An MHD Modeling of Successive X2.2 and X9.3 Solar Flares of 2017 September 6

Details of the NLFFF extrapolation method are described in our previous studies(Inoue et al. 2014a, Inoue et al. 2018a, Inoue et al. 2018b). Especially, the parameters used in this study are the same as those in our previous study(Inoue et al. 2018a). The viscosity and resistivity are given as $\nu_{nlfff}=1.0\times 10^{-3}$ and $\eta_{\rm nlfff}=\eta_{0}+\eta_{1}|\mbox{\boldmath$J$}\times\mbox{\boldmath$B$}||\mbox{\boldmath$v$}|^{2}/|\mbox{\boldmath$B$}|^{2}$, respectively, where $\eta_{0}=5.0\times 10^{-5}$ and $\eta_{1}=1.0\times 10^{-3}$ in the non-dimensional unit. The second term is introduced to enhance reconnection which helps to make highly twisted lines observed in a central area of active regions. Furthermore, since the resistivity is proportional to the Lorentz force, it accelerates the relaxation of the non-equilibrium magnetic field towards the force-free state. The potential field is given as the initial condition, which is extrapolated using the Green function method(Sakurai 1982). During the iteration, three components of the magnetic field are fixed at each boundary, while the velocity is fixed to zero and the von Neumann condition $\partial/\partial n$=0 is imposed on $\phi$. Note that the bottom boundary is fixed according to

where $\mbox{\boldmath$B$}_{\rm bc}$ is the transverse component determined by a linear combination of the observed magnetic field ($\mbox{\boldmath$B$}_{\rm obs}$), and the potential magnetic field($\mbox{\boldmath$B$}_{\rm pot}$). $\zeta$ is a coefficient in the range of 0 to 1. When $R=\int|\mbox{\boldmath$J$}\times\mbox{\boldmath$B$}|^{2}$dV, which is calculated over the interior of the computational domain, falls below a critical value denoted by $R_{\rm min}$, during the iteration, the value of the parameter $\zeta$ is increased to $\zeta=\zeta+{\rm d}\zeta$. In this paper, $R_{\rm min}$ and d$\zeta$ have the values $1.0\times 10^{-2}$ and 0.02, respectively. If $\zeta$ is equal to 1, $\mbox{\boldmath$B$}_{\rm bc}$ is completely consistent with the observed data. Furthermore, the velocity is controlled as follows. If the value of $v^{*}(=|\mbox{\boldmath$v$}|/|\mbox{\boldmath$v_{A}$}|)$ is larger than $v_{\rm max}$(here set to 0.02), then the velocity is modified as follows: $\mbox{\boldmath$v$}\Rightarrow(v_{\rm max}/v^{*})\mbox{\boldmath$v$}$. These processes would help avoid a sudden jump from the boundary into the domain during the iterations.

The NLFFFs are reconstructed from the photospheric magnetic field obtained at 02:36 UT and 08:36 UT on September 6. Figure 1(b) shows the 3D magnetic field of the NLFFF at 08:36 UT. The strongly twisted lines have strong current density while the overlying field lines take a state close to the potential field. The comparison with extreme ultraviolet image and how much the magnetic field satisfies the force-free state are described in Appendix in Bamba et al. (2020).

The zero-beta MHD simulations are carried out using the NLFFFs as the initial conditions which are reconstructed at 08:36UT and 02:36UT on September 6, respectively, hereafter denoted Run A and B. The resistivity and viscosity are set to a uniform $\eta_{\rm mhd}=1.0\times 10^{-5}$ and $\nu_{\rm mhd}=1.0\times 10^{-4}$. The density is initially given as $\rho(\mbox{\boldmath$x$},t=0)=|\mbox{\boldmath$B$}(\mbox{\boldmath$x$},t=0)|$, even during the MHD simulation, we imposed $\rho(\mbox{\boldmath$x$},t)=|\mbox{\boldmath$B$}(\mbox{\boldmath$x$},t)|$; however, the dynamics in the lower corona as focused in this study are not so different from those obtained using a mass conservation law equation(Amari et al. 1999, Inoue et al. 2014b). Note that there is no physical significance of the density obtained in this simulation. The density is fixed at boundaries during the calculation. The normal component $B_{n}$ is fixed at all boundaries but the transverse component, $\mbox{\boldmath$B$}_{\rm t}$, is derived from an induction equation under the velocity equals to zero and $\eta$ is also fixed to zero at each boundary. For instance, at the bottom surface, the time variations of $B_{x}$ and $B_{y}$ are derived from

where $E_{x}=(\eta_{\rm mhd}\mbox{\boldmath$J$}-\mbox{\boldmath$v$}\times\mbox{\boldmath$B$})_{x}$ and $E_{y}=(\eta_{\rm mhd}\mbox{\boldmath$J$}-\mbox{\boldmath$v$}\times\mbox{\boldmath$B$})_{y}$ are x and y components of the electric field. Boundary conditions of the other variables are identical to those in the NLFFF calculation.

To find out the start of the X9.3 flare, the initiation of the eruption of the large MFR should be discussed due to the association of the flare with the eruption of the MFR. We trace the temporal evolution of the field line in black shown in Fig.7(a), which is one of the components of the erupting MFR, to understand the transition from pre- to post-eruption of the MFR. Figure 7(b) shows the temporal evolution of the height of the erupting MFR in red, and the magnetic flux where the magnetic twist($T_{w}$) of the field lines satisfies $T_{w}\leq-1.0$ in blue, respectively. The magnetic twist(Berger & Prior 2006) is defined as

where $dl$ is a line element of a field line. Note that, in this study, the twist is calculated under the condition $|B_{z}|>2.5\times 10^{-3}T$ on the surface. Since $T_{w}$ measures the number of turns of two infinitesimally close field lines, the twist is strictly different from the winding number of the field line around the magnetic axis of the MFR(Liu et al. 2016, Threlfall et al. 2018). In figures 7(b) and (c) $t\sim 0$ corresponds to the onset time of the X2.2 flare. The height profile shows a slow-rising phase, $t\leq 2$, and a fast raising phase, $t\geq 2$, both of which are typical profiles of the erupting MFR. From Figures 7(b) and (c) we can thus deduce that the X9.3 flare starts around $t\sim 2$ when the MFR shifts to the fast-rising mode. During the slow-rising phase, the size of the MFR, which is occupied with $T_{w}\leq-1.0$, increases, i.e., reconnection constantly takes place after the X2.2 flare. Therefore, a pre-eruption MFR is produced through the X2.2 flare and by the continuous reconnection in the slow-rising phase, until the MFR is accelerated.

We also attempt to answer the fundamental question of what mechanism drives the rapid acceleration of the MFR. Figure 7(c) plots the temporal evolution of velocity of the erupting MFR in blue, in addition to the height profile in red, which is the same as the one shown in Fig. 7(b). The velocity is plotted on a log scale, in which it grows linearly during $t=1.5\sim 2.5$. Since this period covers the bifurcation from the slow- to fast-rising phase, an instability drives the eruption. Note that after $t\approx 4$, the velocity gradually decreases since the side boundary suppresses the dynamics of the MFR. Figures 8(a)-(c) present the snapshots of the 3D field lines from the end of the slow-rising phase to the start of the fast rising phase (i.e., initiation of the MFR), respectively, with the decay index plotted on the vertical cross-section. At $t=1.4$, a part of the flux rope enters the region where the instability would be triggered in idealized conditions($n\geq$1.5). At $t=$2.54, further twisted lines, which are enhanced by the reconnection, enter the region of instability. Therefore, this result suggests that the X9.3 flare is driven by the torus instability, which is consistent with results analyzed by the NLFFF extrapolations performed by Zou et al. (2020).

We compare the evolution of the magnetic field at the time of the initiation of the X9.3 flare with SDO/AIA observations. The bottom in Fig. 8(d) is set at the AIA 1600Å mage taken at the very early stage of the X9.3 flare. These results suggest that the brightening in AIA 1600Å is enhanced at the time corresponding to when the erupting MFR is formed through the reconnection and gradually arises because the field lines whose footpoints anchored to the brightening areas are changing to the long field lines through the reconnection. Afterward, as seen in Fig.3, the FUV brightening is strongly enhanced and extended southward. In this simulation, the erupting MFR, named ”MFR1”, in Fig.8(d), reconnects with the twisted field lines named ”TFL” during the eruption. Consequently, the large MFR is formed as shown in Fig.6(a), and one footpoint of the newly created MFR is anchored to the brightening region extending southward, as seen in AIA 1600Å. Therefore, this simulation offers a sound explanation of the behavior of the 1600Å images observed in the X9.3 flare.

In the previous sections, we have demonstrated that the reconnection occurring at a local area, where the negative polarity intrudes into the opposite one (see Fig.1(c)), is important to produce the X2.2 flare. Further continuous reconnection drives the evolution of the MFR and eventually causes the X9.3 flare. We now discuss another scenario to produce the successive flares. Since AR12673 shows a very complex magnetic field distribution, magnetic null points are likely to exist at several points. For example, Fig. 9(a) and (b) (top views in Fig.9(c) and (d)) show the location of the null points in purple field lines and yellow field lines. The location of the null point A corresponds to the one marked by the dashed circle in Fig. 4(c). Our simulation suggested that the X flares are produced here and this was also supported by several previous studies (e.g., Price et al. 2019, Zou et al. 2020, Bamba et al. 2020). On the other hand, another null point B, which is further away form the null point A, is also reported in Mitra et al. (2018), Zou et al. (2019). This might have a potential to cause a flare and an eruption. In this section, we discuss a possibility of flare triggering via reconnection in a different area than where the intruding motion was observed, such as the null point B.

To test this hypothesis, we perform another MHD simulation using the NLFFF reconstructed from the photospheric magnetic field, recorded at 02:36 UT on September 6. This simulation is named ”Run B”, while the previous one is denoted ”run A”. In this study, the flare triggering reconnection in Run A is excited by the numerical resistivity in the strong current region, due to the strong intruding motion of the negative polarity. Figures 10(a) and (b) exhibit the distribution of $B_{z}$ observed at 02:36 UT and 08:36 UT, respectively, from which it is seen that the intrusion of the negative polarity has not fully taken place at 02:36 UT. Therefore, as of 02:36 UT, we expect that the reconnection should be suppressed at the region where the strong intruding motion was observed later. Figures 10(c) and (d) show the results of the magnetic twist, $T_{w}$, obtained from the two NLFFFs reconstructed at $t$=02:36 and 08:36, respectively and a quantitative comparison is shown in Fig.10(e). These distributions of $T_{w}$ are similar in both cases, but the value obtained at 08:36 UT is higher than the one at 02:36 UT. Fig.10(f) shows the field lines at $t$=0.28 obtained from the MHD simulation when the NLFFF reconstructed at 02:36 UT is used as the initial condition, in which we can find the two regions of steep gradient of the magnetic field marked by the dashed yellow and red circles because $|\mbox{\boldmath$J$}|/|\mbox{\boldmath$B$}|$ is strongly enhanced there. These locations are similar to those shown in Fig.9 while the magnetic configuration at 02:36 UT is different from the one at 08:36 UT. From our results, the X2.2 flare appeared at the area indicated by the yellow circle when we used the photospheric magnetic field obtained 6 hours later.

The field lines constituting the MFR and $|\mbox{\boldmath$J$}|/|\mbox{\boldmath$B$}|$ distribution on the vertical-cross section for Runs A and B are plotted in Figs.11(a) and (b), respectively. In both the cases, the MFR is formed and the patterns of $|\mbox{\boldmath$J$}|/|\mbox{\boldmath$B$}|$ are similar to each other. In Run B, however, the location of the current sheet formed under the MFR is slightly deviated from the region where the strong intruding motion is observed, while being enhanced above the region in Run A. Therefore, these results suggest that the reconnection region is different in between Run A and Run B. Specifically, the reconnection in Run B does not take place at the region intruded by the negative polarity. Figures 11(c) and (d) show snapshots of the 3D magnetic field lines obtained in Run B, from top and side views, respectively. The field lines colored according to $|\mbox{\boldmath$J$}|/|\mbox{\boldmath$B$}|$ are traced from the region around the current sheet, following which the reconnection occurs outside of the highly twisted lines. This location corresponds to the region indicated by red dashed circle in Fig.10(f) and the MFR can be created here while it takes longer time to make the MFR compared to Run A.

Interestingly, both Runs A and B exhibited eruptions albeit driven by different reconnections which take place at different locations: one is due to intrusion of the negative polarity in Run A and the other is due to the reconnection at a different location, which is away from the region of the intruding motion of the sunspot, in Run B. In this event, as shown in Figs. 4(e) and (f), the strong brightening was observed at the beginning of the X2.2 flare at the region of the intruding motion of the sunspot. Furthermore, from the Figs.4(a) and (b), the footpoints of the filed lines are anchored well on the strong brightening region of AIA 1600 Å, which are not the field lines connecting close to the null point of the magnetic field indicated by red dashed circle shown in Fig.10(f). If the eruption in Run B occurs, the flare brightening observed in the AIA 1600 Å observations would appear at the location outside of the highly twisted lines because they are surrounded by field lines participating in the reconnection.

From the above, we confirm the possibility of another eruption which is different from the observed one, moreover, it would be possible 6 hours before the X2.2 flare was observed. Yamasaki et al. (2021) pointed out that the highly twisted field lines, which produce the X-flares, are already formed as of 2 days before and suggested that the strong intrusion of the sunspot plays an important role for breaking the stable magnetic field through the reconnection. Since the value of electric resistivity is considered as very small in the solar corona, the reconnection does not happen so easily in that situation unless the MFR becomes unstable or the photospheric motion drives the upper magnetic field connecting the null point. In other words, if these would instead have taken place the area where the null point B exists, a different eruption before the X-flares might have taken place.