# Influence of magnetic reconnection on the eruptive catastrophes of coronal
magnetic flux ropes

Quanhao Zhang 1,2,3,∗, Xin Cheng 4, Rui Liu 2,3,5, Anchuan Song 2,3,6, Xiaolei
Li 2,3,6, and Yuming Wang 1,2,3

###### Abstract

## 1

Large-scale solar eruptive activities have a close relationship with coronal
magnetic flux ropes. Previous numerical studies have found that the
equilibrium of a coronal flux rope system could be disrupted if the axial
magnetic flux of the rope exceeds a critical value, so that the catastrophe
occurs, initiating the flux rope to erupt. Further studies discovered that the
catastrophe does not necessarily exist: the flux rope system with certain
photospheric flux distributions could be non-catastrophic. It is noteworthy
that most previous numerical studies are under the ideal magnetohydrodynamic
(MHD) condition, so that it is still elusive whether there is the catastrophe
associated with the critical axial flux if magnetic reconnection is included
in the flux rope system. In this paper, we carried out numerical simulations
to investigate the evolutions of coronal magnetic rope systems under the ideal
MHD and the resistive condition. Under the ideal MHD condition, our simulation
results demonstrate that the flux rope systems with either too compact or too
weak photospheric magnetic source regions are non-catastrophic versus varying
axial flux of the rope, and thus no eruption could be initiated; if there is
magnetic reconnection in the rope system, however, those flux rope systems
could change to be capable of erupting via the catastrophe associated with
increasing axial flux. Therefore, magnetic reconnection could significantly
influence the catastrophic behaviors of flux rope system. It should be both
the magnetic topology and the local physical parameters related to magnetic
reconnection that determine whether the increasing axial flux is able to cause
flux rope eruptions.

##  2 Keywords:

Sun: filaments – Sun: prominences – Sun: flares – Sun: coronal mass ejections
– Sun: magnetic fields – Sun: activity

##  3 Introduction

Large-scale solar eruptive activities are the principal cause of extreme space
weather in Earth and planetary space environments (Švestka, 2001; Cheng et
al., 2014; Lugaz et al., 2017; Gopalswamy et al., 2018). Different kinds of
large-scale solar eruptions, including prominence/filament eruptions (Li et
al., 2016a; Jenkins et al., 2018; Fan, 2020; Devi et al., 2021), flares (Chen
et al., 2015; Li et al., 2016b; Cheung et al., 2019), and coronal mass
ejections (CMEs, Shen et al., 2014; Lamy et al., 2019; Bemporad et al., 2022),
are generally considered to be different manifestations of the eruptions of
coronal magnetic flux rope (Zhang et al., 2001; Vršnak et al., 2005; Jiang et
al., 2016; Chen et al., 2020; Liu, 2020). Therefore, investigating the
initiation and evolution of flux rope eruptions is crucial not only for
understanding solar eruptions, but also for space weather forecasting. Many
theoretical models have been proposed to shed light on the physical scenario
of coronal flux rope eruptions. These models are based on different kinds of
physical mechanisms, such as ideal magnetohydrodynamic(MHD) instabilities
(Török and Kliem, 2003; Aulanier et al., 2010; Guo et al., 2010; Savcheva et
al., 2012; Keppens et al., 2019), magnetic reconnection (Antiochos et al.,
1999; Chen and Shibata, 2000; Moore et al., 2001; Sterling and Moore, 2004;
Archontis and Hood, 2008; Inoue et al., 2015), and the catastrophes of coronal
flux ropes (Van Tend and Kuperus, 1978; Forbes and Priest, 1995; Lin and van
Ballegooijen, 2002; Kliem et al., 2014; Zhang et al., 2021).

Many theoretical studies suggested that flux rope catastrophes are intriguing
candidates for the source of solar eruptions (e.g., Lin and Forbes, 2000;
Isenberg et al., 1993). In the flux rope catastrophe theory, the onset of the
eruption is approached as the loss of equilibrium of the coronal flux rope
system. Before the onset, the flux rope should be static or quasi-static
(Török et al., 2013; Liu, 2020), indicating that the flux rope system is in
equilibrium. If the equilibrium is not disrupted, the net force on the flux
rope will always be zero, so that its state of motion will remains unchanged,
i.e., no eruption of the flux rope could occur. Therefore, loss of equilibrium
must occur during the onset of the eruption, and the resultant net force
initiates the flux rope to erupt. This is the fundamental scenario of the
catastrophe of coronal flux ropes. The investigation about when and how the
catastrophe occurs is based on the equilibrium manifold (Forbes and Priest,
1995; Isenberg and Forbes, 2007; Kliem et al., 2014), which consists of all
the equilibrium states of the flux rope system. For example, by analytically
deriving the equilibrium manifold as a function of current within the flux
rope, Van Tend and Kuperus (1978) for the first time discovered that there is
a critical value of the current, beyond which there are no neighbouring
equilibrium states so that the catastrophe occurs, which results in a
discontinuous equilibrium manifold; this critical value is referred to as the
catastrophic point. Many more analytical studies have verified that the
catastrophe could occur in various types of coronal flux rope systems,
resulting in the eruption of the flux rope (Priest and Forbes, 1990; Forbes
and Priest, 1995; Lin and Forbes, 2000; Démoulin and Aulanier, 2010; Longcope
and Forbes, 2014; Kliem et al., 2014). In addition, numerical simulations have
also been carried out in many previous studies to investigate the catastrophes
of coronal flux ropes (Forbes, 1990; Chen et al., 2007; Zhang et al., 2017a,
2021). In numerical studies, the equilibrium manifold as a function of a
certain physical parameter is obtained by simulating the equilibrium solutions
of the flux rope system with different values of this parameter. For example,
Su et al. (2011) discovered that the equilibrium manifold versus the axial
magnetic flux of the flux rope is discontinuous at a critical value, beyond
which a catastrophe occurs. Similar results are obtained in many other
numerical studies (Bobra et al., 2008; Su et al., 2009, 2011; Zhang et al.,
2017a, b), indicating that the axial flux of the rope should play an important
role in initiating solar eruptions.

The catastrophe associated with the critical axial flux does not necessarily
exist in coronal flux rope system. Previous numerical studies have
demonstrated that the photospheric flux distributions of the background field
greatly influence the catastrophic behaviors of coronal flux rope systems (Sun
et al., 2007; Zhang et al., 2017a, b). Previous studies found that there are
two types of non-catastrophic flux rope systems: if the photospheric flux is
too concentrated toward the polarity inversion line (hereafter “compact”
cases), or the total magnetic flux originating from the photospheric magnetic
source regions is too weak (hereafter “weak” cases), the equilibrium manifold
as a function of the rope’s axial flux will be continuous so that no
catastrophe could occur. It is still an open question why there is no
catastrophe in these two types of flux rope system; Zhang et al. (2017b)
inferred that the constraint from the background field on the flux rope might
probably plays an considerable role. It is noteworthy that magnetic
reconnection is completely prohibited in those studies, so that their
conclusion about the two types of non-catastrophic flux rope systems is under
the ideal MHD condition. Since the actual solar corona is resistive, there
should be reconnection in actual coronal flux rope systems (Jiang et al.,
2021; Yan et al., 2022; Bian et al., 2022). In fact, magnetic reconnection
could not only change the magnetic topology, resulting in the redistribution
of the Ampère’s force (also known as Lorentz force in many papers), but the
reconnection outflow could also push the flux rope upward (Chen et al., 2004;
Wang et al., 2007; Xue et al., 2016; Cheng et al., 2020). This indicates that
magnetic reconnection could contribute to the force balance of a coronal flux
rope system, implying that the catastrophic behaviors of the rope system might
differ under the ideal MHD and the resistive condition. Therefore, it is
critical to assess the influence of magnetic reconnection on the catastrophic
behaviors of coronal flux rope systems, so as to shed more light on the
catastrophe theory for solar eruptions. To achieve this, we use a
2.5-dimensional MHD numerical model to simulate the catastrophic behaviors of
the coronal flux rope systems with either compact or weak photospheric source
regions. Both the equilibrium manifolds versus the axial magnetic flux under
the ideal MHD and the resistive condition are simulated, based on which the
influence of magnetic reconnection is investigated. The rest of this paper is
arranged as follows: the numerical model and simulating procedures are
introduced in Sect. 4, the simulation results are presented in Sect. 5, and
the conclusion and discussion are given in Sect. 6.

##  4 Numerical model

###  4.1 Basic equations

The numerical model in our simulation is similar as those used in Zhang et al.
(2017a, b). For 2.5-dimensional cases, by assuming all the quantities satisfy
$\partial/\partial z=0$, the magnetic field can be written in form of magnetic
flux function $\psi$:

$\textbf{B}=\triangledown\psi\times\hat{\textbf{\emph{z}}}+B_{z}\hat{\textbf{\emph{z}}}.$
(1)

Here z is the unit vector in $z-$direction. Obviously, the divergence-free
condition ($\triangledown\cdot\textbf{B}=0$) is always satisfied. With the
form given in Equation (1), the MHD equations in our simulations could be
rewritten in the following form:

$\displaystyle\frac{\partial\rho}{\partial
t}+\triangledown\cdot(\rho\textbf{\emph{v}})=0,$ (2)
$\displaystyle\frac{\partial\textbf{\emph{v}}}{\partial
t}+\frac{2}{\rho\beta_{0}}(\vartriangle\psi\triangledown\psi+B_{z}\triangledown
B_{z}+\triangledown\psi\times\triangledown
B_{z})+\textbf{\emph{v}}\cdot\triangledown\textbf{\emph{v}}$
$\displaystyle~{}~{}~{}+\triangledown
T+\frac{T}{\rho}\triangledown\rho+g\hat{\textbf{\emph{y}}}=0,$ (3)
$\displaystyle\frac{\partial\psi}{\partial
t}+\textbf{\emph{v}}\cdot\triangledown\psi-\eta\vartriangle\psi=0,$ (4)
$\displaystyle\frac{\partial B_{z}}{\partial
t}+\triangledown\cdot(B_{z}\textbf{\emph{v}})+(\triangledown\psi\times\triangledown
v_{z})\cdot\hat{\textbf{\emph{z}}}-\eta\vartriangle B_{z}=0,$ (5)
$\displaystyle\frac{\partial T}{\partial t}-\frac{\eta(\gamma-1)}{\rho
R}\left[(\vartriangle\psi)^{2}+|\triangledown\times(B_{z}\hat{\textbf{\emph{z}}})|^{2}\right]$
$\displaystyle~{}~{}~{}+\textbf{\emph{v}}\cdot\triangledown
T+(\gamma-1)T\triangledown\cdot\textbf{\emph{v}}=0,$ (6)

where

$\vartriangle\psi=\frac{\partial^{2}\psi}{\partial
x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}},~{}~{}\vartriangle
B_{z}=\frac{\partial^{2}B_{z}}{\partial
x^{2}}+\frac{\partial^{2}B_{z}}{\partial y^{2}}.$ (7)

Here $\rho$ is the density, v the velocity, $T$ the temperature, $\gamma=5/3$
the polytropic index, $g$ the gravity, $\eta$ the resistivity, and
$\beta_{0}=0.1$ the characteristic ratio of the gas pressure to the magnetic
pressure, which is comparable to the value in quiescent regions (Anzer and
Heinzel, 2007; Hillier et al., 2012; Xia et al., 2012). The $x$, $y$, and
$z-$component of the quantities are denoted by the subscript $x,y,z$. The
radiation and the heat conduction in the energy equation are neglected. The
numerical domain in our simulation is $0<x<200$ Mm, $0<y<500$ Mm; it is
discretized into 200$\times$250 uniform meshes.

###  4.2 Initial state

Figure 1: The magnetic configurations of the background fields and initial
states in the simulation. Panel (a) is the background field for Case I, and
panel (e) is the corresponding initial state; panels (b) and (f), (c) and (g),
(d) and (h) are those for Case II, III, IV, respectively. The black curves are
the magnetic field lines, and the green curves mark the boundary of the flux
rope. The physical parameters for the background fields and initial states are
tabulated in Table 1.

The initial states in our simulations are constructed by numerical procedures.
Here we first use the complex variable method (e.g., Hu and Liu, 2000; Zhang
et al., 2017a) to construct the background field originating from photospheric
magnetic source regions. The background field is a partially open bipolar
field, with a negative surface magnetic charge located at the lower boundary
within $-b<x<-a$, and a positive one within $a<x<b$; the distance between the
two source regions is $D=2a$, and the width of the source region $W=b-a$. The
background field in $x-y$ plane could then be expressed in the complex
variable form as:

$f(\omega)\equiv B_{x}-iB_{y}=\lambda\frac{(\omega+iy_{N})^{1/2}(\omega-
iy_{N})^{1/2}}{F(a,b,y_{N})}\mathrm{ln}\left(\frac{\omega^{2}-a^{2}}{\omega^{2}-b^{2}}\right),$
(8)

where $\omega=x+iy$,

$\displaystyle
F(a,b,y_{N})=\frac{1}{b-a}\int_{a}^{b}(x^{2}+y_{N}^{2})^{1/2}dx=\frac{1}{2(b-a)}\times$
$\displaystyle\left[b(b^{2}+y_{N}^{2})^{1/2}-a(a^{2}+y_{N}^{2})^{1/2}+y_{N}^{2}\mathrm{ln}\left(\frac{b+(b^{2}+y_{N}^{2})^{1/2}}{a+(a^{2}+y_{N}^{2})^{1/2}}\right)\right],$
(9)

and the neutral point of the background field is at ($y=y_{N}$, $x=0$). As
indicated by Equation (8), the magnetic field strength of the background field
is proportional to the dimensionless parameter $\lambda$. The magnetic flux
function within the domain could then be calculated by:

$\psi(x,y)=\mathrm{Im}\left\\{\int f(\omega)d\omega\right\\},$ (10)

With calculated the flux function $\psi$, and by letting $B_{z}=0$ in the
background field, the magnetic configuration of the background field is
obtained (Equation (1)). In particular, the flux function of the background
field at the lower base, $\psi_{i}$, could be calculated as:

$\psi_{i}=\psi(x,0)=\left\\{\begin{array}[]{ll}{\lambda\widehat{\psi_{0}}\pi(b-a)},&{|x|<a}\\\
{\lambda\widehat{\psi_{0}}\pi(b-a)F(|x|,b,y_{N})/F(a,b,y_{N})},&{a\leqslant|x|\leqslant
b}\\\ {0},&{|x|>b}\end{array}\right.$ (11)

and the flux function at the neutral point of the background field is

$\psi_{N}=\psi(0,y_{N})=\frac{\lambda\widehat{\psi_{0}}\pi(b^{2}-a^{2})}{2F(a,b,y_{N})}$
(12)

Here $\psi_{c}=\lambda\pi(b-a)\widehat{\psi_{0}}$
($\widehat{\psi_{0}}=37.25\mathrm{~{}Mx~{}cm^{-2}}$) is the total magnetic
flux emanating upward from the positive photospheric source region per unit
length along $z-$direction, i.e. $\psi_{c}$ represents the intensity of
background magnetic field originating from the photospheric source regions.
The ratio of the background field’s open magnetic flux to its total magnetic
flux could then be calculated by:

$\kappa=\frac{\psi_{N}}{\psi_{c}}.$ (13)

For a given background magnetic field, its characteristic physical parameters
include $D$, $W$, $\psi_{c}$ and $\kappa$, which could be calculated with a
group of given $a$, $b$, $y_{N}$, and $\lambda$. For example, by letting
$a=5.00$ Mm, $b=7.00$ Mm, $y_{N}=4.49$ Mm, $\lambda=1.0$, the calculated
background configuration is plotted in Fig. 1(a), and the characteristic
parameters of this background field are: $D=10$ Mm, $W=2$ Mm,
$\psi_{c}=2.34\times 10^{10}$ Mx cm-1, and $\kappa=0.80$, as tabulated in the
second column (Case I) in Table 1. The initial corona in our simulations is
static and isothermal:

$T_{c}\equiv T(0,x,y)=1\times 10^{6}~{}\mathrm{K},\ \
\rho_{c}\equiv\rho(0,x,y)=\rho_{0}\mathrm{e}^{-gy},$ (14)

where $\rho_{0}=3.34\times 10^{-13}$ kg m-3.

With the background field obtained above, we let a flux rope emerge from the
lower base of the domain with similar simulating procedures as those
introduced in, e.g., Zhang et al. (2017a) and Zhang et al. (2020), and then
relax the flux rope system to equilibrium. The resultant equilibrium state
consisting of a flux rope embedded in the background field is the constructed
initial state. For the background field plotted in Fig. 1(a), the constructed
initial state is illustrated in Fig. 1(e). The magnetic properties of a flux
rope could be characterized by its magnetic fluxes, including the total axial
flux passing through the rope’s cross section, $\Phi_{z}=\iint
B_{z}\mathrm{d}S$, and the annular flux per unit length along $z$-direction of
the rope, $\Phi_{p}$; in 2.5 dimensional cases, the poloidal flux is
calculated as:

$\Phi_{p}=\psi_{rc}-\psi_{rb},$ (15)

where $\psi_{rc}$ and $\psi_{rb}$ are the flux function $\psi$ at the center
and the boundary of the flux rope, respectively. For the initial state
illustrated in Fig. 1(e), $\Phi_{z0}=1.416\times 10^{19}$ Mx and
$\Phi_{p0}=7.450\times 10^{9}$ Mx cm-1. With this initial state, we could
further simulate the catastrophic behaviors of the coronal flux rope system
with the given photospheric magnetic conditions (detailed simulating
procedures are introduced in Sect. 4.3). It is noteworthy that the radius of
the flux rope in our simulation is finite, so that the thin-rope approximation
is not satisfied. Under this circumstance, the initial state could only be
obtained by numerical procedures.

In our simulations, apart from those plotted in Fig. 1(a) and Fig. 1(e), we
also construct another three groups of background field and initial state with
similar procedures introduced above, and their parameters are tabulated in
Table 1; all these 4 cases are marked as Case I$\sim$IV, respectively. The
background field and the initial state in Case II are illustrated in Fig. 1(b)
and Fig. 1(f), respectively. As shown by the the second and the third column
in Table 1, $D$, $W$, and $\kappa$ in Case I and Case II are the same; the
only difference in the characteristic parameters of the background field
between these two cases is that $\psi_{c}$ in Case II is much smaller than
that in Case I (achieved by adjusting $\lambda$ to 0.1 in Case II), i.e. the
intensity of the background magnetic field originating from the photospheric
source regions in Case II is much weaker than those in Case I. For Case III
and Case IV, the corresponding background fields and initial states are
illustrated in Fig. 1(c)-1(d) and Fig. 1(g)-1(h), respectively. As shown in
the last two columns in Table 1, the characteristic parameters of the
background field in these two cases are the same, except that the distance
between the source regions, $D$, is much smaller in Case IV than that in Case
III.

Table 1: The parameters of the background fields and the initial states for
Case I-IV.

| Case I | Case II | Case III | Case IV
---|---|---|---|---
$a$ (Mm) | 5.00 | 5.00 | 10.00 | 2.00
$b$ (Mm) | 7.00 | 7.00 | 30.00 | 22.00
$y_{n}$ (Mm) | 4.49 | 4.49 | 1999.89 | 1199.93
$\lambda$ | 1.00 | 0.10 | 0.10 | 0.10
$D$ (Mm) | 10.00 | 10.00 | 20.00 | 4.00
$W$ (Mm) | 2.00 | 2.00 | 20.00 | 20.00
$\psi_{c}$ (Mx cm-1) | 2.340$\times 10^{10}$ | 0.234$\times 10^{10}$ | 2.340$\times 10^{10}$ | 2.340$\times 10^{10}$
$\kappa$ | 0.80 | 0.80 | 0.01 | 0.01
$\Phi_{z0}$ (Mx) | 1.416$\times 10^{19}$ | 1.118$\times 10^{18}$ | 5.960$\times 10^{19}$ | 2.421$\times 10^{19}$
$\Phi_{p0}$ (Mx cm-1) | $7.450\times 10^{9}$ | $7.450\times 10^{8}$ | $1.490\times 10^{10}$ | $1.490\times 10^{10}$

  * *

Here $a$, $b$, $y_{n}$, and $\lambda$ are the parameters used to construct the
background field; $D$, $W$, $\psi_{c}$, and $\kappa$ are the corresponding
characteristic physical parameters of the background field; $\Phi_{z0}$ and
$\Phi_{p0}$ are the magnetic fluxes of the initial flux rope.

###  4.3 Simulating procedures

Starting from each of the four initial states obtained above, we simulate the
equilibrium states of the corresponding flux rope system with different axial
magnetic fluxes of the rope: from $0\sim 20\tau_{A}$ ($\tau_{A}=17.4$ s is the
typical Alfvén transit time), the axial flux $\Phi_{z}$ and the poloidal flux
$\Phi_{p}$ are adjusted from the initial values to a certain group of target
values ($\Phi_{z}^{1}$, $\Phi_{p}^{t}$), and from $20\sim 300\tau_{A}$, the
magnetic system is relaxed to equilibrium, with the magnetic fluxes fixed at
the target values ($\Phi_{z}^{1}$, $\Phi_{p}^{t}$). We note that the
conservation of the poloidal flux is achieved by fixing the flux function at
the rope center, $\psi_{rc}$, during the relaxation (Equation (15)). The final
state at $t=300\tau_{A}$ is regarded as the equilibrium state of the flux rope
system for $\Phi_{z}^{1}$. During the relaxation, magnetic reconnection is
either included or completely prohibited during the relaxation, corresponding
to the resistive condition and ideal MHD condition, respectively. For the
simulations under the resistive condition, anomalous resistivity is used:

$\displaystyle\eta=\begin{cases}0,&~{}j\leq j_{c}\\\
\eta_{m}\mu_{0}v_{0}L_{0}(\frac{j}{j_{c}}-1)^{2}.&~{}j>j_{c}\end{cases}$ (16)

Here $\eta_{m}$=0.10, $L_{0}=10^{7}$ m, $v_{0}=128.57$ km s-1,
$j_{c}=2.96\times 10^{-5}$ A m-2, and $\mu_{0}$ is the vacuum magnetic
permeability. On the other hand, for the simulations under the ideal MHD
conditions, we use similar simulating procedures as those in Zhang et al.
(2017a, b) during the relaxation to prohibit the reconnection: first set the
resistivity $\eta$ to be $0$, and then reassign the flux function $\psi$ along
the current sheet with the initial values at each time step, so as to keep
$\psi$ invariant along the current sheet. Since any reconnection will reduce
the value of $\psi$ at the current sheet, both physical and numerical
reconnections are prohibited with the simulating procedures introduced above.
For the given target values $\Phi_{z}^{1}$, we are able to simulate two
equilibrium states with the simulating procedures above: if reconnection is
prohibited during the relaxation, the resultant equilibrium state is under
ideal MHD condition, whereas the resultant equilibrium state is under
resistive condition if reconnection is included during the relaxation.

Similar simulations are repeated for different target values (($\Phi_{z}^{2}$,
$\Phi_{p}^{t}$), ($\Phi_{z}^{3}$, $\Phi_{p}^{t}$), …) so as to obtain
equilibrium states with different $\Phi_{z}$. Eventually, all the calculated
equilibrium states without reconnection constitute a equilibrium manifold
under the ideal MHD condition, and all those with reconnection constitute the
equilibrium manifold under the resistive condition.

In our simulations, the quantities at the lower boundary of the domain are
fixed, so that the lower boundary corresponds to the photosphere. Symmetric
boundary condition is used at the left side of the domain ($x=0$). At the
other boundaries, increment equivalent extrapolation (Zhang et al., 2017a, b)
is used to prescribe the boundary quantities:

$U^{n+1}_{b}=U^{n+1}_{b-1}+U^{n}_{b}-U^{n}_{b-1},$ (17)

where $U$ represents the quantities. The superscript $n$ denote the quantities
at the current time step, and $n+1$ the next time step; the subscript $b$
indicates the quantities at the boundary, and $b-1$ those at the grids next to
the boundary.

##  5 Simulation results

###  5.1 Weak cases

The equilibrium manifold as a function of the axial flux $\Phi_{z}$ under the
ideal MHD condition for Case I is plotted in Fig. 2(a). All the equilibrium
states have the same $\Phi_{p}=7.45\times 10^{9}$ Mx cm-1. Obviously, this
equilibrium manifold is discontinuous: the flux rope keeps sticking to the
photosphere (Fig. 2(b)-2(c)) before reaching a critical axial flux at about
$1.527\times 10^{18}$ Mx, which is marked by the black vertical dotted line in
Fig. 2(a); otherwise, the flux rope jumps upward (Fig. 2(d)), indicating that
a catastrophe occurs. Therefore, the flux rope system in Case I is
catastrophic under the ideal MHD condition.

The equilibrium manifold under the ideal MHD condition for Case II is plotted
in Fig. 2(e), and all the equilibrium states have the same
$\Phi_{p}=2.980\times 10^{9}$ Mx cm-1. Different from that in Fig. 2(a), the
equilibrium manifold under the ideal MHD condition for Case II is continuous:
the height of the rope axis, $H$, gradually rises as $\Phi_{z}$ increases, and
the magnetic configurations of some equilibrium states are illustrated in Fig.
2(f)-2(h). The evolution of the flux rope system during the relaxation to
simulate the equilibrium state in Fig. 2(g) is shown in Fig. 3(a1)-3(a4).
After $\Phi_{z}$ is increased to $1.613\times 10^{18}$ Mx at $t=20\tau_{A}$
(Fig. 3(a1)), the flux rope gradually rises and then is suspended in the
corona (Fig. 3(a2)-3(a3)), until it reaches equilibrium (Fig. 3(a4), the same
as Fig. 2(g)). Although a current sheet is formed by the magnetic field of
opposite directions below the rope, the magnetic reconnection is prohibited by
the simulating procedures introduced in Sect. 4.3. These simulation results
demonstrate that the flux rope system in Case II is non-catastrophic so that
no eruption could be initiated under the ideal MHD condition. As introduced in
Sect. 4.2, the only difference in the characteristic parameters of the
background field in Case I and Case II is $\psi_{c}$: the intensity of the
background magnetic field originating from the photospheric source regions in
Case II is much weaker than those in Case I, i.e. Case II is a “weak” case.
This result is consistent with the conclusions in Zhang et al. (2017a, b).

Figure 2: The simulation results for Case I and Case II. Panel (a) is the
equilibrium manifold for Case I under the ideal MHD condition; panels (b)-(d)
are the magnetic configuration of the simulated equilibrium states for Case I,
and the corresponding $\Phi_{z}$ is marked at the top of each panel. Panels
(e)-(h) and panels (i)-(l) are the simulation results for Case II under the
ideal MHD and the resistive condition, respectively; the red vertical dotted
line in panel (i) denote the catastrophic point $\Phi_{zc}^{w}$. The green
curves mark the boundary of the flux rope.

To investigate the influence of magnetic reconnection on the catastrophic
behaviors of flux rope systems with weak photospheric source regions, we also
simulate the equilibrium manifold under the resistive condition for Case II,
as plotted in Fig. 2(i). Obviously, if there is magnetic reconnection in the
flux rope system, the equilibrium manifold is discontinuous: there is a
critical axial magnetic flux $\Phi_{zc}^{w}\approx 1.617\times 10^{18}$ Mx,
which is marked by the red vertical dotted line in Fig. 2(i). As shown in Fig.
2(i), $H$ gradually rises as $\Phi_{z}$ increases before reaching
$\Phi_{zc}^{w}$. Fig. 2(k) illustrate the magnetic configuration of the
equilibrium state with $\Phi_{z}=1.613\times 10^{18}$ Mx, i.e., the
equilibrium state right before $\Phi_{zc}^{w}$ is reached, and the evolution
of the flux rope system to reach this equilibrium state is shown in Fig.
3(b1)-3(b4). Comparing with the simulation results without reconnection in
Fig. 3(a1)-3(a4), it is demonstrated that magnetic reconnection occurs in the
current sheet below the flux rope (Fig. 3(b2)-3(b3)), resulting in closed
arcades below the rope. The flux rope does not further rises but remains
suspended, and eventually evolves to equilibrium (Fig. 3(b4), the same as Fig.
2(g)). As introduced in Sect. 4.2, $B_{z}=0$ in the background field, so that
the reconnection has no effect on the total axial magnetic flux of the rope.
If $\Phi_{z}$ increases to reach $\Phi_{zc}^{w}$, the flux rope is initiated
to erupt, as shown by the evolution of the flux rope system illustrated in
Fig. 3(c1)-3(c4): the flux rope does not remains suspended in the corona, but
keeps rising after $\Phi_{z}$ is increased to $\Phi_{zc}^{w}$, resulting in
the eruption of the flux rope. Obviously, the evolutions of the flux rope
system before and after $\Phi_{z}$ reaches $\Phi_{zc}^{w}$ are quite different
if magnetic reconnection is included, indicating that there is a catastrophe
in the flux rope system, and $\Phi_{zc}^{w}$ is the catastrophic point. We
note that there is no equilibrium state if the catastrophe occurs under the
ressitive condition, so that we illustrate the state at $t=180\tau_{A}$ in
Fig. 2(l) (the same as Fig. 3(c4)) as the characteristic state for
$\Phi_{z}=\Phi_{zc}^{w}$, and the corresponding red point in Fig. 2(i) is
plotted at the top boundary. Our simulation results suggest that the magnetic
flux rope system with weak photospheric source regions could also be
catastrophic if magnetic reconnection is included in the system.

Figure 3: The temporal evolutions of the flux rope for Case II. Panels
(a1)-(a4) illustrate the evolution of the flux rope during the relaxation to
simulate the equilibrium state in Fig. 2(g). Panels (b1)-(b4) illustrate the
evolution of the flux rope during the relaxation to simulate the equilibrium
state in Fig. 2(k), which is the equilibrium state right before
$\Phi_{z}=\Phi_{zc}^{w}$ is reached under the resistive condition; Panels
(b1)-(b4) illustrate the evolution right after $\Phi_{z}=\Phi_{zc}^{w}$ is
reached under the resistive condition.

###  5.2 Compact cases

The simulation results under the ideal MHD condition for Case III and Case IV
are illustrated in Fig. 4(a)-4(d) and Fig. 4(e)-4(h), respectively, in which
all the simulated equilibrium states have the same $\Phi_{p}=1.490\times
10^{10}$ Mx cm-1. As introduced in Sect. 4.2 and Table 1, the distance between
the photospheric source regions is much smaller in Case IV than that in Case
III. As shown in Fig. 4(a), the flux rope system in Case III is catastrophic:
the catastrophe occurs if $\Phi_{z}$ reaches $6.072\times 10^{19}$ Mx, as
marked by the black vertical dotted line in Fig. 4(a). The flux rope system in
Case IV, however, is non-catastrophic; its equilibrium manifold is continuous.
This indicates Case IV is a “compact” case, which is also consistent with
Zhang et al. (2017a, b). The evolution of the flux rope system to reach the
equilibrium state in Fig. 4(g) is shown in Fig. 5(a1)-5(a4).

Figure 4: The simulation results for Case III and Case IV. Panel (a)-(d) are
the simulation results for Case III under the ideal MHD condition. Panels
(e)-(h) and panels (i)-(l) are the simulation results for Case IV under the
ideal MHD condition and the resistive condition, respectively; the red
vertical dotted line in panel (i) denote the catastrophic point
$\Phi_{zc}^{c}$. The green curves mark the boundary of the flux rope.

Similar as those in Sect. 5.1, we also simulate the equilibrium manifold under
the ressitive condition for Case IV, so as to investigate the influence of
magnetic reconnection on the catastrophic behaviors of flux rope systems with
compact photospheric source regions. As plotted in Fig. 4(i), there is a
critical axial magnetic flux $\Phi_{zc}^{c}\approx 2.727\times 10^{19}$ Mx if
magnetic reconnection is included in the simulation, as marked by the red
vertical dotted lines in Fig. 4(i). The evolutions of the flux rope system
before and after $\Phi_{z}$ reaches $\Phi_{zc}^{c}$ are quite different: if
$\Phi_{z}$ is smaller than $\Phi_{zc}^{c}$, the flux rope does not erupt but
evolves to equilibrium (Fig. 5(b1)-5(b4)); if $\Phi_{zc}^{c}$ is reached, the
flux rope erupts(Fig. 5(c1)-5(c4)). This indicates that a catastrophe
associated with the increasing axial magnetic flux of the rope could occur
under the ressitive condition. For comparison, the topologies of the
equilibrium states before and after $\Phi_{z}$ reaches $\Phi_{zc}^{c}$ are
quite similar if there is no reconnection (Fig. 4(g) and Fig. 4(h)). This
further confirms that it is the magnetic reconnection that changes the
catastrophic behaviors of the flux rope system versus varying axial flux.
Therefore, the flux rope system with compact photospheric source regions could
also be catastrophic if there is magnetic reconnection in the rope system.

Figure 5: The temporal evolution of the flux rope for Case IV. Panels
(a1)-(a4) illustrate the evolution of the flux rope during the relaxation to
simulate the equilibrium state in Fig. 4(g). Panels (b1)-(b4) illustrate the
evolution of the flux rope during the relaxation to simulate the equilibrium
state in Fig. 4(k), which is the equilibrium state right before
$\Phi_{z}=\Phi_{zc}^{c}$ is reached under the resistive condition; Panels
(b1)-(b4) illustrate the evolution right after $\Phi_{z}=\Phi_{zc}^{c}$ is
reached under the resistive condition.

##  6 Discussion and Conclusion

In this paper, we investigate the influence of magnetic reconnection on the
catastrophic behaviors of coronal flux rope systems associated with increasing
axial magnetic flux of the rope. Under the ideal MHD condition, our simulation
results demonstrate that coronal flux rope systems with either too compact or
too weak photospheric source regions are non-catastrophic versus varying axial
magnetic flux. Under the resistive condition, however, both the flux rope
system with too compact and too weak photospheric source regions could change
to be catastrophic: the catastrophe occurs when the axial flux of the rope
increases to reach the critical value, so that the eruption of the flux rope
is initiated.

Our simulation results demonstrate that the equilibrium manifolds of a coronal
flux rope system could be quite different under the ideal MHD and the
resistive condition (e.g., Fig. 2(e) and Fig. 2(i)). This indicates that
magnetic reconnection should play a considerable role in determining the force
balance and the corresponding the loss of equilibrium associated with
increasing axial magnetic flux in coronal flux rope systems. We note that the
continuous equilibrium manifold (Fig. 2(e)) is different from the
discontinuous equilibrium manifold (Fig. 2(i)) in essence. It is widely
accepted that solar eruptions should be caused by photospheric and coronal
disturbances (Lin et al., 2003), the spatial range of which, however, is much
smaller than that of solar eruptions (Priest, 2014). If the equilibrium
manifold of a flux rope system is continuous, the variations of $\Phi_{z}$
caused by photospheric or coronal disturbances could only cause flux rope
movements on a spatial scale comparable to the disturbance itself. In
contrast, if the equilibrium manifold is discontinuous, there is a critical
axial flux of $\Phi_{z}$, so that even an infinitesimal increment of
$\Phi_{z}$ to reach this critical value could cause the catastrophe of the
flux rope, resulting in much larger scale of movement of the flux rope than
that of the disturbances. Therefore, solar eruptions could only be initiated
in the coronal flux rope system whose equilibrium manifold is discontinuous.

In previous studies, the forecasting of solar eruptions is generally based on
the photospheric magnetic flux distributions within solar active regions
(e.g., Leka and Barnes, 2007; Bobra and Couvidat, 2015; Toriumi and Wang,
2019; Korsós et al., 2020). However, as revealed by our simulation results,
not only do different photospheric flux distributions result in different
evolutions of the flux rope system, but the catastrophic behaviors of a
particular flux rope system with given photospheric flux distribution could
also differ under the ideal MHD condition and the resistive condition. This
implies that two active regions with similar photospheric flux distribution
might be quite different in their ability to initiate solar eruptions,
provided that the local resistivity and characteristic spatial scale differs
in the two active regions. Therefore, it should not be sufficient to predict
whether a solar active region will be a potential source region for solar
eruptions based solely on its photospheric magnetic flux distributions. Our
simulation results suggest that both the magnetic topology and the local
physical parameters related to magnetic reconnection determine whether
increasing axial flux is able to cause flux rope eruptions.

There are also catastrophes associated with other physical parameters of flux
rope systems, such as increasing shear of the background field (Chen et al.,
2006) and decreasing mass of the rope (Jenkins et al., 2019; Zhang et al.,
2021). It could be inferred that the catastrophic behaviors associated with
those parameters might probably also be determined by both the magnetic
topology and whether the magnetic system is resistive. In our future work, we
will carry out more simulations to further investigate the influence of
magnetic reconnection on different kinds catastrophes.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

Quanhao Zhang carried out the numerical simulation, analyzed the simulation
results, and prepared the manuscript. Xin Cheng contributed to the initial
inspiration of this study, and the improvement of the numerical model. Rui Liu
contributed to the analysis of the simulation results. Anchuan Song, Xiaolei
Li, and Yuming Wang joined in the discussions about the simulation results.

## Funding

This research is supported by the National Natural Science Foundation of China
(NSFC 42174213, 42188101, 42130204, 41804161, 41774178, 41774150), the
Strategic Priority Program of CAS (XDB41000000 and XDA15017300), the
Informatization Plan of Chinese Academy of Sciences, Grant No.CAS-
WX2021PY-0101, and the Key Research Program of the Chinese Academy of
Sciences, Grant NO. ZDBS-SSW-TLC00103. This research is also supported by USTC
Research Funds of the Double First-Class Initiative.

## Acknowledgments

The authors acknowledge for the support from National Space Science Data
Center, National Science `&` Technology Infrastructure of China
(www.nssdc.ac.cn). Quanhao Zhang acknowledge for the support from Young Elite
Scientist Sponsorship Program by the China Association for Science and
Technology (CAST).

## Data Availability Statement

The simulation results in this study are included in the article; if there is
any further inquiry, please contact the corresponding author.

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