# Thermodynamic and Magnetic Topology Evolution of the X1.0 Flare on 2021
October 28 Simulated by a Data-driven Radiative Magnetohydrodynamic Model

J. H. Guo School of Astronomy and Space Science and Key Laboratory of Modern
Astronomy and Astrophysics, Nanjing University, Nanjing 210023, China  Centre
for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven,
Celestijnenlaan 200B, B-3001 Leuven, Belgium  Y. W. Ni School of Astronomy and
Space Science and Key Laboratory of Modern Astronomy and Astrophysics, Nanjing
University, Nanjing 210023, China  Z. Zhong Center for Integrated Research on
Space Science, Astronomy, and Physics, Institute of Frontier and
Interdisciplinary Science, Shandong University, Qingdao 266237, China  Y. Guo
School of Astronomy and Space Science and Key Laboratory of Modern Astronomy
and Astrophysics, Nanjing University, Nanjing 210023, China  C. Xia School of
Physics and Astronomy, Yunnan University, Kunming 650050, China  H. T. Li
School of Astronomy and Space Science and Key Laboratory of Modern Astronomy
and Astrophysics, Nanjing University, Nanjing 210023, China  S. Poedts Centre
for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven,
Celestijnenlaan 200B, B-3001 Leuven, Belgium  Institute of Physics, University
of Maria Curie-Skłodowska, Pl. M. Curie-Skłodowska 5, 20-031 Lublin, Poland
B. Schmieder Centre for mathematical Plasma Astrophysics, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium  LESIA,
Observatoire de Paris, CNRS, UPMC, Université Paris Diderot, 5 place Jules
Janssen, 92190 Meudon, France  P. F. Chen School of Astronomy and Space
Science and Key Laboratory of Modern Astronomy and Astrophysics, Nanjing
University, Nanjing 210023, China

###### Abstract

Solar filament eruptions, flares and coronal mass ejections (CMEs) are
manifestations of drastic release of energy in the magnetic field, which are
related to many eruptive phenomena from the Earth magnetosphere to black hole
accretion disks. With the availability of high-resolution magnetograms on the
solar surface, observational data-based modelling is a promising way to
quantitatively study the underlying physical mechanisms behind observations.
By incorporating thermal conduction and radiation losses in the energy
equation, we develop a new data-driven radiative magnetohydrodynamic (MHD)
model, which has the capability to capture the thermodynamic evolution
compared to our previous zero-$\beta$ model. Our numerical results reproduce
major observational characteristics of the X1.0 flare on 2021 October 28 in
NOAA active region (AR) 12887, including the morphology of the eruption,
kinematic of flare ribbons, extreme-ultraviolet (EUV) radiations, and two
components of the EUV waves predicted by the magnetic stretching model, i.e.,
a fast-mode shock wave and a slower apparent wave due to successive stretching
of magnetic field lines. Moreover, some intriguing phenomena are revealed in
the simulation. We find that flare ribbons separate initially and ultimately
stop at the outer stationary quasi-separatrix layers (QSLs). Such outer QSLs
correspond to the border of the filament channel and determine the final
positions of flare ribbons, which can be used to predict the size and the
lifetime of a flare before it occurs. In addition, the side view of the
synthesized EUV and white-light images exhibit typical three-part structures
of CMEs, where the bright leading front is roughly cospatial with the non-wave
component of the EUV wave, reinforcing the magnetic stretching model for the
slow component of EUV waves.

Radiative magnetohydrodynamics (2009); Solar coronal mass ejections (310);
Solar magnetic fields (1503); Solar prominences (1519); Solar flares (1496)

##  1 Introduction

Solar eruptions, such as filament eruptions, solar flares and coronal mass
ejections (CMEs; Chen, 2011; Webb & Howard, 2012; Toriumi & Wang, 2019),
correspond to the sudden and explosive release of magnetic energy in the
corona via magnetic reconnection, which are usually associated with particle
acceleration, and various types of waves, constituting the prominent features
of many eruptive astrophysical processes. In general, the core field of the
pre-eruption system is either a twisted magnetic flux rope or a sheared arcade
(Ouyang et al., 2015). After being triggered, the eruptions usually excite
waves and wave-like perturbations, including coronal EUV waves and
chromospheric Moreton waves (Vršnak & Cliver, 2008; Warmuth, 2015; Chen,
2016). When a CME propagates in the heliosphere, it is termed interplanetary
coronal mass ejection (ICME) or magnetic cloud (MC; Burlaga et al., 1981;
Tsurutani et al., 2022). The magnetized ejecta following the enhanced flaring
electromagnetic emissions, along with the associated energetic particles, are
the main disturbing sources of the heliospheric environment. Therefore,
studies of solar eruptions not only deepen our understanding on basic
astrophysical processes, but also improve the capacity of predicting adverse
space weather events.

Although these eruptive phenomena appear to be significantly different, they
can be linked by the standard CSHKP flare model (Carmichael, 1964; Sturrock,
1966; Hirayama, 1974; Kopp & Pneuman, 1976). In this model, the core magnetic
field, either a flux rope or a sheared arcade, is triggered to ascend due to
an ideal MHD instability or magnetic reconnection (Chen, 2011), inducing the
formation of a current sheet below the core field. The magnetic reconnection
inside the current sheet produces downward outflows and flaring loops below
the reconnection site and upward flow with concave-upward magnetic field above
the reconnection site. The “slingshot” effect of the magnetic field in the
upward flow pushes the core field, which might host a filament, to move upward
and ultimately evolve into a CME (Jiang et al., 2021b). Furthermore, the
reconnection of magnetic field converts the magnetic energy to thermal and
kinetic energies in a bulk of materials, in addition to a large number of
energetic particles traveling along magnetic field lines. Energetic particles
and heat conduction propagating downward heat the plasma in the lower
atmosphere to form flare ribbons, and cause chromospheric evaporation to form
flare loops simultaneously (Hudson, 2011). As the flux rope keeps rising, more
field lines participate in the reconnection, and newly-formed larger loops
overlie on the previous ones, manifested as the continuous expansion of flare
loops and separation of two main ribbons (Petrasso et al., 1979). Albeit the
key processes are described well by the above physical paradigm, some issues
have not yet been well addressed. For example, where do the separating flare
ribbons stop (Chen et al., 2012)? Besides, a lot of observations showed that a
CME generally presents a three-part structure, that is, bright core, dark
cavity and bright leading front (Chen, 2011). However, it is still not clear
how the CME leading fronts are formed. In contrast to the general belief that
they are due to plasma pileup ahead of the eruption, Chen (2009) proposed that
the CME leading fronts correspond to the density enhancement due to successive
magnetic field line stretching. As for the CME cores, Song et al. (2022)
suggested that both the warm channel and prominence can evolve into the CME
core.

Apart from these, the classical 2-dimensional (2D) model encounters many
difficulties in quantitatively explaining some 3-dimensional (3D) features in
observations. First, in the 2.5D model, there is strict distinction between
the toroidal and poloidal magnetic fields, and the toroidal flux keeps
constant as in the pre-eruptive structure. However, in 3D cases, there is no
strict distinction between toroidal and poloidal fields, and there exists
mutual conversion, for example, observations revealed that reconnection can
contribute a substantial amount of toroidal flux to the flux rope (Qiu et al.,
2007; Wang et al., 2017; Xing et al., 2020a). Second, the footpoints of the
flux rope might drift significantly in real eruptions (Aulanier & Dudík, 2019;
Zemanová et al., 2019; Xing et al., 2020a), which is hard to be explained by
2D models. To address these issues, Aulanier & Dudík (2019) analyzed the
results of a 3D flare model based on a zero-$\beta$ MHD simulation, which
contains three types of reconnection geometries: the reconnection in the
overlying arcades (aa-rf reconnection), the reconnection between ambient
sheared arcades and the flux rope (ar-rf reconnection), and the reconnection
between two flux-rope legs (rr-rf reconnection). This 3D flare model sets the
stage for explaining complex observations (Zemanová et al., 2019; Xing et al.,
2020a), which is worthy of validation and further refinement by comparing
simulations and observations.

In recent years, numerical models based directly on observational data are
drawing ever more attention. In these models, the observational data in the
photosphere, such as the magnetic field (Jiang et al., 2016b), velocity
(Hayashi et al., 2019; Kaneko et al., 2021; Jiang et al., 2021a), the
combination of velocity and magnetic field (Guo et al., 2019b) and electric
field (Cheung & DeRosa, 2012; Hayashi et al., 2018; Pomoell et al., 2019;
Fisher et al., 2020), is taken as the input to drive the evolution of the
coronal magnetic field and plasma. This kind of data-driven or data-
constrained models can be directly compared with the multi-wavelength
observations, presenting great power in quantitatively explaining the
underlying physical mechanisms behind observations (Jiang et al., 2022). For
example, Jiang et al. (2018b) reproduced the corona loops, filaments, and
flare ribbons in the largest solar flare in solar cycle 24. Guo et al. (2019a)
demonstrated that the torus instability (Kliem & Török, 2006; Aulanier et al.,
2010) plays a pivot role in the triggering of a flux rope eruption. Zhong et
al. (2021) found that the Lorentz force component resulting from the non-
axisymmetry of the flux rope can constrain the eruption. However, it is
noticed that, to reduce computing expenses, many authors adopted the time-
dependent magnetofrictional model (Cheung & DeRosa, 2012; Cheung et al., 2015;
Pomoell et al., 2019; Kilpua et al., 2021; Lumme et al., 2022), zero-$\beta$
model (Kliem et al., 2013; Inoue et al., 2018; Guo et al., 2019a; Zhong et
al., 2021; Guo et al., 2021c), or isothermal MHD model (Jiang et al., 2013,
2016a, 2016c, 2018b, 2018a), in which the thermal properties of the plasma are
discarded or simplified. This means that the above models are incapable of
synthesizing the radiations from the density and temperature self-
consistently, and cannot be directly compared with EUV observations. Recently,
some data-based MHD models that take into account the non-adiabatic effects
have been performed to reproduce the CME (Fan, 2022). In short, to better
understand the nature of some emission structures (such as filaments, EUV
waves and coronal loops), we need to develop data-driven radiative MHD models.

Here, we develop and perform a data-driven radiative MHD simulation of the
X1.0 flare on 2021 October 28. The initial magnetic field is obtained with a
non-linear force-free field (NLFFF) reconstruction. Thermal conduction and
radiation losses are considered in the energy equation. We have two goals in
this paper: the first is to reproduce most of the observational features, and
the second is to study thermodynamics and the magnetic topology evolution
during the eruption, especially the flare ribbons and CME leading front. The
event overview and numerical methodology are described in Section 2. Numerical
results are displayed in Section 3, which are followed by discussions in
Section 4. Finally, we summarize our results and discoveries in Section 5.

##  2 Event overview and numerical setup

The event we select is the second GOES X-class flare in solar cycle 25, hosted
in NOAA AR 12887, occurring around 15:35 UT on 2021 October 28. This event has
been reported by other papers, which focused on some observational features,
such as the observations of EUV waves and the corresponding CME (Hou et al.,
2022; Devi et al., 2022), CME three-part structure (Devi et al., 2022),
eruption mechanism (Yamasaki et al., 2022), the Sun-as-a-star spectroscopic
characteristics (Xu et al., 2022), solar energetic particles (Li et al.,
2022), and its geomagnetic effects (Papaioannou et al., 2022). It would be of
great interest to check in what degree such a geoeffective solar eruption can
be reproduced with a data-driven radiative MHD model. In this paper, we mainly
dig out the underlying physical properties, in particular the flare ribbons
and CME leading front with the aid of numerical simulations. Figure 1 shows
the composite images of the event at three EUV wavebands (131 Å, 171 Å, and
304 Å) during eruption, which are provided by the Atmospheric Imaging Assembly
(AIA; Lemen et al., 2012) on board the Solar Dynamics Observatory. There are
two primary observational features. First, a long filament is located in this
active region before the flare onset (Figure 1a), implying that a pre-eruptive
twisted/sheared magnetic structure exists before eruption. Second, the flare
is composed of two main parallel ribbons (marked as R1 and R2) and a remote
ribbon (R3). In particular, ribbon R1 is exceedingly long and narrow. Besides,
one can see that only the eastern filament materials are still partly visible
during eruption, reflecting that there might be deeper dips that are able to
host filament materials for a longer time in the eastern portion of the flux
rope than in the western portion. The above observations provide some
constrained conditions for the following numerical setup of the data-driven
model.

For the numerical simulation, we consider a full MHD model including thermal
conduction, empirical heating, and radiative cooling terms. The following are
the governing equations:

$\displaystyle\frac{\partial\rho}{\partial
t}+\nabla\cdot(\rho\boldsymbol{v})=0,$ (1)
$\displaystyle\frac{\partial(\rho\boldsymbol{v})}{\partial
t}+\nabla\cdot(\rho\boldsymbol{vv}+p_{{}_{tot}}\boldsymbol{I}-\frac{\
\boldsymbol{BB}}{\mu_{0}})=\rho\boldsymbol{g},$ (2)
$\displaystyle\frac{\partial\boldsymbol{B}}{\partial
t}+\nabla\cdot(\boldsymbol{vB-Bv})=0,$ (3)
$\displaystyle\frac{\partial\varepsilon}{\partial
t}+\nabla\cdot(\varepsilon\boldsymbol{v}+p_{{}_{tot}}\boldsymbol{v}-\frac{\boldsymbol{BB}}{\mu_{0}}\cdot\boldsymbol{v})=\rho\boldsymbol{g\cdot
v}+H_{0}e^{-z/\lambda}-n_{\rm e}n_{\rm{}_{H}}\Lambda(T)$ (4)
$\displaystyle+\nabla\cdot(\boldsymbol{\kappa}\cdot\nabla T),$

where $p_{{}_{tot}}\equiv p+B^{2}/(2\mu_{0})$ is the total pressure with the
assumption of full ionization, $\varepsilon=\rho
v^{2}/2+p/(\gamma-1)+B^{2}/(2\mu_{0})$ is the total energy density,
$\boldsymbol{g}=-g_{\odot}r_{\odot}^{2}/(r_{\odot}+z)^{2}\boldsymbol{e_{z}}$
is the gravitational acceleration, $g_{\odot}=\rm 274\ m\ s^{-2}$ is the
gravitational acceleration at the solar surface, $r_{\odot}$ is the solar
radius, $\boldsymbol{\kappa}=\kappa_{\parallel}\boldsymbol{\hat{b}\hat{b}}$ is
field-aligned thermal conduction, $\kappa_{\parallel}=10^{-6}\
T^{\frac{5}{2}}\ \rm erg\ cm^{-1}\ s^{-1}\ K^{-1}$ is the Spitzer heat
conductivity, $H_{0}=10^{-4}\rm\ erg\ cm^{-3}\ s^{-1}$, $n_{{}_{H}}$ is the
number density of protons, $n_{e}$ is the number density of electrons,
$\lambda=60\ \rm Mm$ (Xia & Keppens, 2016), $\Lambda(T)$ is the radiative loss
function, and the other parameters have their general meanings. Equation (4)
mainly takes into account an empirical heating term used to maintain the hot
corona ($H_{0}e^{-z/\lambda}$), optically thin radiation ($n_{\rm
e}n_{\rm{}_{H}}\Lambda(T)$), and thermal conduction
($\nabla\cdot(\boldsymbol{\kappa}\cdot\nabla T)$). The curve of the radiative
loss function for our simulation is similar to that used in Zhao et al.
(2017), where the radiative cooling coefficient was calculated by Colgan et
al. (2008). It is worth noting that, the background heating function in this
paper only varies with the height, which is principally used to compensate the
coronal radiation losses, and widely used in simulations for filament
formation and eruption (Xia & Keppens, 2016; Fan, 2017; Zhao et al., 2017;
Fan, 2020). In future works, we expect to consider a more realistic heating
function inspired by the $\rm Alfv\acute{e}n$ turbulence dissipation (Mok et
al., 2016). In addition, we use the transition region adaptive conduction
(TRAC) method to correct the chromospheric evaporation, which can better
capture the energy exchange between the transition region and corona with
limited spatial resolution (Johnston & Bradshaw, 2019; Zhou et al., 2021).

The initial magnetic field $\boldsymbol{B}$ for the simulation is provided by
the NLFFF model constructed by the Regularized Biot-Savart Laws (RBSL; Titov
et al., 2018) and magneto-frictional (MF; Guo et al., 2016a, b) method. The
RBSL model is utilized to provide a flux rope consistent with the
observational filament, and the MF method is to relax the magnetic field to a
force-free state. The bottom boundary for the NLFFF model is provided by the
corrected vector magnetic field in the photosphere at 15:24 UT observed by
SDO/HMI (Scherrer et al., 2012; Schou et al., 2012; Hoeksema et al., 2014), as
shown in Figure 2a. To be specific, the pretreatment for the vector
magnetogram contains two steps: to correct the projection effects resulting
from the spherical shape of the solar surface (Guo et al., 2017), and to
remove the Lorentz force and torque in order to satisfy the force-free
assumption (Wiegelmann et al., 2006). As introduced by Titov et al. (2018), an
RBSL flux rope is controlled by four parameters, that is, flux-rope path,
minor radius, $a$, toroidal flux, $F$, and electric current, $I$. The minor
radius of the flux rope is taken as twice the filament width, approximately 14
Mm, as a filament might occupy only half of the radial extent of the flux rope
(Guo et al., 2022). Regarding the 3D path of the flux rope, similar to our
previous work (Guo et al., 2021a), we outline the projected path according to
the AIA observations at first, as indicated by the red dots in Figure 2b.
Then, we derive the height distribution of the flux rope using the
triangulation technique (Thompson, 2009) combined with the data from the
_Solar Terrestrial Relations Observatory_ (_STEREO_). However, we do not adopt
the filament height directly as the flux-rope height, for two reasons. First,
the filament is generally located at the bottom of the flux rope so that the
filament is likely to deviate from the flux-rope axis (Guo et al., 2021b,
2022). Second, the angle between SDO and STEREO is relatively small ($\sim
37^{\circ}$), so it is prone to bring large measurement errors. As a result,
we merely use the information of the filament apex as the apex of the flux
rope axis, and the heights of the rest points are fitted with the following
formula (Török et al., 2010; Xu et al., 2020):

$\displaystyle z(x)=\begin{cases}\frac{x(2x_{\rm h}-x)}{x_{\rm h}^{2}}h\quad
0\leq x\leq x_{\rm h},\\\ \frac{(x-2x_{{}_{\rm h}}+1)(1-x)}{(1-x_{\rm
h})^{2}}h\qquad x_{{}_{\rm h}}<x\leq 1\end{cases}$ (5)

where $x=s/l$ is the normalized coordinates along the filament axis,
$x_{{}_{\rm h}}\sim 0.2$ is the normalized position of the filament apex,
$h_{\rm f}=19\;$Mm is the height of the filament apex, $s$ is the distance
from the start point, $l$ is the projection length of the filament, $h=h_{\rm
f}+a$ is the height of the flux-rope apex. For the toroidal flux, similar to
our previous works (Guo et al., 2019b, 2021a), the reference flux is selected
as the average of the unsigned flux of two footprints, namely,
$F_{0}=6.19\times 10^{20}\;$Mx. According to our experience and some tests, we
select $3.5F_{0}$ as the final toroidal flux. Then, the electric current, $I$,
is computed by Equation (12) in Titov et al. (2018). It is noted that the sign
of the electric current determines the magnetic helicity of the flux rope. In
this event, the filament is located in the southern hemisphere, implying that
the chirality of the filament is most probably sinistral according to the
hemispheric rule (Ouyang et al., 2017). Moreover, the axial magnetic field of
the filament points to the left if we stand on the positive side of the
polarity inversion line. These strongly suggest that the chirality of the
filament is sinistral, corresponding to positive helicity of the flux rope and
electric current. So far, we can construct the RBSL flux rope and embed it
into the potential field extrapolated by the Green’s function method (Chiu &
Hilton, 1977), and then relax the magnetic field to a force-free state with
the MF method (Guo et al., 2016a, b). After relaxation, the force-free metric
is $\sigma_{J}=0.29$, and the divergence-free metric is
$\langle|f_{i}|\rangle=1.0\times 10^{-5}$ (see Guo et al., 2016b, for details
of the two metrics). Figure 2c shows the sheared and twisted field lines after
relaxation. One can see that the reconstructed flux rope and sheared arcades
resemble the observed filaments.

To obtain a hydrostatic atmospheric model from the chromosphere to the corona,
similar to Xia & Keppens (2016), the temperature distribution is described by
the following formula:

$\displaystyle T(z)=\begin{cases}T_{{}_{\rm ch}}+(T_{{}_{\rm co}}-T_{{}_{\rm
ch}})(1+T_{{}_{\rm ch}}+{\rm tanh}(z-h_{{}_{\rm tr}}-0.27)/w_{\rm tr})/2\qquad
z<h_{{}_{tr}},\\\ (7F_{\rm c}(z-h_{\rm tr})/(2\kappa)+T_{{}_{\rm
tr}}^{7/2})^{2/7}\qquad z\geq h_{{}_{tr}}\end{cases}$ (6)

where $T_{{}_{\rm ch}}=8000\rm\ K$ is the chromospheric temperature,
$T_{{}_{\rm co}}=1.5\rm\ MK$ is the coronal temperature at the top,
$h_{{}_{\rm tr}}=2\rm\ Mm$ is the height of initial transition region,
$w_{{}_{\rm tr}}=0.2\ \rm Mm$ determines the thickness of initial transition
region, and $F_{\rm c}=2\ \times 10^{5}\ \rm erg\ cm^{-2}\ s^{-1}$ is the
constant thermal conduction flux. Then, with the assumption of the hydrostatic
atmosphere, the density distribution can be derived from the given number
density at the bottom of $1.15\times 10^{15}\rm\ cm^{-3}$. Finally, we need to
insert a filament according to the observation, which can be done by
increasing the density by about 50 times of magnitude but keeping the gas
pressure unchanged (Zhou et al., 2018), where the prominence path is measured
from observations. Figure 2d shows the side view of the 3D magnetic field
lines, temperature distributions and the inserting filament, one can see that
the cold filament materials almost reside in the bottom of the flux rope,
consistent with the simulated filament formed with the thermal instability or
the thermal non-equilibrium process (Xia et al., 2014; Guo et al., 2021b,
2022).

Similar to our former works (Guo et al., 2019a; Zhong et al., 2021), the
$v$–$B$ driven boundary is adopted at the inner ghost cells of the bottom
boundary. Concretely, we implement a sequence of vector magnetograms and
velocity fields derived from the Differential Affine Velocity Estimator for
Vector Magnetograms (DAVE4VM; Schuck, 2008) at the inner ghost cells so that
the bottom boundary is synchronous with the observation. To reduce the small-
scale fluctuations resulting from the uncertainty in the input magnetograms
and to better match the boundary for the initial NLFFF model, we also adopt
the preprocessing technique (Wiegelmann et al., 2006) to process the time
series of the driven magnetograms, where the smoothing term in the iteration
function can effectively decrease the fluctuations of input magnetograms and
the derived velocity fields. At the outer ghost cells of the bottom boundary,
the magnetic field is provided by the fourth-order zero-gradient
extrapolation, namely,
$u(x_{i})=-0.12u(x_{i+4})+0.64u(x_{i+3})-1.44u(x_{i+2})+1.92u(x_{i+1})$.
Additionally, since the cadence of the observational data is 12 min, the
missing data within each 12 min are calculated with linear interpolation in
time. For the other five boundaries, the velocity and magnetic field are
provided by the equivalent extrapolation, $u(x_{i+1})=u(x_{i})$, and the
second-order zero-gradient extrapolation,
$u(x_{i})=(-u(x_{i+2})+4u(x_{i+1}))/3$, respectively. Regarding the density
and gas pressure, they are fixed to be the initial values at the bottom,
flexible at the top according to gravitational stratification (Xia & Keppens,
2016), and provided by the equivalent extrapolation at four side boundaries.
It is noted that, to save computing time and reduce the numerical dissipation,
following previous simulations (Jiang et al., 2016c; Kaneko et al., 2021), we
reduce the magnetic field strength to 1/11 of the original observed data.
Still, the maximum of the driving velocity ($\sim$ 1 km s-1) is less than the
average $\rm Alfv\acute{e}n$ velocity at the bottom ($\sim$10 km s-1).

The 3D MHD equations in the local Cartesian coordinate system are numerically
solved with the Message Passing Interface Adaptive Mesh Refinement Versatile
Advection Code (MPI-AMRVAC111http://amrvac.org, Xia et al., 2018; Keppens et
al., 2020, 2023). For the numerical scheme, we use a three-step Runge-Kutta
time discretization, HLL Riemann solver with the Cada limiter. The
computational domain is $[x_{min},x_{max}]\ \times[y_{min},y_{max}]\
\times[z_{min},z_{max}]=[-242.6,227.9]\ \times[-185.8,185.8]\
\times[1.0,400.0]\;$Mm, resolved by a uniform grid with $330\times 260\times
400$ cells. The time range of the simulation is from 15:24 to 16:00 UT in the
observation, covering the whole process of the eruption.

##  3 Numerical Results

###  3.1 Global evolution and comparison with observations

The global evolution of the simulated flux rope during the eruption is shown
in Figure 3, which is overlaid on observed 304 Å images in the left column and
on the magnetograms in the right column. The morphology and erupting direction
of the flux rope resemble those of the observed filament fairly well,
indicating that the simulation almost reproduces the gross picture of the
observation. Snapshots from the side views show drastic magnetic reconnection
during the eruption, recognized by the hyperbolic flux tube (HFT; Titov et
al., 2002) below the flux rope and the heating along the field lines (Figure
3d). Moreover, the flux rope expands a lot and presents a stratified
characteristic: the core field lines, which hold the cold prominence are
weakly twisted, but the outer newly-formed field lines, which hold hot
plasmas, are highly twisted, as shown in Figure 3f. More intriguingly, the
western footpoint of the flux rope is not fixed in the photosphere, which
appears to drift westward. The drifting of the footpoint strongly implies that
there should exist 3D reconnection geometry beyond the 2D CSHKP model
(Aulanier & Dudík, 2019).

Quasi-separatrix layers (QSLs) are regions where magnetic connectivity changes
drastically, and the squashing factor ($Q$) is larger than 2. They are the
favorite places for magnetic reconnection (Priest & Démoulin, 1995; Demoulin
et al., 1996; Titov et al., 2002). Former static NLFFF models and dynamic MHD
simulations have shown that flare ribbons are almost cospatial with QSLs in
the photosphere (Demoulin et al., 1997; Janvier et al., 2013; Dudík et al.,
2014; Yang et al., 2015; Dalmasse et al., 2015; Zuccarello et al., 2017; Jiang
et al., 2018b; Guo et al., 2019a; Zhong et al., 2021). Accordingly, their
spatial relationship is an effective touchstone to examine the rationality of
the simulation. To this end, we calculate the QSLs at each time step in the
simulation with an open-source code (Liu et al., 2016; Zhang et al., 2022) and
compare them with the AIA 1600 and 304 Å observations, as shown in the top two
panels of Figure 4. Note that the observational images are de-projected to a
top view. We find that the simulated QSLs coincide well with the flare
ribbons, including the two main parallel ribbons (R1 and R2) and the remote
ribbon (R3). As expected, both R1 and R2 show separation motions. To further
study their spatiotemporal relationship, we select a slice along the
separation direction of the two parallel ribbons (Figure 4a). The time-
distance diagram in Figure 4e illustrates that flare ribbons and QSLs show
analogous kinematic characteristics: both the QSLs and the cospatial ribbons
separate quickly and the flare ribbons eventually stop. The difference between
the two ribbons is also remarkable. Ribbon R1 initially moves away from the
polarity inversion line with a speed of $\sim$120 km s-1, and then decelerates
drastically. In contrast, ribbon R2 initially moves away from the polarity
inversion line with a speed of $\sim$19 km s-1, and then decelerates
gradually. The comparability between the simulated QSLs and the observed flare
ribbons strongly indicates that our numerical model reproduces the evolution
of the magnetic topology in the observations very well.

In addition to the evolution of the coronal magnetic field, our numerical
model takes thermal processes into account. Consequently, we have the
capability to compute the synthesized EUV images from the simulated
temperature and density, and compare them with the EUV observations directly.
The optically thin emission in each cell is calculated with the following
formula (Chen, 2006):

$\displaystyle I_{{}_{\lambda}}(x,y,z)=G_{{}_{\lambda}}(T)\ n_{e}^{2}(x,y,z)$
(7)

where $G_{{}_{\lambda}}(T)$ is the instrumental response function for
different EUV wavebands. The synthesized image can be obtained by the
integration along the line of sight. Figure 5 displays the comparisons between
our synthesized EUV images (integration along the $z$-axis) and AIA
observations at different wavelengths. It is found that the synthesized images
show an analogous morphology to the observations, in particular the locations
and the shapes of ribbon R1 and remote ribbon R3. An oval-shaped enhancement
in the synthesized EUV images in panels (d–f) corresponds to the fast-
component EUV wave on the solar surface. A similar wavelike structure is also
discernable in observations as indicated by a slightly larger oval-shaped
structure in Figure 5b. It implies that our simulated EUV wave is slightly
slower than in observations. Figures 6 and 7 display the side and end views of
the synthesized EUV images, which are obtained by the integration along the
$y$\- and $x$-axes, respectively. Many common observational phenomena are
presented in the synthesized EUV images, for example, the helical and eruptive
prominence (Figure 6e and 7e), flare loops (Figures 6d and 7d), cusp-like
structure (Figures 6f and 7f) and CME three-part structure (Figure 7b). As the
eruption goes on, the filament observed in 304 Å waveband becomes fainter
gradually, which might be due to enhanced heating and the density decrease
caused by the expansion of the flux rope. It is seen that the erupting flux
rope can be seen in three wavebands, reflecting that it is a multithermal
structure during the eruption. In addition, one can find that the majority of
filament material stays in the eastern portion of the flux rope during the
eruption (Figure 6e), which is similar to the observations.

In conclusion, many observational features are reproduced well by our data-
driven radiative model, in terms of the morphology of the eruption, dynamics
of the magnetic topology and EUV radiations. These results indicate that the
simulation is reasonable and provide a solid basis for the following
quantitative analyses.

###  3.2 Magnetic reconnection and topology evolution during the eruption

Figure 8 displays the evolution of the distributions of the magnetic field
lines, heating, QSLs, and $J/B$. It is noted that $J/B$ is a scalar metric to
denote the variation of the magnetic field ($J/B\sim|\nabla\times B|/B$),
which is widely used to locate the current sheet and magnetic reconnection
(Gibson & Fan, 2006; Fan & Gibson, 2007; Jiang et al., 2016c, 2018b, 2018a,
2021b). Three types of magnetic reconnection can be identified, as represented
by the three columns, which are described in detail as follows.

Figures 8a, 8d, and 8g show the reconnection between the flux-rope leg and the
ambient sheared arcades, which is also called ar-rf reconnection in the 3D
flare model since one post-reconnection field line merges with the flux rope
and the other line becomes a flare loop (Aulanier & Dudík, 2019). This
reconnection geometry is characterized by the highly curved post-reconnection
field lines and significant heating ($\sim 4\;$MK). The $Q$ and $J/B$
distributions denote the formation of the current layer at the interface
between the western leg of the flux rope and ambient arcades. One can see that
the location of the western footpoint of the flux rope changes after
reconnection. Such reconnection is also reflected by the brightening away from
the flare ribbons in observations, as shown by the blue box in Figure 4a. The
ar-rf reconnection which likely happens in our simulation changes the
connectivity of the field lines and leads to the drifting of the western
footpoint of the flux rope (Aulanier & Dudík, 2019; Zemanová et al., 2019).

Figures 8b, 8e, and 8h show the reconnection between the overlying sheared
arcades, which is proposed as the standard reconnection paradigm in the 2D
CSHKP model, and is called aa-rf reconnection in the 3D flare model (Aulanier
& Dudík, 2019). The narrow and vertical current below the flux rope (Figure
8h) is a proof to demonstrate this reconnection geometry. The ring-like
patterns of the QSLs in Figures 8e reflect the complexity of the flux rope
during the eruption, which might be due to the newly-formed twisted field
lines and the injected flux. Such QSL structures also reflect that internal
reconnection might occur inside the flux rope (Zhong et al., 2021). This type
of magnetic reconnection is also manifested in the temperature distributions
and the post-reconnection field lines, where the flare loops and the overlying
hot cusp-shaped structure revealed in Figure 8b are the direct evidence for
this reconnection geometry (Masuda et al., 1994). Moreover, the border of the
CME bubble is heated to 8 MK by the reconnection, which also appeared in other
2D MHD simulations (Mei et al., 2012; Zhao et al., 2017; Ye et al., 2021).

Figures 8c, 8f, and 8i display the reconnection above the flux rope. As the
flux rope rises, overlying arcades are pushed outward and reconnect with the
ambient antiparallel field lines. This reconnection geometry resembles the
breakout model that occurs at the magnetic null above the quadrupole
polarities (Antiochos et al., 1999). The reconnected field lines related to
this reconnection geometry correspond to the remote ribbon R3 and the southern
part of ribbon R1, accounting for the multi-ribbon characteristic of this
flare.

The above-mentioned magnetic reconnection changes the topology of magnetic
field, and leads to the variation of the twist ($T_{w}$) inside the flux rope,
which is the core structure of the initial magnetic field. In order to check
how magnetic twist changes with time, we calculate the magnetic twist with the
parallel electric current method as described in the formula (16) in Berger &
Prior (2006). According to this method, the initial twist of the flux rope is
around 2.16 turns, which is larger than some well-known thresholds of the kink
instability (Hood & Priest, 1981; Török et al., 2004). In Figures 9a and 9b we
plot two snapshots of the magnetic twist distribution in the photosphere, and
it is seen that the twist of the flux-rope footpoint changes a lot during the
eruption. First, the twist increases as the reconnection goes on, reaching
$\sim$ 2.5 turns at 15:26 UT and $\sim$3 turns at 15:48 UT. Second, the
western footpoint splits into two parts, i.e., regions P1 and P2. Region P1 is
almost along the polarity inversion line, corresponding to outer newly-formed
field lines due to the aa-rf reconnection. Region P2 reflects the drifting of
the western footpoint of the flux rope due to the ar-rf reconnection. The
distribution of field lines in Figure 3 provides credence to the above
physical picture. Figure 9c shows the temporal evolution of the toroidal
($\phi_{T}=\int\boldsymbol{B_{T}}\cdot\boldsymbol{ds}$) and poloidal fluxes
($\phi_{P}=T_{w}\phi_{T}$), which can be divided into three stages. First,
both the toroidal and poloidal fluxes increase, reflecting the aa-rf
reconnection, which injects the poloidal and toroidal fluxes into the flux
rope. Then, the toroidal flux decreases but the poloidal flux still increases,
indicating that the aa-rf reconnection might have stopped and other
reconnection geometries begin to work. In particular, the time when the
toroidal flux starts to decrease almost corresponds to the time when the flare
ribbons cease moving. According to Aulanier & Dudík (2019), the toroidal flux
should be maintained during the ar-rf reconnection, so we infer that internal
reconnection inside the flux rope likely occurs in the second stage. Such a
process transfers some of the toroidal flux to the poloidal flux. Finally,
both the toroidal and poloidal fluxes decrease, which results from the erosion
of the flux rope flux by the null point reconnection above.

###  3.3 Thermal properties of the CME and the flare loops

Figure 10a displays the synthesized SDO/AIA 193 Å emission image viewed from
the $x$-axis (along the flux-rope axis) at 15:32 UT. It is seen that the EUV
image presents the three-component structures of CMEs, i.e., a bright front, a
cavity, and bright core, with the solar flare below the bright core near the
solar surface. In order to show the structures more clearly, we make a
corresponding running-difference image in Figure 10b, where the 193 Å image at
15:30 UT is subtracted. Besides the three components of a typical CME (where
the frontal edge is indicated by the blue solid line), a faint dome-like front
is identified above the EUV frontal front as indicated by the red dashed line,
which should correspond to the piston-driven shock. In order to compare our
simulation results with coronagraph observations, we synthesize the white-
light image, which is shown in Figure 10c. The Thomson scattering of the
white-light intensity is calculated with the eltheory.pro in the Solar
SoftWare, which considers the white-light intensity to be proportional to the
integration of the density along the line of sight. As shown in Figure 10c,
the white-light image of the eruption is characterized by the typical three-
component structures, i.e., a leading front, a dark cavity, and a bright core.
However, it should be mentioned that the leading front is comprised of two
parts, an inner part as indicated by the blue solid line which corresponds to
the EUV frontal edge, and an outer part as indicated by the red dashed line
which corresponds to the suspected piston-driven shock. There is no distinct
gap between the two parts although the intensity in between is slightly weaker
than the leading and trailing fronts. To understand the thermal properties of
the suspected shock and the three components of the CME, we calculate the
averaged temperature of each pixel in Figure 10c by $\bar{T}=\int
n_{H}^{2}Tdx$/$\int n_{H}^{2}dx$ (Cheng et al., 2012; Gou et al., 2015), and
plot the $\bar{T}$ distribution in Figure 10d. It is seen that the suspected
shock, as indicated by the red dashed line, corresponds to a drastic increase
of temperature, and the downstream of the suspected shock is strongly heated,
which is in accordance with the shock theory. In order to confirm the shock
nature of this front, we calculate the propagation velocity at the nose part
based on the synthesized images at 15:32 UT. It is found that the propagation
speed is 590.2 km s-1, which is larger than the local fast-mode MHD wave speed
with 268.3 km s-1. Our result confirms that the dome-like structure as
indicated by the red dashed line in Figure 10b is indeed a fast-mode shock
wave with a fast-mode Mach number of 2.1. It is interesting to see that while
the EUV bright edge, as indicated by the blue solid line, is only slightly
heated, the cavity is moderately heated. The CME bright core has a dichotomous
distribution of temperature, with the central part being cold and the shell
part being warm.

The corresponding magnetic field lines at 15:32 UT are displayed in the left
panel of Figure 11, where the strongly twisted core field and the less twisted
envelope field can be clearly distinguished. The magnetic field geometry
viewed from the same vantage point as Figure 10 are displayed in the right
panel of Figure 11, with the background being the synthesized 193 Å intensity
map. One can see that the bright core, wrapped by twisted field lines,
occupies slightly more than the lower half of the flux rope. As for the dark
cavity, we find that the plasma therein is hot and tenuous, and the magnetic
field lines consist of two portions, inner and outer portions: the inner field
lines are hot and twisted (cyan lines in Figure 11), but the outer portion is
embedded by non-twisted magnetic loops (yellow lines in Figure 11). It implies
that part of the cavity is due to the upper portion of the twisted flux rope
with low density, and another part is due to magnetic field-line stretching as
described by Chen (2011). Regarding the EUV bright front (which is also the
CME frontal edge), the plasma there is moderately heated (2 MK) and compressed
(blue solid lines in Figure 10), corresponding to the outer edge of the
stretching field lines (red lines in Figure 11) according to the magnetic
field-line stretching model (Chen et al., 2002), which is the typical
characteristics of the EUV wave (Liu & Ofman, 2014).

A cusp-shaped structure is widely perceived as important evidence for magnetic
reconnection (Shibata, 1999), which is worthy of exploration in a data-driven
simulation. Figures 12a and 12b display the synthesized 94 Å and 171 Å images,
respectively. It is seen that the cusp-shaped structure can only appear in
high-temperature waveband (94 Å, $\sim 10\;$MK ), and post-flare loops are
more visible in the low-temperature 171 Å image ($\sim 0.8\;$MK). Figures 12c
and 12d depict the number density and temperature distributions, respectively.
We find that the cusp-shaped structure is hot (8 MK), whereas the post-flare
loops are relatively cold ($<1\;$MK) and dense, as expected. Figures 12e and
12f show the velocity distributions, where we can recognize the inflow and
outflow near the reconnection site, and the chromospheric evaporation at the
loop footpoints. Figure 12g illustrates the curl of the velocity, which is
generally used to detect the slow-mode shock (Wang et al., 2009; Mei et al.,
2020). It is seen that the slow-mode shock is located near the hot
reconnection site, confirming that it plays a significant role in heating the
reconnection outflow as claimed in the reconnection model (Priest & Forbes,
1986). Figure 12h illustrates the field lines color-coded in temperature.
Clearly, one can see the highly curved and hot cusp-like structures, as well
as cold and rounded flare loops, which is similar to the scenario proposed by
Shibata (1999).

##  4 Discussions

###  4.1 Stopping positions of flare ribbons

Magnetic topology governs the global dynamics of solar fares (Longcope &
Kankelborg, 2001). According to the magnetic reconnection model (Priest &
Forbes, 1986), magnetic energy is released to produce heat and non-thermal
particles around the reconnection site, which are then transferred to the
solar surface along magnetic separatrices. As a result, two flare ribbons
usually appear at the bases of the separatrices, which run roughly parallel
with the polarity inversion line. As time goes on, the flare loops become
bigger and bigger apparently and the two flare ribbons separate. The moving
speed of each ribbon can be more than 50 km s-1 in the impulsive phase and
then decreases to even less than 1 km s-1. It is noted in passing that the
separating motion of the flare ribbons was frequently attributed to the rising
reconnection point. However, as pointed out by Chen et al. (1999), this is a
misinterpretation, and the flare ribbons would separate even if the
reconnection point is fixed once the reconnection point is high enough.

There were a variety of efforts revealing the cospatiality between flare
ribbons and separatrices (Somov et al., 2002; Demoulin et al., 1993; Dalmasse
et al., 2015; Zuccarello et al., 2017), or in more general cases, QSLs
(Demoulin et al., 1996; Titov et al., 2002), which confirmed the validity of
the magnetic reconnection model for solar flares. In all these works, the
separatrices or QSLs were calculated from the magnetic field extrapolation
derived from the photospheric magnetograms. Chen et al. (2012) argued that the
flare ribbon-associated QSLs are dynamic, whereas the QSLs derived from the
static extrapolated coronal field are fixed. Therefore, they proposed that
there should exist two types of QSLs, i.e., inner and outer QSLs. The inner
QSLs are linked to the reconnection site, and are cospatial with the moving
flare ribbons, whereas the outer QSLs correspond to the location where the
flare ribbon would eventually stop. This argument was reinforced by the fact
that the extrapolated QSLs are cospatial with the flare ribbons at their full
extents (Savcheva et al., 2015).

To simulate the dynamic QSLs, Savcheva et al. (2016) used the
magnetofrictional flux-rope insertion method to generate an unstable coronal
magnetic field, where the iteration of the coronal magnetic field relaxation
leads to continuous reconnection under the flux rope rather than approaching
an equilibrium state. Although the relaxation is not equivalent to dynamic
evolution, it indeed captures the topological variations of CME/flare
eruptions, including the expansion of flaring loops and the separation of
flare ribbons. Of course, a better approach to compare dynamic QSLs and flare
ribbons is to conduct MHD numerical simulations (Jiang et al., 2018b; Guo et
al., 2019a; Zhong et al., 2021). While quantitative comparisons were made by
several research groups, Jiang et al. (2018b) performed the first quantitative
comparison between separation speeds of the simulated QSLs and observed flare
ribbons. However, their simulated QSLs separate with a speed three times
faster than that of observed flare ribbons, which they attributed to a faster
reconnection rate due to high numerical resistivity in their simulations.

In this paper, we performed a data-driven MHD simulation in order to reproduce
the CME/flare eruption event on 2021 Oct 28, and then compared the simulated
QSLs location with the observed flare ribbons. As shown in Figure 4e, the
inner QSLs, which are linked to the reconnection site, and almost cospatial
with the observed flare ribbons during the whole eruption. The simulated
separating speeds of the two QSLs in the early stage are 120 km s-1 and 19 km
s-1, respectively, both of which are consistent with the observed separating
speeds of the two flare ribbons. More importantly, we found that both the
simulated QSLs and observed flare ribbons eventually stop. As seen from Figure
4e, the locations where the simulated QSLs and the observed flare ribbons stop
are at the pre-existing QSLs, which correspond to the outer QSLs as defined by
Chen et al. (2012).

To illustrate the relationship between the inner separating QSLs and the
almost fixed outer QSLs, in Figure 13a we plot the distribution of the
squashing factor $Q$ in the cross-section of the simulation domain at $x=10.6$
Mm at 15:28 UT. The cross-section is in the $y$–$z$ plane, which is roughly
perpendicular to the axis of the flux rope. One can see an erupting flux rope
in the middle, with the outer shell extending down to form an X-shaped
hyperbolic flux tube, which corresponds to the inner QSLs, as indicated by the
red arrows. The inner QSL further extends down, bifurcating into two branches,
and their intersection with the bottom boundary corresponds to the two
separating flare ribbons. Meanwhile, we can see in Figure 13a that beyond the
left and right sides of the flux rope, there stand two QSLs, as indicated by
the pink arrows, which are labeled as $S_{\rm out}^{L}$ for the left QSL and
$S_{\rm out}^{R}$ for the right QSL. Only the magnetic field lines between the
two outer QSLs and the two inner QSLs are available for further reconnection.
Hence, as the dynamic inner QSLs move outward and approach the outer QSLs,
magnetic reconnection stops. As seen from Figures 13b and 13c, the inner QSLs
move outward as the reconnection goes on, while the outer QSLs are almost
fixed in the photosphere. Moreover, the time-distance diagram shown in Figure
13d illustrates that the dynamic inner QSLs eventually stop at the static
outer QSLs, as expected. It is noticed that in our simulation the time when
the two flare ribbons stop is very close to the moment when the toroidal flux
starts to decrease, which implies that the outer QSLs determine where the aa-
rf reconnection stops.

We perceive that this property is profound and might have various applications
in future space weather prediction. First, the co-spatiality between QSLs and
flare ribbons is an effective touchstone to examine the rationality of any
data-driven simulations. Second, the outer QSLs, which can be determined by
the magnetograms before eruption happens, determine directly the size of the
flare ribbons and reconnection regions. It means that we can estimate the
magnetic flux to be injected into the CME due to reconnection even before the
flare, and then infer the total magnetic flux and the twist of an ICME (Qiu et
al., 2007; Wang et al., 2017; Xing et al., 2020b). Moreover, with some
assumptions, we can even estimate the lifetime of a flare before it occurs.
Consequently, we strongly suggest that the outer QSLs and the above-mentioned
properties can be considered as novel indexes of future space weather
forecasting.

###  4.2 Relationship between CME fronts and EUV waves

A typical CME is generally characterized by a frontal edge, with a cavity and
bright core embedded inside. When a CME is fast enough, a piston-driven shock
would straddle over the CME bubble (Chen, 2011), as revealed in observations
(Vourlidas et al., 2003). Albeit this phenomenon has been observed for several
decades, the consensus as to its nature has not been obtained hitherto. The
reason is that it involves numerous elaborated physical processes, in
particular the thermodynamics, some of which are omitted in the traditional
zero-$\beta$ or isothermal MHD models. Among the portions of CME three-part
structures, the most controversial one is the bright leading front. Its nature
and formation mechanism are still elusive.

In earlier times, the CME frontal edge was attributed to be MHD waves or
erupting coronal loops (Nakagawa et al., 1975; Poland & Munro, 1976). Later,
it has been assumed to be plasma pileup ahead of the erupting flux rope (see
Forbes, 2000, for a review). However, Chen (2009) proposed a conjecture to
explain it with the field-line stretching model (Chen et al., 2002). He
investigated the relationship between “EIT wave” and CME leading front by
comparing the white-light coronagraph images in the high corona and low-corona
EUV observational data, where the “EIT wave” corresponds to the slow component
of the coexisting two EUV waves. It is worth noting that, as suggested by Chen
(2016), we are inclined to use “EUV waves” for any kind of wavelike phenomena
observed in EUV wavebands, including fast and slow components in this paper,
and use “EIT waves” for the wavelike phenomena discovered by Thompson et al.
(1998) with the SOHO/EIT telescope, which are typically three times slower
than type II radio bursts and Moreton waves (Klassen et al., 2000), i.e., “EIT
waves” correspond to the slow-component EUV waves in this paper. It was found
that the CME front edge is almost cospatial with the “EIT wave”, and the
following dimming region corresponds to the CME dark cavity. Therefore, he
applied the formation mechanism of “EIT waves”. i.e., magnetic field-line
stretching model, to the CME frontal edge: As a flux rope rises, it pushes the
overlying field lines to stretch successively, producing the density
disturbances that propagate upward with the fast-mode wave velocity.
Simultaneously, such disturbances are also transferred down to the footpoints
along field lines with the Alfv$\acute{\rm e}$n velocity. The density
enhancements at different portions of field lines at a given time compose the
CME front. The enclosing volume increases as the field lines are stretched,
forming the coronal dimming and the CME cavity. As such, the “EIT wave” and
the following dimming correspond to the CME leading front and dark cavity,
respectively (Chen, 2009).

The above theory is verified very well by our data-driven model. As shown in
Figures 10b and 10c, the synthesized EUV images based on our simulation
reproduce two components of EUV waves and the ensuing EUV dimmings. In
particular, it is confirmed that the dimmings and the slow-component EUV wave
correspond to the synthesized CME cavity and CME leading front, respectively,
whereas the fast-component EUV wave corresponds to the CME piston-driven
shock. To further understand the physical nature of the EUV waves, we plot the
density distributions at two moments in Figures 14a along the white dashed
line in Figure 10a. There are two fronts with enhanced density, i.e., a
leading front which travels faster and a following front which travels slower.
Besides, a dimming region is found to be immediately after the slower front.
All these features are analogous to the MHD simulation performed by Chen et
al. (2002). Then, we calculate the velocities of these wave-like structures
and local fast-mode waves. We find that the leading fast front moves faster
than that of the local fast-mode wave ($\sim 2.05\ V_{\rm fw}$), but the
following slow front almost propagates at the local fast-mode wave ($\sim
0.86\ V_{\rm fw}$). As mentioned in Chen (2009), a rising flux rope can
produce two fronts propagating upward, i.e., a piston-driven shock front and a
following dome-shaped front. The latter corresponds to the slow-component EUV
waves and the CME leading front, where the top part travels upward with the
fast-mode wave speed and the legs travel horizontally with a speed about three
times slower than the fast-mode wave speed (Chen et al., 2002, 2005).

Since EUV waves are more frequently observed across the solar disk, we examine
the time-distance evolution of these two waves in the horizontal direction.
The horizontal slice is marked as the red solid line in Figure 10a. The
locations of the legs of the two EUV waves at several moments are displayed in
Figure 14b. It is found that the velocity of the fast-component EUV wave is
about 2.69 times that of the slow component. Consequently, we believe that the
two components EUV waves in our simulation, i.e., the fast and slow
components, correspond to the fast piston-driven shock and the non-wave
component predicted by the field-line stretching model (Chen et al., 2002,
2005). It is noticed that while the fast-component wave decelerates slightly,
the slow-component wave decelerates significantly. According to the magnetic
stretching model (Chen et al., 2002, 2005), the apparent speed of the slow-
component EUV wave is not only related to the magnetic field strength, but
also to the magnetic configuration. If the magnetic field lines become more
and more elongated in the vertical direction, as indicated by Fig. 8 of Chen
et al. (2005), the slow-component EUV wave would decelerate substantially. The
slow-component EUV wave is three times slower than the fast-component EUV wave
only when the magnetic field lines are concentric semi-circles (Chen et al.,
2002). It is noted that recent observational analysis for this event performed
by Devi et al. (2022) distinguished two components of EUV waves in this event,
including the leading fast-mode wave and the following slow non-wave
component, verifying the field-line stretching model of EUV waves. Moreover,
they found that the ratio between the speeds of these two components ranges
from 2.0 to 2.5, which is fairly closer to our simulation result. Therefore,
our data-driven simulation provides a straightforward verification for the
field-line stretching model (Chen et al., 2002, 2005), and confirms that the
slow-component EUV wave is cospatial with the CME frontal edge (Chen, 2009).

To explore the nature of the CME frontal edge, as done in observations (Chen,
2009), we compare the EUV waves in the synthesized EUV images and three-part
structures of the CME in the synthesized white-light image in Figure 15. One
can see that the fast- and slow-component EUV waves are almost cospatial with
the shock and the CME frontal edge, respectively, and the EUV dimmings are
cospatial with the CME cavity as well. To compare the synthesized images with
observations, we plot the composed COR1 white-light and EUVI 195 Å images
observed by STEREO A at two moments in Figures 15c and 15f, where the COR1 and
EUVI observations are roughly simultaneous. Owing to the limited field of view
of the EUVI, we cannot compare the synthesized EUV images with EUVI
observations directly. However, we can compare the synthesized white-light
images with the COR1 observations. As revealed in the panels a, c, d, and f of
Figure 15, the CME frontal edge and the piston-driven shock resemble the
observations very well. Generally it is often taken for granted that the CME
piston-driven shock should be distinguished clearly from the CME frontal edge
as seen in some events. However, it is noted that in other events like this
one, the CME frontal edge and the piston-driven shock above become inseparable
because the density in the downstream (or the sheath) region of the piston-
driven shock is also enhanced, and the CME frontal edge and the CME shock join
together as a unity as revealed in Figures 15c and 15f. This result should be
kept in mind when identifying the CME frontal edges and piston-driven shock
waves in some CME events.

Therefore, our simulation results strongly indicate that the CME frontal edge
corresponds to the density enhancement caused by the successive stretching of
the erupting magnetic field lines rather than the plasma pileup (Chen, 2009).
That is to say, the CME frontal edge is the same as “EIT wave” in the
formation mechanism and physical nature, i.e., both are due to successive
stretching of magnetic loops.

##  5 Summary

In this paper, we perform a data-driven radiative MHD simulation to study the
thermodynamic and magnetic topology evolutions of the X1.0 flare on 2021
October 28. We adopt a non-adiabatic MHD model, including the thermal
conduction and radiative losses. The initial magnetic field with a flux rope
is provided by the RBSL technique, which is then relaxed to a force-free state
with the MF method. The evolution is driven by the changing bottom boundary
conditions, including a sequence of vector magnetic field and velocity field
obtained from the observations. Our simulation reproduces many observational
features in the corona and sheds light on some underlying physical processes,
which are summarized as follows.

  1. 1.

Our simulation results are comparable with observations in many sences. First,
the simulated magnetic flux rope resembles the filament, and possesses the
analogous eruption direction to the observations. Second, the simulated QSLs
in the photosphere are almost cospatial with the observed flare ribbons.
Particularly, the typical separation motion of flare ribbons is reproduced
very well by our numerical model. Third, the synthesized EUV images reproduce
some general observational features in CME/flare eruptions, for example, the
gross morphology of the flare, helical prominence, three-part CME structure,
and flaring loops. Given all of these, we believe that our data-driven
simulation has reproduced the principal observational features and physical
processes, encompassing the thermodynamic and magnetic topology evolutions.

  2. 2.

With the aid of the field-line connectivity, heating, QSLs, $J/B$, and flux
evolution, we identified three main types of reconnection geometries during
the eruption in our simulation, including the reconnection in the overlying
arcades, the reconnection between flux-rope legs and adjacent arcades, and the
breakout-like reconnection above the flux rope. Such reconnection geometries
greatly affect the topology of the flux rope and eruption dynamics. Our data-
driven radiative model strongly suggests that the reconnection in real
observations are far more complicated than described in the 2D CSHKP model.

  3. 3.

Our numerical model provides conclusive validation for the conjecture that the
separating flare ribbons, which correspond to the dynamic QSL, ultimately stop
at the outer stationary QSLs (Chen et al., 2012). The outer QSLs represent the
interface beyond which magnetic field is not involved in magnetic reconnection
associated with the flare. When all field lines inside the outer QSLs have
reconnected, the flare ribbons would stop at this separatrix. This finding
provides a new window to predict the size of the flare and reconnection
regions before eruption, which could be used in future space weather
forecasting.

  4. 4.

Our numerical model reveals the cospatiality between CME piston-driven shock
and the fast-component EUV wave, as well as the cospatiality between the CME
leading front and “EIT wave”, i.e., the slow-component EUV wave. The
simulation reproduces the CME leading front followed by a dark cavity, which
correspond to the “EIT wave” and the dimming region, respectively. Besides,
the “EIT wave” speed is about a third of the piston-driven shock ahead when
propagating along the solar surface, supporting the magnetic field-line
stretching model for the “EIT waves” (Chen et al., 2002).

It is worth pointing out that this event has also been studied by Yamasaki et
al. (2022) with their zero-$\beta$ data-constrained simulation, focusing on
the eruption mechanism and the reconnection process. Some differences emanate
from the comparison between these two models. In their paper, they constructed
the initial NLFFF with the direct extrapolation, so the eastern portion of the
filament located above weak fields fails to be constructed very well.
Accordingly, the flare ribbons in this region are barely reproduced. In our
paper, we construct the initial magnetic field with the RBSL flux-rope
technique and MF relaxation, which is particularly capable to construct the
flux rope that is in agreement with the observational filament located in
weak-field regions. As a result, the corresponding flare ribbons in our
simulations are more accordant with observed ribbons. Therefore, we conclude
that the RBSL flux-rope model is extremely advantageous to construct the
twisted magnetic structure and further study the eruption originating from the
weak-field regions.

Although our simulation has reproduced many observational features and
revealed several physical processes, the deficiencies are also apparent.
First, flare ribbon R2 is not evident in the synthesized EUV images. We
perceive that the primary reason is probably the resolution of our simulation.
Owing to the restrictions of the computing resources and observational
resolutions, the grid spacing in our simulation is about 1 Mm, which is
significantly lower than in other numerical radiative MHD models. For example,
the highest spatial resolution can attain 192 km in Cheung et al. (2019), 64
km in Chen et al. (2022) and 25 km in Chen et al. (2021). Thus, the energy
exchange between the corona and transition region might be difficult to be
captured very well in our simulation. The second reason might be the treatment
of energy terms. Our simulation adopts the optically thin radiative losses and
ignores the radiation transfer in the low atmosphere, which is likely to
affect the morphology of the flare ribbons. Other than these, our MHD model
neglects the contribution of the non-thermal energetic particles, which
enhance the chromsopheric evaporation and the brightening of flare ribbons
(Kontar et al., 2011; Ruan et al., 2020). The contribution of fast electrons
should be more significant for flare ribbon R2 due to the magnetic mirror
effect (Yang et al., 2012), since it has weaker magnetic field than flare
ribbon R1. We believe that future data-driven radiative MHD simulations
combined with fast electron physics and radiation transfer are bound to better
match the observations. In addition to the mismatch of flare ribbon R2, the
second drawback of our simulation is the driving duration. This simulation
only involves the eruption phase of the flux rope, whereas the buildup process
of the magnetic free energy is not simulated, being substituted by the NLFFF
model. In future work, we attempt to develop the long-term evolution of the
active region and pay more attention to the buildup process of the CME source
region.

The numerical calculations in this paper were performed in the cluster system
of the High Performance Computing Center (HPCC) of Nanjing University. The SDO
data are available courtesy of NASA/SDO and the AIA and HMI science teams. The
EUVI data are available courtesy of the STEREO/SECCHI consortium. This
research was supported by National Key Research and Development Program of
China (2020YFC2201200, 2022YFF0503004), NSFC (12127901, 11961131002, and
11773016, 11533005). J.H.G was supported by China Scholarship Council under
file No. 202206190140. SP acknowledges support from the projects C14/19/089
(C1 project Internal Funds KU Leuven), G.0D07.19N (FWO-Vlaanderen), SIDC Data
Exploitation (ESA Prodex-12), and Belspo project B2/191/P1/SWiM.

## References

  * Antiochos et al. (1999) Antiochos, S. K., DeVore, C. R., & Klimchuk, J. A. 1999, ApJ, 510, 485, doi: 10.1086/306563
  * Aulanier & Dudík (2019) Aulanier, G., & Dudík, J. 2019, A&A, 621, A72, doi: 10.1051/0004-6361/201834221
  * Aulanier et al. (2010) Aulanier, G., Török, T., Démoulin, P., & DeLuca, E. E. 2010, ApJ, 708, 314, doi: 10.1088/0004-637X/708/1/314
  * Berger & Prior (2006) Berger, M. A., & Prior, C. 2006, Journal of Physics A: Mathematical and General, 39, 8321, doi: 10.1088/0305-4470/39/26/005
  * Burlaga et al. (1981) Burlaga, L., Sittler, E., Mariani, F., & Schwenn, R. 1981, J. Geophys. Res., 86, 6673, doi: 10.1029/JA086iA08p06673
  * Carmichael (1964) Carmichael, H. 1964, in NASA Special Publication, Vol. 50, 451
  * Chen et al. (2022) Chen, F., Rempel, M., & Fan, Y. 2022, ApJ, 937, 91, doi: 10.3847/1538-4357/ac8f95
  * Chen (2006) Chen, P. F. 2006, ApJ, 641, L153, doi: 10.1086/503868
  * Chen (2009) —. 2009, ApJ, 698, L112, doi: 10.1088/0004-637X/698/2/L112
  * Chen (2011) —. 2011, Living Reviews in Solar Physics, 8, 1, doi: 10.12942/lrsp-2011-1
  * Chen (2016) —. 2016, Washington DC American Geophysical Union Geophysical Monograph Series, 216, 381, doi: 10.1002/9781119055006.ch22
  * Chen et al. (1999) Chen, P. F., Fang, C., Ding, M. D., & Tang, Y. H. 1999, ApJ, 520, 853, doi: 10.1086/307477
  * Chen et al. (2005) Chen, P. F., Fang, C., & Shibata, K. 2005, ApJ, 622, 1202, doi: 10.1086/428084
  * Chen et al. (2012) Chen, P. F., Su, J. T., Guo, Y., & Deng, Y. Y. 2012, Chinese Science Bulletin, 57, 1393, doi: 10.1007/s11434-011-4829-9
  * Chen et al. (2002) Chen, P. F., Wu, S. T., Shibata, K., & Fang, C. 2002, ApJ, 572, L99, doi: 10.1086/341486
  * Chen et al. (2021) Chen, Y., Przybylski, D., Peter, H., et al. 2021, A&A, 656, L7, doi: 10.1051/0004-6361/202140638
  * Cheng et al. (2012) Cheng, X., Zhang, J., Saar, S. H., & Ding, M. D. 2012, ApJ, 761, 62, doi: 10.1088/0004-637X/761/1/62
  * Cheung & DeRosa (2012) Cheung, M. C. M., & DeRosa, M. L. 2012, ApJ, 757, 147, doi: 10.1088/0004-637X/757/2/147
  * Cheung et al. (2015) Cheung, M. C. M., De Pontieu, B., Tarbell, T. D., et al. 2015, ApJ, 801, 83, doi: 10.1088/0004-637X/801/2/83
  * Cheung et al. (2019) Cheung, M. C. M., Rempel, M., Chintzoglou, G., et al. 2019, Nature Astronomy, 3, 160, doi: 10.1038/s41550-018-0629-3
  * Chiu & Hilton (1977) Chiu, Y. T., & Hilton, H. H. 1977, ApJ, 212, 873, doi: 10.1086/155111
  * Colgan et al. (2008) Colgan, J., Abdallah, J., J., Sherrill, M. E., et al. 2008, ApJ, 689, 585, doi: 10.1086/592561
  * Dalmasse et al. (2015) Dalmasse, K., Chandra, R., Schmieder, B., & Aulanier, G. 2015, A&A, 574, A37, doi: 10.1051/0004-6361/201323206
  * Demoulin et al. (1997) Demoulin, P., Bagala, L. G., Mandrini, C. H., Henoux, J. C., & Rovira, M. G. 1997, A&A, 325, 305
  * Demoulin et al. (1996) Demoulin, P., Henoux, J. C., Priest, E. R., & Mandrini, C. H. 1996, A&A, 308, 643
  * Demoulin et al. (1993) Demoulin, P., van Driel-Gesztelyi, L., Schmieder, B., et al. 1993, A&A, 271, 292
  * Devi et al. (2022) Devi, P., Chandra, R., Awasthi, A. K., Schmieder, B., & Joshi, R. 2022, arXiv e-prints, arXiv:2211.07438.  https://arxiv.org/abs/2211.07438
  * Dudík et al. (2014) Dudík, J., Janvier, M., Aulanier, G., et al. 2014, ApJ, 784, 144, doi: 10.1088/0004-637X/784/2/144
  * Fan (2017) Fan, Y. 2017, ApJ, 844, 26, doi: 10.3847/1538-4357/aa7a56
  * Fan (2020) —. 2020, ApJ, 898, 34, doi: 10.3847/1538-4357/ab9d7f
  * Fan (2022) —. 2022, ApJ, 941, 61, doi: 10.3847/1538-4357/aca0ec
  * Fan & Gibson (2007) Fan, Y., & Gibson, S. E. 2007, ApJ, 668, 1232, doi: 10.1086/521335
  * Fisher et al. (2020) Fisher, G. H., Kazachenko, M. D., Welsch, B. T., et al. 2020, ApJS, 248, 2, doi: 10.3847/1538-4365/ab8303
  * Forbes (2000) Forbes, T. G. 2000, J. Geophys. Res., 105, 23153, doi: 10.1029/2000JA000005
  * Gibson & Fan (2006) Gibson, S. E., & Fan, Y. 2006, Journal of Geophysical Research (Space Physics), 111, A12103, doi: 10.1029/2006JA011871
  * Gou et al. (2015) Gou, T., Liu, R., & Wang, Y. 2015, Sol. Phys., 290, 2211, doi: 10.1007/s11207-015-0750-8
  * Guo et al. (2021a) Guo, J. H., Ni, Y. W., Qiu, Y., et al. 2021a, ApJ, 917, 81, doi: 10.3847/1538-4357/ac0cef
  * Guo et al. (2022) Guo, J. H., Ni, Y. W., Zhou, Y. H., et al. 2022, A&A, 667, A89, doi: 10.1051/0004-6361/202244253
  * Guo et al. (2021b) Guo, J. H., Zhou, Y. H., Guo, Y., et al. 2021b, ApJ, 920, 131, doi: 10.3847/1538-4357/ac17e8
  * Guo et al. (2017) Guo, Y., Cheng, X., & Ding, M. 2017, Science China Earth Sciences, 60, 1408, doi: 10.1007/s11430-017-9081-x
  * Guo et al. (2016a) Guo, Y., Xia, C., & Keppens, R. 2016a, ApJ, 828, 83, doi: 10.3847/0004-637X/828/2/83
  * Guo et al. (2019a) Guo, Y., Xia, C., Keppens, R., Ding, M. D., & Chen, P. F. 2019a, ApJ, 870, L21, doi: 10.3847/2041-8213/aafabf
  * Guo et al. (2016b) Guo, Y., Xia, C., Keppens, R., & Valori, G. 2016b, ApJ, 828, 82, doi: 10.3847/0004-637X/828/2/82
  * Guo et al. (2019b) Guo, Y., Xu, Y., Ding, M. D., et al. 2019b, ApJ, 884, L1, doi: 10.3847/2041-8213/ab4514
  * Guo et al. (2021c) Guo, Y., Zhong, Z., Ding, M. D., et al. 2021c, ApJ, 919, 39, doi: 10.3847/1538-4357/ac10c8
  * Hayashi et al. (2018) Hayashi, K., Feng, X., Xiong, M., & Jiang, C. 2018, ApJ, 855, 11, doi: 10.3847/1538-4357/aaacd8
  * Hayashi et al. (2019) —. 2019, ApJ, 871, L28, doi: 10.3847/2041-8213/aaffcf
  * Hirayama (1974) Hirayama, T. 1974, Sol. Phys., 34, 323, doi: 10.1007/BF00153671
  * Hoeksema et al. (2014) Hoeksema, J. T., Liu, Y., Hayashi, K., et al. 2014, Sol. Phys., 289, 3483, doi: 10.1007/s11207-014-0516-8
  * Hood & Priest (1981) Hood, A. W., & Priest, E. R. 1981, Geophysical and Astrophysical Fluid Dynamics, 17, 297, doi: 10.1080/03091928108243687
  * Hou et al. (2022) Hou, Z., Tian, H., Wang, J.-S., et al. 2022, ApJ, 928, 98, doi: 10.3847/1538-4357/ac590d
  * Hudson (2011) Hudson, H. S. 2011, Space Sci. Rev., 158, 5, doi: 10.1007/s11214-010-9721-4
  * Inoue et al. (2018) Inoue, S., Kusano, K., Büchner, J., & Skála, J. 2018, Nature Communications, 9, 174, doi: 10.1038/s41467-017-02616-8
  * Janvier et al. (2013) Janvier, M., Aulanier, G., Pariat, E., & Démoulin, P. 2013, A&A, 555, A77, doi: 10.1051/0004-6361/201321164
  * Jiang et al. (2021a) Jiang, C., Bian, X., Sun, T., & Feng, X. 2021a, Frontiers in Physics, 9, 224, doi: 10.3389/fphy.2021.646750
  * Jiang et al. (2022) Jiang, C., Feng, X., Guo, Y., & Hu, Q. 2022, The Innovation, 3, 100236, doi: 10.1016/j.xinn.2022.100236
  * Jiang et al. (2018a) Jiang, C., Feng, X., & Hu, Q. 2018a, ApJ, 866, 96, doi: 10.3847/1538-4357/aadd08
  * Jiang et al. (2013) Jiang, C., Feng, X., Wu, S. T., & Hu, Q. 2013, ApJ, 771, L30, doi: 10.1088/2041-8205/771/2/L30
  * Jiang et al. (2016a) Jiang, C., Wu, S. T., Feng, X., & Hu, Q. 2016a, Nature Communications, 7, 11522, doi: 10.1038/ncomms11522
  * Jiang et al. (2016b) Jiang, C., Wu, S. T., Yurchyshyn, V., et al. 2016b, ApJ, 828, 62, doi: 10.3847/0004-637X/828/1/62
  * Jiang et al. (2016c) —. 2016c, ApJ, 828, 62, doi: 10.3847/0004-637X/828/1/62
  * Jiang et al. (2018b) Jiang, C., Zou, P., Feng, X., et al. 2018b, ApJ, 869, 13, doi: 10.3847/1538-4357/aaeacc
  * Jiang et al. (2021b) Jiang, C., Feng, X., Liu, R., et al. 2021b, Nature Astronomy, 5, 1126, doi: 10.1038/s41550-021-01414-z
  * Johnston & Bradshaw (2019) Johnston, C. D., & Bradshaw, S. J. 2019, ApJ, 873, L22, doi: 10.3847/2041-8213/ab0c1f
  * Kaneko et al. (2021) Kaneko, T., Park, S.-H., & Kusano, K. 2021, ApJ, 909, 155, doi: 10.3847/1538-4357/abe414
  * Keppens et al. (2023) Keppens, R., Popescu Braileanu, B., Zhou, Y., et al. 2023, arXiv e-prints, arXiv:2303.03026, doi: 10.48550/arXiv.2303.03026
  * Keppens et al. (2020) Keppens, R., Teunissen, J., Xia, C., & Porth, O. 2020, arXiv e-prints, arXiv:2004.03275.  https://arxiv.org/abs/2004.03275
  * Kilpua et al. (2021) Kilpua, E. K. J., Pomoell, J., Price, D., Sarkar, R., & Asvestari, E. 2021, Frontiers in Astronomy and Space Sciences, 8, 35, doi: 10.3389/fspas.2021.631582
  * Klassen et al. (2000) Klassen, A., Aurass, H., Mann, G., & Thompson, B. J. 2000, A&AS, 141, 357, doi: 10.1051/aas:2000125
  * Kliem et al. (2013) Kliem, B., Su, Y. N., van Ballegooijen, A. A., & DeLuca, E. E. 2013, ApJ, 779, 129, doi: 10.1088/0004-637X/779/2/129
  * Kliem & Török (2006) Kliem, B., & Török, T. 2006, Phys. Rev. Lett., 96, 255002, doi: 10.1103/PhysRevLett.96.255002
  * Kontar et al. (2011) Kontar, E. P., Brown, J. C., Emslie, A. G., et al. 2011, Space Sci. Rev., 159, 301, doi: 10.1007/s11214-011-9804-x
  * Kopp & Pneuman (1976) Kopp, R. A., & Pneuman, G. W. 1976, Sol. Phys., 50, 85, doi: 10.1007/BF00206193
  * Lemen et al. (2012) Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17, doi: 10.1007/s11207-011-9776-8
  * Li et al. (2022) Li, X., Wang, Y., Guo, J., & Lyu, S. 2022, ApJ, 928, L6, doi: 10.3847/2041-8213/ac5b72
  * Liu et al. (2016) Liu, R., Kliem, B., Titov, V. S., et al. 2016, ApJ, 818, 148, doi: 10.3847/0004-637X/818/2/148
  * Liu & Ofman (2014) Liu, W., & Ofman, L. 2014, Sol. Phys., 289, 3233, doi: 10.1007/s11207-014-0528-4
  * Longcope & Kankelborg (2001) Longcope, D. W., & Kankelborg, C. C. 2001, Earth, Planets and Space, 53, 571, doi: 10.1186/BF03353272
  * Lumme et al. (2022) Lumme, E., Pomoell, J., Price, D. J., et al. 2022, A&A, 658, A200, doi: 10.1051/0004-6361/202038744
  * Masuda et al. (1994) Masuda, S., Kosugi, T., Hara, H., Tsuneta, S., & Ogawara, Y. 1994, Nature, 371, 495, doi: 10.1038/371495a0
  * Mei et al. (2012) Mei, Z., Shen, C., Wu, N., et al. 2012, MNRAS, 425, 2824, doi: 10.1111/j.1365-2966.2012.21625.x
  * Mei et al. (2020) Mei, Z. X., Keppens, R., Cai, Q. W., et al. 2020, MNRAS, 493, 4816, doi: 10.1093/mnras/staa555
  * Mok et al. (2016) Mok, Y., Mikić, Z., Lionello, R., Downs, C., & Linker, J. A. 2016, ApJ, 817, 15, doi: 10.3847/0004-637X/817/1/15
  * Nakagawa et al. (1975) Nakagawa, Y., Wu, S. T., & Tandberg-Hanssen, E. 1975, Sol. Phys., 41, 387, doi: 10.1007/BF00154076
  * Ouyang et al. (2015) Ouyang, Y., Yang, K., & Chen, P. F. 2015, ApJ, 815, 72, doi: 10.1088/0004-637X/815/1/72
  * Ouyang et al. (2017) Ouyang, Y., Zhou, Y. H., Chen, P. F., & Fang, C. 2017, ApJ, 835, 94, doi: 10.3847/1538-4357/835/1/94
  * Papaioannou et al. (2022) Papaioannou, A., Kouloumvakos, A., Mishev, A., et al. 2022, arXiv e-prints, arXiv:2202.07927.  https://arxiv.org/abs/2202.07927
  * Petrasso et al. (1979) Petrasso, R. D., Nolte, J. T., Gerassimenko, M., et al. 1979, Sol. Phys., 62, 133, doi: 10.1007/BF00150140
  * Poland & Munro (1976) Poland, A. I., & Munro, R. H. 1976, ApJ, 209, 927, doi: 10.1086/154791
  * Pomoell et al. (2019) Pomoell, J., Lumme, E., & Kilpua, E. 2019, Sol. Phys., 294, 41, doi: 10.1007/s11207-019-1430-x
  * Priest & Démoulin (1995) Priest, E. R., & Démoulin, P. 1995, J. Geophys. Res., 100, 23443, doi: 10.1029/95JA02740
  * Priest & Forbes (1986) Priest, E. R., & Forbes, T. G. 1986, J. Geophys. Res., 91, 5579, doi: 10.1029/JA091iA05p05579
  * Qiu et al. (2007) Qiu, J., Hu, Q., Howard, T. A., & Yurchyshyn, V. B. 2007, ApJ, 659, 758, doi: 10.1086/512060
  * Ruan et al. (2020) Ruan, W., Xia, C., & Keppens, R. 2020, ApJ, 896, 97, doi: 10.3847/1538-4357/ab93db
  * Savcheva et al. (2016) Savcheva, A., Pariat, E., McKillop, S., et al. 2016, ApJ, 817, 43, doi: 10.3847/0004-637X/817/1/43
  * Savcheva et al. (2015) —. 2015, ApJ, 810, 96, doi: 10.1088/0004-637X/810/2/96
  * Scherrer et al. (2012) Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207, doi: 10.1007/s11207-011-9834-2
  * Schou et al. (2012) Schou, J., Scherrer, P. H., Bush, R. I., et al. 2012, Sol. Phys., 275, 229, doi: 10.1007/s11207-011-9842-2
  * Schuck (2008) Schuck, P. W. 2008, ApJ, 683, 1134, doi: 10.1086/589434
  * Shibata (1999) Shibata, K. 1999, Ap&SS, 264, 129, doi: 10.1023/A:1002413214356
  * Somov et al. (2002) Somov, B. V., Kosugi, T., Hudson, H. S., Sakao, T., & Masuda, S. 2002, ApJ, 579, 863, doi: 10.1086/342842
  * Song et al. (2022) Song, H., Li, L., & Chen, Y. 2022, ApJ, 933, 68, doi: 10.3847/1538-4357/ac7239
  * Sturrock (1966) Sturrock, P. A. 1966, Nature, 211, 695, doi: 10.1038/211695a0
  * Thompson et al. (1998) Thompson, B. J., Plunkett, S. P., Gurman, J. B., et al. 1998, Geophys. Res. Lett., 25, 2465, doi: 10.1029/98GL50429
  * Thompson (2009) Thompson, W. T. 2009, Icarus, 200, 351, doi: 10.1016/j.icarus.2008.12.011
  * Titov et al. (2018) Titov, V. S., Downs, C., Mikić, Z., et al. 2018, ApJ, 852, L21, doi: 10.3847/2041-8213/aaa3da
  * Titov et al. (2002) Titov, V. S., Hornig, G., & Démoulin, P. 2002, Journal of Geophysical Research (Space Physics), 107, 1164, doi: 10.1029/2001JA000278
  * Toriumi & Wang (2019) Toriumi, S., & Wang, H. 2019, Living Reviews in Solar Physics, 16, 3, doi: 10.1007/s41116-019-0019-7
  * Török et al. (2010) Török, T., Berger, M. A., & Kliem, B. 2010, A&A, 516, A49, doi: 10.1051/0004-6361/200913578
  * Török et al. (2004) Török, T., Kliem, B., & Titov, V. S. 2004, A&A, 413, L27, doi: 10.1051/0004-6361:20031691
  * Tsurutani et al. (2022) Tsurutani, B. T., Zank, G. P., Sterken, V. J., et al. 2022, arXiv e-prints, arXiv:2209.14545.  https://arxiv.org/abs/2209.14545
  * Vourlidas et al. (2003) Vourlidas, A., Wu, S. T., Wang, A. H., Subramanian, P., & Howard, R. A. 2003, ApJ, 598, 1392, doi: 10.1086/379098
  * Vršnak & Cliver (2008) Vršnak, B., & Cliver, E. W. 2008, Sol. Phys., 253, 215, doi: 10.1007/s11207-008-9241-5
  * Wang et al. (2009) Wang, H., Shen, C., & Lin, J. 2009, ApJ, 700, 1716, doi: 10.1088/0004-637X/700/2/1716
  * Wang et al. (2017) Wang, W., Liu, R., Wang, Y., et al. 2017, Nature Communications, 8, 1330, doi: 10.1038/s41467-017-01207-x
  * Warmuth (2015) Warmuth, A. 2015, Living Reviews in Solar Physics, 12, 3, doi: 10.1007/lrsp-2015-3
  * Webb & Howard (2012) Webb, D. F., & Howard, T. A. 2012, Living Reviews in Solar Physics, 9, 3, doi: 10.12942/lrsp-2012-3
  * Wiegelmann et al. (2006) Wiegelmann, T., Inhester, B., & Sakurai, T. 2006, Sol. Phys., 233, 215, doi: 10.1007/s11207-006-2092-z
  * Xia & Keppens (2016) Xia, C., & Keppens, R. 2016, ApJ, 823, 22, doi: 10.3847/0004-637X/823/1/22
  * Xia et al. (2014) Xia, C., Keppens, R., Antolin, P., & Porth, O. 2014, ApJ, 792, L38, doi: 10.1088/2041-8205/792/2/L38
  * Xia et al. (2018) Xia, C., Teunissen, J., El Mellah, I., Chané, E., & Keppens, R. 2018, ApJS, 234, 30, doi: 10.3847/1538-4365/aaa6c8
  * Xing et al. (2020a) Xing, C., Cheng, X., & Ding, M. D. 2020a, The Innovation, 1, 100059, doi: 10.1016/j.xinn.2020.100059
  * Xing et al. (2020b) Xing, C., Cheng, X., Qiu, J., et al. 2020b, ApJ, 889, 125, doi: 10.3847/1538-4357/ab6321
  * Xu et al. (2022) Xu, Y., Tian, H., Hou, Z., et al. 2022, arXiv e-prints, arXiv:2204.11722.  https://arxiv.org/abs/2204.11722
  * Xu et al. (2020) Xu, Y., Zhu, J., & Guo, Y. 2020, ApJ, 892, 54, doi: 10.3847/1538-4357/ab791b
  * Yamasaki et al. (2022) Yamasaki, D., Inoue, S., Bamba, Y., Lee, J., & Wang, H. 2022, arXiv e-prints, arXiv:2210.14563.  https://arxiv.org/abs/2210.14563
  * Yang et al. (2015) Yang, K., Guo, Y., & Ding, M. D. 2015, ApJ, 806, 171, doi: 10.1088/0004-637X/806/2/171
  * Yang et al. (2012) Yang, Y.-H., Cheng, C. Z., Krucker, S., Hsieh, M.-S., & Chen, N.-H. 2012, ApJ, 756, 42, doi: 10.1088/0004-637X/756/1/42
  * Ye et al. (2021) Ye, J., Cai, Q., Shen, C., et al. 2021, ApJ, 909, 45, doi: 10.3847/1538-4357/abdeb5
  * Zemanová et al. (2019) Zemanová, A., Dudík, J., Aulanier, G., Thalmann, J. K., & Gömöry, P. 2019, ApJ, 883, 96, doi: 10.3847/1538-4357/ab3926
  * Zhang et al. (2022) Zhang, P., Chen, J., Liu, R., & Wang, C. 2022, ApJ, 937, 26, doi: 10.3847/1538-4357/ac8d61
  * Zhao et al. (2017) Zhao, X., Xia, C., Keppens, R., & Gan, W. 2017, ApJ, 841, 106, doi: 10.3847/1538-4357/aa7142
  * Zhong et al. (2021) Zhong, Z., Guo, Y., & Ding, M. D. 2021, Nature Communications, 12, 2734, doi: 10.1038/s41467-021-23037-8
  * Zhou et al. (2021) Zhou, Y.-H., Ruan, W.-Z., Xia, C., & Keppens, R. 2021, A&A, 648, A29, doi: 10.1051/0004-6361/202040254
  * Zhou et al. (2018) Zhou, Y.-H., Xia, C., Keppens, R., Fang, C., & Chen, P. F. 2018, ApJ, 856, 179, doi: 10.3847/1538-4357/aab614
  * Zuccarello et al. (2017) Zuccarello, F. P., Chandra, R., Schmieder, B., Aulanier, G., & Joshi, R. 2017, A&A, 601, A26, doi: 10.1051/0004-6361/201629836

Figure 1: Multi-wavelength composite images of the SDO/AIA at 15:24, 15:32,
15:35 and 15:42 UT on 2021 October 28, showing the evolution of the filament,
flare ribbons and post-flare loops. The blue/red contour lines in panel (b)
correspond to the positive/negative magnetic polarities. Figure 2: (a)
Remapped HMI vector magnetogram as viewed from the top. The $B_{z}$ is plotted
in the background (gray scale), and the purple (red) arrows denote the
horizontal magnetic fields with positive (negative) polarities. (b) AIA 131 Å
image at 15:24 UT on 2021 October 28, where the red dots outline the path of
the inserted flux rope. (c) Selected core field lines of the NLFFF model
viewed from the top, colored in temperature. The temperature is obtained from
the inserted filament and hydrostatic model. (d) Side view of the extrapolated
field lines. (e) The constructed flux rope and inserting prominence with a
zoomed-in view of the black rectangle in panel d, where the pink isosurface
represents the prominence material where the temperature is lower than 20000
K. Figure 3: Snapshots of the simulated magnetic field lines colored in
temperature. The left column shows the evolution of the simulated flux ropes,
which are overlaid on the 304 Å images at (a) 15:26, (c) 15:32 and (e) 15:50
UT. The right column shows the evolution of the flux ropes from the side view
at the same time as the left side. Figure 4: Comparison between photospheric
QSLs (lg $Q>$ 2) and the emission in AIA 1600 Å (a–b) and 304 Å (c–d). QSLs
with the positive (negative) vertical magnetic component are colored in orange
and chartreuse (cyan). The black dashed line in panel (a) illustrates a slice
crossing the flare ribbons. The blue box shows the observation evidence of the
3D reconnection in Section 3.2. (e) Time-distance diagram of the QSLs and 1600
Å image for slice S in panel (a). Figure 5: Comparisons between AIA
observations (a–c) and synthesized EUV radiative images (d–f) observed from
the top at 15:32 UT. Panels (a) and (d) show the 131 Å channel images, panels
(b) and (e) show the 193 Å channel images, and panels (c) and (f) show the 335
Å channel images. Figure 6: Side views of the synthesized EUV radiations at
15:28 (a–c) and 15:44 UT (d–f). Panels from the left to right show the 304 Å
(a,d), 171 Å (b,e) and 94 Å (c,f) images, respectively. Figure 7: Same as the
Figure 6 but for the end views. Figure 8: Reconnection geometries recognized
by the field-line connectivity (a–c), $Q$ distribution (d–f) and $J/B$
distribution (g–i). Columns from the left to right represent the ar-rf
geometry at 15:30 UT, aa-rf geometry at 15:32 UT and breakout-like geometry at
15:34 UT, respectively. (a–c): Pre-reconnection (cyan) and post-reconnection
(purple) field lines, where the transparent surfaces represent the plasma
heated by the reconnection. (d–f): $Q$ distributions near the current sheet,
which are correspond to three types of reconnection geometries displayed in
panels (a–c). (g–i): $J/B$ distributions at the same plane of the second row.
Figure 9: Twist distributions on the photosphere at (a) 15:26 UT and (b) 15:48
UT. (c) Temporal evolution of the toroidal (black line) and poloidal (red
line) flux of the flux rope. The measurement is repeated 10 times to estimate
the errors originating from the uncertainties of the selected regions for the
toroidal flux calculation. And the uncertainty of the poloidal flux is
calculated by the product of the toroidal flux error and the standard
deviation of the twist. The error bars at the bottom represent the mean value
of the uncertainty during the whole process. Three reconnection stages are
divided by two vertical dashed lines. Figure 10: (a) Synthesized EUV 193 Å
image at 15:32 UT. (b) Running-difference image of the EUV 193 Å radiation at
15:32 UT, where the blue solid line outlines the CME leading edge, and the red
dashed line outlines the piston-driven shock. (c) Synthesized white-light
image at 15:32 UT. (d) Averaged temperature distribution at 15:32 UT. Figure
11: Field lines passing through the asterisks in Figure 10a (red: CME leading
edge; cyan and yellow: CME cavity; blue: CME core) on the plane of $x=$10 Mm.
Panel (a) shows the side view, and panel (b) shows the end view overlaid on
the 193 Å image. Figure 12: Panels (a) and (b) respectively show the
synthesized 94 Å and 171 Å images integrated from the $x=7$ Mm to 21 Mm zoomed
in the rectangle region in Figure 10 a. (c) Number density distribution on the
plane of $x=$14 Mm. (d) Temperature distribution on the plane of $x=$14 Mm.
(e) $v_{y}$ distribution on the plane of $x=$14 Mm. (f) $v_{z}$ distribution
on the plane of $x=$14 Mm. (g) The curl of the velocity distribution on the
plane of $x=14$ Mm. (h) Magnetic field lines colored in temperature near the
cusp-shaped structure, where the background transparent slice represents the
number density distribution in panel (c). Figure 13: (a) $Q$ distribution on
the plane of $x=$10.6 Mm at 15:28 UT. Panels (b) and (c) illustrate the Q
distributions zoomed in the purple rectangle region in panel (a) at 15:26 and
15:28 UT, respectively. (d) Time-distance diagram of $Q$ along the slice (red
solid line) in panel (a). Figure 14: Panels (a) illustrates the number density
distributions at 15:32 UT and 15:34 UT along the white dashed line in Figure
10a, respectively. Panel (b) illustrates the time-distance diagram of two
wave-like structures along the red solid line in Figure 10a. Figure 15: Panels
(a) and (d) respectively show the side views of the synthesized white-light
images at 15:30 UT and 15:36 UT. Panels (b) and (e) respectively show the side
views of the synthesized EUV 193 Å running-difference images at 15:30 UT and
15:36 UT. The pink arrows in panels (d) and (e) indicate the CME leading front
and “EIT wave”, respectively. Panels (c) and (f) show the STEREO/COR1 images
with the superposition of EUVI 195 Å running-difference images at 15:46 and
15:51 UT, respectively. The blue (red) dashed lines outline the CME leading
front (forward shock wave), and the solid lines outline the two components of
EUV waves.