# Comparison of the Hall Magnetohydrodynamics and Magnetohydrodynamics
evolution of a flaring solar active region

K. Bora  Udaipur Solar Observatory, Physical Research Laboratory, Dewali, Bari
Road, Udaipur-313001, India  Discipline of Physics, Indian Institute of
Technology, Gandhinagar-382355, India  R. Bhattacharyya  Udaipur Solar
Observatory, Physical Research Laboratory, Dewali, Bari Road, Udaipur-313001,
India  Avijeet Prasad  Center for Space Plasma & Aeronomic Research, The
University of Alabama in Huntsville, Huntsville, AL 35899, USA  Institute of
Theoretical Astrophysics, University of Oslo, Postboks 1029, Blindern No-0315
Oslo, Norway  Rosseland Centre for Solar Physics, University of Oslo, PO Box
1029, Blindern 0315, Oslo, Norway  Bhuwan Joshi  Udaipur Solar Observatory,
Physical Research Laboratory, Dewali, Bari Road, Udaipur-313001, India  Qiang
Hu  Center for Space Plasma & Aeronomic Research, The University of Alabama in
Huntsville, Huntsville, AL 35899, USA  Department of Space Science, The
University of Alabama in Huntsville, Huntsville, AL 35899, USA

(Accepted on November, 20 2021)

###### Abstract

This work analyzes the Hall magnetohydrodynamics (HMHD) and
magnetohydrodynamics (MHD) numerical simulations of a flaring solar active
region as a testbed while idealizing the coronal Alfvén speed to be of two
orders of magnitude lesser. HMHD supports faster magnetic reconnection and
shows richer complexity in magnetic field line evolution compared to the MHD.
The magnetic reconnections triggering the flare are explored by numerical
simulations augmented with relevant multi-wavelength observations. The initial
coronal magnetic field is constructed by non-force-free extrapolation of
photospheric vector magnetic field. Magnetic structure involved in the flare
is identified to be a flux rope, with its overlying magnetic field lines
constituting the quasi-separatrix layers (QSLs) along with a three-dimensional
null point and a null line. Compared to the MHD simulation, the HMHD
simulation shows a higher and faster ascend of the rope together with the
overlying field lines, which further reconnect at the QSL located higher up in
the corona. The foot points of the field lines match better with the
observations for the HMHD case with the central part of the flare ribbon
located at the chromosphere. Additionally, field lines are found to rotate in
a circular pattern in the HMHD, whereas no such rotation is seen in the MHD
results. Interestingly, plasma is also observed to be rotating in a co-spatial
chromospheric region, which makes the HMHD simulation more credible. Based on
the aforementioned agreements, HMHD simulation is found to agree better with
observations and, thus, opens up a novel avenue to explore.

##  1 Introduction

The astrophysical plasmas characterized by high Lundquist number $S\equiv
Lv_{A}/\eta$ ($L\equiv$ length scale of the magnetic field B variability,
$v_{A}\equiv$ Alfvén speed, and $\eta\equiv$ magnetic diffusivity) satisfy the
Alfvén’s flux-freezing theorem in presence of laminar plasma flow, ensuring
magnetic field lines to be tied to fluid parcels (Alfvén, 1942). The scenario
is different in a turbulent magnetofluid, see Lazarian & Vishniac (1999,
2000); Eyink & Aluie (2006) for details. An inherent large $L$ implies large
$S$ and ensures the flux freezing in the astrophysical plasmas. Particularly,
the solar corona with global $L\approx 100~{}\rm Mm$, $v_{A}\approx 10^{6}$
ms-1, B$\approx 10$ G, and $\eta\approx 1$ m2s-1 (calculated using Spitzer
resistivity) has a $S\approx 10^{14}$ (Aschwanden, 2005). However, the coronal
plasma also exhibits diffusive behavior in the form of solar transients—such
as solar flares, coronal mass ejections (CME), and jets. All of these are
manifestations of magnetic reconnections that in turn lead to dissipation of
magnetic energy into heat and kinetic energy of plasma flow, accompanied by a
rearrangement of magnetic field lines (Choudhuri, 1998). The magnetic
reconnections being dissipative processes, their onset is due to the
generation of small scales in consequence of large-scale dynamics, ultimately
increasing the magnetic field gradient and thereby resulting in intermittently
diffusive plasma. The small scales may naturally occur as current sheets (CSs)
(Parker, 1994), magnetic nulls (Parnell et al., 1996; Nayak et al., 2020) and
quasi-separatrix layers (QSLs) (Démoulin, 2006; Prasad et al., 2020), or can
develop spontaneously during the evolution of the magnetofluid. Such
spontaneous developments (owing to discontinuities in the magnetic field) are
expected from Parker’s magnetostatic theorem (Parker, 1994) and have also been
established numerically by MHD simulations (Nayak et al., 2020; Kumar &
Bhattacharyya, 2011; Kumar et al., 2015; Kumar et al., 2016; Kumar et al.,
2017; Prasad et al., 2017; Prasad et al., 2018; Nayak et al., 2019; Kumar et
al., 2021). Identification of the small (viz. the dissipation) scale depends
on the specific physical system under consideration. For example, the length
scale at which the reconnection occurs is found to be
$L_{\eta}\equiv\sqrt{\tau_{d}\eta}\approx$32 m, based on $\eta\approx 1$ m2s-1
and the magnetic diffusion time scale $\tau_{d}$ approximated by the impulsive
rise time of hard X-ray flux $\approx 10^{3}$ s (Priest & Forbes, 2000) during
a flare. Consequently, the estimated ion inertial length scale
$\delta_{i}\approx 2.25$ m in the solar corona (Priest & Forbes, 2000)
suggests that the order of the dissipation term, $1/S\approx 10^{-5}$
(approximated with $L_{\eta}$), is smaller than the order of the Hall term,
$\delta_{i}/L_{\eta}\approx 10^{-2}$, in standard dimensionless induction
equation (Westerberg & Åkerstedt, 2007; Bora et al., 2021)

$\frac{{\partial\bf{B}}}{\partial
t}=\nabla\times\left({\bf{v}}\times{\bf{B}}\right)-\frac{1}{S}\nabla\times{\bf{J}}-\frac{\delta_{i}}{L_{\eta}}\nabla\times\left({\bf{J}}\times{\bf{B}}\right)~{},$
(1)

where ${\bf{J}}(=\nabla\times{\bf{B}})$ and ${\bf{v}}$ are the volume current
density and the plasma flow velocity, respectively. This difference in the
order of magnitude irrefutably indicates the importance of the Hall term in
the diffusive limit  (Birn et al., 2001; Bhattacharjee, 2004) of the solar
coronal plasma which, further signifies that the HMHD can play a crucial role
for coronal transients as the magnetic reconnections are their underlying
mechanism. Importantly, the aforesaid activation of the Hall term only in the
diffusive limit is crucial in setting up a HMHD based numerical simulation,
invoked latter in the paper.

Important insight into magnetic reconnection can be gained by casting (1) in
absence of dissipation, as

$\frac{{\partial\bf{B}}}{\partial
t}=\nabla\times\left({\bf{w}}\times{\bf{B}}\right)~{},$ (2)

following Hornig & Schindler (1996). The velocity
${\bf{w}}={\bf{v}}-\delta_{i}/L_{\eta}{\bf{J}}$, which is also the electron
fluid velocity, conserves magnetic flux (Schindler et al., 1988) and topology
(Hornig & Schindler, 1996) since field lines are tied to it. Consequently,
field lines slip out from the fluid parcels advecting with velocity v to which
the lines are frozen in ideal MHD. Importantly, the resulting breakdown of the
flux freezing is localized to the region where current density is large and
the Hall term is effective. Because of the slippage, two fluid parcels do not
remain connected with the same field lines over time—a change in field line
connectivity. Quoting Schindler et al. (1988), such localized breakdown of
flux freezing along with the resulting change in connectivity can be
considered as the basis of reconnection (Axford, 1984). Additional slippage of
field lines occur in presence of the dissipation term, but with a change in
magnetic topology. The present paper extensively relies on this interpretation
of reconnection as the slippage of magnetic field lines and the resulting
change in magnetic connectivity.

The importance of HMHD is by no means limited to coronal transients. For
example, HMHD is important in the Earth’s magnetosphere, particularly at the
magnetopause and the magnetotail where CSs are present (Mozer et al., 2002).
Generally, the HMHD is expected to support faster magnetic reconnections, yet
without directly affecting the dissipation rate of magnetic energy and
helicity by the Hall term in the induction equation (Priest & Forbes, 2000;
Shi et al., 2019). The faster reconnection may be associated with a more
effective slippage of field lines in HMHD compared to the resistive MHD,
compatible with the arguments presented earlier. Nevertheless, these unique
properties of the HMHD are expected to bring subtle changes in the dynamical
evolution of plasma, particularly in the small scales dominated by magnetic
reconnections, presumably bringing a change in the large scales as a
consequence. Such subtle changes were found in the recent HMHD simulation
(Bora et al., 2021), performed by extending the computational model EULAG-MHD
(Smolarkiewicz & Charbonneau, 2013) to include the Hall effects. Notably, the
faster reconnection compared to MHD led to a breakage of a magnetic flux rope,
generated from analytically constructed initial bipolar magnetic field lines
(Kumar et al., 2016). In turn, the flux rope breakage resulted in the
generation of magnetic islands as theorized by Shibata & Tanuma (2001).
Clearly, it is compelling to study the HMHD evolution in a more realistic
scenario with the initial magnetic field obtained from a solar magnetogram. To
attain such an objective, we select the recently reported active region (AR)
NOAA 12734 by Joshi et al. (2021) that produced a C1.3 class flare.

In absence of reliable direct measurement of the coronal magnetic field,
several extrapolation models such as nonlinear force-free field (NLFFF)
(Wiegelmann, 2008; Wiegelmann & Sakurai, 2012) and non-force-free field (non-
FFF) (Hu & Dasgupta, 2008; Hu et al., 2010) have been developed to construct
the coronal magnetic field using photospheric magnetograms. The standard is
the NLFFF, and the recent data-based MHD simulations initialized with it have
been reasonably successful in simulating the dynamics of various coronal
transients (Jiang et al., 2013; Amari et al., 2014; Inoue et al., 2014;
Savcheva et al., 2016). However, the NLFFF extrapolations require to treat the
photosphere as force-free, while it is actually not so (Gary, 2001). Hence, a
“preprocessing technique” is usually employed to minimize the Lorentz force on
the photosphere in order to provide a boundary condition suitable for NLFFF
extrapolations (Wiegelmann et al., 2006; Jiang & Feng, 2014) and thereby
compromising the reality. Recently, the non-force-free-field (non-FFF) model,
based on the principle of minimum energy dissipation rate (Bhattacharyya &
Janaki, 2004; Bhattacharyya et al., 2007), has emerged as a plausible
alternative to the force-free models (Hu & Dasgupta, 2008; Hu et al., 2010,
2008). In the non-FFF model, the magnetic field B satisfies the double-curl-
Beltrami equation (Mahajan & Yoshida, 1998) and the corresponding Lorentz
force on the photosphere is non-zero while it decreases to small values at the
coronal heights (Prasad et al., 2018; Nayak et al., 2019; Prasad et al.,
2020)—concurring with the observations. In this paper, we use non-FFF
extrapolation (Hu et al., 2010) to obtain the magnetic field in corona using
the photospheric vector magnetogram obtained from the Helioseismic Magnetic
Imager (HMI) (Schou et al., 2012) onboard the Solar Dynamics Observatory (SDO)
(Pesnell et al., 2012).

The paper is organized as follows. Section 2 describes the flaring event in AR
NOAA 12734, section 3 presents magnetic field lines morphology of AR NOAA
12734 along with the preferable sites for magnetic reconnections such as QSLs,
3D null point, and null-line found from the non-FFF extrapolation. Section 4
focuses on the numerical model, numerical set-up and the evolution of magnetic
field lines obtained from the extrapolation along with their realizations in
observations. Section 5 highlights the key findings.

##  2 Salient features of the C1.3 class flare in AR NOAA 12734

The AR NOAA 12734 produced an extended C1.3 class flare on March 08, 2019
(Joshi et al., 2021). The impulsive phase of the flare started at 03:07 UT as
reported in the Figure 3 of Joshi et al. (2021), which shows the X-ray flux in
the 1-8 Å and 0.5-4 Å detected by the Geostationary Operational Environmental
Satellite (GOES) (Garcia, 1994). The flux evinces two subsequent peaks after
the onset of the flare, one around 03:19 UT and another roughly around 03:38
UT. Joshi et al. (2021) suggested the eruptive event to take place in a
coronal sigmoid with two distinct stages of energy release. Additional
observations using the multi-wavelength channels of Atmospheric Imaging
Assembly (AIA) (Lemen et al., 2012) onboard SDO are listed below to highlight
important features pertaining to simulations reported in this paper. Figure 1
illustrates a spatio-temporal observational overview of the event. Panel (a)
shows the remote semicircular brightening (C1) prior to the impulsive phase of
the flare (indicated by the yellow arrow). Panels (b) to (d) indicate the
flare by yellow arrow and the eruption by the white arrow in the 94 Å, 171 Å,
and 131 Å channels respectively. Notably, the W-shaped brightening appears in
panels (b) to (d) along with the flare in different wavelength channels of
SDO/AIA. Panel (e) shows the circular structure of the chromospheric material
(C2) during the impulsive phase of the flare. It also highlights the developed
W-shaped flare ribbon (enclosed by the white box) which has a tip at the
center (marked by the white arrow). Panel (f) depicts the post-flare loops in
171 Å channel, indicating the post-flare magnetic field line connectivity
between various negative and positive polarities on the photosphere.

##  3 non-FFF Extrapolation of the AR NOAA 12734

As stated upfront, the non-FFF extrapolation technique proposed by Hu &
Dasgupta (2008) and based on the minimum dissipation rate theory (MDR)
(Bhattacharyya & Janaki, 2004; Bhattacharyya et al., 2007) is used to obtain
the coronal magnetic field for the AR NOAA 12734. The extrapolation
essentially solves the equation

$\displaystyle\nabla\times\nabla\times\nabla\times\textbf{B}+a_{1}\nabla\times\nabla\times\textbf{B}+b_{1}\nabla\times\textbf{B}=0~{},$
(3)

where parameters $a_{1}$ and $b_{1}$ are constants. Following (Hu et al.,
2010), the field is constructed as

$\displaystyle\textbf{B}=\sum_{i=1,2,3}\textbf{B}_{i}~{},~{}~{}\nabla\times\textbf{B}_{i}=\alpha_{i}\textbf{B}_{i}~{},$
(4)

where $\alpha_{i}$ is constant for a given $\textbf{B}_{i}$. The subfields
$\textbf{B}_{1}$ and $\textbf{B}_{3}$ are linear force-free having
$\alpha_{1}\neq\alpha_{3}$, whereas $\textbf{B}_{2}$ is a potential field with
$\alpha_{2}=0$. An optimal pair of $\alpha=\\{\alpha_{1},\alpha_{3}\\}$ is
iteratively found by minimizing the average deviation between the observed
transverse field ($\textbf{B}_{t}$) and the computed ($\textbf{b}_{t}$)
transverse field, quantified by

$E_{n}=\left(\sum_{i=1}^{M}|\textbf{B}_{t,i}-\textbf{b}_{t,i}|\times|\textbf{B}_{t,i}|\right)/\left(\sum_{i=1}^{M}|\textbf{B}_{t,i}|^{2}\right)~{},$
(5)

on the photosphere. Here, $M=N^{2}$ represents the total number of grid points
on the transverse plane. The grid points are weighted with respect to the
strength of the observed transverse field to minimize the contribution from
weaker fields, see (Hu & Dasgupta, 2008; Hu et al., 2010) for further details.

Since (3) involves the evaluation of the second-order derivative,
$(\nabla\times\nabla\times\textbf{B})_{z}=-(\nabla^{2}\textbf{B})_{z}$ at
$z=0$, evaluation of B requires magnetograms at two different values of $z$.
In order to work with the generally available single-layer vector
magnetograms, an algorithm was introduced by Hu et al. (2010) that involves
additional iterations to successively fine-tune the potential subfield
$\textbf{B}_{2}$. The system is reduced to second order by taking initial
guess $\textbf{B}_{2}=0$, which makes it easier to determine the boundary
condition for $\textbf{B}_{1}$ and $\textbf{B}_{3}$. If the calculated value
of $E_{n}$ turns out unsatisfactory—i.e., overly large—then a potential field
corrector to $\textbf{B}_{2}$ is calculated from the difference in the
observed and computed transverse fields and subsequently summed with the
previous $\textbf{B}_{2}$ to further reduce $E_{n}$. Notably, recent
simulations initiated with the non-FFF model have successfully explained the
circular ribbon-flares in AR NOAA 12192 (Prasad et al., 2018) and AR NOAA
11283 (Prasad et al., 2020) as well as a blowout jet in AR NOAA 12615 (Nayak
et al., 2019), thus validating non-FFF’s credibility.

The vector magnetogram is selected for 2019 March 08, at 03:00 UT ($\approx$ 7
minutes prior to the start of flare). The original magnetogram cut out of
dimensions 342$\times$195 pixels with pixel resolution 0.5 arcsec per pixel
having an extent of $124~{}\rm Mm\times 71$ Mm from
“hmi.sharp$\\_$cea$\\_$720s" series is considered, which ensures an
approximate magnetic flux balance at the bottom boundary. To optimize the
computational cost with the available resources, the original field is re-
scaled and non-FFF-extrapolated over a volume of 256$\times$128$\times$128
pixels while keeping the physical extent same and preserving all magnetic
structures throughout the region. The reduction, in effect, changes the
conversion factor of 1 pixel to $\approx 0.484$ Mm along x and $\approx 0.554$
Mm along y and z directions of the employed Cartesian coordinate system.

Panel (a) of Figure 2 shows $E_{n}$ in the transverse field, defined in (5),
as a function of number of iterations. It shows that $E_{n}$ tends to saturate
at the value of $\approx$0.22. Panel (b) of Figure 2 shows logarithmic decay
of the normalized horizontally averaged magnetic field, current density, and
Lorentz force with height. It is clear that the Lorentz force is appreciable
on the photosphere but decays off rapidly with height, agreeing with the
general perception that the corona is force-free while the photosphere is not
(Liu et al., 2020; Sarp Yalim et al., 2020). Panel (c) shows that the
Pearson-r correlation between the extrapolated and observed transverse fields
is $\approx$0.96, implying strong correlation. The direct volume rendering of
the Lorentz force in panel (d) also reveals a sharp decay of the Lorentz force
with height, expanding on the result of panel (b).

To facilitate description, Figure 3 (a) shows the SDO/AIA 304 Å image at 03:25
UT, where the flare ribbon brightening has been divided into four segments
marked as B1-B4. Figure 3 (b) shows the initial global magnetic field line
morphology of AR NOAA 12734, partitioned into four regions R1-R4,
corresponding to the flare ribbon brightening segments B1-B4. The bottom
boundary of panel (b) comprises of $B_{z}$ maps in grey scale where the
lighter shade indicates positive polarity regions and the darker shade marks
the negative polarity regions. The magnetic field lines topologies and
structures belonging to a specific region and contributing to the flare are
documented below.

Region R1: The top-down view of the global magnetic field line morphology is
shown in the panel (a) of Figure 4. To help locate QSLs, the bottom boundary
is overlaid with the $\log Q$ map of the squashing factor $Q$ (Liu et al.,
2016) in all panels of the figure. Distribution of high $Q$ values along with
$B_{z}$ on the bottom boundary helps in identifying differently connected
regions. The region with a large $Q$ is prone to the onset of slipping
magnetic reconnections (Démoulin, 2006). Foot points of magnetic field lines
constituting QSL1 and QSL2 trace along the high $Q$ values near the bottom
boundary. QSL1, involving the magnetic field lines Set I (green) and Set II
(maroon), is shown in panel (b). Particularly, magnetic field lines Set I
(green) extends higher in the corona forming the largest loops in R1. Panel
(c) illustrates a closer view of QSL2 (multicolored) and the flux rope (black)
beneath, situated between the positive and negative polarities P1, P2 and N1,
respectively. In panel (d), the flux rope (constituted by the twisted black
magnetic field lines) is depicted using the side view. The twist value $T_{w}$
(Liu et al., 2016) in the three vertical planes along the cross section of the
flux rope is also overlaid. Notably, the twist value is 2 at the center of the
rope and decreases outward (cf. vertical plane in middle of the flux rope in
panel (d)).

Region R2: Figure 5 (a) shows the side view of a 3D null point geometry of
magnetic field lines and the bottom boundary $B_{z}$ overlaid with log $Q$
ranging between 5 and 10. Panel (b) depicts an enlarged view of the 3D null
location, marked black. The height of the null is found to be $\approx$ 3 Mm
from the photosphere. The null is detected using the bespoke procedure (Kumar
& Bhattacharyya, 2011; Nayak et al., 2020) that approximates the Dirac delta
on the grid as

$n(B_{i})=\exp\big{[}-\sum_{i=x,y,z}{(B_{i}-B_{o})^{2}}/{d_{o}^{2}}\big{]}~{},$
(6)

where small constants $B_{o}$ and $d_{o}$ correspond to the isovalue of
$B_{i}$ and the Gaussian spread. The function $n(B_{i})$ takes significant
values only if $B_{i}\approx 0~{}\forall i$, whereupon a 3D null is the point
where the three isosurfaces having isovalues $B_{i}=B_{o}$ intersect.

Region R3: Side view of the magnetic field line morphology in region R3 is
shown in Figure 5 (c), where the yellow surface corresponds to $n=0.9$. Panel
(d) highlights a “fish-bone-like” structure, similar to the schematic in
Figure 5 of Wang et al. (2014). To show that in the limiting case $n=0.9$
reduced to a null line, we plot corresponding contours in the range $0.6\leq
n\leq 0.9$ on three pre-selected planes highlighted in panel (e). The size
reduction of the contours with increasing $n$ indicates the surface converging
to a line. Such null lines are also conceptualized as favorable reconnection
sites (Wang et al., 2014).

Region 4 Figure 5 (f) shows magnetic field lines relevant to plasma rotation
in B4. Notably, the null line from the R3 intrudes into R4 and the extreme
left plane in R3 (Figure 5 (e)) is also shared by the R4.

##  4 HMHD and MHD simulations of AR NOAA 12734

###  4.1 Governing Equations and Numerical Model

In the spirit of our earlier related works (Prasad et al., 2018; Nayak et al.,
2019; Prasad et al., 2020), the plasma is idealized to be incompressible and
thermodynamically inactive as well as explicitly nonresistive. While this
relatively simple idealization is naturally limited, it exposes the basic
dynamics of magnetic reconnections unobscured by the effects due to
compressibility and heat transfer. Albeit the latter are important for coronal
loops (Ruderman & Roberts, 2002), they do not directly affect the magnetic
topology—in focus of this paper. Historically rooted in classical
hydrodynamics, such idealizations have a proven record in theoretical studies
of geo/astrophysical phenomena (Rossby et al., 1938; Dahlburg et al., 1991;
Bhattacharyya et al., 2010; Bora et al., 2021). Inasmuch as their cognitive
value depends on an a posteriori validation against the observations, the
present study offers yet another opportunity to do so.

The Hall forcing has been incorporated (Bora et al., 2021) in the
computational model EULAG-MHD (Smolarkiewicz & Charbonneau, 2013) to solve the
dimensionless HMHD equations,

$\displaystyle\frac{\partial{\bf v}}{\partial t}+({\bf v}\cdot\nabla){\bf v}$
$\displaystyle=$ $\displaystyle-\nabla p+(\nabla\times{\bf B})\times{\bf
B}+\frac{1}{R_{F}^{A}}\nabla^{2}{\bf v}~{},$ (7)
$\displaystyle\frac{\partial{\bf B}}{\partial t}$ $\displaystyle=$
$\displaystyle\nabla\times(\textbf{v}\times{\bf
B})-d_{H}\nabla\times((\nabla\times{\bf B})\times{\bf B})~{},$ (8)
$\displaystyle\nabla\cdot{\bf v}$ $\displaystyle=$ $\displaystyle 0~{},$ (9)
$\displaystyle\nabla\cdot{\bf B}$ $\displaystyle=$ $\displaystyle 0~{},$ (10)

where $R_{F}^{A}=(v_{A}L/\nu)$, $\nu$ being the kinematic viscosity—is an
effective fluid Reynolds number, having the plasma speed replaced by the
Alfvén speed $v_{A}$. Hereafter $R_{F}^{A}$ is denoted as fluid Reynolds
number for convenience. The transformation of the dimensional quantities
(expressed in cgs-units) into the corresponding non-dimensional quantities,

${\bf{B}}\longrightarrow\frac{{\bf{B}}}{B_{0}},\quad{\bf{x}}\longrightarrow\frac{\bf{x}}{L_{0}},\quad{\bf{v}}\longrightarrow\frac{\bf{v}}{v_{A}},\quad
t\longrightarrow\frac{t}{\tau_{A}},\quad
p\longrightarrow\frac{p}{\rho_{0}{v_{A}}^{2}}~{},$ (11)

assumes arbitrary $B_{0}$ and $L_{0}$ while the Alfvén speed $v_{A}\equiv
B_{0}/\sqrt{4\pi\rho_{0}}$. Here $\rho_{0}$ is a constant mass density, and
$d_{H}$ is the Hall parameter. In the limit of $d_{H}=0$, (7)-(10) reduce to
the MHD equations (Prasad et al., 2018).

The governing equations (7)-(10) are numerically integrated using EULAG-MHD—a
magnetohydrodynamic extension (Smolarkiewicz & Charbonneau, 2013) of the
established Eulerian/Lagrangian comprehensive fluid solver EULAG (Prusa et
al., 2008) predominantly used in atmospheric research. The EULAG solvers are
based on the spatio-temporally second-order-accurate nonoscillatory forward-
in-time advection scheme MPDATA (for multidimensional positive definite
advection transport algorithm) (Smolarkiewicz, 2006). Importantly, unique to
MPDATA is its widely documented dissipative property mimicking the action of
explicit subgrid-scale turbulence models wherever the concerned advective
field is under-resolved; the property known as implicit large-eddy simulations
(ILES) (Grinstein et al., 2007). In effect, magnetic reconnections resulting
in our simulations dissipate the under-resolved magnetic field along with
other advective field variables and restore the flux freezing. These
reconnections being intermittent and local, successfully mimic physical
reconnections.

###  4.2 Numerical Setup

The simulations are carried out by mapping the physical domain of $256\times
128\times 128$ pixels on the computational domain of $x\in\\{-1,1\\}$,
$y\in\\{-0.5,0.5\\}$, $z\in\\{-0.5,0.5\\}$ in a Cartesian coordinate system.
The dimensionless spatial step sizes are $\Delta x=\Delta y=\Delta z\approx
0.0078$. The dimensionless time step is $\Delta t=5\times 10^{-4}$, set to
resolve whistler speed—the fastest speed in incompressible HMHD. The rationale
is briefly presented in the Appendix A. The corresponding initial state is
motionless ($\textbf{v}=0$) and the initial magnetic field is provided from
the non-FFF extrapolation. The non-zero Lorentz force associated with the
extrapolated field pushes the magnetofluid to initiate the dynamics. Since the
maximal variation of magnetic flux through the photosphere is only 2.28$\%$ of
its initial value during the flare (not shown), the $\text{B}_{z}$ at the
bottom boundary (at $z=0$) is kept fixed throughout the simulation while all
other boundaries are kept open. For velocity, all boundaries are set open. The
mass density is set to $\rho_{0}=1$.

The fluid Reynolds number is set to $500$, which is roughly two orders of
magnitude smaller than its coronal value $\approx 25000$ (calculated using
kinematic viscosity $\nu=4\times 10^{9}~{}\rm m^{2}s^{-1}$ (Aschwanden, 2005)
in solar corona). Without any loss in generality, the reduction in $R_{F}^{A}$
can be envisaged to cause a reduction in computed Alfvén speed,
$v_{A}|_{\text{computed}}\approx 0.02\times v_{A}|_{\text{corona}}$ where the
$L$ for the computational and coronal length scales are set to 71 Mm and 100
Mm respectively. This diminished Alfvén speed reduces the requirement of
computational resources and also relates it with the observation time. The
results presented herein pertain to a run for 1200$\Delta t$ which along with
the normalizing $\tau_{A}\approx 3.55\times 10^{3}$ s roughly corresponds to
an observation time of $\approx$ 35 minutes. For the ease of reference in
comparison with observations, we present the time in units of 0.005$\tau_{a}$
(which is 17.75 s) in the discussions of the figures in subsequent sections.

Although the coronal plasma idealized to have reduced Reynolds number is
inconsequential here, in a comparison of MHD and HMHD evolution, we believe
the above rationale merits further contemplation. Undeniably such a coronal
plasma is not a reality. Nevertheless, the reduced $R_{F}^{A}$ does not affect
the reconnection or its consequence, but slows down the dynamics between two
such events and importantly—reduces the computational cost, making data-based
simulations realizable even with reasonable computing resources. A recent work
by Jiang et al. (2016) used homologous approach toward simulating a realistic
and self-consistent flaring region.

In the present simulations, all parameters are identical for the MHD and the
HMHD except for the $d_{H}$, respectively set to 0 and 0.004. The value 0.004
is motivated by recognizing ILES dissipation models intermittent magnetic
reconnections at the ${\mathcal{O}}(\parallel\Delta{\bf x}\parallel)$ length
scales, consistent with the thesis put forward in Introduction, we specify an
appreciable Hall coefficient as $d_{H}=0.5\Delta z/L\approx 0.004$, where
$L=1\equiv$ smallest extent of the computational volume, having $\Delta
y=\Delta z\approx 0.0078$ as the dissipation scales because of the ILES
property of the model. Correspondingly, the value is also at the lower bound
of the pixel or scale order approximation and, in particular, an order of
magnitude smaller that its coronal value valid at the actual dissipation
scale. An important practical benefit of this selection is the optimization of
the computational cost while keeping magnetic field line dynamics tractable.
Importantly, with dissipation and Hall scales being tied, an increased current
density at the dissipation scale introduces additional slippage of field lines
in HMHD over MHD (due to the Hall term) and, may be responsible for more
effective and faster reconnections found in the Hall simulation reported
below.

###  4.3 Comparison of the HMHD and MHD simulations

The simulated HMHD and MHD dynamics leading to the flare show unambiguous
differences. This section documents these differences by comparing
methodically simulated evolution of the magnetic structures and topologies in
the AR NOAA 12734—namely, the flux rope, QSLs, and null points—identified in
the extrapolated initial data in the regions R1-R4.

####  4.3.1 Region R1

The dynamics of region R1 are by far the most complex among the four selected
regions. To facilitate future reference as well as to outline the organization
of the discussion that follows, Table 1 provides a brief summary of our
findings—in a spirit of theses to be proven by the simulation results.

Table 1: Salient features of magnetic field lines dynamics in R1 Magnetic field lines structure |  HMHD |  MHD
---|---|---
QSL1 |  Fast reconnection followed by a significant rise of loops, eventually reconnecting higher in the corona. |  Slow reconnection followed by a limited rise of loops.
QSL2 |  Fast reconnection causing the magnetic field lines to entirely disconnect from the polarity P2. |  Due to slow reconnection magnetic field lines remain connected to P2.
Flux rope |  Fast slipping reconnection of the flux-rope foot points, followed by the expansion and rise of the rope envelope. |  Slow slipping reconnection and rise of the flux-rope envelope; the envelope does not reach the QSL1.

The global dynamics of magnetic field lines in region R1 is illustrated in
Figure 6; consult Figure 4 for the initial condition and terminology. The
snapshots from the HMHD and MHD simulations are shown in panels (a)-(d) and
(e)-(f), respectively. In panels (a) and (b), corresponding to $t=19$ and
$t=46$, the foot points of magnetic field lines Set II (near P2, marked
maroon) exhibit slipping reconnection along high values of the squashing
factor $Q$ indicated by black arrows. Subsequently, between $t=80$ and 81 in
panels (c) and (d), the magnetic field lines Set II rise in the corona and
reconnect with magnetic field lines Set I to change connectivity. The MHD
counterpart of the slipping reconnection in panels (e) and (f), corresponds to
magnetic field lines Set II between t=19 and t=113. It lags behind the HMHD
displays, thus implying slower dynamics. Furthermore, the magnetic field lines
Set II, unlike for the HMHD, do not reach up to the magnetic field lines Set I
constituting QSL1 and hence do not reconnect. A more informative visualization
of the highlighted dynamics is supplemented in an online animation. The decay
index is calculated for each time instant for both the simulations and is
found to be less than 1.5 above the flux rope, indicating an absence of the
torus instability (Kliem & Török, 2006). For more detail, Figures 7 and 8
illustrate evolution of QSL2 and flux rope separately.

Figure 7 panels (a)-(b) and (c)-(d) show, respectively, the instants from the
HMHD and MHD simulations of QSL2 between P1, P2 and N1. The HMHD instants show
magnetic field lines that were anchored between P2 and N1 at $t=10$ have moved
to P1 around t=102, marked by black arrows in both panels. The magnetic field
lines anchored at P2 moved to P1 along the high $Q$ values—signifying the
slipping reconnection. The MHD instants in panels (c)-(d) show the
connectivity changes of the violet and white colored magnetic field lines. The
white field line was initially connecting P1 and N1, whereas the violet field
line was connecting P2 and N1. As a result of reconnection along QSL, the
white field line changed its connectivity from P1 to P2 and violet field line
changes the connectivity from P2 to P1 (marked by black arrows). Notably, in
contrast to the HMHD evolution, all magnetic field lines initially anchored in
P2 do not change their connectivity from P2 to P1 during the MHD evolution,
indicating the slower dynamics.

The flux rope has been introduced in panels (c) and (d) of Figure 4,
respectively, below the QSL2 and in enlargement. Its HMHD and MHD evolutions
along with the twists on three different vertical cross sections are shown in
panels (a)-(f) and (g)-(i) of Figure 8, respectively. Magnetic field lines
constituting the rope, rise substantially higher during the HMHD evolution as
a result of slipping reconnection along the high $Q$ in panels (c)-(f). In
panel (c) at $t=32$, the foot points of the rope that are anchored on right
side (marked by black arrow) change their connectivity from one high $Q$
regime to another in panel (d) at t=33; i.e., the foot points on the right
have moved to the left side (marked by black arrow). Afterwards, the magnetic
field lines rise because of the continuous slipping reconnection, as evidenced
in panels (e) to (f) and the supplemented animation. Comparing panels (a) with
(g) at $t=10$ and (c) with (h) at t=32, we note that the twist value $T_{w}$
is higher in the HMHD simulation. Panels (h)-(i) highlight the displaced foot
points of flux rope due to slipping reconnection at t=32 and t=120 (cf. black
arrow). The rope is preserved throughout the HMHD and MHD simulations.

The rise and expansion of the flux-rope envelope owing to slipping
reconnection is remarkable in the HMHD simulation. Dudík et al. (2014) have
already shown such a flux-rope reconnection along QSL in a J-shaped current
region, with slipping reconnection causing the flux rope to form a sigmoid
(S-shaped hot channel observed in EUV images of SDO/AIA) followed by its rise
and expansion. Further insight is gained by overlaying the flux rope evolution
shown in Figure 8 with direct volume rendering of $|{\bf J}|/|{\bf B}|$
(Figures 9 and 10) as a measure of magnetic field gradient for the HMHD and
MHD simulations. In the HMHD case, appearance of large values of $|{\bf
J}|/|{\bf B}|>475$ inside the rope (panels (a) to (c)) and foot points on left
of the rope (panels (d) to (e)) are apparent. The development of the large
$|{\bf J}|/|{\bf B}|$ is indicative of reconnection within the rope.
Contrarily, MHD simulation lacks such high values of $|{\bf J}|/|{\bf B}|$ in
the same time span (panels (a)-(b)) and the field lines show no
slippage—agreeing with the proposal that large currents magnify the Hall term,
resulting into more effective slippage of field lines.

####  4.3.2 Region R2

To compare the simulated magnetic field lines dynamics in region R2 with the
observed tip of the W-shaped flare ribbon B2 (Figure 3 (a)) during the HMHD
and MHD evolution, we present the instants from both simulations at t=70 in
panels (a) and (b) of Figure 11 respectively. Importantly, the lower spine
remains anchored to the bottom boundary during the HMHD simulation (evident
from the supplemented animation along with Figure 11). Further, Figure 12
shows the evolution of the lower spine along with the
$|\textbf{J}|/|\textbf{B}|$ on the bottom boundary for the HMHD (panels (a) to
(d)) and MHD (panels (e) to (h)) cases. In the HMHD case, noteworthy is the
slipping motion of lower spine (marked by the black arrows) tracing the
$|\textbf{J}|/|\textbf{B}|>350$ regions on the bottom boundary (panels (a) to
(b)). Whereas, in the MHD such high values of $|\textbf{J}|/|\textbf{B}|$ are
absent on the bottom boundary—suggesting the slippage of the field lines on
the bottom boundary to be less effective in contrast to the HMHD. The finding
is in agreement with the idea of enhanced slippage of field lines due to high
current densities as conceptualized in the introduction. The anchored lower
spine provides a path for the plasma to flow downward to the brightening
segment B2. In the actual corona, such flows result in flare brightening
(Benz, 2017). In contrast, the lower spine gets completely disconnected from
the bottom boundary (Figure 11 (b)) in the MHD simulation, hence failing to
explain the tip of the W-shaped flare ribbon in B2. The anchored lower spine
in the HMHD simulation is caused by a complex series of magnetic field lines
reconnections at the 3D null and along the QSLs in R2, as depicted in the
animation.

####  4.3.3 Region R3

HMHD and MHD simulations of magnetic field lines dynamics around the null-line
are shown in Figures 13 and 14 respectively. Figure 13 shows the blue magnetic
field lines prior and after the reconnections (indicated by black arrows)
between t=4 to 5 (panels (a)-(b)), t=52 to 53 (panels (c)-(d)), and t=102 to
103 (panels (e)-(f)) during the HMHD simulation. Figure 14 shows the same blue
magnetic field lines prior and after the reconnections (indicated by black
arrows) between t=12 to 13 (panels (a)-(b)), t=59 to 60 (panels (c)-(d)), and
t=114 to 115 (panels (e)-(f)) during the MHD simulation. Comparison of the
panels (a)-(f) of Figure 13 with the same panels of Figure 14 reveals earlier
reconnections of the blue magnetic field lines in the HMHD simulation. In both
figures, green velocity vectors on the right represent the local plasma flow.
They get aligned downward along the foot points of the fan magnetic field
lines, as reconnection progresses. Consequently, the plasma flows downward and
impacts the denser and cooler chromosphere to give rise to the brightening in
B3. The velocity vectors pointing upward represent a flow toward the null-
line. The plasma flow pattern in R3 is the same in the HMHD and in the MHD
simulation. The vertical $yz-$plane passing through the cross section of the
null-line surface (also shown in Figure 5 (d)) in all the panels of Figures 13
and 14 shows the variation of $n$ with time. It is evident that the null is
not destroyed throughout the HMHD and MHD evolution. Structural changes in the
field lines caused by reconnection is near-identical for both the simulations,
indicating inefficacy of the Hall term. This inefficacy is justifiable as
$|\textbf{J}|/|\textbf{B}|$ remains small $\approx 10$ (not shown) in both
HMHD and MHD evolution.

####  4.3.4 Region R4

The development of the circular motion of magnetic field lines in region R4
during the HMHD simulation is depicted in Figure 15. It shows the global
dynamics of magnetic field lines in R4 and the inset images show the zoomed
view of magnetic field lines in R4 to highlight the circular motion of
magnetic field lines. The bottom boundary is $B_{z}$ in the main figure while
the inset images have the $z-$component of the plasma flow at the bottom
boundary (on $xy-$plane). The red vectors represent the plasma flow direction
as well as magnitude in all the panels of Figure 15 where the anticlockwise
pattern of the plasma flow is evident. The global dynamics highlight
reconnection of the loop anchored between positive and negative polarities at
t=60 in Figure 15 as it gets disconnected from the bottom boundary in panels
(c)-(d) of Figure 15. The animation accompanying Figure 15 highlights an
anticlockwise motion of foot points in the same direction as the plasma flow,
indicating field lines to be frozen in the fluid. The trapped plasma may cause
the rotating structure B4 in the observations (c.f. Figure 3 (a)). However, no
such motion is present during the MHD evolution of the same magnetic field
lines (not shown). An interesting feature noted in the animation is the
clockwise slippage of field lines after the initial anticlockwise rotation.
Further analysis of R4 using the direct volume rendering of
$|\textbf{J}|/|\textbf{B}|$ is presented in Figure 16. The figure shows
$|\textbf{J}|/|\textbf{B}|$ attains high values $\geq 225$ (enclosed by the
blue rectangles) within the rotating field lines from t$\approx$86 onward.
This suggests the slippage of field lines is, once again, related to the high
magnetic field gradients.

For completeness, we present the snapshots of an overall magnetic field lines
morphology including the magnetic structures and topology of regions R1, R2,
R3, and R4 together, overlaid with 304 Å and 171 Å from the HMHD and MHD
simulations. Figure 17 (a) shows an instant (at t=75) from the HMHD simulation
where the topologies and magnetic structures in R1, R2, R3, and R4, plus the
additionally drawn locust color magnetic field lines between R2 and R3 are
shown collectively. It shows an excellent match of the magnetic field lines in
R2 with the observed tip of W-shaped flare ribbon at B2, which is pointed out
by the pink arrow in panel (a). Foot points of the spine-fan geometry around
the 3D null orient themselves in the same fashion as the observed tip of the
W-shaped flare ribbon at B2 as seen in 304 Å channel of SDO/AIA. The rising
loops indicated by the white arrow correspond to the same evolution as shown
in Figure 6. An overall magnetic field lines morphology mentioned in Figure 15
(a) is given at the same time (t=75) during the MHD simulation overlaid with
304 Å image in Figure 15 (b). Importantly, unlike the HMHD simulation, the MHD
simulation does not account for the anchored lower spine and fan magnetic
field lines of the 3D null at the center of the B2. Also, the significant rise
of overlying maroon magnetic field lines and the circular motion of the
material in B4 is captured in the HMHD simulation only. In panel (c) magnetic
field lines overlaid with 171 Å image shows the magnetic field lines (higher
up in the solar atmosphere) have resemblance with the post-flare loops during
the HMHD. Overall, the HMHD evolution seems to be in better agreement with the
observations in comparison to the MHD evolution.

##  5 Summary and Discussion

The paper compares data-based HMHD and MHD simulations using the flaring
Active Region NOAA 12734 as a test bed. Importance of the HMHD stems from the
realization that the Hall term in the induction equation cannot be neglected
in presence of the magnetic reconnection—the underlying cause of solar flares.
The selected event is the C1.3 class flare on March 08, 2019 around 03:19 UT
for the aforementioned comparison. Although the event is analyzed and reported
in the literature, it is further explored using the multi-wavelength
observations from SDO/AIA. The identified important features are: an elongated
extreme ultraviolet (EUV) counterpart of the eruption on the western side of
the AR, a W-shaped flare ribbon and circular motion of cool chromospheric
material on the eastern part. The magnetic field line dynamics near these
features are utilized to compare the simulations. Notably, the simulations
idealize the corona to have an Alfvèn speed which is two orders of magnitude
smaller than its typical value. Congruent to the general understanding, the
Hall parameter is selected to tie the Hall dynamics to the dissipation scale
$\mathcal{O}(\Delta\textbf{x})$ in the spirit of the ILES carried out in the
paper. The magnetic reconnection here is associated with the slippage of
magnetic field lines from the plasma parcels, effective at the dissipation
scale due to local enhancement of magnetic field gradient. The same
enhancement also amplifies the Hall contribution, presumably enhancing the
slippage and thereby making the reconnection faster and more effective than
the MHD.

The coronal magnetic field is constructed by extrapolating the photospheric
vector magnetic field obtained from the SDO/HMI observations employing the
non-FFF technique (Hu et al., 2010). The concentrated distribution of the
Lorentz force on the bottom boundary and its decrease with the height justify
the use of non-FFF extrapolation for the solar corona. The initial non-zero
Lorentz force is also crucial in generating self-consistent flows that
initiate the dynamics and cause the magnetic reconnections. Analyses of the
extrapolated magnetic field reveal several magnetic structures and topologies
of interest: a flux rope on the western part at flaring location, a 3D null
point along with the fan-spine configuration at the centre, a “Fish-bone-like
structure" surrounding the null-line on the eastern part of the AR. All of
these structures are found to be co-spatial with the observed flare ribbon
brightening.

The HMHD simulation shows faster slipping reconnection of the flux rope foot
points and overlying magnetic field lines (constituting QSLs above the flux
rope) at the flaring location. Consequently, the overlying magnetic field
lines rise, eventually reaching higher up in the corona and reconnecting to
provide a path for plasma to eject out. The finding is in agreement with the
observed elongated EUV counterpart of the eruption on western part of the AR.
Contrarily, such significant rise of the flux rope and overlying field lines
to subsequently reconnect higher up in the corona is absent in the MHD
simulation—signifying the reconnection to be slower compared to the HMHD.
Intriguingly, rise and expansion of the flux rope and overlying field lines
owing to slipping reconnection on QSLs has also been modelled and observed in
an earlier work by Dudík et al. (2014). These are typical features of the
“standard solar flare model in 3D”, which allows for a consistent explanation
of events which are not causally connected (Dudík et al., 2014). It also
advocates that null-points and true separatrices are not required for the
eruptive flares to occur—concurring the results of this work. HMHD evolution
of the fan-spine configuration surrounding the 3D null point is in better
agreement with the tip of W-shaped flare ribbon at the centre of the AR. The
lower spine and fan magnetic field lines remain anchored to the bottom
boundary throughout the evolution which can account for the plasma flowing
downward after the reconnection and cause the brightening. Whereas in the MHD,
the lower spine gets disconnected and cannot account for the brightening. The
reconnection dynamics around the null-line and the corresponding plasma flow
direction is same in the HMHD as well as the MHD simulation and agrees with
the observed brightening. Nevertheless, reconnection is earlier in the HMHD.
HMHD evolution captures an anti-clockwise circular motion of magnetic field
lines in the left part of the AR which is co-spatial with the location of the
rotating chromospheric material in eastern side of the AR. No such motion was
found in the MHD simulation. Importantly, the simulations explicitly associate
generation of large magnetic field gradients to HMHD compared to MHD,
resulting in faster and more efficient field line slippage because of the
enhanced Hall term.

Overall, the results documented in the paper show the HMHD explains the flare
brightening better than the MHD, prioritizing the requirement to include HMHD
in future state-of-the-art data-based numerical simulations.

##  6 Acknowledgement

The simulations are performed using the 100TF cluster Vikram-100 at Physical
Research Laboratory, India. We wish to acknowledge the visualization software
VAPOR (www.vapor.ucar.edu), for generating relevant graphics. Q.H. and A.P.
acknowledge partial support of NASA grants 80NSSC17K0016, 80NSSC21K1671, LWS
80NSSC21K0003 and NSF awards AGS-1650854 and AGS-1954503. This research was
also supported by the Research Council of Norway through its Centres of
Excellence scheme, project number 262622, as well as through the Synergy Grant
number 810218 (ERC-2018-SyG) of the European Research Council.

## Appendix A

The dimensionless time step is obtained by employing the Hall induction
equation

$\frac{\partial\textbf{B}}{\partial
t}=-d_{H}{\nabla}\times(({\nabla}\times\textbf{B})\times\textbf{B}),$ (A1)

for a stationary fluid. The aforementioned equation is linearized over an
equilibrium magnetic field $\textbf{B}_{0}$ to obtain

$\frac{\partial\delta\textbf{B}}{\partial
t}=-d_{H}[\nabla\times(\nabla\times\delta\textbf{B})\times\textbf{B}_{0}],$
(A2)

$\delta\textbf{B}$ being the perturbation. To obtain the wave modes, the
perturbation is assumed to be periodic along $x$ and $y$ of a Cartesian
coordinate system

$\delta B_{x}=\delta B_{y}\propto\exp[i(k_{z}z-\omega t)].$ (A3)

where the equilibrium field is selected as $\textbf{B}_{0}=B_{0}\hat{e}_{z}$.
Straightforward mathematical manipulations yield the dispersion relation for
the Whistler wave as

${\omega=d_{H}B_{0}{k_{z}}^{2}}.$ (A4)

The wave number is selected as $k_{z}=(2\pi)/{\Delta z}$, $\Delta z$ is the
dissipation scale in the computational domain, making the choice harmonious
with the philosophy used extensively in the paper. since the dimensionless
$\rho_{0}=1$ in the numerical model, (A4) can be written

$\left(\frac{\Delta z}{\Delta
t}\right)_{whis}=4\pi^{\frac{3}{2}}d_{H}\left(\frac{\Delta z}{\Delta
t}\right)_{Alf}{\left(\frac{1}{\Delta z}\right)_{whis}}.$ (A5)

With $\Delta z_{whis}=\Delta z_{Alfven}=0.0078$, namely the dissipation scale
in the present model along with $d_{H}$=0.004, while $\Delta t_{Alf}\approx
10^{-3}$ from previous numerical experiments; in the present model $\Delta
t=\Delta t_{whis}\approx 10^{-4}$.

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Figure 1: Panels (a)-(f) are SDO/AIA images showing the multi-wavelength
observations of the flaring active region AR NOAA 12734. Panel (a) shows the
quasi circular brightening at the western part of AR prior to the flare
(marked by C1). Panels (b)-(d) show the initiation of the flare followed by
eruption (indicated by yellow arrow). Panel (e) shows the circular structure
after eruption at the eastern part of AR (marked by C2) and the W-shaped flare
ribbon (enclosed by white box). Panel (f) shows the post-flare loops. Figure
2: Panel (a) shows the variation of the deviation $E_{n}$ with number of
iterations in non-FFF extrapolation. Panel (b) shows the logarithmic variation
of horizontally averaged magnetic field (Y=B), the current density (Y=J), and
the Lorentz force (Y=L) with height $z$ in pixels. All the quantities plotted
in panel (b) are normalized with their respective maximum values. Panel (c)
shows the scatter plot of the correlation between the observed and
extrapolated magnetic field. The red line is the expected profile for perfect
correlation. Distribution of the magnitude of the Lorentz-force for initial
extrapolated field is shown in panel (d) using direct volume rendering (DVR).
The distribution clearly shows that the Lorentz-force is maximum at bottom
boundary and decreasing with the height in computational volume. The red,
green and blue arrows on the bottom left corner represent $x$, $y$ and $z-$
directions respectively here and hereafter. The color bars on the right side
of the panel represent the magnitude of the strength of Lorentz-force . Figure
3: Panel (a) shows the SDO/AIA 304Å image where the flare ribbon brightening
has been divided into four parts B1, B2, B3, and B4 (enclosed by boxes). Panel
(b) shows an overall extrapolated magnetic field lines morphology of AR NOAA
12734 with the $B_{z}-$component of magnetogram at the bottom boundary. Foot
points of the magnetic structures contained in regions R1, R2, R3, and R4
correspond to the brightening B1, B2, B3, and B4 respectively. Figure 4: Panel
(a) shows magnetic field lines morphology of region R1 between positive and
negative polarities P1, P2, N1, N2, and N3 respectively. Panel (b) highlights
the structure of QSL 1 comprised of magnetic field lines Set I (green) and Set
II (maroon). Panel (c) shows the zoomed top view of the flux rope structure
(black) and an overlying QSL 2 (multi color arrowed magnetic field lines),
between the positive and negative polarities P1, P2, and N1 respectively.
Panel (d ) shows the side view of the flux rope where three vertical planes
along the cross section of the flux rope show the twist value $T_{w}$ at
different locations along the flux rope. In all the panels the log $Q$ between
5 and 10, is overlaid on $B_{z}-$component of magnetogram at the bottom
boundary. Figure 5: Panel (a) shows a 3D null spine-fan configuration in
region R2 with the $B_{z}$ as the bottom boundary overlaid with log $Q$
between 5 and 10. Panel (b) is zoomed view of (a), highlighting the 3D null-
point (in black)—an iso surface of $n=0.6$ indicated by an arrow. Panel (c)
shows the side view of magnetic field lines structure in region R3 along with
the yellow surface representing the null-line corresponding to $n=0.9$. Panel
(d) shows the top-down view of red magnetic field lines of (c) forming fish-
bone-like structure. In panel (e) we show the value of $n$ on the three
different vertical planes passing through the cross-sections of the null-line
surface. Notably, the planes show circles in the cross-section at different
locations which indicates that the yellow surface is a null line. Panel (f)
depicts magnetic field lines morphology in region R4 along with the value of
$n$ on a vertical plane where the green circular contour corresponds to
$n=0.6$ suggesting the right part of magnetic field lines morphology may be a
part of the null-line geometry (shown in panel (c)). Figure 6: Snapshots of
the global dynamics of magnetic field lines in region R1 during the HMHD and
MHD simulations are shown in panels (a)-(d) and panels (e)-(f) respectively.
Panels (a) and (b) show the departure of foot points of magnetic field lines
Set II (maroon) away from polarity P2 between t=19 and 46 on the bottom
boundary (marked by black arrow). Panel (c) depicts the rising magnetic field
lines Set II (maroon) higher up in the solar corona at t=80 (marked by black
arrow) and (d) shows subsequent connectivity change of rising magnetic field
lines at t= 81, due to reconnection with the Set I (green) magnetic field
lines. Panels (e) and (f) depict the departure of foot points of magnetic
field lines Set II (maroon) away from P2 between t= 19 to 113 which is similar
to the HMHD but delayed in time—indicating the slower dynamics in the MHD.
Notably, significant rise of magnetic field lines Set II and consequent
reconnection of it with magnetic field lines Set I higher up in the solar
corona is absent in the MHD simulation. An animation of this figure is
available. The video shows the evolution of magnetic field lines in region R1
from t=0 to 109 for the HMHD and from t=0 to 120 for the MHD simulations
respectively. The realtime duration of the video is 12 s. Figure 7: Snapshots
of the HMHD and MHD evolution of QSL2 (Figure 4 (c)) are shown in panels
(a)-(b) and panels (c)-(d) respectively. Panels (a)-(b) show magnetic field
linesanchored in the positive polarity P2 at t=10 have moved to the polarity
P1 by t=102 and changed their connectivity (marked by black arrow) due to
reconnection along QSL during the HMHD. Panels (c)-(d) show the connectivity
changes of the violet and white color magnetic field lines during the MHD
evolution. The white field line was initially connecting the polarities P1 and
N1 whereas the violet field line was connecting P2 and N1. As a result of
reconnection along QSL the white field line changes its connectivity from P1
to P2 and violet field line changes the connectivity from P2 to P1 (marked by
black arrows). Notably, unlike the HMHD simulation not all magnetic field
lines move to P1 from P2 due to reconnection along QSL during the MHD which
indicates the slower dynamics. An animation of this figure is available. The
video shows the evolution of QSL2 (shown in Figure 4) in region R1 from t=0 to
120 for the HMHD and MHD simulations respectively. The realtime duration of
the video is 12 s. Figure 8: Time sequence showing the HMHD (panels (a)-(f))
and MHD (panels (g)-(i)) evolution of the flux rope (shown in Figure 4(c))
along with the twist $T_{w}$. Panel (a) shows the twist on the middle and the
right plane on flux rope is higher than initial values (c.f. Figure 4(d)) and
reduced with time in panel (b). Panels (c)-(d) depict the connectivity change
of the foot point of rope from right to left (indicated by black arrow) due to
reconnection along QSL. Panels (e)-(f) show the connectivity change of
magnetic field lines on left hand side (indicated by black arrow). Panels
(g)-(i) depict the dynamic rise of the flux rope between t=10 and t=120 during
the MHD simulation. Notably, the foot points of the rope on the right side
(marked by black arrow) at t=32 in (h) have moved towards left by t=120 in (i)
as a result of reconnection along QSL. An animation of this figure is
available. The video shows the evolution of magnetic flux rope in region R1
from t=0 to 92 for the HMHD and from t=0 to 120 for the MHD simulations
respectively. The realtime duration of the video is 12 s. Figure 9: Temporal
variation of the direct volume rendering of $(|\textbf{J}|/|\textbf{B}|)$
along with the flux rope is shown during the HMHD simulation. Noticeably, the
high magnetic field gradient regions with $(|\textbf{J}|/|\textbf{B}|)\geq
475$ develop within (panels (a) to (c)) and on the left side of the flux rope
(panels (d) to (f)). The values $(|\textbf{J}|/|\textbf{B}|)\geq 475$ in each
panel are enclosed within the black rectangular boxes. Figure 10: Temporal
variation of the direct volume rendering of $(|\textbf{J}|/|\textbf{B}|)$
along with the flux rope is shown during the MHD simulation. Panels (a) and
(b) shows the absence of high values of $(|\textbf{J}|/|\textbf{B}|)$ within
the rope at $t=28$ and $t=32$ but in later panels (c) to (f)
$(|\textbf{J}|/|\textbf{B}|)\geq 475$ appears (enclosed by the black
rectangular boxes). Notably, as compared to the HMHD case (Figure 9), the
development of $(|\textbf{J}|/|\textbf{B}|)$ is not significant in the region
R1 during the MHD. Figure 11: Panels (a) and (b) show the comparison of
magnetic field lines topology in region R2 at t=70 with the flare ribbons
observed in the SDO/AIA 304 Å channel (side views) during the HMHD and MHD
simulations respectively. The inset images on the top left corner in each
panel show the top view of the same magnetic field lines topology. Notably,
the spine is anchored in the HMHD while it is not connected to the bottom
boundary in the MHD at t=70 (marked by white arrow in inset images). An
animation of this figure is available. The video shows the evolution of
magnetic field lines in region R2 with the AIA 304 Å image at the bottom
boundary from t=0 to 117 for the HMHD and from t=0 to 120 for the MHD
simulations respectively. The realtime duration of the video is 12 s. Figure
12: Panels (a) to (d) depict the slipping motion of the lower spine field
lines (also shown in the Figure 5) overlaid with the
$|\textbf{J}|/|\textbf{B}|$ on the bottom boundary during the HMHD evolution.
The motion is marked by the black arrows in all the panels indicating the
successive change in the location of field lines on the bottom boundary. A and
B (in panels (a) and (b)) are the two regions with
$|\textbf{J}|/|\textbf{B}|>350$ on the bottom boundary (just below the lower
spine). Notably, the field lines follow the high values of
$|\textbf{J}|/|\textbf{B}|$ on the bottom boundary and remain anchored. Panels
(e) to (h) show the evolution of the same lower spine field lines during the
MHD simulation. The large values of $|\textbf{J}|/|\textbf{B}|$ do not appear
below the lower spine (on the bottom boundary) and it does not remain anchored
from $t\approx 55$ onward (panels (f) to (h)). Figure 13: Time sequence
showing the blue field line prior and after reconnection (indicated by back
arrow) in region R3 during the HMHD simulation. Evolution of the flow vectors
is depicted by green arrows (on the right side)—mimicking the direction of the
plasma flow. The plane along the cross section of magnetic field lines
morphology in R3, showing the blue circular contours represent the value of
$n$ (also shown in Figure 5(d)). Figure 14: Time sequence showing the blue
field line prior and after reconnection (indicated by back arrrow) in region
R3 during the MHD simulation. Evolution of the flow vectors is depicted by
green arrows (on the right side)—mimicking the direction of the plasma flow.
The plane along the cross section of magnetic field lines morphology in R3,
showing the blue circular contours represent the value of $n$ (also shown in
Figure 5(d)). Notably, reconnection of the blue magnetic field lines is
slightly delayed in comparison to its HMHD counterpart. A combined animation
of Figure 13 and this figure is available. The video shows the evolution of
magnetic field lines of region R3 along with the flow vectors (green) from t=0
to 120 for the HMHD and MHD simulations respectively. The realtime duration of
the video is 12 s. Figure 15: Panels (a)-(d) show the global dynamics of
magnetic field lines in region R4 during the HMHD simulation. Inset images in
each panel (on right) depict the time sequence of the zoomed top-down view of
the rotational motion of magnetic field lines. The background shows the
variation of the z-component of flow $\in$ [-0.00022,0.00032] in all inset
images. The red vectors represent the plasma flow and change its direction in
an anticlockwise manner in panels (a)-(d). The rotational motion of magnetic
field lines coincides with the circular part of the flow. An animation of this
figure is available. The video shows the evolution of magnetic field lines in
region R4 from t=0 to 100 for the HMHD simulation. The development of an
anticlockwise rotational motion of the foot points from t$\approx$33 onward
and the clockwise slippage of field line from t$\approx$76 onward is evident
in the video. The realtime duration of the video is 10 s. Figure 16: Panels
(a) to (f) show the side view of the rotating magnetic field lines structure
in the region R4 overlaid with $|\textbf{J}|/|\textbf{B}|$. The figure depicts
the temporal development of strong magnetic field gradient regions of
$|\textbf{J}|/|\textbf{B}|>225$ (enclosed in the blue rectangular boxes)
within the rotating magnetic structure. Figure 17: Top-down view of an overall
magnetic field lines morphology overlaid on the SDO/AIA 304 Å (panels (a) and
(b)) and 171 Å images (panel (c)). Anchored magnetic field lines foot points
in central part match well with the observed tip of the W-shaped flare ribbon
(marked by pink arrow in panel (a)) in the HMHD while magnetic field lines
foot points are completely disconnected from the bottom boundary in the MHD
(panel (b)). Loops rising higher up in the corona is remarkable in the HMHD
(indicated by white arrow in panel (a)). An animation of panels (a) and (b) is
available. The video shows the evolution of magnetic field lines in AR along
with the AIA 304 Å image at the bottom from t=0 to 120 for the HMHD and MHD
simulations respectively. The realtime duration of the video is 12 s.