EVOLUTION OF A MAGNETIC FLUX ROPE AND ITS OVERLYING ARCADE BASED ON NONLINEAR FORCE-FREE FIELD EXTRAPOLATIONS

A twisted magnetic flux bundle, i.e., a magnetic flux rope, is a crucial configuration for hosting chromospheric material showing as a filament. Eruptive filaments and flares are the manifestations of three-dimensional (3D) magnetic reconnection process in the solar atmosphere (Priest & Forbes 2002), and remain the subject of intense research on both observational and theoretical studies. To a great extent, our knowledge of the magnetic reconnection/eruption scenario depends on how well we can characterize in quantitative detail the 3D structure and topology of a magnetic flux rope and its embedding field. However, routine measurements of chromospheric and coronal vector fields are not yet available with present tools. Thus, most of our understanding of the magnetic reconnection process is actually based on interpretation of sets of surface and/or coronal observations combined with theoretical/numerical models.

When observing the photospheric layer, one of the most striking flare-associated magnetic changes is the enhanced horizontal field across the magnetic polarity inversion line (PIL; e.g., Wang & Liu 2010; Sun et al. 2012; Wang et al. 2012). One recent example of such a change is reported in detail by Wang et al. (2013, hereafter Paper I). Specifically, taking advantage of the high resolution and high cadence of the New Solar Telescope (NST) at the Big Bear Solar Observatory (BBSO), Paper I presents the observation of the rapid formation of sunspot penumbra at the PIL that is closely associated with the C7.4 flare of NOAA active region (AR) 11515. The formation of penumbra is substantiated by the transformation of the solar granulation pattern into penumbral fibril structure, and is accompanied by persistent penumbral darkening and horizontal field enhancement.

On the theoretical side, the abovementioned phenomenon could be explained in terms of magnetic reconnection after which the magnetic field relaxes to a more potential configuration. A flare loop system formed by reconnection decreases in height, referred to as "shrinkage," to restore equilibrium (Forbes & Acton 1996). Similar to the term "shrinkage" that is generally used during the gradual phase of a flare, the "magnetic implosion" conjectured by Hudson (2000) specifically refers to the contraction motion of coronal loops during the impulsive phase of a flare. Both magnetic field shrinkage and implosion imply that a part of the reconnected magnetic flux tubes moves downward and the accompanying plasma motion may cause abrupt impacts on the photosphere, such as a stepwise enhancement of horizontal fields (Hudson et al. 2008; Fisher et al. 2012). To understand the physical scenario, we need a fuller investigation of the magnetic topology in a spatially and temporally resolved manner.

Thanks to the new generation of instruments such as the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamic Observatory (SDO) and the recent advances in the nonlinear force-free field (NLFFF) extrapolation methods, the ability to probe the 3D magnetic field from photosphere to corona has been improved dramatically. Some studies have been conducted to examine 3D NLFFFs, in terms of magnetic energy (Sun et al. 2012), relative magnetic helicity (Jing et al. 2012; Dalmasse et al. 2013), pre- to post-flare magnetic connectivity change (Inoue et al. 2011; Liu et al. 2013), etc. However, NLFFF modeling of the magnetic flux rope structure and its evolution along with the overlying arcade field during the reconnection process appear to be fairly rare (Jiang et al. 2014).

In this Letter, we present the NLFFF modeling of the flux rope in NOAA AR 11515 which aligns well with an observed Hα filament. We follow the evolution of this magnetic flux rope and its overlying arcade over a five hour period. The extrapolation presented here indicates an interesting restructuring process. The elongated, highly sheared flux rope splits into two shorter segments which then pass and move away from each other. The dramatic separation of the two shorter segments is followed by the C7.4 flare and seemingly facilitates the subsequent subsidence of the overlying arcade. The descent of the arcade matches, spatially and temporally, with the penumbral darkening and horizontal field enhancement described in Paper I, confirming the conjecture of magnetic implosion.

AR 11515 is located at (S17°, E03°) on 2012 July 2, and produced five C-class flares and one M-class flare during the five hour period covered by this study. They are C2.0 (peaked at 16:44 UT in the GOES 1–8 Å soft X-ray flux), C3.7 (17:50 UT), C7.4 (18:56 UT), C1.4 (19:31 UT), C2.5 (19:43 UT), and M3.8 (20:07 UT). In particular, the C7.4 flare is the key event that is associated with the formation of penumbra and enhancement of the horizontal field at the PIL (see Paper I).

We use the latest version of the HMI photospheric vector magnetograms in the Space weather HMI Active Region Patches (Turmon et al. 2010) as the boundary conditions for the NLFFF extrapolation. The data series is from 16:24 UT to 22:00 UT at a 12 minute cadence and a pixel size of ∼05. Each vector magnetogram was disambiguated to determine the azimuth angle in the full range of 0°–360° using the minimum energy method (Metcalf 1994; Leka et al. 2009), de-rotated to the disk center, re-mapped using the Lambert equal area projection (Calabretta & Greisen 2002; Thompson 2006), and transformed to heliographic coordinates with projection effects removed.

To extrapolate the 3D NLFFF, we rebin the magnetograms to 1'' pixel intervals, preprocess the data toward the force-free conditions (Wiegelmann et al. 2006), and use the weighted optimization method of Wiegelmann (2004), which is an implementation of the original work of Wheatland et al. (2000). The extrapolation was performed within a computational domain of 232 × 232 × 200 pixels, corresponding to ∼ 168 × 168 × 145 Mm³. The visual comparison between some extrapolated NLFFF lines (Figures 1(a)–(c)) and the observed coronal loops seen in the 171 Å image (Figures 1(d)–(f)) taken by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) shows an overall similarity between the NLFFF model and the observation.

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To evaluate the performance of the NLFFF extrapolation, we compute 〈CWsin θ〉 metric and 〈|f_i|〉 metric. The former is the current-weighted average of sin θ, where θ is the angle between the magnetic field B and electric current density J. The latter is the average of |f_i| = |(∇ · B)_i|/(6|B|_i/▵x) over all pixels i, where ▵x is the grid spacing (Metcalf et al. 2008; Schrijver et al. 2008; DeRosa et al. 2009). Briefly, 〈CWsin θ〉 and 〈|f_i|〉 respectively reach an optimal value of zero for a perfect force-free and divergence-free field, so that a smaller value indicates that the field is closer to the force-free and divergence-free state. From the boundary data from 16:24 UT to 22:00 UT, we extrapolate a time series of 29 NLFFFs. Their 〈CWsin θ〉 and 〈|f_i|〉 metrics range from 0.38 to 0.44 and from 6.7×10⁻⁴ to 7.5×10⁻⁴, respectively, suggesting a moderately satisfied force-free and divergence-free condition.

We focus our analysis on a small field of view (FOV) of 34×30 Mm² as marked by the boxes in Figure 1, where the flares mentioned above occurred. This small region of interest (ROI) is also the target of the NST diffraction-limited observation on that day. Figures 2(a)–(c) present the close-up views of the photospheric horizontal field B_h = (B_x²+B_y²)^1/2, azimuthal angle tan⁻¹(B_y/B_x), and vertical field B_z of this ROI at 16:24 UT. The NLFFF at this time shows the presence of an elongated, twisted flux bundle (–1.1 turns; Inoue et al. 2011), suggestive of a flux rope configuration. The flux rope is represented by 36 field lines and their mean force-free parameter 〈α〉 (∇ × B = αB) is about –0.4 Mm⁻¹. This flux rope is strongly sheared over the PIL and generally aligned with the filament as seen in the NST Hα line center image (Figure 2(d)). Projections of the flux rope on the x–y and x–z planes are shown in Figures 2(e) and (f), respectively. For better illustration, the side view of the flux rope as well as the overlying arcade is presented in the top left panel of Figure 4. In general, the magnetic structure can be characterized as a highly sheared core field enclosed within an envelope of less-sheared arcade field.

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Prior to laying out more extrapolation results, we recapitulate the key features in the observation presented in Paper I, which are important in the comparison with the extrapolation. First, there is magnetic cancellation on the photosphere starting from ∼18 UT (Figures 3(a) and (b); also see movie 1). Second, the filament lying along the PIL erupts outward starting from ∼18:33 UT (animation 1(a) in Paper I), followed by the C7.4 flare. The NST Hα−0.75 Å off-band image at the flare peak time 18:56 UT is given in Figure 3(c), clearly showing the two flare ribbons and the erupting material. Third, with the onset of the C7.4 flare, a significant section of penumbra newly forms rapidly across the PIL (animation 1(b) in Paper I). Figures 3(d) and (e) are the NST TiO images taken about 1 hr before and after the C7.4 flare. The newly formed penumbra is pointed out by the arrow. Finally, the horizontal field across the PIL starts to increase gradually from ∼18 UT, then exhibits the highest increase rate simultaneous with the peak of the C7.4 flare, and continues to increase gradually afterward (Figure 5(d), and animation 3 in Paper I).

Now to proceed with the extrapolation, the flux rope shown in the NLFFF at 16:24 UT is represented by 36 field lines (Figure 2(e)). For the subsequent NLFFF sequence, we extrapolated field lines from the same starting points (labeled FP1 and FP2 in Figure 2), i.e., the locations of the two footpoints of the initial flux rope, and see how the overall magnetic structure evolves with time. Even though setting the starting points of field line extrapolation to co-move with the photospheric magnetic field would be more physically meaningful, here we keep the starting points fixed in light of practical concerns: FP1 was located in a positive magnetic polarity patch in the beginning, which then collided with a southward fast-moving negative magnetic patch. This collision led to magnetic cancellation, by which the magnetic patch at FP1 changed from positive to negative (18–20 UT, movie 1). The magnetic cancellation dominating the magnetic apparent motion makes it difficult to track the starting points with time. On the other hand, FP2 is always anchored to a weak negative magnetic polarity patch where the apparent motion is insignificant (within the spatial dimension of FP2). Keeping starting point FP2 fixed is unlikely to affect the modeling results.

The evolution is shown in Figure 4 and the online animation (movie 2). It turns out that the pink field lines that originate from FP1 and land at FP2 at 16:24 UT do not land at the same place at a different time. The landing points calculated in subsequent times are systematically displaced away from FP2. The same is true for the yellow field lines extrapolated from FP2. As a whole, it looks like the two moving ends move away from each other over time. More specifically, the initial elongated flux rope appears to split itself sideways into two shorter field line bundles (FLBs; between 16:48 and 17:36 UT), which are sheared, pass each other, and separate further (Figure 4). For the eastern FLB (colored pink in Figure 4), the whole configuration relaxes to a nearly potential state (〈α〉 ≈ −0.03 and –0.002 Mm⁻¹ at 18 UT and 20 UT, respectively). For the western FLB (colored yellow in Figure 4), the flux rope structure remains over the period covered by this study (〈α〉 ≈ −0.2 Mm⁻¹).

We limit the following velocity study to the western FLB to avoid complications associated with the polarity change in FP1. A quantitative comparison between the velocity of the moving end of the western FLB and the velocity of the local magnetic field at the same moving end is shown in Figure 5(b). The latter is derived with the differential affine velocity estimator (DAVE; Schuck 2006) and with the full resolution vertical magnetic field B_z at 05 pixel⁻¹. The window size used in DAVE is 19 pixels. Apparently, there are two stages in the motion of the moving end of the FLB: a fast-moving stage (with a velocity of ∼1–4 km s⁻¹) before the C7.4 flare and a slower stage (with a velocity of ∼0.3–1 km s⁻¹) after the flare. The velocity in both stages is generally much higher than that of the magnetic field traced by DAVE. It readily suggests that the movement of the western FLB is not due to the systematic drift motion of the photospheric magnetic field, but due to the continuous rearrangements of the magnetic field line connectivity.

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As the two shorter FLBs move away from each other, the magnetic arcade (blue field lines in Figure 4) arching over the initial flux rope starts to subside. At the location where the initial flux rope resides magnetic flux is evacuated by the separation of the two shorter FLBs, allowing the arcades to subside toward the surface all the way. As shown in Figure 5(c), the arcade starts to subside at ∼18 UT, drops steeply during the impulsive phase of the C7.4 flare from ∼4.5 to ∼2 Mm with a maximum speed of 1.8 km s⁻¹, and continues dropping more gradually to a height lower than 0.5 Mm. The subsidence of the arcade is, both spatially and temporally, coincident with the observed penumbral darkening (Figure 3(f)) and horizontal field enhancement (Figure 5(d)).

We also searched for observational features consistent with our model-based prediction of the flux rope reconstruction. In Hα, it was hard to tell because the penumbra extending from the nearby sunspot obstructs the view. In the extreme ultraviolet the coronal loops in this small FOV are too faint to be detected over the time of the flare brightening. Only the AIA 171 Å images taken at non-flaring or later post-flare phases show the signatures of the modeled flux rope. As shown in Figures 1(g)–(i), the long strand of coronal loops at 16:24 UT breaks into two shorter ones at 18:00 UT, and only the western one could be identified at 21:30 UT. This is in agreement with our modeled results.

In this Letter, we presented the evolution of a magnetic flux rope and its overlying arcade, using NLFFF modeling. The major results and their significance are summarized below.

First, this study offers a unified view which relates the surface observation described in Paper I to the implosion (Hudson 2000) and shrinkage (Forbes & Acton 1996) of the magnetic field. The Hα filament is associated with the initial magnetic flux rope enclosed within the magnetic arcades. The filament eruption preceding the C7.4 flare is probably the sign of the flux rope eruption. As the flux rope escapes upward, the arcade system shrinks after reconnection as a consequence of the magnetic field relaxing into a more potential state (Priest & Forbes 2002). In addition, it was conjectured that the photospheric magnetic field may become more horizontal during flares (Hudson et al. 2008; Fisher et al. 2012). Paper I reported the formation of new penumbra and the enhancement of the photospheric horizontal field across the PIL, which argued for magnetic implosion in the solar atmosphere. In this Letter, the subsidence of the magnetic arcade is actually demonstrated by the extrapolated NLFFFs, which is indeed consistent with the penumbral darkening and horizontal field enhancement described in Paper I. The arcade subsidence presented here shows the highest rate of change during the impulsive phase of the C7.4 flare, and then slows down over time. This result gives a more quantitative picture for the previous understanding of the magnetic implosion and subsequent shrinkage.

Second, a new type of flare-associated magnetic restructuring found in this study is the fast-moving footpoints of NLFFF lines, therefore implying a type of magnetic reconnection. Because this motion is along the PIL, we suggest that it could be either a phenomenon of slow field line slippage during magnetic field diffusion or similar to the slipping–running reconnection (Aulanier et al. 2006; Janvier et al. 2013). The derived footpoint (FP) speed here is 1–4 km s⁻¹, suggesting that the reconnection is in the regime of slow slipping rather than the slip–running at super-Alfvénic speed (Aulanier et al. 2006). The slow slipping reconnection is a relatively slow process and accompanies an implosion as found in this study. The strong-to-weak shear transition of flare loops shown in the slip–running reconnection (Aulanier et al. 2006; Janvier et al. 2013) is also qualitatively consistent with the arcade relaxation here. Of course, our model results are subject to the force-free field assumption; nonetheless, we consider it worthwhile to point out the similarity/dissimilarity between ours and the slipping–running model with the hope that the time-dependent boundary condition used in our NLFFF model may somehow reflect the change of coronal magnetic field due to some dynamics.

Finally, we note that the enhancement of the horizontal field and the subsidence of the arcade field are more obvious during the C7.4 flare rather than the M3.8 flare. Although the exact physical mechanism is still unclear, it seems that these magnetic changes are facilitated by the dramatic separation of two shorter FLBs preceding the C7.4 flare. The magnetic restructuring and energy release processes via successive separations deserve further theoretical and observational investigation.

J.J., C.L., S.W., Y.X., and H.W. were supported by NASA under grants NNX11AQ55G, NNX13AG13G, and NNX13AF76G and by NSF under grants AGS 1153226, AGS 1153424, and AGS 1250374. J.L. was supported by the international scholarship from Kyung Hee University. T.W. was supported by DLR-grant 50 OC 0501. We thank the NASA/SDO–HMI team for the photospheric magnetogram data, the BBSO team for the NST Hα and TiO data, P. Schuck for the DAVE code, and the referees for valuable comments.