Velocities of flare kernels and the mapping norm of field line connectivity

TL;DR

This study analyzes the velocities of flare ribbon kernels during a solar eruption, linking their motion to magnetic field strength and the norm of quasi-separatrix layers, supporting the interpretation of kernel motions as signatures of magnetic reconnection.

Contribution

The paper introduces a combined observational and modeling approach to connect flare kernel velocities with the norm of QSLs, providing new insights into magnetic reconnection during eruptions.

Findings

Kernel velocities up to 450 km/s observed along flare ribbons.

Kernel velocities are roughly anti-correlated with magnetic field strength.

Modelled norm N of QSLs matches the observed kernel dynamics.

Abstract

We report on observations of flare ribbon kernels during the 2012 August 31 filament eruption. In the 1600\,\AA{} and 304\,\AA{} channels of the Atmospheric Imaging Assembly, flare kernels were observed to move along flare ribbons at velocities $v_\parallel$ of up to $450$ km\,s$^{-1}$. Kernel velocities were found to be roughly anti-correlated with strength of the magnetic field. Apparent slipping motion of flare loops was observed in the 131\,\AA{} only for the slowest kernels moving through strong-$B$ region. In order to interpret the observed relation between $B_{\text{LOS}}$ and $v_\parallel$, we examined distribution of the norm $N$, a quantity closely related to the slippage velocity. We then calculated the norm $N$ of the quasi-separatrix layers (QSLs) in MHD model of a solar eruption adapted to the magnetic environment which qualitatively agrees to that of the observed event. We found that both the modelled $N$ and velocities of kernels reach their highest values in the same weak-field regions, one located in the curved part of the ribbon hook and the other in the straight part of the conjugate ribbon located close to a parasitic polarity. Oppositely, lower values of the kernel velocities are seen at the tip of the ribbon hook, where the modelled $N$ is low. Since the modelled distribution of $N$ matches the observed dynamics of kernels, this supports that the kernel motions can be interpreted as a signature of QSL reconnection during the eruption.

Tables

Table 1. Table 1: Summary of kernel observations showing velocities v ∥ subscript 𝑣 parallel-to v_{\parallel} of flare kernels analysed in Sections 2.3 and 2.4 and B LOS subscript 𝐵 LOS B_{\mathrm{LOS}} measured in locations, where the kernels moved. Both v ∥ subscript 𝑣 parallel-to v_{\parallel} and B LOS subscript 𝐵 LOS B_{\mathrm{LOS}} have been averaged along the paths of the respective kernels.

	$v_{\parallel }$ [km s-1]	$B_{\mathrm{LOS}}$ [G]	location
Case 1a	46 $\pm$ 7	–29	NRH
Case 1b	40 $\pm$ 6	–133	NRH, tip
Case 2	254 $\pm$ 86	–5	NRH, elbow
Case 3	447 $\pm$ 151	9	PR, near parasitic polarity

Equations

$$v_{\text{slip}}=v_{\text{slipproj}}sin(\theta ),$$

Full text

magnetic reconnection – Sun: flares – Sun: transition region – Sun: UV radiation – Sun: X-rays, gamma rays

1 Introduction

Solar flares are among the most energetic phenomena occuring in the solar system, producing radiation across the whole domain of the electromagnetic radiation (e.g. Fletcher et al., 2011). During a flare, magnetic energy of up to $10^{33}$ ergs (see e.g. Aulanier et al., 2013) is converted by magnetic reconnection into other forms of energy such as partile acceleration and plasma heating. Flare emission can be traced from the solar corona down to the transition region and chromosphere, where foot-points of newly formed flare loops are observed as flare kernels within flare ribbons.

There have been numerous studies concerning flare kernels and flare ribbons with respect to the photospheric magnetic field. Li & Zhang (2009) presented a study of motion of flare ribbons in 190 M- and X-class solar flares using the 1600 Å passband observations of TRACE (Handy et al., 1999). They reported on measurements of speed of ribbon separation, i.e. the motion perpendicular to the polarity inversion line (PIL), which was up to 15 km s*-1*. Velocities reaching 39 km s*-1* were reported for ribbon propagation (elongation), i.e. the motion parallel to the PIL. Velocities of the same order of magnitude were obtained in numerous studies of flare ribbons (e.g. Tripathi et al., 2006; Lee & Gary, 2008; Qiu, 2009; Qiu et al., 2017, and references therein). Velocities of ribbon elongation as high as hundreds of kilometers per second were suggested e.g. by Vorpahl (1976); Kawaguchi et al. (1982), or Qiu et al. (2010). The highest velocities associated with ribbon dynamics from current EUV imaging instrumentation are about 200 km s*-1* (Joshi et al., 2018; Li et al., 2018). Note that during the Carrington’s famous white-light event, motions with similar velocities might have been observed (Carrington, 1859).

The dynamics of individual bright kernels was investigated using observations of H$\alpha$ (Asai et al., 2002; Qiu et al., 2002) and TRACE (Fletcher et al., 2004). These studies report on kernel velocities ranging between $\approx$15 km s*-1* and 120 km s*-1*. It was however not distinguished whether the kernel velocities were measured along or across the PIL. Similar velocities were reported for parallel motion of HXR sources observed by RHESSI in Cheng et al. (2012). There, authors associated moving kernels with moving UV fronts observed in flare ribbons. Kernel motions along the PIL were seen in ground-based observations of flare ribbons in the Ca II H filter of the GREGOR telescope (Sobotka et al., 2016), carried out at high spatial and temporal resolution of 0.1$\arcsec$ and 1 s, respectively. Observed bright kernels were found to be either stationary or moving along ribbons at velocities of $7-11$ km s*-1*, and were interpreted as a signature of slipping reconnecion.

The relation of the underlying photospheric magnetic fields and the ribbon elongation or separation, both occuring at typical rates of tens of kilometers per second, were studied by many authors. For example, weak anti-correlations between ribbon separation and magnetic field in the photosphere were found by Jing et al. (2007, 2008). Qiu et al. (2017) reported on anti-correlations between speed of ribbon elongation and photospheric magnetic field. The slower elongation in stronger magnetic field was suggested to result from an asymmetry between the magnetic field strength in the two footpoints of reconnecting magnetic field lines, so as to balance the reconnection fluxes.

In 2D, the reconnection rate is defined as $E=\partial{A}/\partial{t}=v_{\perp}B$, where $v_{\perp}$ is the horizontal motion of a ribbon perpendicularly to the PIL and $B$ is the underlying vertical component of the magnetic field (see e.g. Forbes & Lin, 2000). This classical definition of the reconnection rate uses velocity of ribbon separation $v_{\perp}$ away from the PIL (see e.g. Fletcher et al., 2004; Hinterreiter et al., 2018) instead of $v_{\parallel}$ along it. The ribbon separation is the only type of ribbon motion contained in the standard 2D model of solar flares (CSHKP, see e.g. Shibata & Magara, 2011). Motions parallel to the PIL, such as ribbon propagation, elongation, or motion of kernels along existing flare ribbons are governed by mechanisms of a three-dimensional nature and occur along the dimension missing in the 2D model. Therefore, using the kernel velocities along PIL to calculate the reconnection rate may not be a correct procedure. Attempts to theoretically interpret the aforementioned definition of the reconnection rate in 3D have been made by Lee & Gary (2008). However, the reconnection rate depended on the coronal reconnection region, which is difficult to measure.

The flare reconnection in three dimensions is described by the Standard model of solar flares and eruptions in 3D (Aulanier et al., 2012; Janvier et al., 2013). In this model, magnetic reconnection takes place in the absence of null-points in quasi-separatrix layers (QSLs, Priest & Démoulin, 1995; Démoulin et al., 1996b) where magnetic connectivity has strong gradients but is still continuous. Magnetic reconnection in QSLs manifests itself in the form of apparent slipping motion of magnetic field lines, with their connectivity changing sequentially from one to another. This apparent motion of field lines occurs along photospheric QSL footprints, which are known to correspond to flare ribbons (Démoulin et al., 1997; Aulanier et al., 2006; Zhao et al., 2016). As field lines reconnect in different parts of QSLs, the apparent slipping velocity may vary by more than a factor of hundred (see Figure 6 and 7 in Janvier et al., 2013). When the apparent slipping motion is slower than the coronal Alfvén speed, lines undergo so-called slipping reconnection (see for example a review of Janvier, 2017). If the velocity of slippage is super-Alfvénic, the reconnection is called slip-running and in this case, field lines are changing their connections nearly instantaneously. Observations of apparent slipping motion were reported for flare loops seen in EUV, see e.g. Dudík et al. (2014, 2016); Li & Zhang (2014, 2015). However, due to limitations of current instruments, especially their cadence, observations of only the sub-Alfvénic regime have been reported so far. Typical observed velocities of this motion are of the order of $10-100$ km s*-1* (Dudík et al., 2014, 2016).

Flare kernels accompanied with the apparent slipping motion of flare loops were first studied by Dudík et al. (2014, 2016) who reported on a good correspondence between the 131 Å footpoints of flare loops exhibiting the apparent slipping motion of flare loops, and bright kernels observed in 304 Å and 1600 Å filter channels. However, only the slipping velocities of flare loops have been measured, and no further attention was paid to the kernels themselves. Li et al. (2018) reported on a progressive elongation of circular ribbon at high $v_{\parallel}$, but no flare loops were observed to originate from the elongating ribbon. Fast kernels together with apparently slipping flare loops were observed in elongating ribbon by Joshi et al. (2018) (Figure 7a and Figure 8c there), yet slipping velocities were not measured. To summarize, while the slipping hot flare loops were observed to correspond well to ribbon kernels at their footpoints, the reverse may not always be true, as the relationship between kernels and slipping loops were not yet studied in detail.

In this work, we report on observations of motion of flare ribbon kernels and elongation of flare ribbons occuring at high velocities. We analyze observations of an eruption of a quiescent filament and accompanying C8.4-class flare performed by Solar Dynamics Observatory (SDO). In order to investigate association of these events with distribution of $B$, we also study variations of the norm $N$ of field line connectivity in three-dimensional simulations of solar eruption.

The paper is organized as follows. Data and observations of the event are introduced in Section 2. This Section contains further analysis of apparent slipping motion of flare loops (Section 2.2), flare kernels (Section 2.3), and ribbon elongation (Section 2.4). Section 3 contains analysis of theoretical models performed in order to associate dynamics of flare ribbons to variations in strength of the ambient magnetic field. In Section 4 we summarize our results.

2 Data analysis

We focus on an eruption of a quiescent filament from 2012 August 31 identified as SOL2012-08-31T19:45:00. The eruption and the accompanying flare were observed by the Atmospheric Imaging Assembly (AIA, Lemen et al., 2012) onboard Solar Dynamics Observatory. AIA creates full-disk images of 4096 $\times$ 4096 pixels with a pixel size of 0.6$\arcsec$ and a spatial resolution of 1.5$\arcsec$ at 12 s or 24 s cadence. AIA observes the solar photosphere in the 1700 Å and 4500 Å filter channels, solar chromosphere in the 304 Å filter channel, and the transition region in the 1600 Å filter channel. The 171 Å, 193 Å, 211 Å, and the 335 Å filter channels are used for imaging hot plasma in the solar corona and the 94 Å, 131 Å for flares (Lemen et al., 2012). AIA data were first converted into level 1.5 via the standard aia_prep.pro routine. Stray light was then deconvolved using the method of Poduval et al. (2013). The 1600 Å filter channel data were manually corrected for shifts in the coordinate systems compared to the other filter channels. Structure of the photospheric magnetic field was analyzed using the Helioseismic and Magnetic Imager (HMI, Scherrer et al., 2012) onboard SDO. According to HMI observations, two active regions NOAA 11562 and 11563 (Figure 1b) as well as regions of weaker magnetic field (network) were located in vicinity of the studied eruption. We note that the uncertainty in $B_{\text{LOS}}$ due to the photon noise is $\pm 7$ G (Couvidat et al., 2016).

2.1 Overview of the eruption

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 1a) 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 investigated also the hard X-ray emission sources related to this event observed by the Reuven-Ramaty High Energy Solar Spectroscopic Imager (RHESSI, Lin et al., 2002a) and were only able to reconstruct the footpoint sources at $\approx$19:47 UT (Figure 1c). X-ray emission 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 $\approx$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 (Figure 1b–d). We denote the hook of NR as NRH. 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 1b). The conjugate ribbon, located to the south-west is spatially coincident with regions of the positive polarity and we denote it PR. After $\approx$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 1b. Since 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 $\leq$15 km s*-1*. We also found discontinuities in the separation of ribbons. For example, straight portion of NR moved quicky (”jumped”) through a supergranule located at $\approx$[–680$\arcsec$,–350$\arcsec$] at around 20:15 UT. The conjugate ribbon PR jumped through several supergranules located at $\approx$[–620$\arcsec$,–470$\arcsec$] between 19:45 UT – 20:30 UT (see animation accompanying Figure 1d). 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 overal dynamics of kernels.

The NR and PR were connected by an arcade of flare loops (see animation accompanying Figure 1c). First flare loops appeared at about 19:45 UT in the 94 Å 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). Formation of arcade of flare loops was accompanied by ribbon elongation and apparent slipping motion of flare loops, which are analyzed in sections 2.2, 2.3, and 2.4.

2.2 Slipping flare loops in NRH

In the period between $\approx$19:41–19:52 NRH underwent a complex evolution. While propagating towards west, it slightly expanded and its tip elongated towards 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 $\approx$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 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 flare loops was observed shortly after the onset of the eruption at $\approx$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 2a). One patch of slipping loops is anchored in the lower kernel 1b at the tip of the ribbon (Figre 2a–h). Another patch of slipping loops is anchored in the upper kernel 1a, which propagates in the opposite direction with respect to the elongation and is broader (Figure 2d–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 2f. 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 2a). The time-distance diagram of the 131 Å filter channel data is shown in Figure 3. In panel b), different linestyles were used to distinguish between the apparent slipping motion of flare loops and ribbon emission visible in Figure 3a. Dotted lines in Figure 3b denote apparent slipping motion of flare loops. Velocity of the apparent slipping motion of flare loops was found to be between $\approx$27 and 85 km s*-1*. Apparent slipping velocities of this order are typical (see e.g. Dudík et al., 2014; Li & Zhang, 2014, 2015; Dudík et al., 2016; Jing et al., 2017). Solid lines fit apparent motion of flare loops tracked in a proximity to their foot-points. Velocities were found to range between 24 and 66 km s*-1*, again of the same order of magnitude. The bright feature in the bottom-right corner fitted with dashed lines is NRH, which after $\approx$19:48 UT crossed the cut as it entered its contracting phase.

Panel c) of this figure shows the time-distance diagram constructed using the 304Å  filter channel data along the same cut as in the case of the 131Å  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 3b) and the propagating ribbon (dashed lines in Figure 3b), but not the slipping flare loops (dotted lines in Figure 3b), which are only visible in the 131Å  channel. On the contrary, since the features fitted with solid lines are present in time-distance diagrams from both filter channels, foot-points of flare loops are strongly multi-thermal.

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 2a).

2.3 Observations of flare kernels in NRH

We now analyze observations of flare kernels and flare ribbons carried out with the 1600 Å filter channel of the instrument (Figure 1d, animation avilable 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 (Figure 4c–k). In the remainder of this paper, we will refer to the motion of these kernels as Case 1. Another bright apparently bright kernel, from now on referred to as Case 2, moved along the straight part of NRH. There, kernel appeared at $\approx$[–720$\arcsec$,–325$\arcsec$] and moved in a northern direction between 19:43:28 UT till 19:44:40 UT (Figure 4f–i).

To estimate velocities of motion of individual flare kernels, we employed a method which consisted 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 so that it does not include bright portions of the network, while 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. Length of the trajectory along which kernels moved was measured as the sum of lengths of lines joining these positions (hereafter, ’intensity centers’). Velocity of kernels was then calculated by using the distance between the first and the last exposure, in which the motion was visible. Uncertainties of measured velocities were calculated using an uncertainty of length of the trajectory and an uncertainty in time. Since the lengths of trajectories are dependent on positions of centroids of intensity enhancements, uncertainties in their measurements were taken as 1.5$\arcsec$, what corresponds to the resolution of AIA (Lemen et al., 2012; Boerner et al., 2012). The upper limit of the uncertainty in time was estimated as a cadence of the 1600 Å filter channel, namely 24 seconds.

2.3.1 Case 1

We first analyze in detail Case 1, i.e. the motion of kernels in a vicinity of the tip of NRH. In Figure 5a, contours corresponding to 500 DN s*-1* px*-1*in the 1600 Å channel are plotted in eleven exposures between 19:41:52 UT till 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 5b shows HMI $B_{\text{LOS}}$ data plotted with crosses marking intensity centers and Figure 5c shows the HMI background image corresponding to 19:43:22 UT.

Within Case 1, two individual kernels are analyzed. Motion of the kernel 1a was found to be ordered and continuous. It appeared close to regions of stronger magnetic field (red plus sign in Figure5b). The kernel then moved through a region of weaker field of $B_{\text{LOS}}\approx-35$ G (averaged between the orange to blue + signs) and ended its motion before entering a strong field region of $B_{\text{LOS}}$ $\leq-200$ G (dark-blue and violet + signs) at 19:45:52 UT (Figure 2l, Figure 4l). Motion of the kernel 1b started and continuously proceeded through a region with $B_{\text{LOS}}$ ranging between $\approx$–50 G and –200 G (red to light-blue + signs), then ”jumped over” a weak field region ($B_{\text{LOS}}$ $\approx-13$ G) of a size of few arc seconds (see Figure5b, Figure 4j), and then arrived to another region of $B_{\text{LOS}}\leq-200$ G at which point its motion stopped at 19:45:52 UT. Approximately two minutes later (Figure 2p), this kernel fragmented and was further difficult to trace.

Velocities of kernels between individual intensity centers as well as $B_{\text{LOS}}$ averaged between two following intensity centers are detailed in Figure 5d–e. While there is an apparent anti-correlation between the velocity and $B_{\text{LOS}}$ in the case of the kernel 1a, velocity of the kernel 1b remained relatively constant despite the gradual increase of $|B_{\text{LOS}}|$. However, a sudden drop of $|B_{\text{LOS}}|$ resulted in accelerating the kernel up to velocity of $\approx 150$ km s*-1*. Finally, the product of $B_{\text{LOS}}$ and $v_{\parallel}$ for both kernels is typically a few V cm*-1*, when expressed in units of electric fields. We however note that in some positions the magnetic field strength was comparable or only a few times stronger than the noise of HMI. Average velocity of the kernel 1a was found to be $46\pm 6.5$ km s*-1*. The kernel 1b moved at velocity of $40\pm 6$ 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 (see 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 $24\pm 4$ km s*-1* and $75\pm 19$ km s*-1* for the kernel moving at 40 and 46 km s*-1*, respectively. Apparent slipping motion observed at the lower velocity can be traced to the motion of the kernel 1b through the strong-field region until $\approx$19:45:04 UT (red to blue contours in Figure 5a and segments in Figure 5e), i.e. until it ”jumps” over the weak-field region described apriori.

The apparent slipping motion observed at $75\pm 19$ km s*-1*, attributed to the kernel 1a was observed while the kernel moved in regions between 19:42:40 UT – 19:43:52 UT (red to green contours in Figure 5a and segments in Figure 5d). 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 (see Figure 2f that these flare loops are not parallel to the cut). The observed slipping velocity of inclined flare loops can be converted into slipping velocity parallel to the cut by using an approximate correction factor

[TABLE]

where $\theta$ is the inclination between the patch of slipping flare loops and the cut, which is also parallel to ribbon. For $\theta\approx 45^{\circ}$ estimated from Figure 2f, the velocity of the apparent slipping motion is $\approx$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 since 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${}_{\text{LOS}}$ data observed at 19:44:07 UT.

The kernel first appeared at 19:43:28 UT (red contour in Figure 6a). Then, it proceeded through weak-field region ($B_{\text{LOS}}$ $\approx-5$ G, orange and green contours) towards north, until it stopped in a region of $B_{\text{LOS}}$ $\approx-50$ G (blue contour). Since 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 $\pm$ 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 (Figure 6c–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 $\approx$19:43 UT. It first moved a distance of about 5$\arcsec$ at velocity of 70 $\pm$ 34 km s*-1*. The kernel then accelerated to 294 $\pm$ 131 km s*-1* and moved further $\approx$15$\arcsec$ along the cut (until $\approx$ 19:44:40 UT). After this time and position, no motion of this kernel can be distinguished in either 1600 Å (Figure 6a) or 304 Å (Figure 6d). Finally, it seems that another kernel moved between positions $\approx$20$\arcsec$–25$\arcsec$ two minutes earlier with velocity 103 $\pm$ 68 km s*-1* (dashed line in Figure 6c). Therefore, the intensity enhancement in the uppermost position of the motion (orange upper contour in Figure 6a–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 of Li et al. (2018) and Joshi et al. (2018), 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 low electric field of $Bv_{\parallel}\approx$1 V cm*-1*, close to those of Case 1.

2.4 Case 3: Observations of flare kernels in PR

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

A portion of PR first appeared during the onset of the flare around 19:40 UT between [$-600\arcsec$,$-450\arcsec$] and [$-550\arcsec$,$-450\arcsec$]. The ribbon then evolved towards NW, encircled a supergranule, and underwent a continuous development, as a series of kernels moved towards SE along the straight part of the ribbon (Figure 7c–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 7a) located close to a plage region.

The motion of kernels is detailed in Figure 7 and schematically shown also in Figure 8a, 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 (Figure 7c–d, also Figure 8a, red contour). In the subsequent image, a new contour has appeared towards SE, while the original one has slightly shifted towards S (Figure 7e, Figure 8a, orange contours). The situation has repeated itself in another subsequent image (Figure 7f and 8a, green contours). At 19:47:28 UT (Figure 7g and 8a, blue contours), the kernels at the original position disappeared, while bright kernels now appeared even further towards SE.

The relationship of this apparent motion to the photospheric magnetic field is analyzed in Figure 8b. There, the same contours as in panel a) are plotted over HMI $B_{\text{LOS}}$ data. It is seen that the motion of the kernels analyzed within this case occured in weak-field regions, until it halted in a vinicity 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). 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 $B_{\text{LOS}}$ of $\pm 20$ G.

We note that the method of tracking intensity centers used in Case 1 and Case 2 can not be used here to estimate the velocity of the elongation of PR, since the previously lit-up 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 8a). These positions were then used to calculate velocities of kernels and it was found to be 447 $\pm$ 151 km s*-1*. This velocity is more than a factor of 2 larger than previous reports from space-borne observations. Given the low magnetic field of $\approx$9 G averaged along the path of the kernel, the electric field was found to be $\approx$4 V cm*-1*, which is similar to the values obtained for Cases 1 and 2.

However, we emphasize this result ought to be taken with a grain of salt, since 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 severly limited by the low cadence of the 1600 Å channel. There are no clear observations of this part of the ribbon in the 304Å  as well as the EUV filter channels, as it was obscured by the erupting filament at this time (Figure 1c).

We note that RHESSI (Lin et al., 2002b) sources of hard X-ray emission were observed around 19:47 UT further along the direction of elongation of PR (compare Figure 1c 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 1c). This hints at the ribbon elongation being connected to local heating by accelerating particles.

As a last remark, Figure 8 also displays examples of stationary kernels. Between 19:46:16 UT – 19:46:40 UT, two large kernels (red and orange contours) located at $\approx$[$-600\arcsec$,$-460\arcsec$] were present in regions of higher $B_{\text{LOS}}$ (Figure 8b). One exposure later, kernels slightly elongated (green contours) until they formed one large kernel (blue contour) at 19:47:28 UT.

2.5 Summary of the kernel observations