# A Data-constrained Magnetohydrodynamic Simulation of the X1.0 Solar Flare of
2021 October 28

Daiki Yamasaki  Astronomical Observatory, Kyoto University
Kitashirakawaoiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan  Center for Solar-
Terrestrial Research, New Jersey Institute of Technology,
University Heights, Newark, NJ 07102-1938, USA  Satoshi Inoue  Center for
Solar-Terrestrial Research, New Jersey Institute of Technology,
University Heights, Newark, NJ 07102-1938, USA  Yumi Bamba  Institute for
Advanced Research (IAR), Nagoya University,
Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan  Institute for Space-Earth
Environmental Research (ISEE), Nagoya University,
Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan  Jeongwoo Lee  Center for Solar-
Terrestrial Research, New Jersey Institute of Technology,
University Heights, Newark, NJ 07102-1938, USA  Haimin Wang  Center for Solar-
Terrestrial Research, New Jersey Institute of Technology,
University Heights, Newark, NJ 07102-1938, USA

###### Abstract

The solar active region NOAA 12887 produced a strong X1.0 flare on 2021
October 28, which exhibits X-shaped flare ribbons and a circle-shaped erupting
filament. To understand the eruption process with these characteristics, we
conducted a data constrained magnetohydrodynamics simulation using a nonlinear
force-free field of the active region about an hour before the flare as the
initial condition. Our simulation reproduces the filament eruption observed in
the H$\alpha$ images of GONG and the 304 $\mathrm{\AA}$ images of SDO/AIA and
suggests that two mechanisms can possibly contribute to the magnetic eruption.
One is the torus instability of the pre-existing magnetic flux rope (MFR), and
the other is upward pushing by magnetic loops newly formed below the MFR via
continuous magnetic reconnection between two sheared magnetic arcades. The
presence of this reconnection is evidenced by the SDO/AIA observations of the
1600 Å brightening in the footpoints of the sheared arcades at the flare
onset. To clarify which process is more essential for the eruption, we
performed an experimental simulation in which the reconnection between the
sheared field lines is suppressed. In this case too, the MFR could erupt, but
at a much reduced rising speed. We interpret this result as indicating that
the eruption is not only driven by the torus instability, but additionally
accelerated by newly formed and rising magnetic loops under continuous
reconnection.

Sun: flares - Sun: magnetic fields - Magnetohydrodynamics(MHD)

††journal: ApJ

##  1 Introduction

Solar flares are the rapid energy release phenomena in solar atmosphere
(Priest & Forbes, 2002). The energy source of solar flares is widely
considered as magnetic energy accumlated in solar active regions (ARs; Toriumi
& Wang, 2019). Some of the free magnetic energy are converted into kinetic
energy of erupting plasmas through magnetic reconnection (Coppi & Friedland,
1971; Spicer, 1982; Shibata & Magara, 2011). These erupting plasmas are often
observed as filament eruptions in the H$\alpha$ line (Parenti, 2014; Seki et
al., 2017, 2019). Solar filaments are cool and dense plasma compared to the
surrounding plasma, they are widely considered to be supported by helical
coronal magnetic field structure, magnetic flux ropes (MFRs; Xu et al., 2012;
Hanaoka & Sakurai, 2017; Gibson, 2018).

Chintzoglou et al. (2019) proposed that emerging dipole flux become strongly
sheared due to photospheric motion and result in the formation of a bundle of
helical magnetic field lines. Yan et al. (2016) presented an observation that
magnetic reconnection between pre-existing sheared magnetic arcades forms
MFRs. This process is consistent with the tether-cutting reconnection scenario
by Moore et al. (2001) and the flux-cancellation model by van Ballegooijen &
Martens (1989). In solar eruptions, several triggering processes are proposed,
for instance, the MHD instability such as the kink instability (Fan & Gibson,
2003; Török & Kliem, 2004) and double-arc instability (Ishiguro & Kusano,
2017; Kusano et al., 2020), or magnetic reconnection (Moore et al., 2001;
Antiochos et al., 1999). To accelerate the solar eruption, the torus
instability (Kliem & Török, 2006) plays an important role.

In order to understand the initiation of MFR eruptions, three-dimensional (3D)
coronal magnetic field provides crucial information because the free magnetic
energy is released in the solar corona and topology of the coronal magnetic
field changes associated with a flare. Observational limitation cannot allow
us for direct observation of coronal magnetic field, and we need to
extrapolate them by using a numerical technique such as potential field and
nonlinear force-free field (NLFFF) modeling (Wiegelmann & Sakurai, 2012;
Inoue, 2016). There are several studies using a time series of NLFFF to reveal
the formation process of MFRs (Su et al., 2009; Savcheva & van Ballegooijen,
2009; Inoue et al., 2013; Kang et al., 2016; Kawabata et al., 2017; Muhamad et
al., 2018; Yamasaki et al., 2021). However, NLFFF is assumed as an equilibirum
state only considering Lorentz force balance, and we cannot discuss dynamical
evolutions of 3D magnetic field during eruptive phase of flares. Since a data-
constrained magnetohydrodynamic (MHD) simulation show time evolving MHD
processes of the coronal magnetic field that are free from a force-free
condition, we can extend our understanding to the dynamics of an erupting
precess. Some of the previous studies on the data-constrained MHD simulations
successfully produced the eruptions that are driven by a reconnection or
instability and these results are in good agreement with observations (Jiang
et al., 2013; Inoue et al., 2015, 2018b; He et al., 2020; Inoue & Bamba,
2021). Therefore, the data-based MHD simulation is helpful to understand
dynamics of the coronal magnetic field in a realistic magnetic environment.

In this study, in order to understand the initiation and dynamics of the MFR
eruption associated with X1.0 flare on October 28 2021, we performed a data-
constrained MHD simulation using a NLFFF as the initial condition. The NLFFF
is extrapolated with observed photospheric vector magnetic field at 14:00 UT
October 28, which is 1.5 hours before the onset of the flare. The rest of this
paper is structured as follows: the observations and methods of analysis are
described in Section 3, results are presented in Section 4, discussions on the
eruption mechanism of the MFR are shown in Section 5, and our conclusions are
summarised in Section 6.

Figure 1: The 2021 October 28 flare in GOES soft X-ray lightcurves and HMI
vector magnetograms. (a) Soft X-ray fluxes from the $GOES$ 13 satellite in the
$1-8\mathrm{\AA}$ (red) and $0.5-4.0\mathrm{\AA}$ (blue) passband. The
vertical black line shows the time of the initial condition set for the
present simulation. (b) Radial component of the photospheric magnetic field in
the AR 12887 at 14:00 UT on October 28. Color of the field lines represent the
electric current density. (c) Temporal evolution of the photospheric vector
magnetic field. Gray-scaled background and red arrows show the radial and
horizontal component of the magnetic field, respectively. (An animation of the
HMI photospheric magnetic field images is available. The duration of the
animation is 4 seconds and it provides information of the temporal evolution
of the photospheric vector magnetic field including an intruding motion of a
negative patch into a positive patch from 00:00 UT to 23:48 UT 2021 October
28.)

##  2 Observation

The GOES class X1.0 flare occurred on 2021 October 28 in NOAA active region
(AR) 12887. According to the $GOES$ X-ray lightcurves in Figure 1 (a), the
onset time was 15:17 UT and the soft X-ray flux reached its maximum at 15:35
UT. The black solid line at 14:00 UT indicates the time that we selected for
calculating the bottom boundary to be used for the MHD simulation. Figure 1
(b) shows the radial magnetogram and the coronal magnetic field lines
extrapolated with NLFFF around the AR 12887 at 14:00 UT. The photospheric
vector magnetic field data are taken by the Helioseismic and Magnetic Imager
(HMI; Scherrer et al., 2012) onboard the $Solar~{}Dynamics~{}Observatory$
($SDO$; Pesnell et al., 2012). The vector magnetograms which we use have been
released as the Spaceweather HMI Active Region Patch (SHARP; Bobra et al.,
2014) data series (hmi.sharp_cea_720s series). Details of the vector magnetic
field data reduction and other related information about HMI data products can
be found in Hoeksema et al. (2014) and Bobra et al. (2014). In Figure 1 (c),
we show the temporal evolution of the photospheric vector magnetic field in
the region of interest, which is the same as the region indicated by the red
box in Figure 1 (b). Background grayscale and the red arrows show the radial
and the horizontal component of the magnetic field, respectively. According to
Figure 1 (c), we can find an intruding motion of the negative polarity towards
the positive polarity on October 28 in the region indicated by the green box
in the panels.

Figure 2 (a-c) show the temporal evolution of the H$\alpha$ (6562.8
$\mathrm{\AA}$) images observed with the Global Oscillation Network Group
(GONG; Harvey et al., 1996) around the AR 12887 during the X1.0 flare. Details
of the instrument can be found in Harvey & GONG Instrument Team (1995). Figure
2 (d-i) show the temporal evolution of the 304 $\mathrm{\AA}$ images observed
by the Atmospheric Imaging Assembly (AIA; Lemen et al., 2012) on board the
$SDO$ during the X1.0 flare. Green and blue lines in panels (d-f) indicate the
contours of 250 and -250 G of radial component of the photospheric magnetic
fields observed by HMI, respectively. We can find the J-shaped dark filament
in the early phase of the flare (Figure 2 (a,d,g)) and the erupting filament
and the X-shaped flare ribbons in the main flare phase (Figure 2 (b,e,h)). The
H$\alpha$ flare ribbons last until the later phase of the flare (Figure 2
(c)). The post flare arcades can be seen in the 304 $\mathrm{\AA}$ images
(Figure 2 (f,i)).

Figure 2: Temporal evolution of AR 12887 during the 2021 October 28 flare.
(a-c) H$\alpha$ (6562.8 $\mathrm{\AA}$) observation with GONG shows the flare
ribbons and a dark filament that later erupts. (d-i) SDO/AIA 304
$\mathrm{\AA}$ images show the flare loops, the erupting filament, and post-
flare arcades. Green and blue lines in panels (d-f) indicate the contours of
250 and -250 G of radial component of the photospheric magnetic fields,
respectively. (An animation of the AIA 304 images is available. The duration
of the animation is 5 seconds and it provides information on the temporal
evolution of dark filament eruption, flare enhacements, and post flare arcades
from 15:00 UT to 16:00 UT 2021 October 28.)

##  3 Methods

We construct a nonlinear force-free field and then use it as the initial
condition for the data-constrained MHD simulation. Once the 3D magnetic field
is obtained, we calculate the magnetic twist number and the decay index to
study the instability leading to eruption.

###  3.1 Nonlinear Force-Free Field Extrapolation

In order to perform the NLFFF extrapolation, we first calculate the potential
field (Sakurai, 1982) which is used as the initial condition of the MHD
relaxation (Inoue et al., 2014; Inoue, 2016). We solve zero-beta MHD equations
to obtain the force-free field because the gas pressure and gravity are
neglected approximately compare to the magnetic pressure in the solar corona
(Gary, 2001). We solve the following MHD equations,

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle|\bm{B}|,$ (1)
$\displaystyle\frac{\partial\bm{v}}{\partial t}$ $\displaystyle=$
$\displaystyle-(\bm{v}\cdot{\bm{\nabla}})\bm{v}+\frac{1}{\rho}\bm{J}\times\bm{B}+\nu_{i}\bm{\nabla}^{2}\bm{v},$
(2) $\displaystyle\frac{\partial\bm{B}}{\partial t}$ $\displaystyle=$
$\displaystyle\bm{\nabla}\times(\bm{v}\times\bm{B}-\eta_{i}\bm{J})-\bm{\nabla}\phi,$
(3) $\displaystyle\bm{J}$ $\displaystyle=$
$\displaystyle\bm{\nabla}\times\bm{B},$ (4)
$\displaystyle\frac{\partial\phi}{\partial t}$ $\displaystyle+$ $\displaystyle
c_{\mathrm{h}}^{2}\bm{\nabla}\cdot\bm{B}=-\frac{c_{\mathrm{h}}^{2}}{c_{\mathrm{p}}^{2}}\phi,$
(5)

where the subscript $i$ of $\nu$ and $\eta$ corresponds to different values
used in NLFFF and MHD. $\rho$, $\bm{B}$, $\bm{v}$, $\bm{J}$, and $\phi$ are
plasma pesudo-density, magnetic flux density, velocity, electric current
density, and a conventional potential to reduce errors derived from
$\bm{\nabla}\cdot\bm{B}$ (Dedner et al., 2002). The pseudo-density in equation
(1) is assumed to be proportional to $|\bm{B}|$. In these equations, the
length, magnetic field, density, velocity, time, and electric current density
are normalized by $L^{*}=254.8$ $\mathrm{Mm}$, $B^{*}=3000$ $\mathrm{G}$,
$\rho^{*}=|B^{*}|$, $V_{\mathrm{A}}^{*}\equiv B^{*}/(\mu_{0}\rho^{*})^{1/2}$,
where $\mu_{0}$ is the magnetic permeability, $\tau_{\mathrm{A}}^{*}\equiv
L^{*}/V_{\mathrm{A}}^{*}$, and $J^{*}=B^{*}/\mu_{0}L^{*}$, respectively. In
equation (2), $\nu_{\mathrm{NLFFF}}$ is a viscosity fixed at $1.0\times
10^{3}$. The coefficients $c_{\mathrm{h}}^{2},c_{\mathrm{p}}^{2}$ in equation
(5) are fixed to the constant values $0.04$ and $0.1$. The resistivity in
equation (3) is given as
$\eta_{\mathrm{NLFFF}}=\eta_{0}+\eta_{1}|\bm{J}\times\bm{B}||\bm{v}|^{2}/|\bm{B}|^{2}$,
where $\eta_{0}=5.0\times 10^{-5}$ and $\eta_{1}=1.0\times 10^{-3}$ in non-
dimensional units. As for the boundary conditions, 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 $\phi$
during the iteration. Here we note that we fixed the bottom boundary according
to

$\displaystyle\bm{B_{\mathrm{bc}}}=\gamma\bm{B_{\mathrm{obs}}}+(1-\gamma)\bm{B_{\mathrm{pot}}},$
(6)

where $\bm{B_{\mathrm{bc}}}$ is the transversal component determined by a
linear combination of the observed magnetic field ($\bm{B}_{\mathrm{obs}}$)
and the potential magnetic field ($\bm{B_{\mathrm{pot}}}$). $\gamma$ is a
coefficient in the range of 0 to 1. The value of the parameter $\gamma$ is
increased to $\gamma=\gamma+d\gamma$ if $R=\int|\bm{J}\times\bm{B}|^{2}dV$,
which is integrated over the computational domain, becomes smaller than a
critical value which is denoted by $R_{\mathrm{min}}$ during the iteration. In
this paper, we set $R_{\mathrm{min}}$ and $d\gamma$ the values of $5.0\times
10^{-3}$ and $0.02$, respectively. When $\gamma$ reaches to 1,
$\bm{B_{\mathrm{bc}}}$ is completely consistent with the observed data.
Furthermore, we controll the velocity as follows. If the value of
$v^{*}(=|\bm{v}|/|\bm{v_{A}}|)$ is larger than $v_{\mathrm{max}}$ (here we set
to $0.04$), then we modify the velocity from $\bm{v}$ to
$(v_{\mathrm{max}}/v^{*})\bm{v}$. We adopted these processes because they
would help avoid a sudden jump from the boundary into the domain during the
iterations.

###  3.2 Data-constrained MHD Simulation

Next we performed the MHD simulation using NLFFF as an initial condition. We
name this simulation RUN A. The equations we solve are identical to those in
the NLFFF extrapolation. However, the handling of the bottom boundary
condition is different between the NLFFF extrapolation and the data-
constrained simulation. In the data-constrained simulation, the bottom
$B_{\mathrm{x}}$ and $B_{\mathrm{y}}$ follow an induction equation while the
normal component is fixed with time. Although the boundary magnetic field
evolve with time in a physically consistent manner, these evolutions are
inconsistent with the observations (Inoue & Bamba, 2021). We set resistivity
and viscosity as $\eta_{\mathrm{MHD}}=1.0\times 10^{-4}$ and
$\nu_{\mathrm{MHD}}=1.0\times 10^{-3}$, respectively, which are different from
those in the NLFFF. The coefficients $c_{\mathrm{h}}^{2}$ and
$c_{\mathrm{p}}^{2}$ in equation (5) are fixed to the constant values $0.04$
and $0.1$, respectively.

For both calculations, the numerical domain has dimensions of $255\times
195\times 191$ $\mathrm{Mm}^{3}$, or $1.00\times 0.77\times 0.75$ in non-
dimensional units. The region is devided into $352\times 270\times 264$ grid
points.

###  3.3 Analysis of Magnetic Fields

Once the 3D magnetic field is obtained, we calculate the magnetic twist number
and the decay index to study the instability leading to eruption. The magnetic
twist of each field line ($e.g.$ Inoue et al., 2011; Liu et al., 2016, etc.)
is calculated using the following definition (Berger & Prior, 2006),

$\displaystyle
T_{\mathrm{w}}=\int_{L}\frac{\bm{\nabla}\times\bm{B}\cdot\bm{B}}{4\pi
B^{2}}~{}dl,$ (7)

where $dl$ is a line element. The decay index, $n$, the proxy criterion for
the torus instability (Kliem & Török, 2006) is calculated as

$\displaystyle
n=-\frac{z}{|\bm{B_{\mathrm{ex}}}|}\frac{\partial|\bm{B_{\mathrm{ex}}}|}{\partial
z}$ (8)

Here $\bm{B_{\mathrm{ex}}}$ denotes the horizontal component of the external
field, which is assumed to be the potential field in this study.

##  4 Results

###  4.1 Evolution of the MFR

In Figure 3, we display three snapshots of the MHD simulation to show temporal
evolution of the coronal magnetic field. Figure 3 (a) and (d) show the initial
condition of the simulation viewed in two different perspective angles. Figure
3 (b,e) and (c,f) show the magnetic field structure at $4.26$ and $t=10.2$ at
the same angle as above, respectively. The colored field lines are selected in
the criteria of strong twist built up during the evolution. For this purpose,
we first calculated the magnetic twist of every field line starting from each
pixel using equation (7), which yields a map of magnetic twist on the bottom
boundary. We then select the regions with $T_{\mathrm{w}}>1.0$ at the final
time step ($t=10.2$) of the simulation from which we perform the field line
tracing. P1, P2, and P3 correspond to the footpoints of the pink, green, and
yellow field lines in the regions of positive magnetic polarity, and N1, N2,
N3 are their counterparts in the negative polarity. In our NLFFF, only the
elbow part of the J-shaped dark filament located at the west side of the AR
(Figure 2 (a)) could be well reconstructed by the yellow field lines in Figure
3. We think that this is because the straight part of the J-shaped filament
formed at the east side of the AR was located above the relatively weak field
region and failed to build a highly twisted structure. We define a bundle of
field lines with $T_{\mathrm{w}}>1.0$ as an MFR and that with
$0<T_{\mathrm{w}}<1.0$ as a sheared magnetic arcade. In this definition, the
yellow lines are identified with an MFR, and pink and green lines, a pair of
sheared magnetic arcades at the initial time. In Figure 3 (c,f), we can see
the MFR (yellow lines) undergo eruption and expansion. We suggest that the
continuous magnetic reconnection taking place between the pink and green field
lines led to the formation of magnetic loops below the MFR, see the online
animation.

Figure 3: Three snapshots of the simulation of RUN A, showing temporal
evolution of the coronal magnetic fields. (a) and (d) show top and side views
of the 3D magnetic fields at $t=0$. (b) and (e) show those at $t=4.26$. (c)
and (f) show those at $t=10.2$. The pink, green, and yellow lines are the
field lines with footpoints at P1-N1, P2-N2, and P3-N3, respectively. The
yellow lines are identified as an MFR that erupts. (An animation of the RUN A
simulation is available. The duration of the animation is 1 second and it
provides information on the temporal evolution of the 3D magnetic field in our
RUN A simulation including the MFR eruption and magnetic reconnection between
pre-existing sheared magnetic arcades.)

###  4.2 Decay index distribution

In Figure 4 we plot the height distribution of the decay index calculated
using equation (8) on a vertical $x-z$ plane at $y=0.34$. Figure 4 (a) shows
that the MFR (yellow lines) was already located in the region of high decay
index $n>2.0$ at the start. In panels (b) and (c), we plot, in azure color,
the potential magnetic field lines surrounding the MFR in 3D and 2D,
respectively. A magnetic null like region ($B\approx 0$) is found on the plane
at a low latitude $z=0.09$ (Figure 4 (c,d)). Since magnetic field rapidly
decreases toward the null, a large value of the decay index was realized in
the low latitude (Figure 4 (d)). We, therefore, expect that the MFR (yellow
lines) could easily become unstable to the torus instability. In addition, as
shown in panel (d), we find a torus stable region with $n<1.3$ above the
initial location of the MFR ($0.09<z<0.46$). Kliem et al. (2021) pointed out
that in the case of such cubic functional decay index distribution, an
eruption starts but fails because of torus stable region. However, some
numerical studies reported that the MFRs could erupt if they were located at
the torus unstable region in the initial condition (Inoue et al., 2018a; Zhong
et al., 2021; Joshi et al., 2021). In such a case, even if the decay index
above the MFRs is low enough to be torus stable, the MFRs could erupt. Our
result supports the latter scenario.

Figure 4: Decay index distribution in 3D. (a) The index is plotted in vertical
cross section of $x$–$z$ plane at $y=0.34$. The initial magnetic field lines
($t=0$) in the simulation are plotted with the same color convention as used
in Figure 3 (b). (b) Azure lines are the potential field lines surrounding the
MFR. (c) The field lines of the potential field are projected on the $x$–$z$
plane at $y=0.34$. The green line is the vertical line of $(x,y)=(0.50,0.34)$
selected for plotting 1D distribution of the decay index and the magnetic
field strength in (d).

###  4.3 Comparison of the flare ribbon structures of observation and
simulation

Figure 5 shows the flare ribbons observed in H$\alpha$ and EUV (a-c) and
calculated from the MHD model (d) in the main phase of the flare. The
H$\alpha$ and EUV images show bright ribbons not only at two footpoints of the
MFR (P3 and N3), but also at other four footpoints of sheared magnetic arcades
formed below the MFR (P1, P2, N1, and N2) (Figure 5 (a-c)). The model flare
ribbon (red features in Figure 5 (d)) is calculated simply based on the
distance between two footpoints of a field line per pixel. Since the only way
that this length changes significantly is via reconnection, we regard those
regions as ribbons. Specifically those pixels where field lines with the
distance changing by more than $3.6$ $\mathrm{Mm}$ are marked red. These
computed flare ribbons are found not only at the footpoints of the MFR, but
also at the four footpoints of sheared magnetic arcades consistent with the
observation. This result supports the scenario of the reconnection between two
sheared magnetic arcades below the pre-existing MFR during the X1.0 flare.

Although the computed flare ribbons in the northern part of the AR fairly well
agree with the observed ribbons in both location and intensity, those in the
southern part of the AR do not. The observed ribbon in the location of
$-$540${}^{\prime\prime}<y<-$530′′ is barely predicted by the model, while the
observed ribbon in $-$560${}^{\prime\prime}<y<-$550′′ is very weak compared
with the model prediction. We consider two possibilities for the partial
success of our ribbon prediction. One is that our model prediction for flare
ribbons is designed mainly for location but not intensity, and our criterion
for ribbons based on the field line length change does not work well for the
southern ribbons. The other possibility is that the topology of the magnetic
field in the southern area is not correctly reproduced under the NLFFF
extrapolation to limit the accuracy of our ribbon prediction in that area.

Figure 5: Observed and computed flare ribbons. Green lines in all the panels
show the polarity inversion lines. (a) Flare ribbons in the GONG H$\alpha$
line image at 15:37:10 UT. (b) The radial component of the photospheric
magnetic field overlaid with the enhanced H$\alpha$ intensity as red contours
at 15:37:10 UT. (c) Flare ribbons in the AIA 1600 Å channel at 15:37:02 UT.
(d) The computed flare ribbons from the MHD simulation of RUN A (red) plotted
over the magnetogram (grayscale).

##  5 Discussion

The results of simulation RUN A indicate two possibilities for the driving
mechanism of the MFR eruption. One is the torus instability of the MFR (Figure
4). According to the result of the statistical study of an eruption mechanism
done by Jing et al. (2018), the decay index value of $n>2.0$ is large enough
to produce eruptions via torus instability. The other possibility is that the
MFR is pushed upward by the magnetic loops formed by the magnetic reconnection
between two sheared magnetic arcades below the pre-existing MFR (Figure 3).
This process can also drive the MFR eruption. The newly formed loops are
created through the reconnection, therefore it can be interpreted that the
reconnection drives the MFR eruption.

In order to find out which process is essential for the eruption of the MFR,
we conduct another MHD simulation named RUN B. To clarify what role is played
by the reconnection under the erupting filament, we halt the motion in the
specific region where strong current density is formed; it lies between the
sheared field lines in pink and green (see orange box in Figure 6 (b)). This
process is expected to partially suppress the magnetic reconnection between
two sheared magnetic arcades. We thus prevent in this experiment, the large
loops from forming that is suspected to push the erupting MFR.

In Figure 6, we compare the field lines found in simulation RUN A with those
in RUN B. The same color code is used for both. The results of RUN B show how
the field line structure differs when the reconnection between pink and green
lines is suppressed. However, the MFR with yellow lines moved upwards in RUN B
as well as in RUN A (Figure 6 (a,b)). Therefore, the MFR can ascend without
being pushed up by the large arcade.

Figure 7 compares simulation RUN B with RUN A in terms of the vertical
velocity ($V_{\mathrm{z}}$) and normalized current density,
$|\bm{J}|/|\bm{B}|$. The distribution of $V_{\mathrm{z}}$ in an $x$-$z$ plane
is plotted at $y=0.34$ at a time in the early phase, $t=0.85$, for both
simulations. RUN A shows that the enhanced upward $V_{\mathrm{z}}$ extends
down to a lower height than in RUN B (Figure 7 (a,b)). In RUN A, the MFR and
the newly formed magnetic loops exist in the region of enhanced velocity,
whereas, in RUN B, the velocity is enhanced only in the location of the MFR.
The height-variation of $V_{z}$ plotted in Figure 7 (c) clearly shows that the
velocity is much more enhanced when the reconnection between the sheared field
lines is allowed.

In a region of strongly enhanced $|\bm{J}|/|\bm{B}|$, the magnetic field
topology may change rapidly and thus it could be a boundary between two
topologically different regions. For instance, the region indicated by the
white arrow in Figure 7 (d,e) can indicate a boundary between the MFR and the
overlying field lines, thus roughly the edge of the MFR. In Figure 7 (f), we
show the one-dimensional plot of $|\bm{J}|/|\bm{B}|$. The edge of the MFR in
RUN A is higher than that in RUN B, while $V_{\mathrm{z}}$ of RUN A is still
larger than that of RUN B. These results suggest that the MFR could erupt even
without being pushed by the magnetic loops newly formed below the MFR.
However, the push-up from below can be important to help the acceleration of
the MFR. And according to the results of the observed and the computed flare
ribbons, they were observed not only at the two footpoints of the erupting MFR
but also at the four footpoints of the reconnecting pre-existing magnetic
arcades, the scenario of RUN A is more consistent with the observation rather
than RUN B.

As shown in Figure 6, observation supports the scenario with magnetic
reconnection between pink and green field lines. By comparing the photospheric
magnetic field structure before and after the X1.0 flare, intruding motion of
the negative polarity (N1 in Figure 5) into the positive polarity (P2 in
Figure 5) was observed at the location where we found the magnetic
reconnection in RUN A (see the green box of Figure 1 (c)). Inoue & Bamba
(2021) pointed out that the intruding motion can be a trigger of magnetic
reconnection concerning to the MFR eruption. We suggest that the intruding
motion we found in this case of X1.0 flare could be a trigger of the magnetic
reconnection.

Figure 6: Comparison of magnetic fields in RUN A and RUN B. The magnetic
fields at $t=4.26$ in RUN A (a) and those at $t=4.22$ in RUN B (b). The same
color convention as in Figure 3 is used to identify the three groups of field
lines. Orange box indicated by orange arrow in (b) shows the region where we
set the velocity to zero in RUN B. It lies in $0.45\leq x\leq 0.54$, $0.31\leq
y\leq 0.40$, and $0.02\leq z\leq 0.06$, satisfying $|{\bm{J}}|\geq 10$. Figure
7: Comparison of RUN A and RUN B in terms of the vertical velocity, $V_{z}$
(a–c) and current density normalized by magnetic field, $|\bm{J}|/|\bm{B}|$
(d–f). (a,b) show 2D distribution of $V_{z}$ in a vertical $x-z$ plane at
$y=0.34$, obtained from RUN A and RUN B at $t=0.85$, respectively. (c) shows
their 1D distributions at the location of $(x,y)=(0.50,0.34)$ indicated by the
white line marked in other panels. Likewise, (d, e) show the 2D distribution,
and (f), the 1D distribution of $|\bm{J}|/|\bm{B}|$ obtained from RUN A at
$t=5.11$ and RUN B at $t=5.06$, respectively. Regions of enhanced
$|\bm{J}|/|\bm{B}|$ are regarded as edges of the MFR.

##  6 Conclusion

We presented a data-constrained MHD simulation to understand the formation and
erupting process of an MFR associated with the X1.0 flare that occurred on
2021 October 28, which is characterized by an X-shaped flare ribbon and a
nearly circular J-shaped filament. Our simulation is meant to reproduce the
observed MFR eruption with the shape of the filament and the ribbons as
constraint. According to the initial condition obtained from a nonlinear
force-free field extrapolation, the MFR was initially in a region of
sufficient decay index. Therefore the MFR could erupt under the torus
instability alone. However, we found that the magnetic reconnection between
two sheared magnetic arcades took place under the pre-existing MFR during the
erupting phase of the MFR, and paid attention to the possible role of this
reconnection of the underlying loops in facilitating the eruption of the MFR.

By performing an experimental simulation in which this reconnection is
suppressed, we found that the MFR is still able to erupt but at a reduced
speed. The MFR erupts faster when the large magnetic loops form underneath.
The reconnection of the magnetic loops formed below the MFR is thus essential
for accelerating the MFR. In addition, this reconnection cannot be overlooked
because flare ribbons were observed not only at the footpoints of MFR but also
at those of the newly formed magnetic loops during the filament eruption. From
these results, we propose that the initial driving mechanism of the filament
eruption associated with this GOES X1.0 flare was facilitated by the combined
action of the torus instability and the formation of the magnetic loops
through the reconnection below the pre-existing MFR. A similar idea was
presented by Inoue et al. (2018b).

This event produced a strong CME which has been well studied in many papers
(Xu et al., 2022; Hou et al., 2022; Papaioannou et al., 2022; Li et al.,
2022). It is thus of new interest how this MFR grows into the CME, although
the present simulation domain is not large enough to fully trace the MFR in a
long range. It is also worthwhile to investigate how the pre-erupting MFR is
formed. We plan to address these issues in future with an extended time
coverage.

$SDO$ is a mission of NASA’s Living With a Star Program. This work utilizes
data from the National Solar Observatory Integrated Synoptic Program, which is
operated by the Association of Universities for Research in Astronomy, under a
cooperative agreement with the National Science Foundation and with additional
financial support from the National Oceanic and Atmospheric Administration,
the National Aeronautics and Space Administration, and the United States Air
Force. The GONG network of instruments is hosted by the Big Bear Solar
Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur
Solar Observatory, Instituto de Astrofísica de Canarias, and Cerro Tololo
Interamerican Observatory. This work was supported by MEXT/JSPS KAKENHI Grant
Number JP21J14036 and 21K20379. This work was also supported by JSPS Overseas
Challenge Program for Young Researchers. This study is partially supported by
National Science Foundation AGS-2145253, AGS-1954737 and AST-2204384. JL
acknowledges support by NASA grants, 80NSSC18K1705 and 80NSSC21K1671.
Visualization of magnetic field lines are produced by VAPOR
(www.vapor.ucar.edu), a product of the Computational Information Systems
Laboratory at the National Center for Atmospheric Research (Li et al., 2019).

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