, a kernel appeared at ≈[–720'', –325''] and moved in a northern direction between 19:43:28 UT until 19:44:40 UT (Figures 4(f)–(i)).

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To estimate velocities of motion of individual flare kernels, we employed a method consisting of tracking contours of the same intensity level through series of images. In a reference image, intensity enhancements in the flare ribbons were first encompassed with contours. The appropriate value of intensity for the contour was determined manually: it had to be large enough not to include bright portions of the network, while still capturing the morphology of the evolving kernel. The typical values used are 300–500 DN s⁻¹ px⁻¹, on a case-by-case basis.

The kernel motion was then tracked in subsequent images. Positions of centroids of individual intensity enhancements were derived by calculating the average intensity-weighted position within the contour. The length of the trajectory along which kernels moved was measured as the sum of lengths of lines joining these positions (hereafter, "intensity centers"). The velocity of a kernel was then calculated by using the distance between the first and the last exposure in which the motion was visible. The uncertainties of these measured velocities were calculated using an uncertainty of length of the trajectory and an uncertainty in time. Because the lengths of trajectories are dependent on positions of centroids of intensity enhancements, uncertainties in their measurements were taken as 15, which corresponds to the resolution of AIA (Boerner et al. 2012; Lemen et al. 2012). The upper limit of the uncertainty in time was estimated as a cadence of the 1600 Å filter channel, namely 24 s.

2.3.1. Case 1

Our first detailed analysis is of Case 1, i.e., the motion of kernels in a vicinity of the tip of NRH. In Figure 5(a), contours corresponding to 500 DN s⁻¹ px⁻¹ in the 1600 Å channel are plotted in 11 exposures from 19:41:52 UT to 19:45:52 UT. The background image corresponds to 19:43:28 UT. The black lines join intensity centers derived using the method described before. Figure 5(b) shows HMI B_LOS data plotted with crosses marking intensity centers, and Figure 5(c) shows the HMI background image corresponding to 19:43:22 UT.

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Within Case 1, two individual kernels are analyzed. The motion of kernel 1a was found to be ordered and continuous. It appeared close to regions of stronger magnetic field (red "+" sign in Figure 5(b)). The kernel then moved through a region of weaker field of B_LOS ≈ −35 G (averaged between the orange to blue "+" signs) and ended its motion before entering a strong field region of B_LOS ≤ −200 G (violet and dark blue "+" signs) at 19:45:52 UT (Figures 2(l), 4(l)). The motion of kernel 1b started and continuously proceeded through a region with B_LOS ranging between ≈–50 G and −200 G (red to light blue "+" signs), then "jumped over" a weak field region (B_LOS ≈ −13 G) a few arcseconds in size (see Figures 5(b), 4(j)), and then arrived to another region of B_LOS ≤ −200 G, at which point its motion stopped at 19:45:52 UT. Approximately two minutes later (Figure 2(p)), this kernel fragmented and became difficult to trace thereafter.

Velocities of kernels between individual intensity centers as well as B_LOS averaged between two following intensity centers are detailed in Figure 5(d)–(e). While there is an apparent anticorrelation between the velocity and B_LOS in the case of the kernel 1a, the velocity of the kernel 1b remained relatively constant despite the gradual increase of $| {B}_{\mathrm{LOS}}|$. However, a sudden drop of $| {B}_{\mathrm{LOS}}|$ resulted in the kernel accelerating up to a velocity of ≈150 km s⁻¹. Finally, the product of B_LOS and v_∥ for both kernels is typically a few V cm⁻¹, when expressed in units of electric fields. In some positions, however, the magnetic field strength was comparable to—or only a few times stronger than—the noise of HMI. Average velocity of the kernel 1a was found to be 46 ± 6.5 km s⁻¹. The kernel 1b moved at velocity of 40 ± 6 km s⁻¹, which is (within the uncertainty) comparable to the velocity of the kernel 1a. Even though the velocity profiles of both kernels are complicated and depend on the intensity of magnetic field, the averaged velocities are rather typical (e.g., Fletcher et al. 2004; Cheng et al. 2012).

Velocities of the apparent slipping motion of flare loops anchored in the kernels analyzed in Case 1 (Section 2.2) are 24 ± 4 km s⁻¹ and 75 ± 19 km s⁻¹ for the kernel moving at 40 and 46 km s⁻¹, respectively. The apparent slipping motion observed at the lower velocity can be traced to the motion of the kernel 1b through the strong-field region until ≈19:45:04 UT (red to blue contours in Figure 5(a) and segments in Figure 5(e)), i.e., until it "jumps" over the weak-field region described previously.

The apparent slipping motion observed at 75 ± 19 km s⁻¹, attributed to the kernel 1a, was observed while the kernel moved in regions between 19:42:40 UT and 19:43:52 UT (red to green contours in Figure 5(a) and segments in Figure 5(d)). We note that the discrepancy between the velocity of the apparent slipping motion and the kernel 1a could originate in the inclination of the flare loop and the cut (Figure 2(f) shows that these flare loops are not parallel to the cut). The observed slipping velocity of inclined flare loops can be converted into a slipping velocity parallel to the cut by using an approximate correction factor

$$\begin{eqnarray}&&{v}_{\mathrm{slip}}={v}_{\mathrm{slip}\mathrm{proj}}\ \sin (\theta ),\end{eqnarray} \tag{ 1 }$$

where θ is the inclination between the patch of slipping flare loops and the cut, which is also parallel to ribbon. For θ ≈ 45°, estimated from Figure 2(f), the velocity of the apparent slipping motion is ≈50 km s⁻¹, which nearly corresponds to the velocity of the kernel within the respective uncertainty.

2.3.2. Case 2

The motion of a bright kernel along the outer portion of NRH, denoted as Case 2, is shown in Figure 6. Panel (a) shows the 1600 Å filtergram taken at 19:43:52 UT, with four contours corresponding to 410 DN s⁻¹ px⁻¹ plotted at four times. There are only four frames because the motion of the kernel is fast and lasts only a short time. The black arrow joins intensity centers derived in the same manner as in Case 1. Panel (b) shows the same arrow and contours plotted over HMI B_LOS data observed at 19:44:07 UT.

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The kernel first appeared at 19:43:28 UT (red contour in Figure 6(a)). It then proceeded through a weak-field region (B_LOS ≈ −5 G, orange and green contours) toward the north, until it stopped in a region of B_LOS ≈ −50 G (blue contour). Because emission at the terminal point in this strong-field region was observed at all times with varying intensity, it is likely that a spurious brightening observed at 19:43:52 UT (orange upper contour) is not associated with the motion.

The velocity found using the tracing of the intensity center is 254 ± 86 km s⁻¹. To verify this finding, the motion of the kernel was investigated in the 304 Å data taken at higher cadence along the track found in the 1600 Å channel. Tracking of kernels via time–distance diagrams itself is complicated due to ribbon separation and the erupting filament, which can obscure kernels. Nevertheless, a fast-moving structure was found in the time–distance diagram produced using 304 Å (Figures 6(c)–(d)). The uncertainties of velocities shown in the panel (c) were calculated using Equation (1) of Dudík et al. (2017).

According to the time–distance diagram, the motion of the kernel started at ≈19:43 UT. It first moved a distance of about 5'' at a velocity of 70 ± 34 km s⁻¹. The kernel then accelerated to 294 ± 131 km s⁻¹ and moved a further ≈15'' along the cut (until ≈ 19:44:40 UT). After this time and position, no motion of this kernel can be distinguished in either 1600 Å (Figure 6(a)) or 304 Å (Figure 6(d)). Finally, it seems that another kernel moved between positions ≈20''–25'' two minutes earlier, with velocity 103 ± 68 km s⁻¹ (dashed line in Figure 6(c)). Therefore, the intensity enhancement in the uppermost position of the motion (orange upper contour in Figures 6(a)–(b)) is indeed not associated with the motion of the kernel studied within Case 2.

Several comments on these measurements are in order. First, the large uncertainty of the velocity originates in a small number of exposures taken into calculations of the kernel velocity itself. Second, large velocities of flare kernels or ribbon elongation were reported in recent works by Li et al. (2018) and Joshi et al. (2019), so our observation of a fast-moving kernel is likely not unique in this regard. Third, these velocities, combined with the weak magnetic field in which the motion was observed, result in a low electric field of Bv_∥ ≈ 1 V cm⁻¹, close to those of Case 1.

We have also searched for evidence of ordered and continuous motion of flare kernels in the conjugate ribbon PR (see Figures 1(b) and (d)). Its evolution is detailed in Figure 7.

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A portion of PR first appeared during the onset of the flare around 19:40 UT between [−600'', −450''] and [−550'', −450'']. The ribbon then evolved toward NW, encircled a supergranule, and underwent a continuous development, as a series of kernels moved toward SE along the straight part of the ribbon (Figures 7(c)–(g)). We note that this phenomenon is reminiscent of the elongation of flare ribbon, although the motion of bright kernels can be preceded by much fainter intensity enhancements (black arrows in Figure 7(a)) located close to a plage region.

The motion of kernels is detailed in Figure 7 and schematically shown also in Figure 8(a), where 1600 Å filter channel contours corresponding to 320 DN s⁻¹ px⁻¹ are shown in four different times. The kernel appeared at 19:45:52 UT and started to move at 19:46:16 UT (Figures 7(c)–(d), also Figure 8(a), red contour). In the subsequent image, a new contour has appeared toward SE, while the original one has slightly shifted toward S (Figures 7(e) and 8(a), orange contours). The situation has repeated itself in another subsequent image (Figures 7(f) and 8(a), green contours). At 19:47:28 UT (Figures 7(g) and 8(a), blue contours), the kernels at the original position have disappeared, while bright kernels now appear even further toward SE.

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The relationship of this apparent motion to the photospheric magnetic field is analyzed in Figure 8(b). There, the same contours as in panel (a) are plotted over HMI B_LOS data. It can be seen that the motion of the kernels analyzed within this case occurred in weak-field regions, until it halted in the vicinity of a strong-field region spatially coincident with the plage observed in the 1600 Å channel. Panel (c) shows a detailed view of the supergranule encircled by PR, in which three small concentrations of negative polarity were found to be present (blue contours). The portion of PR shown in this panel is indicated using 1600 Å filter channel contours corresponding to 200 DN s⁻¹ px⁻¹ (yellow dashed contours). Blue and red contours correspond to HMI B_LOS of ±20 G.

We note that the method of tracking intensity centers used in Case 1 and Case 2 cannot be used here to estimate the velocity of the elongation of PR, because the previously illuminated portions of PR remain bright. Instead, we had to resort to manual selection of the leading edges of the contours (see black arrows shown in Figure 8(a)). These positions were then used to calculate the kernel velocity, which was found to be 447 ± 151 km s⁻¹. This velocity is more than a factor of two larger than previous reports from space-borne observations. Given the low magnetic field of ≈9 G averaged along the path of the kernel, the electric field was found to be ≈4 V cm⁻¹, which is similar to the values obtained for Cases 1 and 2.

However, we emphasize that this result ought to be taken with a grain of salt, because the derivation of the velocity was based on leading edges of a series of kernels identified by a certain contour level. Moreover, this analysis is severely limited by the low cadence of the 1600 Å channel. There are no clear observations of this part of the ribbon in the 304 Å as well as the EUV filter channels, as it was obscured by the erupting filament at this time (Figure 1(c)).

We note that RHESSI (Lin et al. 2002) sources of hard X-ray emission were observed around 19:47 UT further along the direction of elongation of PR (compare Figures 1(c) and 8). RHESSI images in the 6–12 keV and 12–25 keV range show a structure co-spatial with the emission of flare loops observed in the 131 Å filter channel of the instrument (Figure 1(c)). This hints at the ribbon elongation being connected to local heating by accelerating particles.

As a final remark, we note that Figure 8 also displays examples of stationary kernels. Between 19:46:16 UT and 19:46:40 UT, two large kernels (red and orange contours) located at ≈[−600'',−460''] were present in regions of higher B_LOS (Figure 8(b)). One exposure later, kernels slightly elongated (green contours) until they formed one large kernel (blue contour) at 19:47:28 UT.

The observed motion of kernels analyzed in Cases 1–3 is summarized in Table 1. This table shows velocities of individual kernels v_∥ as well as magnetic field strength B_LOS, both averaged along the entire paths of the respective kernels. Locations where the motions occurred are indicated in the right column.

Results indicate an apparent relationship between v_∥ and B_LOS, that is; the higher B_LOS, the lower v_∥. This relation does not hold in detail for the two kernels studied within Case 1, which moved at comparable velocities, but through regions where the averaged B_LOS differed for more than a factor of ≈4.5. However, detailed frame-by-frame analysis presented in Section 2.3.1 shows that the motion of the kernel 1a was ordered in the region of B_LOS ≈ –30 G, but the kernel halted before entering the strong-field region of B_LOS ≈ –200 G located nearby. For comparison, note in Figure 5(e) that the kernel 1b moved in B_LOS of this order of magnitude at a low velocity of about 40 km s⁻¹, until it "jumped over" a weak-field region between two consecutive exposures. In analogy to the motion of the kernel 1a, the fast kernel analyzed in Case 2 stopped its motion through the weak-field region of B_LOS ≈ –5 G before entering a region with stronger magnetic field of about –50 G. The situation has repeated itself in Case 3, where the kernel halted again before entering the plage region with B_LOS ≈ 170 G.

To summarize, the analysis of kernel motion in Sections 2.3 and 2.4 show that inhomogenities in the distribution of the photospheric magnetic field as small as a few arcsec affect the motion of kernels. Therefore, we suggest that the deviance from the apparent anticorrelations between v_∥ and B_LOS observed between kernels 1a and 1b within Case 1 (Table 1) is only due to the averaging of these quantities along paths of these kernels.

Our observations of the motion of the selected kernels are in general agreement with the results of other studies focused on kernels and ribbon motions (e.g., Fletcher et al. 2004; Jing et al. 2007, 2008; Qiu et al. 2017), even though the authors used the reconnection rate v_⊥B_z = cst as predicted by the Standard model in 2D to interpret the anticorrelations between B and v.

We highlight that, in our observations, v_∥ for kernels is measured only along the ribbon, while the standard model involves the ribbon motion perpendicular to the PIL. It is unclear how the standard model and the v_⊥B_z = cst property can be used to study motions along ribbons or ribbon hooks. On one hand, magnetic-field variations along the path of a slipping loop (and its footpoint) in 3D may naturally explain its changes in velocity of the slipping motion of the loop, just like in the 2D model. Indeed, magnetic field lines naturally concentrate more in stronger than in weaker field regions, irrespective of the 2D or 3D geometry (e.g., Figure 5 in Restante et al. 2009). On the other hand, some purely 3D topological effects may explain the relative differences in slippage velocities as we measured in the three parts of the ribbons. For example, in the 3D extensions of the standard model for eruptive flares and for a given reconnection rate, the local velocity of slipping field lines v_slip is proportional to the local gradient of field line connectivity as measured by the norm N of QSLs (Aulanier et al. 2006; Janvier et al. 2013). In order to test the latter hypothesis, hereafter we use a generic eruptive flare model.

We use run D2 of a numerical MHD model of a solar eruption from Zuccarello et al. (2015). The model was calculated with the visco-resistive OHM code (Aulanier et al. 2005). The calculations were done in Cartesian geometry, in the zero-β regime, and line-tied boundary conditions were applied at the photospheric plane at z = 0. The initial conditions and time evolution were similar to those of an earlier model (Aulanier et al. 2010). In short, a pre-eruptive flux rope was first formed in a sheared and asymmetric photospheric bipole. Flux cancellation then led the flux rope to grow in size until it passed the threshold of the torus instability, which made it erupt. During the eruption, a current sheet formed in the wake of the flux rope, within which magnetic reconnection occurs. Field lines were reconnecting in the slipping and slip-running regimes across QSL with J-shaped moving footprints (Aulanier & Dudík 2019).

Because the model is generic, it can be applied to interpret a variety of observed behaviors. However, its photospheric magnetic-field distribution is much smoother than that of the 2012 August 31 flare (Figure 1(b)), so it could not have been applied to investigate the dynamics of the flare ribbons analyzed in Sections 2.3 and 2.4. Because of this limitation, we restrict our model analysis to one time only during the eruption. We selected a time interval around $t=190\mbox{--}210{t}_{{\rm{A}}}$ (in the time unit of the model), during which a relatively good match can be found between the model and the observations in terms of the ratio between the area encircled by the hook elbow of the ribbon and the distance covered by the straight part of both ribbons away from the PIL. We then chose the time t = 204t_A in order to be able to relate the present analyses with those already reported in Aulanier & Dudík (2019).

The erupting flux rope at t = 204t_A is plotted in Figure 9(a). There, it is projected such the model is roughly oriented as the eruption is observed with SDO. We used the TOPOTR code (Démoulin et al. 1996a; Pariat & Démoulin 2012) to calculate the squashing degree Q(z = 0) and the norm N(z = 0) of the QSL footprints at t = 204t_A, which are both related with connectivity gradients. The outcome of the calculations was that N and Q show very similar patterns. We focus on the former, because it is directly related to the slippage velocity of reconnecting field lines (Janvier et al. 2013).

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The norm of the QSL at the photosphere at t = 204t_A is plotted in Figure 9(b). When discarding high-N features that are unrelated to the flare reconnection, such as a long bald patch (BP) along the PIL (as described in Aulanier & Dudík 2019), Figure 9(b) shows that the highest-N regions (N ≃ 10⁴ shown in red) are located in the elbows of the two QSL hooks, particularly in the broadest one located in a weak field area at the edge of the negative polarity. Contrariwise, lower N values (N ≃ 10^2–3 shown in green-yellow) are located at the tip of the two hooks and along the straight parts of both QSL footprints parallel to the PIL.

Observations show an important feature not incorporated in the model, namely that the straight part of the southern ribbon PR is not moving away from the PIL in smooth positive flux concentrations. Instead the ribbon moves across fragmented positive flux concentrations, at some places right above a network polarity, and at others in weak-field internetwork regions. One of the later weak-field regions even comprises a small negative parasitic polarity at the time when slippage is observed along PR (see Figure 8(c)). Such parasitic polarities are known to strongly influence the topology of the magnetic field within active regions (Mandrini et al. 1996; Masson et al. 2009) and below coronal flux ropes (Aulanier et al. 1998; Dudík et al. 2012). Therefore, the model had to be modified to match this complex distribution of flux.

To include a parasitic polarity in the model, we chose the simplest approach: a single radial subphotospheric magnetic source was added to the originally calculated magnetic field at t = 204t_A. The position of the source was chosen where the straight part of the original QSL footprint in the positive polarity is located at t = 204t_A. The flux and depth of the source were then adjusted to obtain a small parasitic polarity fully surrounded by a closed PIL well-separated from the main PIL of the main bipole. The resulting modifications in the flux distribution at the line-tied photosphere and in the shape of the coronal flux rope t = 204t_A are plotted in Figure 9(c). Comparison with the flux rope in Figure 9(a) shows that the erupting flux rope in the modified model does not differ much from that in the original model. The norm of the QSL at the photosphere at t = 204t_A in the modified model is shown in Figure 9(d).

After inclusion of the parasitic polarity, QSL footprints were found to be affected only slightly. The same distribution of low-N versus high-N regions as reported above in the original model is present in the modified model. Some local departures from the original model can still be seen in Figure 9(d). For example, the tips of both hooks are slightly less curved in the original model, and elongated low-N branches are located at the tip of the hook in the modified model. However, plotting field lines originating in these two regions does not show any significant differences between the two models.

However, there is a new feature that significantly contributes to the J-shaped structure in the modified model: a circular high-N region surrounding the parasitic polarity. The straight part of one of the J-shaped QSL footprints is deviated around its lower part, where they merge. Investigation of connectivity reveals that the circular high-N region is a new BP separatrix-surface (e.g., Titov et al. 1993; Aulanier et al. 1998; Müller & Antiochos 2008). The existence of this separatrix surface is robust to reasonable changes in the depth, flux, and position of the source. Finally, this circular BP is also associated with a short low-N region in the negative polarity, extending from the straight part of the J-shaped QSL footprint to the PIL.

The BP is associated with a true separatrix surface, and thus the norm of the associated QSL should be $N=\inf$. However, Figure 9(d) shows finite-N values (orange colors) along most of this feature. The TOPOTR code results in finite values there because the code is designed to stop the calculations when the increase in N between two iterations becomes smaller than some predefined threshold, which is what likely happened in case of the very flat BP field lines.

In summary, the small parasitic polarity below the straight part of the QSL results in generating a local curvature in the QSL. There, at a new BP separatrix, the norm N has increased from its original value of ≃10^2–3 up to infinity. Note that all separatrix surfaces are also surrounded by a so-called QSL halo (Masson et al. 2009), i.e., the regions where $N=\inf$ are surrounded by finite N values that decrease away from the separatrix. The existence of the separatrix, though, must allow the N values to be much higher than those of the original model with no parasitic polarity.

The generic eruptive magnetic-field model that we consider predicts that the magnetic-field connectivity gradients (as measured by the norm $N(z=0)$ of the QSL footprints) are stronger in the elbows of the QSL hooks than they are at the tip of the hooks and in the straight parts of the QSLs. However, our present modification of the model highlights that a parasitic polarity (with a sign opposite to that of the dominant surrounding field) can locally increase the connectivity gradients, and even make them infinite, when it is swiped by the straight part of the QSL.

Combining these results with the v_slip ∝ N relation (Aulanier et al. 2006; Janvier et al. 2013) with the known associations between QSL footprints and flare ribbons (Démoulin et al. 1996a; Aulanier et al. 2012) and between kernels and flare loop footpoints (Dudík et al. 2014, 2016) leads to two predictions:

• 1.  
  Relatively slower-moving kernels can be found at the tips of ribbon hooks and in the straight portions of the ribbons parallel to the PIL in an unipolar field.
• 2.  
  Relatively faster-moving kernels can be found in the elbows of ribbon hooks—and where and when the ribbon crosses a parasitic polarity, that locally creates a multipolar field.

To further illustrate the variations of kernel velocities along the ribbons, Figure 10 plots several systems of field lines within the QSL footprints. The blue and green field lines are plotted from a small area in the straight positive-polarity QSL footprint not affected by the inclusion of the parasitic polarity. The green field lines map to the tip of the hook of the negative-polarity QSL footprint. Their footpoints there are relatively nearby, indicating weaker N and consequently lower slipping velocities. Contrary to that, the conjugate footpoints of the blue field lines are spread over the outer portion (elbow) of the hook, due to the locally high N. The same applies for the footpoints of the pink field lines within the BP: their footpoints are fixed in NR, and the conjugate ones spread over the BP surrounding the parasitic polarity (Figure 10) in the modified positive-polarity QSL footprint.

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Summarized in the two points above, the numerical analysis corresponds well to our observations of the 2012 August 31 event in which we identified moving kernels (see Sections 2.3, 2.4 and Table 1) and loops apparently slipping along flare ribbons (Section 2.2). Kernels described within Case 1 and associated slipping flare loops (see Section 2.3.1) located at the tip of NRH (see Figures 4 and 5) moved at velocity of ≃50 km s⁻¹. These kernels were slower than those analyzed within Case 2 (see Section 2.3.2) located in the elbow of NRH (see Figure 6) and Case 3 (see Section 2.4) located in the straight part of PR near the parasitic polarity (see Figure 8), with velocities reaching ≃300 and ≃450 km s⁻¹, respectively. The distribution of N(z = 0) along the QSL hook in the model, which increases as one moves along the straight part of the QSL toward the hook (see Figures 9(b), (d)), is also consistent with the increasing velocity of kernels from ≃50 (Case 1) to ≃300 km s⁻¹ (Case 2) measured along NRH.