HOW DID A MAJOR CONFINED FLARE OCCUR IN SUPER SOLAR ACTIVE REGION 12192?

Solar flares are sudden releases of excess magnetic energy in the solar corona, a plasma environment dominated by the magnetic field (Shibata & Magara 2011). Magnetic reconnection is believed to be the central mechanism that converts free magnetic energy into radiation, energetic particle acceleration, and kinetic energy of plasma (Forbes et al. 2006). Consequently, revealing the magnetic structures associated with reconnection and their evolution during flares is essential for understanding flare dynamics (Priest & Forbes 2002).

Due to the lack of direct measurements of coronal magnetic fields, it is a prevailing way to postulate the flare magnetic evolution from the observed variations of flare plasma emission. This is because the plasma emission reflects the geometry of the invisible magnetic field, as in most parts of the corona the plasma is "frozen" with the magnetic fields. Early studies of typical eruptive flares have converged to a standard flare model (CSHKP, Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976), which describes the essence of flare physics. The standard model mainly concerns a magnetically bipolar source region, the simplest form of solar active regions (ARs), proposing that a twisted magnetic flux rope (corresponding to a filament) rises above the polarity inversion line (PIL), stretches the overlying closed field lines (manifested as coronal loop expansion), and produces a vertical current sheet (CS) underneath where reconnection sets in and results in two parallel chromospheric flare ribbons on both sides of the PIL. The flare ribbons are suggested to be the footprints of the reconnecting field lines. They gradually move apart from one another as the reconnection continues. Meanwhile, the ejecting flux rope eventually travels into solar wind as a coronal mass ejection (CME), leaving behind bright flaring loops that correspond to the re-closed magnetic arcades after the reconnection. Such a dynamic picture inferred from observations is usually represented by simple cartoons.⁶

Recent observations with high spatial/time resolution and multi-wavelength imagers show that numerous solar flares are characterized by complex processes that are not present in the standard model, such as multi-stage and multi-place filament ejections in the same event (e.g., Liu et al. 2009; Schrijver & Title 2011; Shen et al. 2012; Romano et al. 2015), escape of homologous flux ropes (e.g., Li & Zhang 2013), slipping motions of flare loops (e.g., Aulanier et al. 2007; Li & Zhang 2015; Dudík et al. 2016; Gou et al. 2016), flare ribbons of unusual shapes (e.g., quasi-circular and even tri-linear shapes, Masson et al. 2009; Wang & Liu 2012; Wang et al. 2014), multiple ribbons like remote flare ribbons distinct from the eruptive core site (or the secondary ribbon, e.g., Zhang et al. 2014), and the EUV late phase which occurs after the main (impulsive) phase in certain flares (Woods et al. 2011; Dai et al. 2013; Liu et al. 2013a). There are also flares without CMEs, which are usually called confined flares. Some confined flares occur with filament eruptions but fail to escape their overlying field (Ji et al. 2003; Török & Kliem 2005; Guo et al. 2010). Others, simply without any eruption, are the hardest to interpret solely from observations because very small changes of the coronal configuration can be detected in these flares (e.g., Jiang et al. 2012; Dalmasse et al. 2015).

Understanding the mechanisms of the various complex or atypical flares requires us to characterize the realistic magnetic configurations and their evolution associated with flares. Also, from the point of view of prediction of space weather, which is heavily influenced by solar eruptions, a much more accurate understanding and reproducing of the eruption process beyond the standard or theory model is strongly required. Existing techniques to this end include static, nonlinear force-free field (NLFFF) reconstruction (see review papers by Wiegelmann & Sakurai 2012; Régnier 2013), the data-constrained/driven magneto-frictional (MF) evolution method (e.g., Cheung & DeRosa 2012; Yeates 2014; Fisher et al. 2015; Savcheva et al. 2015), data-constrained magnetohydrodynamic (MHD) simulations (e.g., Jiang et al. 2012, 2013; Kliem et al. 2013; Amari et al. 2014; Inoue et al. 2014, 2015), and, more generally, data-driven MHD simulations (e.g., Wu et al. 2006).

Among the available techniques, the NLFFF reconstruction is used most frequently because a variety of approaches and codes for solving NLFFF have been developed within in a relatively long history (e.g., Grad & Rubin 1958; Sakurai 1981; Yang et al. 1986; Wu et al. 1990; Amari et al. 1997; Yan & Sakurai 2000; He & Wang 2006; Wheatland 2006; Wiegelmann & Neukirch 2006; Valori et al. 2010; Jiang & Feng 2012) and prove to be successful for studying snapshots of the coronal fields before and after flares. The signature of a flare mechanism can usually be suggested from analysis of the pre-flare magnetic fields. For example, through studying the magnetic topology, critical magnetic structures relevant to flares, such as magnetic flux ropes, magnetic null points, bald patches (Titov et al. 1993), and quasi-separatrix layers (QSLs, Demoulin et al. 1996; Titov et al. 2002) could be revealed. However, analyzing the pre-flare fields cannot tell directly why and how the flares occur. The lack of dynamics is a major limitation of NLFFF reconstruction, which cannot be used to "reconstruct" the magnetic field during flares. The limitation exists similarly in the MF methods. Although in such methods a dynamic velocity is included to make the magnetic field "evolve," this velocity is actually pseudo since it is determined only by the Lorentz force (i.e., the velocity ${\boldsymbol{v}}={\boldsymbol{J}}\times {\boldsymbol{B}}/\nu$ where ν is the frictional coefficient), while the inertia and pressure of the plasma are neglected (Yang et al. 1986). As a result, the MF approach is still limited for the quasi-static evolution phase of the corona field. When used for modeling the magnetic field evolution, the MF method is essentially similar to using a time sequence of NLFFF or MHD models reconstructed independently from a series of vector magnetograms along time to mimic the coronal evolution, although in the MF method, the magnetic field for each time snapshot is treated to be dependent on the preceding one. It is still problematic to use the MF method to simulate the flare and eruption phase in which the plasma is in extremely dynamic evolution and often associated with magnetic reconnections, although such a way has been used in analysis of the evolution of flare ribbons (e.g., Savcheva et al. 2015, 2016; Janvier et al. 2016). The data-driven MHD model of Wu et al. (2006), however, just uses the line-of-sight magnetograms, and thus the non-potentiality of the coronal field cannot be fully recovered. There are models (Jiang et al. 2013; Kliem et al. 2013; Amari et al. 2014; Inoue et al. 2014) using the NLFFF reconstructed or MF calculated coronal field immediately preceding eruption (thus the unstable nature of the field is already well developed) as the initial condition for the MHD simulation; such models prove to be able to reproduce the fast dynamic phase of the erupting field (e.g., Jiang et al. 2013). However, these kinds of simulations do not self-consistently show how the pre-eruptive field is formed and disrupted, and thus may not be used to identify the true triggering mechanism. Also, such models might not be able to reproduce confined flares, which are not likely triggered by the large-scale instability of the pre-flare magnetic field.

To self-consistently and realistically simulate the coronal evolution from its pre-flare to flare phases, we have developed a new data-driven 3D MHD AR evolution (DARE) model. The DARE model is based on the full MHD equation with its lower boundary driven directly by the solar vector magnetograms, which is unique among all the aforementioned models that attempt to simulate the realistic coronal magnetic evolution. In the first application of this model (Jiang et al. 2016), we modeled the evolution of a complex multi-polar AR with flux emergence over two days leading to an eruptive, also atypical flare. The simulation reasonably recreated the whole process from a long quasi-static evolution to the eruptive stage of extreme dynamics. It was shown that the field morphology resembles the sequence of the corresponding EUV images from SDO/AIA for such a process, in addition to the successful match of the timing of the flare onset.

In this paper, we use the DARE model to study a distinctly different event: a confined major X-class flare occurred in the super NOAA AR 12192 on 2014 October. This AR is "super" because of its size, which is the largest of all ARs during the last 25 years. It produced a series of X-class flares without eruptions, and the strongest one in the series reached X3.1, which sets a record for flare energy for CME-less events (Thalmann et al. 2015) since the confined flares observed were predominantly below X class (Yashiro et al. 2006). A series of studies have been attempted to explain why these extremely powerful flares did not lead to eruptions (Chen et al. 2015; Jing et al. 2015; Sun et al. 2015; Inoue et al. 2016). Here we are curious why and how these flares occurred, or more specifically, what is the evolution of the magnetic fields underlying these non-eruptive flares in AR 12192? We attempt to answer this question by simulating the coronal magnetic field evolution of the AR leading to the X3.1 flare. The rest of this paper is organized as follows. The flare event to be studied is described in Section 2. Then the DARE model is briefly presented in Section 3. Results are given in Section 4, and finally discussions appear in Section 5.

Since an overview of AR 12192 and its unusual features, that is, its extremely large size, its richness in X-class flares but CME-poor quality, have been described well in the literature (Chen et al. 2015; Jing et al. 2015; Sun et al. 2015; Thalmann et al. 2015), we focus on the X3.1 event and the relevant information with our modeling. In the period of our interest from 2014 October 23 to 24, AR 12192 was close to the central meridian. Two major sunspots were well separated by a distance of roughly 100 Mm (see Figure 1(a)). The target flare occurred around 21:00 UT, October 24, and lasted for an unusually long duration of more than one hour with a GOES X-ray flux above X class. Preceding the major flare are relatively small ones of C and M class with shorter durations. When inspecting the SDO/AIA images (for instance, Figure 1(b)) one only sees a series of brightening of coronal loops without much change in their shape. The flare loops are seen connecting the boundaries of the strong magnetic polarities. Interestingly, there is also a set of rather long loops with remote connections to the southwest corner of the field of view as shown (e.g., Chen et al. 2015; Sun et al. 2015). Chromospheric ribbons of the flare (Figure 1(a)) consist of mainly two bands on both sides of the central part of the PIL. Distinct from typical two-ribbon flares, these two ribbons showed barely any separation motion. As such, the coronal configuration changes are not easy to interpret from these EUV observations.

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In the DARE model (Jiang et al. 2016), 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 (SHARP) vector magnetogram data series (Bobra et al. 2014; Hoeksema et al. 2014). With a cadence of 12 minutes and a spatial resolution of 1 arcsec, the SHARP data are adequate to track a relatively long-term evolution (hours to days) of magnetic structures of the typical AR scale. To set up the model, we considered a local Cartesian coordinate system with its origin at the surface center of the AR, which is defined in the cylindrical equal area (CEA) re-mapped SHARP magnetogram. The 3D computational volume extends approximately $[-450,450]$ Mm in both x and y axes and 900 Mm in z axis, which is sufficient to include the large-scale magnetic field related to the flare. Furthermore, all the external boundaries (except the bottom surface) are non-reflective. We note that here the AR spans $\sim 60^\circ$ of the solar sphere, much larger than typical ARs, and the curvature of the associated area should not be ignored. Thus the current modeling in the Cartesian box might not appropriately characterize the geometry for the AR, and we keep in mind the possible influence in discussing our modeled results.

Based on the HMI vector magnetogram, we first constructed an approximately force-free coronal magnetic field (Jiang & Feng 2013) corresponding to a pre-flare instance of 00:00 UT on 2014 October 23. Then, with this field as the initial condition, we numerically solved the full set of time-dependent, 3D MHD equations in the modeling volume. 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. Then the evolving solar surface magnetic fields from observation provides the time-dependent bottom boundary conditions for the simulation domain. We smoothed the original SHARP data before inputting them into the numerical model. This is necessary since the magnetic structures are broadened from the photosphere to the coronal base. We simulated such broadening using Gauss smoothing of the data with a Gaussian window of $\sigma =2$ arcsec as suggested by Yamamoto & Kusano (2012). We further smoothed the data in time with a Gaussian window of $\sigma =4\times 12\,{\rm{minutes}}$ to remove short-term temporal oscillations and mitigate the problem of data spikes due to bad pixels. Interpolation in time was employed to fill small data gaps in the two days of observations.

In addition to the magnetic field, we also need to give a model of plasma in the computation. Here, the 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. No explicit resistivity is included in the magnetic induction equation, and magnetic reconnection is still allowed due to numerical diffusion if any CS forms and becomes thin enough with a thickness close to the grid resolution (i.e., the smallest grid). A small kinematic viscosity ν is used with its value corresponding to the viscous diffusion time as $\sim {10}^{2}$ of the Alfvén time in strong-field regions. This is usually necessary for the sake of numerical stability in the long-term computation. The units of length and time in the model are L = 23 Mm (approximately 32 arcsec on the Sun) and $\tau =L/{c}_{S}=180\,{\rm{s}}$, respectively.

Solution of the MHD equations is implemented by an advanced spacetime high-accuracy scheme (AMR–CESE–MHD, Jiang et al. 2010). We use a non-uniform grid based on the magnetic flux distribution for the sake of saving computational resources. The smallest grid ${\rm{\Delta }}x={\rm{\Delta }}y={\rm{\Delta }}z=1.4$ Mm (approximately 2 arcsec on the Sun) is made around the AR core region (approximately $[-100,100]\times [-100,100]\times [0,100]$ Mm³), where the magnetic fields are strong and evolve actively. Grid size is increased gradually to $4{\rm{\Delta }}x$ near the side and top boundaries.

To further save computing time, the cadence of the input HMI data into the MHD model was increased by 20 times. By this, the model run of a realistic two-day AR evolution can be finished within about 10 hours of wall time when parallelized with a medium number (for example, 100) of CPUs (3 GHz). Compressing of the time in HMI data is justified by the fact that the speed of photospheric flows as measured from the photospheric field evolution is about $0.1\sim 1$ km s⁻¹ (Welsch et al. 2004; Liu et al. 2013b). So in our model settings, the evolution speed of the boundary field, even enhanced by a factor of 20, is still sufficiently small compared with the coronal Alfvén speed (∼Mm s⁻¹), and the basic reaction of the coronal field to the bottom changes should not be affected. As a result, one hour in the HMI data equals one τ in the simulation. When comparing the simulation with the observations of the corona, such scaling of time (a τ equals an hour) also applies to the quasi-static evolution phase without major flares or eruptions. This is because in such a phase, the corona evolves in the same pace as the boundary since any change in the bottom boundary is reflected almost instantly in the corona, which reaches its equilibrium very fast. But this is not justified for the dynamic phases with major eruptions or flares, in which the evolution of the corona is determined by itself rather than by the boundary. Thus, the time unit should be the original one (a τ equals 180 s) in the flare phase.

Coupling of the coronal evolution with the continuous change of the surface magnetic field (i.e., the HMI data) is implemented by a time-dependent bottom boundary condition using the projected-characteristic method. Based on the wave-decomposition principle of the full MHD system (Nakagawa et al. 1987; Wu et al. 2006), the method can naturally mimic the transferring of magnetic energy and helicity to the corona from below (Wu et al. 2006) by self-consistently calculating the surface flow field (Wang et al. 2008), which otherwise would have to be derived by local correlation tracking or similar techniques (Welsch et al. 2004; Schuck 2008). As in our settings, the cadence of HMI data is $\tau /5=36\,{\rm{s}}$. However, the time step in the numerical model is set as ${\rm{\Delta }}t={\rm{CFL}}\min ({\rm{\Delta }}x/{v}_{{\rm{A}}})=0.2\,{\rm{s}}$ according to the Courant–Friedrichs–Lewy (CFL) stability condition (Courant et al. 1967) with a CFL number of 0.5. We thus linearly interpolate the HMI data in time to produce a data set with a cadence matching the time step of the MHD model.

We followed the evolution of the MHD system for two days from 00:00 UT of October 23 (t = 0) to 00:00 UT of October 25 (t = 48).

4.1. The Initial State

In Figure 2 we show the magnetic configuration derived from the near force-free model for the initial state (t = 0) of the MHD simulation. On the large scale, the AR exhibits a bipolar magnetic configuration consisting of two main sunspots with a relatively strong-sheared core field embedded in a less-sheared envelope fields. The core fields carry a relatively much stronger electric current (e.g., current density higher than 20 times of the average value of the whole model box; see Figures 2(b), (d) and (e)) that is concentrated within a narrow vertical layer roughly along the central part of the PIL separating the two major polarities. Such an association of an intense current layer with the PIL of the AR core might be common for flare-productive ARs (e.g., Sun et al. 2015). Here the current layer extends from the bottom to a relatively large height (∼35 Mm, see Figure 2(d)). From a visual comparison with the EUV images (Figures 2(a)–(c)), the simulated magnetic field lines show good agreement with the observed coronal loops. In particular, it can be seen that the weakly sheared envelope fields resemble the long cool loops imaged in the AIA 171 Å channel (about 1 MK), and the strongly sheared core fields resemble the short hot loops imaged in the AIA 94 Å channel (about 6 MK). This is likely due to heating by dissipation of the strong current in the core region, making the plasma there hotter than the surroundings. In the following we show how this coronal field evolved when driven by the photospheric magnetic evolution.

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4.2. Dynamic Evolution

Figure 3 and its supplementary animation show how the photospheric magnetic field evolved over the two-day time period. Continuous movements of the polarities can be seen, which could stress the coronal field. For instance, a horizontal flow map derived for the time moment of 00:00 UT on 2014 October 24 using the DAVE4VM method (Schuck 2008) demonstrates clearly a diverging motion of the two main sunspots. Driven by the magnetic field evolution at the bottom, the MHD system continuously evolved in response. The basic configuration of the coronal field shows no significant changes when we trace the magnetic field lines (thus not shown in the figures here), and the total kinetic energy maintains in a rather low level (less than 0.5% of the total magnetic energy) without significant variation in the whole time interval, indicating that no eruption occurs in the simulation. However, evolution of the distribution of electric current, particularly the CS, a thin layer of intense current as defined below, provides instructive information that might be associated with the flare processes.

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Here we use a method following Gibson & Fan (2006) to locate the CS. It is defined as the volume in which the ratio of the current density to the magnetic field strength, i.e., J/B, is greater than $C/{\rm{\Delta }}x$, where ${\rm{\Delta }}x$ is the local grid size and the constant number C ∼ 1. Such a definition is reasonable because $J/B\sim ({\rm{\Delta }}B/B)/{\rm{\Delta }}x$, where ${\rm{\Delta }}B$ denotes the gradient of magnetic field vectors in adjacent grid points, and ${\rm{\Delta }}B/B\ll 1$ in the smooth region of magnetic field (except regions with $B\sim 0$, for example, near magnetic null points). While a large value of ${\rm{\Delta }}B/B\sim 1$ indicates significant change of magnetic field (as large as the local field strength) between adjacent grid points. This means that the magnetic field vectors of distinctly different directions are squeezed extremely close to each other, forming a narrow interface with a strong current, which is a CS in the context of our numerical model. When the gradient of magnetic field vectors across the CS is steepened sufficiently, numerical diffusion will take effect and "reconnection" could occur in the MHD model. In other words, for such a case, the inversely directed magnetic components on both sides of the CS are brought so close to each other that they "merge" within the CS, resulting in new connections of the corresponding magnetic field lines, and thus related to topological changes of the field. By inspecting the value of J/B, we find a suitable value of C = 0.2 which can well exclude the region of weak current, and such definition gives the width of the CS of $\sim 3{\rm{\Delta }}x$. It should be noted here that the CS is different from that in the theoretical view, i.e., an infinitely thin or simply a 2D surface rather than a finite volume.

The 3D shapes of the CS at different times are shown in Figure 4(a) (and also in its supplementary animation). A horizontal slice of the volume is shown in Figure 4(b), and a vertical slice in Figure 4(c). Initially (t = 0) the CS had not yet formed, thus there are only small-scale structures near the bottom surface. Then, successive formation, expansion, and shrinkage of the CS volume are seen. From the plasma flows and the Lorentz force vectors around the CS volume, we can see that the initial current layer is gradually squeezed from both sides by the magnetic stress. Consequently, it becomes thinner and more intense, leading to increasing of the value J/B and thus the formation of the CS as we defined ($J/B\gt 0.2{\rm{\Delta }}x$). At time t = 36 (close to the peak stage of the CS development, see below), the CS extends from the bottom of the simulated domain up to heights of 50 Mm. With the slow evolution of the basic configuration, the CS also moves slowly from north to south, but overall its location is roughly the same for the whole period. Interestingly, the pattern of reconnection-like plasma flow, i.e., horizontal inflow at both sides of the CS and vertical outflow to up and down, can be seen (Figures 4(b) and (c), see t = 24 and 36, for example), suggesting that reconnection might occur in the simulation.

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We then calculated the following parameters to characterize the CS evolution and to see whether reconnection occurred within the CS.

• 1.  
  Since in the evolution process the CS grows and decays dynamically, for each time snapshot, we integrate J/B for the full volume V of the CS as defined above, which gives a value ${I}_{{\rm{CS}}}=\int J/{BdV}$ in units of area, and can be used to quantify the size or intensity of the evolving CS.
• 2.  
  Furthermore, we calculated the rate of the magnetic energy injection into the CS by ${F}_{{\rm{CS}}}=-{\int }_{S}{\boldsymbol{P}}{dS}=-{\int }_{V}{\rm{\nabla }}\cdot {\boldsymbol{P}}{dV}$, where ${\boldsymbol{P}}=\tfrac{1}{4\pi }{\boldsymbol{B}}\times ({\boldsymbol{v}}\times {\boldsymbol{B}})$ is the Poynting flux vector with ${\boldsymbol{v}}$ as the plasma velocity, and S is the surface area of the CS volume V.
• 3.  
  The total magnetic energy injected into the CS, i.e., ${E}_{F}={\int }_{0}^{t}{F}_{{\rm{CS}}}{dt}$ at time t.
• 4.  
  As here the CS is not a 2D surface but has a finite volume, we would like to know the magnetic energy (${E}_{{\rm{CS}}}={\int }_{V}\tfrac{{B}^{2}}{8\pi }{dV}$) in the CS volume. So the reconnection can be indicated if the magnetic energy stored in this volume is less than that injected into it.

The results are shown in Figure 5. If omitting the small fluctuations of the CS size profile during the whole process, we find that its evolution consists of two phases of distinct behaviors: in the first phase from t = 0 to approximately 30, it keeps a relatively small value (∼200 Mm²); in the second phase from t = 30 to the end of the simulation, it first increases impulsively and reaches the peak value of 1600 Mm² at nearly t = 40, and then decreases rapidly to a value similar to that in the first phase. The evolution of the rate of magnetic energy injected into the CS, ${F}_{{\rm{CS}}}$, shows a similar trend. In the later phase (i.e., from t = 30 to the end), the total magnetic energy input is about 10³³ erg, while the magnetic energy in the CS volume is smaller by nearly two orders in magnitude and can be negligible. This indicates that the amount of the magnetic energy injected into the CS is mostly released via reconnection (converted into other forms), and the energy release rate can be approximated by ${F}_{{\rm{CS}}}$. If we regard the impulsive increase and decrease of the energy release rate in the CS as a simulated "flare" process, we estimate the released energy by the flare is about 10³³ erg and the energy conversion rate is on the order of 10²⁹ erg s⁻¹, and can reach an order higher in the peak time. For a reference, the potential energy of this AR is approximately $1.5\times {10}^{34}$ erg during our studied period, over an order higher than typical-size ARs, e.g., AR 11158 (Sun et al. 2015). On the other hand, the first phase (t = 0–30) can be regarded as a quasi-static evolution duration for which the time unit τ can be scaled as being one hour, as mentioned in Section 3. This means that our simulated flare (t = 30) began 15 hr ahead of the real X3.1 flare (t ∼ 45). However, for the simulated flare phase, the time duration of nearly 20 τ, and thus equaling one hour (since $\tau =180$ s), is close to the real one that has relatively long X-ray duration of about one hour.

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The above analysis of the CS evolution based on the modeled results suggests a reasonable picture of how the real flare was produced: photospheric field evolution stressed the coronal field and built up a large-scale CS in the core region, then magnetic reconnection was triggered immediately and resulted in impulsive release of magnetic energy. As the kinetic energy is very low even during the impulsive phase of energy release, the majority of the released magnetic energy should be converted into energy of accelerated electrons (ions also possible), and subsequently in the form of radiation at various wavelengths. However, our simulation cannot reproduce this process because we did not include the related physics in the model. Nevertheless, the modeled results are in an agreement with the fact that this was a non-eruptive flare, during which no significant disruption of the coronal field was observed.

It is worthy noting that there were small fluctuations, i.e., short episodes of relatively small-scale CS formation and dissipation before the major one. These fluctuations reflect the energy build up and might be the simulation counterparts of the small flares that occurred before the main one (but a one-to-one correspondence was not reproduced). In fact, it signifies that the magnetic field is stressed and from time to time and the process is interrupted by episodic small-scale reconnection. However, preceding to the major flare, the CS is not large enough for the global-scale reconnection to set in.

4.3. Reconnection Configuration and Comparison with Observations

In Figures 6(a) and (b) we trace sampled field lines to illustrate the reconnection configuration. These four field lines are selected from the model at the peak time t = 40 of reconnection. The yellow and white lines are traced from the core site of the CS (shown by the pink object) and they represent the pre-reconnection field lines. These field lines (yellow and white) are sheared and pass each other at their inner footpoints, while the CS was formed at the interface between the crossing field lines. These field lines later reconnected and changed their connectivity, forming a longer field line (blue) connecting the two outer-most footpoints and a shorter line (cyan) connecting the two inner footpoints. By the magnetic tension force, both of the post-reconnection field lines relax. The longer field line expands upward but only slightly, and the shorter line contracts downward. Further evidence of this type of reconnection are the plasma inflows and outflows associated with the CS (i.e., diffusive region), horizontal and vertical slices of which made near the reconnection point are shown in Figures 6(c) and (d). Although the reconnection configuration is in full 3D, shown in the vertical slice they exhibit a typical X-shaped two-dimensional (2D) reconnection picture. Thus, locally such reconnection can be considered in a 2D framework with a strong guide field (i.e., out-of-the-plane component).

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To support the association of the modeled reconnection configuration with the real flare event, observed signatures of the flare emission can be compared with the modeled results, in an indirect way. Among these are the location and shape of the chromospheric flare ribbons, which are recognized to be an indicator of the footpoint locations of those magnetic field lines that underwent reconnection (Qiu 2009). This is mainly due to the non-thermal particles accelerated at the reconnection site traveling down along these field lines toward the photosphere and colliding with the dense chromosphere, causing enhanced heating (Reid et al. 2012).

As the sampled field lines shown in Figure 6 are traced from the core site of the CS (where J/B is the strongest), they can be regarded as the "first" field lines that reconnect in the whole simulated flare process (however, it is difficult to precisely locate the first reconnection point and field lines in the model). These first reconnected field lines have four footpoints at the bottom, and each of them, from left to right as shown in Figure 6(a), roughly corresponds to the four brightening patches as shown in Figure 7, from left to right, respectively, which are identified in AIA 1700 Å channel at the flare beginning. Both the simulation and the observation show that the two inner flaring points are separated by a relatively large distance of approximately 50 Mm. The simulation also suggests that the initial reconnection point is already rather high (∼30 Mm) in the corona.

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Following the initial reconnection, the reconnection site extends horizontally and forms a line of reconnection points with a similar X-shaped configuration at their vertical slice. Along the central line of the CS (see the regions with $J/B\gt 0.3/{\rm{\Delta }}x$ shown in Figure 6(c)), we can see the horizontal reconnection inflow vectors are almost perpendicular to the CS. This central line is a hint of such a reconnection line. When all the field lines along the reconnection line are involved, the two ribbons form (see Figure 8(b)). To simulate such ribbons, we identify all the reconnecting field lines as those that are in contact with or pass through the CS (see Figure 9(a)), which means that these field lines were undergoing reconnection or formed immediately after reconnection. The footpoints of these field lines, when mapped on the photosphere, appear to form two curved narrow areas separated by the central PIL (Figure 8(a)). A strikingly good match in both the location and the shape of these simulated footpoint areas with the observed chromospheric flare ribbons is evident (Figures 8(b) and (c)), including even the relatively weak ribbon extending far in the southwest direction.

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The flaring loop in the hot channels (e.g., AIA-94) can also be compared with our simulated field (see Figures 9(a) and (b)). For a better visual comparison, we generated a synthetic EUV image using a method similar to that used by Cheung & DeRosa (2012). We first trace a large number ($3\times {10}^{5}$) of field lines from their footpoints uniformly distributed at the bottom. Then we assign for each field line a value of emission assumed to be $J/{{BJ}}^{2}$ at the peak value point of J/B along the field line. Here J/B can be regarded as the dissipation rate of current, and by selecting its maximum, we can emphasize the emission from the reconnecting field lines. Finally the total emission is obtained by integration through the volume along the line of sight (here simply along the z axis), and forms the synthetic image. As shown in Figures 9(c) and (d), a good morphological similarity is achieved between the simulated emission and the AIA-94 images of the hot flaring loops.

As can be seen from the comparison, the ribbons in the core region correspond to the footpoints of the short field lines there, which reveal themselves as the short contracting loops. The far southwest ribbon is due to the long field lines that connect the negative-polarity sunspot and the far southwest plage region next to the positive-polarity sunspot, and these long field lines correspond to the long flaring loops. This might explain why flares frequently involve the brightening of these long side loops. We note that the shape of long loops appear not to be very well reproduced. This is possibly because they are close to the side boundaries of the simulation volume and thus the modeling of the field there is subject to the influence of the numerical boundary conditions, especially during such a long-term computation. A more important factor could be that our model is based on Cartesian geometry, thus failed to accurately reproduce the very long loops for which the curvature of the Sun must be considered. Nonetheless, the mapping of the field lines to the bottom surface is still accurate.

We simulated the magnetic dynamics of AR 12192 in a two-day period using the DARE model with the SDO/HMI vector magnetograms as evolving boundary input. Analysis of the modeled results shows that a large-scale CS is developed in the AR core field due to stressing by photospheric driving, and then reconnection is triggered within the CS, resulting in an impulsive release of magnetic energy, which could correspond to an X3.1 flare occurred near the end of the period. The reconnection configuration exhibits signatures of the tether-cutting reconnection (Moore et al. 2001). Comparison with the AIA observations shows that the model almost reproduced exactly the location of the chromospheric flare ribbons, and the morphology of the reconnecting field lines and the simulated EUV image resembles well the flaring coronal loops. Such an agreement of the simulation with observations suggests that our model correctly captured the essentials of the MHD process of this flare. Observations (Chen et al. 2015) show that the X-class flares in the same AR exhibited very similar flaring structures, indicating that these flares were homologous flares with an analogous magnetic mechanism. This might indicate multiple recurrence of the process from slow formation to fast dissipation of the intensive current layer between the sheared arcades, while the large-scale configuration of the AR does not change much.

Analysis of the magnetic decay index (Jing et al. 2015; Sun et al. 2015; Inoue et al. 2016) seems to explain why the flare failed to erupt, as the overlying closed magnetic field is sufficiently strong to confine the eruption from below (see another failed eruption shown in Wang et al. 2015). Here the DARE simulation provides additional and direct explanations. As can be seen from a direct look at the field lines (see Figure 6), the pre-flare magnetic arcades are twisted (or stressed) by a rather weak extent. A quantitative study of the magnetic twist has been performed by Inoue et al. (2016) for the same flare event using an NLFFF model. They found that the magnetic twists before the flare are mostly less than a half turn. Consequently, the sheared magnetic arcades do not show a well-formed two-J shape like those in many sigmoid ARs with eruptive events. Inoue et al. (2016) also showed that after the flare, the magnetic twists are almost reserved without release. This is consistent with our modeled results that the post-reconnection field lines are still sheared arcades, without forming an escaping magnetic flux rope. In our simulation, the reconnected long field lines on top of the CS only expands slightly without propagating further to make the overlying arcades open. Thus, in the context of tether-cutting scenario, only the first stage of tether-cutting occurred, while the second stage, formation of a flux rope and reconnection below the flux rope, did not happen.

Another unusual fact of this flare is that the enhancements of the horizontal field at the photosphere is very weak (Sun et al. 2015). This is unlike in many other large flares, as an evident enhancement of photospheric horizontal field after flare is commonly observed (Wang 2006; Wang & Liu 2010; Petrie 2012; Wang et al. 2012). In the proposed coronal "implosion" mechanism (Hudson 2000), such an enhancement of the photospheric field is due to the downward contraction of the short magnetic arcades formed after the reconnection and their push on the photosphere. As we found in the simulation here, the reconnection site is rather high above the photosphere, which is also suggested by Thalmann et al. (2015) from the study of the limited variation of the flare ribbon separations. As a result, the reconnected short field lines below the CS expanded still rather highly in the corona. So during its contraction, the effect of its push on the bottom might be reduced significantly by the strong corona field below. Thus the large altitude of the reconnection sites and the long extension of the reconnected arcades provide a plausible explanation for the weak "implosion" effect. A further quantitative analysis of this effect will be considered in future works.

It has been found that the magnetic energies from NLFFF models usually underestimate the related flare energies (e.g., Sun et al. 2012; Feng et al. 2013). For the present X3.1 flare, Sun et al. (2015) gives a result of $0.9\times {10}^{32}$ erg from a NLFFF extrapolation code. As a reference, Thalmann et al. (2015) estimated that the non-thermal electron energy for an earlier, confined X1 flare as $1.6\times {10}^{32}$ erg, so the X3.1 flare energy is likely to be much larger than this value. On the other hand, from our simulation we have estimated a total released magnetic energy of 10³³ erg, which is an order higher than that derived from the NLFFF model (Sun et al. 2015) and appears to be sufficient for powering the X3.1 flare. Such an unusually large value of energy release (compared with typical ARs) is still reasonable when considering the unusually large size of this AR with potential energy of about $1.5\times {10}^{34}$, a order higher than typical ARs. While the NLFFF model calculates the drop of total magnetic energy of the whole modeling volume from pre-flare to post-flare states, we can directly calculate the magnetic energy lost in the CS due to the "reconnection" within the CS, which should be more relevant with the flare-released energy. As such, we did not perform energy analysis for the full modeling volume.

Recently, Savcheva et al. (2015) claimed that based on MF or NLFFF models and searching of QSLs, they were able to predict the locations of the flare ribbon. However, we note that the QSLs are only possible sites for reconnection, while they cannot tell where the reconnection occurs specifically in a flare. Moreover, by inspecting the map of QSLs, one usually sees much more complex structures than that of target flare ribbons (see Figure 5 in Savcheva et al. 2015). Thus without knowing ribbon locations in advance, it is still problematic to identify from all the QSLs the particular flare-related one. Here with the MHD model we simulated the flare reconnection process in a self-consistent way and directly identified the reconnecting field lines, and thus we are able to almost reproduce precisely the flare ribbon locations.

We find that the modeled "flare" began (at $t\approx 30$) well before the photospheric magnetic field input at the bottom boundary evolving to the time of the real flare (at $t\approx 45$). A perfect model of reproducing the reality should produce a flare at the exact time when the photospheric field reaches the flare onset time. The mismatch of our model with the reality is probably in a large part due to the over-simplification of the magnetic reconnection process, which might be much more complex since it is related to the microscopic behavior of plasma. However, as in many other solar MHD codes, the modeled reconnection here is simply resulted by numerical viscosity in the CS region, and its behavior depends on the numerical aspects of the model. For example, the thickness of the modeled CS and the numerical viscosity are often sensitive to the grid resolution (as well as the specific numerical scheme). A much thinner CS can develop if using a smaller grid size, and the onset of reconnection in the CS might be postponed. Namely, with a smaller grid size, a CS can sustain even stronger current and thus even larger gradient of magnetic field, which might needs more time to form. Further experiments using different grid resolutions will be required to quantify this effect.

Comparative study of this flare with eruptive ones may provide insight in the different magnetic natures of eruptive and confined flares. Here we refer to our previous study in which we use the DARE model to simulate an eruption event in AR 11283 (Jiang et al. 2016). For that event, the eruption onset time is matched by the model much better, with less than a time lag of $2\tau$ during a whole simulated period of $60\tau$. Unlike this event, the simulated eruption in AR 11283 is due to the formation of a jet-like magnetic configuration that favors breakout-like reconnection (Antiochos et al. 1999). So in that event, the critical condition causing eruption is the formation of a reconnection-favorable magnetic topology, which can be regarded as a macroscopic behavior. Whereas in AR 12192, cause of its flare depends more on the triggering of the reconnection rather than the formation of a favorable topology, since the basic topology is simply a configuration of sheared arcades and it does not change for days. In other words, the cause of the flare relies more on the microscopic behavior of the plasma and thus is more subtle. The MHD model seems to be able to characterize well the macroscopic structures and evolutions, while it may not appropriately simulate the microscopic aspects of the plasma, and thus, the triggering of the reconnection. Such difference might be common and even fundamental between eruptive flares and confined flares, in particular, those events without noticeable changes of the coronal structures.

Finally, the reconnection-related quantities derived from the model should be taken with cautions because of the simplification of reconnection. A deeper understanding of the flare dynamics requires knowledge of the true nature of magnetic reconnection and is beyond the scope of this paper. In addition, the quantitative results suffered uncertainties from several aspects including HMI data and model settings. For example, the evolution speed of the photospheric magnetic field is increased in the model for saving computing time, and this might affect the coronal evolution, most likely at the time around the flare.

In summary, the present data-driven simulation study can provide important insight into understanding why and how solar flares occur, particularly for those events in which the dynamic change is elusive in observations. Further advancements, including more realistic plasma models, reconnection realization and thermodynamics, as well as extension to spherical geometry, are necessary for even more sophisticated modeling of real solar flares. With these improvements, the DARE model will hopefully become a useful tool to the communities of solar physics and space weather.

This work is supported by NSF AGS-1153323, AGS-1062050, and in addition C.J. and X.F. are also supported by the 973 program under grant 2012CB825601, the Chinese Academy of Sciences (KZZD-EW-01-4), the National Natural Science Foundation of China (41204126, 41231068, 41274192, 41531073, 41374176, 41574170, and 41574171), and the Specialized Research Fund for State Key Laboratories and Youth Innovation Promotion Association of CAS (2015122). V.Y. acknowledges support from AFOSR FA9550-15-1-0322 and NSF AGS-1250818 grants and Korea Astronomy and Space Science Institute. HW acknowledges support from NSF AGS-1348513, AGS-1408703. Data from observations are courtesy of NASA SDO/AIA and the HMI science teams. We thank the anonymous referee for help of improving the manuscript. We thank International Space Science Institute (ISSI) for enabling interesting discussions.