Reconstruction of a Large-scale Pre-flare Coronal Current Sheet Associated with a Homologous X-shaped Flare

The Sun has a complex dynamo that generates electric currents and magnetic fields, which provide energy to heat the corona, power solar flares, and eruptions. The currents in solar corona are typically evolved into two forms, volumetric channels (manifested as twisted magnetic flux ropes) and narrow sheets across which the magnetic field vector is almost discontinuous. Current sheets (CSs) are essential for energy dissipation in the solar corona, in particular, by enabling magnetic reconnection. For instance, the large-scale vertical CS is a basic building block in the standard flare model (Lin et al. 2015), which extends from the top of post-flare loops to the bottom of an erupting flux rope, and in which reconnection continuously occurs. A CS not only exists during a flare/eruption, but also forms in the quasi-static evolution of the corona subjected to the slow photospheric motions. The Parker theory demonstrates that the evolution of coronal magnetic fields in response to slow photospheric footpoint motions in general produces states with CSs rather than smooth force-free equilibria, that is, such a spontaneous formation of CSs is a basic magnetohydrodynamics (MHD) process (Parker 1972, 1994; Low 1996; Low & Petrie 2005; Low 2010). This means that the presence of CSs should be ubiquitous in the corona, like the presence of flux ropes. CSs can form in either magnetic separatix surfaces, which define the boundary of topologically separated domains (i.e., the magnetic field-line mapping is discontinuous), or quasi-separatrix layers (QSLs), where magnetic field-line mapping has a steep yet finite gradient (Aulanier et al. 2006; Démoulin 2006).

However, in reconstructing realistically the coronal magnetic field using existing models, even including the up-to-date most sophisticated nonlinear force-free field (NLFFF) models (e.g., DeRosa et al. 2009; Wiegelmann & Sakurai 2012), the reconstruction of CSs is rarely reported in the literature. On the other hand, it is well known that many NLFFF models are able to reconstruct magnetic flux ropes in the corona (Cheng et al. 2017; Guo et al. 2017). Considering the ubiquitous presence of CSs in the corona, there might be some problems (for example, see Low 2013) with available NLFFF reconstruction codes, which make them fail to reproduce CS. Regarding that the formation of CS is very natural in the MHD-relaxation process, it might be more suitable to use an MHD-relaxation method to reconstruct the coronal field containing CSs than many NLFFF codes that are based on mainly non-MHD approaches. From a theoretical point of view, the CS is a discontinuity, which constitutes a weak solution of the MHD equation, and thus numerical reconstruction of CS is a task of obtaining such a weak solution. Since the MHD-relaxation method is based on well-developed computational-fluid-dynamics (CFD) theory and codes, weak solutions are allowed by the CFD codes and can be reproduced correctly. However, the weak solution theory (i.e., solution contains discontinuities) is not yet established for typical NLFFF codes like the optimization and Grad–Rubin ones. So, it is problematic whether those NLFFF codes can produce solutions with CS discontinuities.

In this paper, we show that CSs in the corona could indeed be reconstructed using an MHD-relaxation model constrained by vector magnetograms. We investigated a homologous X-shaped flare in AR 11967 and found that the MHD equilibrium prior to the flares includes a large-scale CS situated vertically in the center of the flare site, which has a magnetic quadrupolar configuration. Furthermore, the photospheric footpoints of the field lines traced from the CS match strikingly well with the X-shaped flare ribbons, which provides strong evidence that reconnection in the CS produces the flares.

We seek MHD equilibrium consistent with a given snapshot of the magnetic field observed in the photosphere (i.e., one single magnetogram). Such an equilibrium is assumed to exist when the corona is not in the eruptive stage. This is because the photospheric driving motions are so slow that at any instant the coronal field has enough time to relax to a new equilibrium. We solve the full set of MHD equations with the magnetic field on the bottom boundary constrained by the vector magnetogram from SDO/HMI. Starting from a potential-field model and an initially hydrostatic plasma, we change the transverse field on the bottom boundary incrementally until it matches the vector magnetogram. This will drive the coronal magnetic field to evolve away from a potential state. Once the bottom field is identical to the vector magnetogram, the system is then left to relax to equilibrium with the bottom field fixed. The basic settings in this paper are similar to our previous works (Jiang et al. 2012, 2013, 2016b, 2017). Both viscosity and a small amount of friction are used for the aid of the relaxation process (see Jiang et al. 2012). Here, the background plasma is initialized in a hydrostatic, isothermal state with $T={10}^{6}$ K (sound speed ${c}_{S}=128$ km s⁻¹) in solar gravity. Its density is configured to make the plasma β as small as $2\times {10}^{-3}$ (the maximal Alfvén ${v}_{{\rm{A}}}$ is 4 Mm s⁻¹) to mimic the coronal low-β and highly tenuous conditions. The plasma thermodynamics are simplified as an adiabatic energy equation since we focus on the evolution of the coronal magnetic field. The bottom boundary of the model is assumed to be the coronal base, thus the magnetic field measured on the photosphere is used as a reasonable approximation of the field at the coronal base. We used the solar surface magnetic field data from the SDO/HMI (Schou et al. 2012), in particular, the Space weather HMI Active Region Patches vector magnetogram data series (Bobra et al. 2014; Hoeksema et al. 2014). Before input to the MHD model, a Gaussian smoothing of the original data with an FWHM of 2 arcsec (i.e., 1.4 Mm) is used to remove the very small-scale features that cannot be properly resolved by the MHD calculation.

The numerical scheme is an AMR-CESE-MHD code described in detail by Jiang et al. (2010). We use a computational volume much larger than the region of interest for the purpose of reducing the influences from the side and top boundaries, where all the variables are fixed as their initial values. At the bottom boundary, the velocity is fixed as zero. With an adaptive-mesh-refinement (AMR) technique, the computational time is significantly saved while the resolutions for important structures are still preserved. In particular, the AMR is designed to automatically capture narrow layers with strong currents as well as resolve strong magnetic field regions (e.g., sunspot regions) with strong gradients, which is illustrated in Figure 1. Here we would like to emphasize that in the numerical sense, a CS is not a 2D surface but a narrow current layer with thickness close to the grid resolution. For a typical resolution of 1 Mm, the thickness of a CS should be less than a few megameters. For capturing such an intense current layer, the value of J/B is used to guide the refinement of the mesh. J/B is proven to be a better indicator that can highlight current sheet-like distribution than the current density J itself (Gibson & Fan 2006; Fan & Gibson 2007; Jiang et al. 2016a). This is because in numerical realization, the CSs usually have both a larger current density and a weaker field than their neighborhoods. As can be seen in Figure 1, the J/B value in the CS abruptly increases over its neighborhoods, and thus it is captured by the mesh points with the highest resolution. During the calculation, any location with J/B becoming larger than $0.2/{\rm{\Delta }}$ (where Δ is the local grid size) will be refined by a factor of two. In addition, any place with a strong magnetic field gradient or a strong current will also be refined, and the criteria are, respectively, given by $| {\rm{\nabla }}({B}^{2}/2)| {\rm{\Delta }}/\rho \gt 100$ and $| ({\boldsymbol{B}}\cdot {\rm{\nabla }}){\boldsymbol{B}}| {\rm{\Delta }}/\rho \gt 100$ (where ρ is the plasma density, and all the variables are in normalized values). The mesh shown in Figure 1 used six levels of refinement with the highest resolution of 0.36 Mm in the x–y plane and 0.18 Mm in the z direction.

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Three X-shaped homologous flares occurred in AR 11967 on 2014 February 2, when the region was close to the central meridian. As an example, Figure 2 shows SDO observations of the third flare at 18:10 UT. All of these flares occurred in the same location of the AR and demonstrated very similar morphology, i.e., X-shaped ribbons and X-shaped brightening loops. As can be seen in Figure 2, the X extends long "arms" and has "legs" resembling those of a running man. The flare ribbons barely exhibit any movement during the whole flaring phase. All of these flares are M-class and they are confined without being associated with a coronal mass ejection or a jet.

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From the HMI magnetogram taken immediately prior to the flare, it is suggested that a quadrupole magnetic configuration is responsible for producing the X-shaped flares (Liu et al. 2016; Kawabata et al. 2017). As shown in Figure 2(c), the two legs of the X-shaped ribbon extended into two major sunspots with the inverse magnetic polarities (labeled as P1, N1) while its arms extended into the plage regions (labeled as P2, N2). Moreover, after the main phase of the flares, hot loops of all four types of connections, i.e., P1–N1, P2–N2, P1–N2, and P2–N1 were seen in AIA 131 and 94 channels, suggesting the magnetic reconnection in the coronal magnetic structures formed above the quadrupole magnetic polarity distribution.

In its lowest energy state, the coronal magnetic field can be modeled by the potential field. Based on a potential-field extrapolation from the vertical component of the vector magnetogram, we find that there is a double-null magnetic topology of the quadrupole, which confirmed the finding of Liu et al. (2016), who attempted to elaborate on the magnetic topology accounting for the same flare with both potential-field and NLFFF extrapolations. In Figures 2(e) and (f) we show the locations of the null and the skeleton magnetic field lines that delineate the magnetic separatrix. A well-defined X shape of the field lines is seen, but obviously it does not match the observed one, and both the nulls are situated far away from the center of the observed X-shaped structure (with a distance of at least 30 Mm). This indicates that the nonpotentiality, that is, a stress of the potential field and modification of the magnetic topology by the electric currents in the corona, plays a key role in shaping the flare (Kawabata et al. 2017). However, as shown in Liu et al. (2016), an NLFFF extrapolation produces a result even worse than the potential-field model in matching the flare topology.

We use the vector magnetogram of the same time 18:00 UT as input to the MHD model. Different from the potential model, the MHD model identifies no coronal null but reveals a CS in the corona. Figure 3 shows the structure of the CS. In 3D, if defined by the iso-surface of $J/B=1$ Mm⁻¹, the CS is a thin current layer with a thickness of ∼1.4 Mm. It is located in the center of the quadrupolar configuration, and extends vertically all the way from the lower boundary of the model box to a height of 40 Mm. The presence of the CS is also prominent in the distribution of the current density J (see Figure 3(c)), where the volumetric currents are distributed much more smoothly in significantly larger space than the CS. The CS separates field lines into two distinct connections, since when tracing field lines from the middle of the iso-surface, they clearly fall into two groups on either side of the layer (see Figure 3(b)), one connecting P1–N2, and the other connecting P2–N1. The field lines naturally form an X-shaped configuration. Moreover, by tracing all the field lines that are in contact or pass through the CS, their footpoints on the bottom surface delineate a sound X shape. This X shape, with its center, arms, and legs, almost coincides with the observed one in different AIA channels (see Figure 4). This strongly suggests that reconnection triggered at the CS produces the flare, since the locations of chromospheric flare ribbons are well recognized to correspond to the footpoint locations of those magnetic field lines that undergo reconnection in the CS (Qiu 2009). We further compute a magnetic squashing-factor map (Titov et al. 2002) at the bottom surface (see Figure 4(c) and (d)), which is a useful tool to reveal all the magnetic separatrix and QSLs. The X-shaped footpoints are co-spatial with two intersecting QSLs, which means there is an HFT and the CS is formed at the HFT (see also Liu et al. 2016). We note that the squashing-factor map can locate all the potential places for reconnection, but it cannot tell where the reconnection will actually take place for a flare. As can be seen, the structure of QSLs is much more complex than that of the flare ribbons (Savcheva et al. 2015; Inoue et al. 2016). Thus, if ribbon locations are not known in advance, it is still problematic to identify from all the QSLs the particular flare-related one. Here, our model very directly shows the location of the reconnection site for the flare.

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Capturing of the CS is robust by our MHD model. For the other two X-shaped flares, we can achieve similar results, i.e., the presence of CS and the good match of the flare ribbons with footpoints of the field lines contacting the CS (not shown here, but see X. L. Yan & C. W. Jiang 2017 in preparation). Furthermore, the CS's location is not sensitive to the resolution of the grid, but its thickness and intensity depend on the grid size. As shown in Figure 5, which gives results for three different grid resolutions, the location of the large value of J/B (or J) is almost the same, while the peak value increases approximately in proportion to the increasing of resolution. Presumably, the current-density distribution will approach to a δ function center at the CS, i.e., a magnetic tangential discontinuity, if the grid size approaches infinitesimal, but before that the microscopic behavior of plasma must be considered, which is beyond the scope of this paper.

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In this paper, we demonstrated the power of the data-constrained MHD model in reconstructing CS in the corona that is associated with flares. The studied event is an atypical confined flare with X shape occurring in a magnetically quadrupolar region. Neither the potential-field model nor the nonlinear force-free model can sufficiently reproduce the magnetic topology in correspondence with the geometry of the flare (see Liu et al. 2016). Our results show that there is actually a large-scale CS formed prior to the flares, and the field lines traced from the CS to the photosphere form an X shape that rather precisely reproduces the geometry of the flare ribbons. We thus suggest that the observed X-shaped flare is a product of the dissipation of the CS through reconnection. The recurrence of the X-shaped flares might be attributed to the repetitive formation and dissipation of the CS, while the formation of CS is driven by the photospheric footpoint motions (Santos et al. 2011). Such a dynamic process will be further investigated by a data-driven MHD model (e.g., Jiang et al. 2016a, 2016b).

This work is supported by the National Natural Science Foundation of China (41574170, 41574171, 41531073, 41374176, 41231068, and 11373066), Shenzhen Technology Project JCYJ20170307150645407, and the Specialized Research Fund for State Key Laboratories. C. J. and Q. H. also acknowledge partial support from NSF grant AGS-1650854. Data from observations are courtesy of NASA SDO/AIA and the HMI science teams. We thank Rui Liu and Jun Chen for providing us with the code for calculating the magnetic squashing factor.