# Changes of Magnetic Energy and Helicity in Solar Active Regions from Major
Flares

Yang Liu  W. W. Hansen Experimental Physics Laboratory, Stanford University,
Stanford, CA 94305-4085, USA  Brian T. Welsch  Natural & Applied Sciences,
University of Wisconsin-Green Bay, 2420 Nicolet Drive, Green Bay, WI 54311,
USA  Gherardo Valori  Max-Planck-Institut für Sonnensystemforschung, Justus-
von-Liebig-Weg 3, 37077, Göttingen,Germany  Manolis K.Georgoulis  Research
Center for Astronomy and Applied Mathematics of the Academy of Athens, 11527
Athens, Greece  Yang Guo  School of Astronomy and Space Science, Nanjing
University, Nanjing 210023, China  Etienne Pariat  Sorbonne Université, École
polytechnique, Institut Polytechnique de Paris, Université Paris Saclay,
Observatoire de Paris-PSL, CNRS, Laboratoire de Physique des Plasmas (LPP),
75005 Paris, France  Sung-Hong Park  W. W. Hansen Experimental Physics
Laboratory, Stanford University, Stanford, CA 94305-4085, USA  Julia K.
Thalmann  University of Graz, Institute of Physics/IGAM, Graz, Austria

(Received June 1, 2019; Revised January 10, 2019; Accepted November 15, 2022)

###### Abstract

Magnetic free energy powers solar flares and coronal mass ejections (CMEs),
and the buildup of magnetic helicity might play a role in the development of
unstable structures that subsequently erupt. To better understand the roles of
energy and helicity in large flares and eruptions, we have characterized the
evolution of magnetic energy and helicity associated with 21 X-class flares
from 2010 to 2017. Our sample includes both confined and eruptive events, with
6 and 15 in each category, respectively. Using HMI vector magnetic field
observations from several hours before to several hours after each event, we
employ (a) the Differential Affine Velocity Estimator for Vector Magnetograms
(DAVE4VM) to determine the photospheric fluxes of energy and helicity, and (b)
non-linear force-free field (NLFFF) extrapolations to estimate the coronal
content of energy and helicity in source-region fields.

Using Superposed Epoch analysis (SPE), we find, on average: (1) decreases in
both magnetic energy and helicity, in both photospheric fluxes and coronal
content, that persist for a few hours after eruptions, but no clear changes,
notably in relative helicity, for confined events; (2) significant increases
in the twist of photospheric fields in eruptive events, with twist
uncertainties too large in confined events to constrain twist changes (and
lower overall twist in confined events); and (3) on longer time scales (event
time +12 hours), replenishment of free magnetic energy and helicity content to
near pre-event levels for eruptive events. For eruptive events, magnetic
helicity and free energy in coronal models clearly decrease after flares, with
the amounts of decrease proportional to each region’s pre-flare content.

Sun: Flares and CMEs; Sun: Magnetic Helicity and Energy; Sun: Active Regions

††journal: ApJ

##  1 Introduction

Solar flares and coronal mass ejections (CMEs) are driven by the release of
free magnetic energy stored in coronal electric currents. Beyond the basic
outline of this “storage and release” paradigm, however, key aspects of how
magnetic energy builds up in coronal magnetic fields and the processes that
trigger its impulsive release in flares and CMEs are poorly understood.
Improved knowledge of these build-up and triggering processes, however, could
benefit efforts to forecast these drivers of space weather disturbances
(Baker, 1998), suggest possible physical processes at work in stellar flares
and ejections, and aid understanding of magnetic energy release from plasmas
generally.

Electric currents in the corona are the source of free magnetic energy — i.e.,
energy above the current-free magnetic field (also known as a potential field)
with the same normal field below the base of the corona, in the solar
photosphere. This potential field is unique, given conditions of isolated
magnetic structures and extension to infinity (i.e., Sakurai, 1989), and
possesses the minimum possible magnetic energy for a given set of boundary
conditions. Using this potential field as a “reference field,” the coronal
field’s relative magnetic helicity is a gauge-invariant measure of the coronal
field’s connectivity and complexity (i.e., twist, writhe, and linkage - for a
brief review of magnetic helicity, see, for instance, Berger 1998.) A nonzero
relative magnetic helicity thus signifies the presence of non-potential,
current-carrying magnetic structures, such as flux systems with twist (helical
winding of field lines around each other within a flux system) or writhe
(helical deformation of the axis of a flux system), and sheared fields (i.e.,
horizontal fields that point along a polarity inversion line [PIL] of the
normal magnetic field, rather than across the PIL as a generic potential field
would). For simplicity, we use the term helicity to refer to relative helicity
throughout the paper, unless otherwise specified.

The availability of routine measurements of the photospheric magnetic field
vector in active regions (ARs) from the Helioseismic and Magnetic Imager (HMI;
Scherrer et al. 2012; Hoeksema et al. 2014) aboard the Solar Dynamics
Observatory (SDO; Pesnell et al. 2012) has enabled estimating the time
evolution of (i) fluxes of magnetic energy (the Poynting flux) and helicity
across the photosphere (e.g., Liu & Schuck 2012a; Kazachenko et al. 2015; Park
et al. 2020; Liokati et al. 2022) and (ii) magnetic energy and helicity in
coronal volumes where, typically, the magnetic field is extrapolated by
assuming that it is force-free (e.g., Wiegelmann et al. 2012; Thalmann et al.
2016; Moraitis et al. 2019).

Variations in the photospheric fluxes or coronal content of magnetic energy
and helicity associated with the occurrence of flares and CMEs have been
investigated in several studies. Particular attention has been focused on AR
magnetic properties that differ between eruptive and confined flares.
Tziotziou et al. (2012, 2014) found that the eruptive events appear fairly
segregated from confined events in both free energy and relative helicity.
When the helicity is broken down into two components — the magnetic helicity
of the non-potential component of the magnetic field (the current-carrying
field), $H_{j}$, and the mutual helicity between potential field and current-
carrying field, $H_{pj}$ (Berger, 2003; Pariat et al., 2017; Linan et al.,
2018) — changes in the ratio between these components could mark the trigger
of an eruptive flare (e.g., Pariat et al., 2017; Zuccarello et al., 2018).
This ratio effectively divides the eruptive and confined events well (Thalmann
et al., 2019b; Gupta et al., 2021). Thus far, comparisons between the two
types of helicity estimation have been limited to individual case studies
(see, e.g., Thalmann et al., 2021).

In this study, we systematically analyze and compare the photospheric fluxes
and coronal content of magnetic helicity and energy in a set of 13 ARs that
produced 21 X-class flares during solar cycle 24. Unlike previous studies, we
focus specifically on changes of these quantities associated with these major
eruptive and confined flares. In addition, we seek and discuss possible
differences in the behavior of time series for these two flare categories.

In the following Section 2, we describe our event sample and methods for
characterizing the evolving non-potentiality of these flare-productive active
regions. In Section 3, we present quantitative results of our analyses. In
Section 4, we recap our most significant findings, consider their
implications, and discuss future research directions of research.

##  2 Data and Methodology

We first describe our procedures for event selection, and then describe the
observational data for our event sample. We then describe our procedures for
analyzing magnetic energy and helicity derived from these observations.

###  2.1 Analysed Data Set

####  2.1.1 Event Sample

We evaluated all X-class flares in the time period, ranging between 2010 to
2017, for which HMI observations are available. We found 28 ARs that produced
X-class flares during this period, out of which we selected 11 ARs that
produced an event associated with a CME (i.e., an eruptive flare) and two ARs
that produced confined events. The remaining ARs only produced X-class events
that occurred relatively close to the limb ($>50^{\circ}$ from the central
meridian) or were located in relatively weak fields at the periphery of an
evolving AR (such as the 2014 January 17 X1.2 flare in AR 11944). Because the
HMI vector data are less reliable in such cases, we excluded these events from
our sample. We also excluded the 2017 September 6 X2.2 flare in AR 12673
because it is unclear if this flare is eruptive or confined. This flare was
reported to be an eruptive event by Mitra et al. (2018), who relied on the
Solar Eruptive Event Detection System (SEEDS) CME
catalog111http://spaceweather.gmu.edu/seeds/monthly.php?a=2017&b=09: it
determined that a CME occurred near that time, with an angular width of 44∘
and a speed of 279 km s-1. However, Liu et al. (2018) reported this flare as a
confined event because they did not observe separation of flare ribbons or
clear CME signatures in He II 304Å images. They also reported that no dimmings
were visible, though we observed minor dimmings in our review of AIA images.
The uncertainty regarding the eruptive nature of this event precluded us from
placing it in either the confined or the eruptive category.

Overall, our sample includes 21 X-class flares from 13 different ARs. Six
flares from two ARs in the sample are confined. All events are listed in Table
1. Because the two X-class flares in AR 11429 occurred about 20 minutes apart,
they are deemed to be one event in our analysis.

Table 1:  List of X-class flares in 13 ARs in our sample, all of which are within $50^{\circ}$ from the central meridian and occurred after May 2010. Six flares in the sample of 21 events are confined, i.e., they have no CMEs associated to them.  Flare ID | Flare Start | Peak | End | Flare | CME | AR | Position
---|---|---|---|---|---|---|---
1 | 2011-02-15T01:44:00 | 2011-02-15T01:56:00 | 2011-02-15T02:06:00 | X2.2 | YES | 11158 | S21W28
2 | 2011-03-09T23:13:00 | 2011-03-09T23:23:00 | 2011-03-09T23:29:00 | X1.5 | NO | 11166 | N11E13
3 | 2011-09-06T22:12:00 | 2011-09-06T22:20:00 | 2011-09-06T22:24:00 | X2.1 | YES | 11283 | N14W18
4 | 2011-09-07T22:32:00 | 2011-09-07T22:38:00 | 2011-09-07T22:44:00 | X1.8 | YES | 11283 | N14W18
5 | 2012-03-07T00:02:00 | 2012-03-07T00:24:00 | 2012-03-07T00:40:00 | X5.4 | YES | 11429 | N17E27
6 | 2012-03-07T01:05:00 | 2012-03-07T01:14:00 | 2012-03-07T01:23:00 | X1.3 | YES | 11429 | N17E26
7 | 2012-07-12T15:37:00 | 2012-07-12T16:49:00 | 2012-07-12T17:30:00 | X1.4 | YES | 11520 | S15W01
8 | 2013-11-05T22:07:00 | 2013-11-05T22:12:00 | 2013-11-05T22:15:00 | X3.3 | YES | 11890 | S09E36
9 | 2013-11-08T04:20:00 | 2013-11-08T04:26:00 | 2013-11-08T04:29:00 | X1.1 | YES | 11890 | S09W20
10 | 2013-11-10T05:08:00 | 2013-11-10T05:14:00 | 2013-11-10T05:18:00 | X1.1 | YES | 11890 | S09W30
11 | 2014-03-29T17:35:00 | 2014-03-29T17:48:00 | 2014-03-29T17:54:00 | X1.0 | YES | 12017 | N11W32
12 | 2014-09-10T17:21:00 | 2014-09-10T17:45:00 | 2014-09-10T18:20:00 | X1.6 | YES | 12158 | N12E29
13 | 2014-10-22T14:02:00 | 2014-10-22T14:28:00 | 2014-10-22T14:50:00 | X1.6 | NO | 12192 | S12E20
14 | 2014-10-24T21:07:00 | 2014-10-24T21:41:00 | 2014-10-24T22:13:00 | X3.1 | NO | 12192 | S12W15
15 | 2014-10-25T16:55:00 | 2014-10-25T17:08:00 | 2014-10-25T18:11:00 | X1.0 | NO | 12192 | S12W28
16 | 2014-10-26T10:04:00 | 2014-10-26T10:56:00 | 2014-10-26T11:18:00 | X2.0 | NO | 12192 | S12W35
17 | 2014-10-27T14:12:00 | 2014-10-27T14:47:00 | 2014-10-27T15:09:00 | X2.0 | NO | 12192 | S12W45
18 | 2014-11-07T16:53:00 | 2014-11-07T17:26:00 | 2014-11-07T17:34:00 | X1.6 | YES | 12205 | N15E33
19 | 2014-12-20T00:11:00 | 2014-12-20T00:28:00 | 2014-12-20T00:55:00 | X1.8 | YES | 12242 | S18W29
20 | 2015-03-11T16:11:00 | 2015-03-11T16:22:00 | 2015-03-11T16:29:00 | X2.1 | YES | 12297 | S16E13
21 | 2017-09-06T11:53:00 | 2017-09-06T12:02:00 | 2017-09-06T12:10:00 | X9.3 | YES | 12673 | S09W38

####  2.1.2 Data

We use vector magnetic field data taken by HMI, a filtergraph instrument with
full-disk coverage over 4096$\times$4096 pixels. Its spatial resolution is
about 1”, with a 0.5” pixel size. The spectral line observed is the Fei 6173Å
absorption line formed in the photosphere (Norton et al., 2006). There are two
CCD cameras in the instrument, the “front camera” and the “side camera.” The
front camera acquires filtergrams at 6 wavelengths along the line Fei 6173Å in
two polarization states with 3.75 seconds between the images. The width of the
filter profile is 76 mÅ. It takes 45 seconds to acquire a set of 12
filtergrams. This set of data is used to derive Dopplergrams and line-of-sight
magnetograms. The side camera is dedicated to measuring the vector magnetic
field. It takes 135 seconds to obtain the filtergrams in 6 polarization states
at 6 wavelength positions. After April 2016, the observation mode changed: the
side camera now only takes linear polarization states at 6 wavelengths in 90
seconds, and measurements from both cameras are combined. The Stokes
parameters [I, Q, U, V] are computed from these measurements.

In order to suppress the p-modes and increase the signal-to-noise ratio,
usually the Stokes parameters are derived from filtergrams averaged over 1215
seconds using a cosine-apodized, moving-boxcar weighting function with a FWHM
of 720 seconds. Averages are computed at a 720-second cadence. They are then
inverted to produce the vector magnetic field using the Very Fast Inversion of
the Stokes Vector (VFISV) algorithm (Borrero et al., 2011; Centeno et al.,
2014). The 180∘ degree ambiguity in the transverse-field azimuth is resolved
based on the “minimum energy” algorithm (Metcalf et al., 1994; Leka et al.,
2009). Photospheric patches containing ARs are automatically identified and
bounded by a feature recognition model (Turmon et al., 2010), and the
disambiguated vector magnetic field data of ARs are deprojected to
heliographic coordinates (Bobra et al., 2014). Here we use the Lambert
(cylindrical equal area) projection method, centered on each region, for the
remapping.

Figure 1:  Temporal profiles of magnetic flux (top, total unsigned flux
divided by 2), energy flux with total energy (middle), and helicity flux and
total helicity (bottom) in AR 11158 from 2011 February 13–17. Right axes on
middle and bottom panels show magnetic energy and helicity, respectively,
calculated from NLFFF models (see Section 2.2 in the text). Vertical blue
lines denote the occurrence of an X-class flare that started at 01:44 UT, 2011
February 15. One-$\sigma$ uncertainties are shown at representative data
points.

###  2.2 Analysis Methodology

For each AR in this study, we used the time series of its photospheric
observations to estimate time series of relative helicity and magnetic energy
injection rates through the solar surface, as well as the relative helicity
and energy in extrapolated coronal fields. We also calculate time series of
magnetic twist in each AR. We use force-free field parameter, $\alpha$, to
represent the magnetic twist in AR (See details in Section 3.3).

The relative magnetic helicity in a volume $V$ can be expressed by (Finn &
Antonsen, 1985),

$H=\displaystyle\int_{V}(\mathbf{A}+\mathbf{A}_{p})\cdot(\mathbf{B}-\mathbf{B}_{p})\,dV~{},$
(1)

where $\mathbf{B}$ and $\mathbf{B_{p}}$ are the magnetic field and the
potential (reference) field, respectively, and $\mathbf{A}$ and
$\mathbf{A_{p}}$ are their respective vector potentials. Details of helicity
calculation are given by Valori et al. (2012). Specifically in this
computation, the vector potentials $\mathbf{A}$ and $\mathbf{A_{p}}$ are
computed assuming that the vertical component of the vector potentials vanish
everywhere in the considered volume $V$ (DeVore gauge, see in particular
Equations 11, 24, and 25 in Valori et al. (2012)). Note that this is different
from the gauge used for the helicity flux calculation, as described below. The
volume helicity is gauge independent, so the choice of gauge does not impact
the helicity calculation. The difference between gauges was also tested
numerically (Valori et al., 2016), and no significant difference was found.
The method chosen here to compute the volume helicity is computationally
faster.

Relative helicity can be broken down into two gauge-invariant components, the
helicity of the current-carrying field, $H_{j}$, and the mutual helicity
between potential field and current-carrying field, $H_{pj}$, expressed as
(Berger, 2003; Pariat et al., 2017),

$H=H_{j}+H_{pj}~{},$ (2)

where

$H_{j}\equiv\displaystyle\int_{V}(\mathbf{A}-\mathbf{A}_{p})\cdot(\mathbf{B}-\mathbf{B}_{p})dV~{},$
(3)

and

$H_{pj}\equiv
2\times\displaystyle\int_{V}\mathbf{A}_{p}\cdot(\mathbf{B}-\mathbf{B}_{p})dV~{}.$
(4)

The magnetic energy in a volume $V$ is computed as

$E=\displaystyle\int_{V}\frac{B^{2}}{8\pi}\,dV~{}.$ (5)

The coronal magnetic field for each AR can be extrapolated from the observed
vector magnetic field at the photosphere by assuming that the Lorentz force
vanishes in the coronal field. We use an optimization-based non-linear force-
free field (NLFFF) algorithm (Wiegelmann, 2004) to extrapolate the coronal
field in a Cartesian volume above a given photospheric magnetogram. A
preprocessing step alters the magnetogram field to remove most of the net
force and torque from the photospheric data, thereby making the boundary more
consistent with the force-free assumption (Wiegelmann et al., 2006).

The flux of magnetic helicity across a surface S is expressed by (Berger &
Field, 1984; Chae, 2001),

$\frac{dH}{dt}\Bigg{|}_{S}=2\displaystyle\int_{S}(\mathbf{A}_{p}\cdot\mathbf{B}_{t})V_{\perp
n}\,dS-2\displaystyle\int_{S}(\mathbf{A}_{p}\cdot\mathbf{V}_{\perp
t})B_{n}\,dS,$ (6)

where: Bt and Bn denote the observed tangential and normal magnetic fields,
respectively; and V⟂t and V⟂n are the tangential and normal components of
${\mathbf{V}_{\perp}}$, the velocity perpendicular to the magnetic field. The
integral is done over the surface imaged in the magnetogram. The vector
potential Ap for the potential field on the photosphere is uniquely determined
by the observed photospheric normal magnetic field and Coulomb gauge with the
following equations (Berger, 1997; Berger & Ruzmaikin, 2000),

$\mathbf{\nabla}\times\mathbf{A}_{p}\cdot\hat{\mathbf{n}}=B_{n},\mathbf{\nabla}\cdot\mathbf{A}_{p}=0,\mathbf{A}_{p}\cdot\hat{\mathbf{n}}=0.$
(7)

Similarly, the magnetic energy (Poynting) flux can be calculated by

$\frac{dE}{dt}\Bigg{|}_{S}=\frac{1}{4\pi}\displaystyle\int_{s}{B}_{t}^{2}V_{\perp
n}\,dS-\frac{1}{4\pi}\displaystyle\int_{S}(\mathbf{B}_{t}\cdot\mathbf{V}_{\perp
t})B_{n}\,dS.$ (8)

The vector velocity field in the photosphere is derived from the Differential
Affine Velocity Estimator for Vector Magnetograms (DAVE4VM; Schuck, 2008),
applied to the time-series of reprojected, co-registered vector magnetic field
data. The window size used in DAVE4VM is 19 pixels, which is about 9.5 arcsec.
DAVE4VM derives velocities at time $t_{i+1/2}$ from (i) the difference
$\Delta\mathbf{B}$ between HMI’s measurements of $\mathbf{B}$ at times $t_{i}$
and $t_{i+1}$ and (ii) the average field from these two measurements. Nominal
cadence between HMI measurements is 720 s, so the nominal $\Delta t$ between
flow estimates is also 720 s.

This window size was chosen based on a combined evaluation of how well the
recovered flows satisfied the ideal MHD induction equation’s vertical
component, and how stable the calculated helicity and energy fluxes were, as
described by Schuck (2008) and Liu & Schuck (2012b). Tests of DAVE4VM with
simulation data by Schuck (2008) showed that the helicity and energy fluxes
became insensitive if the chosen window size exceeded a certain value, and
that the helicity flux from DAVE4VM velocities was about 94% of the true flux.
With HMI data for AR 11158, Lumme et al. (2019) showed the shapes of the time
series of energy and helicity fluxes from different choices of $\Delta t$ and
window sizes were similar, though they differed in magnitude. This suggests
that different choices of window size could still show flare-related changes
in helicity and energy fluxes, but the sizes of the changes found might
differ.

We remark that the cumulative (time-integrated) fluxes of energy and helicity
may not be identical with estimates of the coronal content for various
physical reasons. First, the coronal models contain only a finite subdomain of
the corona, therefore, discrepancies could occur due to fields that close
outside coronal models’ domains. For energy, differences between integrated
Poynting fluxes and coronal models’ energies could be due to losses from
coronal heating and flares smaller than the X-class events that are not our
focus. In contrast, for helicity, which is expected to be approximately
conserved even when magnetic reconnection occurs in plasmas with a low
magnetic Reynolds number (Berger, 1984, 1999), CMEs or smaller-scale ejections
could bodily remove helicity from the coronal volume. Finally, the helicity
flux in Equation (6) is intrinsically derived assuming an infinite plane,
while the relative helicity defined in Equation (1) supposes a finite volume.
This can lead to discrepancies between the two estimates. Comparisons between
estimates of energy and helicity from fluxes versus extrapolations are
underway in a separate study (Pariat et al. 2022, in preparation).

##  3 Results

As an example output of our calculation methods, Figure 1 shows temporal
profiles of magnetic flux (top), magnetic energy and Poynting flux (middle),
and magnetic helicity and helicity injection rate (bottom) for AR 11158 from
13–18 February 2011. The blue vertical lines denote the occurrence of the X2.2
flare that was associated with a CME. The magnetic flux plotted is taken as
half of the total unsigned flux, which is computed by summing the absolute
value of radial field over the AR. The pixels with total field strength
greater than 300 G, which is about three times the estimated 1-$\sigma$
transverse-field noise level for HMI vector magnetograms (Hoeksema et al.,
2014), are included in these calculations.

The 1-$\sigma$ error bars for each quantity are given at representative
points. To estimate the uncertainties in helicity and energy fluxes, we adopt
the results in Liu & Schuck (2012a) that are estimated by employing the Monte
Carlo method: for each run, the magnetograms are perturbed by their estimated
errors, velocities are derived from the perturbed magnetograms, and the fluxes
are computed from the perturbed fields and derived velocities. This
uncertainty estimate does not include errors in the velocity inversion from
DAVE4VM. As a measure of the uncertainty in helicity and energy estimates for
this AR, we overplotted the standard deviations of helicity and energy
calculations from 5 consecutive NLFFF models, derived from vector magnetograms
at 720-second cadence. Assuming that the five consecutive observations
(spanning 48 minutes) are repeated measurements of the same quantity, their
standard deviation characterizes measurement uncertainty.

For NLFFF models, detailed discussions and careful investigations of the
sources of uncertainty in calculating helicity in ARs have been undertaken,
including uncertainties such as solenoidality (Thalmann et al., 2019a)
parameter settings (Thalmann et al., 2020), methods to compute helicity
(Thalmann et al., 2019b) and spatial sampling of modeled NLFFF data (Thalmann
et al., 2022b).

The increase in magnetic flux with time (black curve in top panel of Figure 1)
indicates that this is an developing AR, with new flux emerging. This has been
noted by previous studies, too (i.e., Tziotziou et al., 2013). While the
coronal magnetic energy does not change significantly after the flare (red
curve in the middle panel), the coronal helicity decreases beyond error bars
(red curve in the bottom panel). Note that the magnetic energy in the AR
includes both potential and free energy. The helicity injection rate also
decreases after the event, per the black curve in the bottom panel.

It is essential to combine information from multiple events when considering
of flare-associated changes in $E$, $dE/dt$, $H$, and $dH/dt$. Changes in an
individual case could lead to misleading inferences. For instance, the changes
in coronal helicity like that in AR 11158 ($\sim 2-3\times 10^{42}\;Mx^{2}$ in
Figure 1) might be typical following eruptions, or might arise from some
unique property of this region. Figure 1 also demonstrates that the fluxes of
energy and helicity regularly exhibit substantial fluctuations. Consequently,
any change in these fluxes, even if statistically significant, could be
incidental and not necessarily related to a given flare or eruption.

To synthesize information from all events in our sample, we employ superposed
epoch analysis (SPE; e.g., Chree 1913; Wilcox & Ness 1967; Singh & Badruddin
2006; Mason & Hoeksema 2010). This approach can highlight variations in time
series of a given variable (e.g., $dH/dt$) that are linked with occurrence of
some event (e.g., flares). In SPE, the events’ epochs are superposed: time
series of the variable of interest are summed after each series is shifted in
time to align all event times (the “key times”). In this summation, random
variations in the variable of interest should cancel, but recurring variations
will be reinforced. Below, we present results of SPE applied to our time
series of $E$, $dE/dt$, $H$, $dH/dt$, and twist for our event sample.

###  3.1 Flare-induced Changes in Helicity and Energy Fluxes

Because an AR’s helicity, $H$, scales as the square of the magnetic flux,
$\Phi$, in the AR, combining events from different-sized ARs for SPE will
benefit from rescaling their helicities. Similarly, larger ARs will, all else
equal, have larger Poynting fluxes. For $dH/dt$ and $dE/dt$, we can simply
normalize each time series in the time interval around each flare by each AR’s
maximum value during the interval considered. We also take the absolute value
of $dH/dt$, to combine unsigned helicity fluxes.

Figure 2 shows normalized plots of helicity flux $|dH/dt|$ (top panel) and
energy flux $dE/dt$ (bottom panel) for both eruptive (blue curves) and
confined (red curves) flares within $\pm$5 hours of each flare, with all flare
times co-aligned. For eruptive flares, a pattern of decreases shortly after
flares can be discerned in both fluxes, though it is more clear in the
$|dH/dt|$ plot (top panel). Confined events do not exhibit obvious changes
near the flare times. This suggests that eruptions, unlike confined flares,
are able to impact the injection process, inhibiting the injection of energy
and helicity. Decreases in helicity flux after flares has been reported before
(Wang et al., 2016; Bi et al., 2018).

Two physical scenarios might explain decreases in energy and helicity flux
following eruptive flares: submergence of non-potential / helical magnetic
flux associated with the magnetic restructuring occuring in eruptions; and the
downward propagation of helicity (e.g., twist propagation) along fields that
thread the photosphere. Submergence might occur as fields near the flaring PIL
that shallowly arch over the PIL become more horizontal immediately after the
eruptive flare. Several observations of strong flares causing fields to become
more horizontal have been reported (e.g., Hudson et al. 2008; Wang & Liu 2010;
Barczynski et al. 2019a). Because radial flux only submerges (or emerges)
along PILs, areas with negative Poynting flux should be near PILs if flare-
associated submergence is responsible for changes in energy fluxes.

Figure 2:  Normalized, unsigned helicity flux (top) and energy flux (bottom)
for all the X-class flares listed in Table 1. Each AR’s time series is shifted
to align the start time of its flare to hour zero, which is marked with a
vertical black line in each panel. The curves span 10 hours: 5 hours before
and after each region’s flare time. Blue curves correspond to eruptive events;
red to confined events. In order to better show evolution, a one-hour running
average has been applied to each curve.  Figure 3:  Superposed Epoch Analysis
(SPE) plots for unsigned helicity flux (top) and energy flux (bottom) for all
the events. The fluxes are not smoothed. The black curves are averages from
all eruptive events; red curves are averages from all confined events. To
better compare the results for the two types of the events, the SPE curves for
confined events are scaled by multiplying by 0.1 for helicity flux and 0.2 for
energy flux. (The much larger values for confined events occur because most of
these events occur in AR 12192, which contained an unusually large amount of
magnetic flux, and both helicity and energy fluxes tend to be larger in
larger-flux active regions.) Vertical blue lines denote the flare onset time.
One-$\sigma$ uncertainties by propagating individual uncertainties are
overplotted.

To better show the general trend of the evolution of helicity and energy
fluxes for our events, we average the superposed, not normalized fluxes for
the eruptive events, and, separately, for the confined events. Figure 3 shows
the averaged unsigned helicity flux (top) and energy flux (bottom) for
eruptive (black) and confined (red) events. Uncertainties for each region were
propagated in this averaging. For the eruptive events, both helicity and
energy fluxes decrease substantially after the events, followed by a return
toward pre-event levels during the first hours after the event. The onset of
change in the helicity flux here precedes the onset of change in the energy
flux by roughly one 720-s step. Given our cadence and measurement
uncertainties, however, this difference in onsets is not observationally
significant. Nevertheless, we note that changes in helicity slightly preceded
changes in magnetic energy in the eruption simulations of Pariat et al.
(2015). A follow-up study with larger sample size and higher-cadence data
(e.g., the 135-s HMI magnetograms discussed by Sun et al. 2017) might find
this time difference to be significant. For the confined events, no
significant changes in the helicity and energy injection rates can be seen.

Figure 4:  Temporal profiles of helicity (top) and energy (bottom) flux for
135-second-cadence vector magnetograms of AR 11158. No running average was
applied. Bad pixels due to the X-class flare are corrected using the procedure
suggested by Sun et al. (2017). The red vertical line denotes the flare start
time; 1-$\sigma$ uncertainties are given at representative data points.

It is plausible that the magnitudes of the flare-related changes in helicity
and energy fluxes that we find would differ if a different tracking method
were used to estimate these fluxes. For instance, comparing cumulative
helicity and energy fluxes for AR 11158 from various studies, Kazachenko et
al. (2015) reported differences of as much as $\sim$30% and $\sim$25%,
respectively, between DAVE4VM and other methods; see their Table 2. By
comparison, the drops in helicity and energy fluxes in Figure 3 are close to
50%. Possible differences between tracking methods’ results should not obscure
our key point, though, which is that significant changes occur; while
different methods might yield different magnitudes, we still expect they would
detect such changes. Replication of our analyses using other methods or other
tracking parameter choices would be valuable, to assess the sensitivity of our
results to these factors.

During major flares, the spectral line profile observed by HMI can be greatly
deformed. In magnetogram pixels where this occurs, data are unreliable (Sun et
al., 2017). This effect could impact our calculation of the helicity and
energy fluxes near flare times. We evaluate its possible impact by using the
135-second-cadence vector data for AR 11158 with the bad pixels corrected by
applying a step-like function to fit the time-series good data for each pixel
before and after the flares, suggested by Sun et al. (2017). The result, shown
in Figure 4, is consistent with the aforementioned conclusion that the
helicity flux decreases suddenly during the flares, followed by a recovery.
This test suggests that the observed decrease in helicity and energy fluxes
from the flares is real. Further, the lack of any obvious dip in fluxes for
confined events, which would be equally susceptible to any spectral-line
artifacts, also implies this result is not spurious.

Figure 5: An example flare mask, for the X1.4 flare in AR 11520, is shown by
the orange contour plotted over a magnetogram of $B_{z}$ (background
grayscale) taken at 15:24 UT July 12, 2012. The flare mask’s boundary is
determined by combining the region’s NLFFF-derived $Q$-map and flare
ribbon/arcade emission.

The decreases in helicity and energy fluxes after eruptive X-class flares must
be related to changes in the magnetic and velocity fields upon which these
fluxes depend. Indeed, it has been shown that the horizontal field in flare
areas increases substantially after major flares (see, e.g., Wang & Liu 2010;
Sun et al. 2012; Petrie 2019 for observational analysis, and, e.g., Barczynski
et al. 2019a for modeling). Changes in velocities associated with large flares
have also been reported (e.g., Deng et al. 2006). To understand how these
changes impact helicity and energy fluxes, we analyzed the distributions of
magnetic fields, velocity fields, and energy flux densities from areas that
participated in eruptive flares. While the flux of helicity is gauge-
invariant, the helicity flux density is not (e.g., Pariat et al., 2005), so we
do not investigate changes in this quantity.

As a first step, we produced a so-called flare mask for each event, to isolate
areas associated with the flare. Eruptive ARs, especially complex ones, often
contain substantial flux that does not erupt, and connects to other polarities
within the AR, to neighboring ARs, or to remote areas on the Sun. Because
random changes from such irrelevant flux systems could conceal the variations
associated with flares that interest us, we restrict our study of changes in
${\bf B},{\bf V},$ and energy flux densities (Poynting fluxes) as much as
possible to the areas from which fields erupt. To do so, we selected areas for
each flare mask by visual inspection of both maps of the squashing factor $Q$
(Titov, 2007; Titov et al., 2011) and the observed locations of flare arcades
and ribbons. We first calculate $Q$ at the lower boundary using the NLFFF
model. We then choose the high-$Q$ contours that enclose the flare patterns of
arcades and ribbons. Figure 5 shows a flare mask for the X1.4 flare in AR
11520 at 15:37 UT, July 12, 2012. The flare mask, denoted by the orange
contour, is plotted on an $B_{z}$ magnetogram taken at 15:24 UT. Details about
procedures used to generate a flare mask, as well as the masks for well-known
ARs 11158, 11429, and 12192, are discussed by Liu et al. (2017). In most
cases, this provides a good proxy for the area containing erupting flux. In
each flare mask, we set the selected areas to unity and remaining areas to
zero. We then applied each flare mask to maps of the pre- and post-flare
magnetic fields, velocity fields, and energy flux densities to extract all
flare-involved pixels for all the eruptive events. Pre-event maps were the
last pre-flare observation; post-event maps were the first observation at
least 30 minutes after the flare end time in the GOES catalog.

Figure 6:  Left column: Distributions of the logarithm of pixel counts in the
2-variable phase space of magnetic field versus pre-flare velocity field for
all eruptive events. Right: Distribution of differences in pixel counts in the
same 2-variable space, given by $\log$(post-flare counts) - $\log$(pre-flare
counts). Only pixels within the flare masks are included in these
distributions.

In the left column of Figure 6, we show the pre-flare distributions of the
logarithm of pixel counts in the 2-variable phase space of magnetic field
strength and total velocity (top), horizontal velocity (middle), and vertical
velocity (bottom), before the flares. In the right column, we show
distributions of differences in pixel counts, i.e., $\log$(post-flare counts)
- $\log$(pre-flare counts), in each 2-variable space. In the right column, red
indicates that more pixels occupy a given location after the flare in this
2-variable space, blue indicates that fewer do (and hence that more pixels
occupied this given location before the flare). The color bars saturate at
logarithmic values of -1.2 to +1.2 for the post-to-pre ratio, corresponding to
changes in occupancy by factors of exp(-1.2) $\simeq 0.30$ and exp(+1.2)
$\simeq 3.3$. In these distributions of changes associated with flares, we see
clear increases in strong-field pixels with relatively low $|{\bf V}_{h}|$ and
$|V_{z}|$ (red features near 3000 G) and decreases in moderate-strength pixels
with moderate speeds (blue features near 2000 G). An increase in weak-field
pixels with larger $|{\bf V}_{h}|$ can also be seen. Because stronger fields
suppress convection more effectively than weaker ones (e.g., Berger et al.
1998; Welsch et al. 2012), it is plausible that the increases in horizontal
field strengths that often occur in strong flares shifted pixels from the
central blue patches to the strong-field red patches.

Figure 7: Left two columns: Distributions of the logarithm of pixel counts in
the 2-variable phase space of magnetic field versus Poynting flux for all
eruptive events (first column), and distribution of differences in pixel
counts in the same 2-variable space, given by $\log$(post-flare counts) -
$\log$(pre-flare counts) (second column). The top row shows the logarithm of
counts and the logarithm of change in counts for $|{\bf B}_{h}|$, while the
bottom row shows these quantities for $|B_{z}|.$ Right two columns:
Distributions of the logarithm of pixel counts in the 2-variable phase space
of velocity field versus Poynting flux for all eruptive events (third column),
and distribution of differences in pixel counts in the same 2-variable space
(fourth column). The top row shows the logarithm of counts and the logarithm
of change in counts for $|{\bf V}_{h}|$, while the bottom row shows these
quantities for $V_{z}.$ Only pixels within the flare masks are included in
these distributions.

As shown in Figure 7, we also analyzed pre- and post-flare distributions of
pixel counts in 2-variable phase spaces of Poynting flux versus $|{\bf
B}_{h}|$ (top row) and $|B_{z}|$ (bottom row) in the first and second columns,
and Poynting flux versus $|{\bf V}_{h}|$ (top row) and $V_{z}$ (bottom row) in
the third and fourth columns. For each 2-variable space, the first and third
columns show the logarithm of pre-flare counts, and the second and fourth
columns show the logarithm of the difference in counts, post-flare minus pre-
flare. Panels in the second column show decreases in upward Poynting flux from
moderate-strength pixels (blue areas around 1000 G) and increases in downward
Poynting flux in higher-strength pixels (red areas around 2000 G). We note
three features in the top panel of the fourth column: decreases in both
positive and negative Poynting flux from pixels with moderate horizontal flows
(blue areas around 0.3 km s-1); a decrease in positive Poynting flux from
pixels with weak horizontal flows (blue area near $|{\bf V}_{h}|\simeq$ 0);
and an increase in negative Poynting flux from pixels with similarly weak
horizontal flows (red area near $|{\bf V}_{h}|\simeq$ 0). The bottom panel of
the fourth column shows a large decrease in positive Poynting flux from pixels
with upward flows.

The dominant shift of the pixel distributions in Figure 6 was an approximate
doubling of some horizontal field strengths and an approximate halving in flow
speeds for that shifted population. In principle, these shifts might have
offset each other, leading to little change in the energy flux. Figure 7
demonstrates, however, that there is a change in the character of the Poynting
flux, not just its magnitude. First, more post-flare energy flux is downward.
Second, the field strengths for which most changes in the Poynting flux occur
(around 1000 – 2000 G for both $B_{h}$ and $|B_{z}|$) differ from the field
strengths at which most changes in flows occurred in Figure 6 (around 1500 –
3000 G for both $B_{h}$ and $|B_{z}|$). Because Poynting fluxes depend upon
flow direction in addition to speed, these changes indicate that both
directions and magnitudes of flows change after flares. Our analysis thus far,
however, has not yet revealed the physical basis for such changes. In a
separate study, currently underway, we are further analyzing the nature of
these changes in flows and fields.

As remarked earlier, the helicity flux density is a gauge-dependent quantity,
and thus does not have a clear physical meaning. It is therefore uncertain
what causes the decrease of helicity flux after flares. The decrease in
helicity flux seen in Figure 3 accords with the decrease in strong fields’
horizontal speeds after flares shown in the right column of Figure 6. Like the
Poynting flux, however, the helicity flux depends on flows’ directions in
addition to their magnitudes, so we cannot rule out the possibility that the
same changes in flows’ directions that increased downward energy transport
also reduced the net injection of helicity into the corona.

###  3.2 Flare-Induced Changes in Coronal Helicity and Energy

Having analyzed changes in helicity and energy fluxes associated with our
sample of X-class flares, we next study evolution in the coronal content of
helicity and energy associated with these flares. As shown in Figure 1, the
helicity present in our model of the coronal field of AR 11158 decreases after
the eruptive X2.2 flare. To determine if coronal models of other ARs in our
sample tend to exhibit similar flare-associated changes, we created SPE plots,
shown in Figure 8, of magnetic helicity (top), free energy (middle) and total
energy (bottom) for the eruptive (black) and confined (red) events. The left-
column SPE plots were made combining events’ time series in physical units.
Because these quantities for each AR scale with that AR’s total unsigned flux,
evolution in a subset of high-flux ARs could dominate these curves. To check
this, we also created the right-column SPE plots by first normalizing each
AR’s time series by its highest value over the entire interval that it was
observed and then combining the regions’ time series.

In each plot, we show five data points before and five data points after the
event times, i.e., the GOES flare start and end times in Table 1. The data
cadence for each five-point block is 720 seconds. The $t0$ in the x-axis
refers to the flare start time. To avoid bad data immediately after these
major flares, the post-event data start with the first HMI observation at
least 30 minutes after the end time of each flare. The $t1$ in the x-axis
refers to this time. The green vertical line denotes the flare time, but its
placement with respect to pre- and post-flare data points should not be viewed
as physical: due to differences in flares’ start times and durations with
respect to HMI’s nominal, 720-second cadence for vector magnetograms, the
spacing between the last pre-flare and first post-flare data points shown does
not correspond to 1440 seconds. Our use of variable-sized time windows is
appropriate, given the physical basis for our study: we believe that magnetic
reconnection is the ultimate cause of both (i) the GOES emission that
determines the events’ start and stop times and (ii) the changes in magnetic
energy and helicity (and their fluxes) that we analyze.

Flare-associated changes are noticeable for eruptive flares in all six plots,
and for confined events in the two versions of the free energy plot. To assess
the significance of these changes, a uniform uncertainty was computed for the
pre-flare epoch, and, separately, for the post-flare epoch, estimated from the
standard deviations of the corresponding five data points. The error bars
shown in the SPE plots are calculated from the propagated errors, i.e., as the
square root of the mean of the total variance (the sum of squared
uncertainties in the quantities being averaged). The change in each quantity
(in physical units) in the left-column plots was computed as the average of
the five post-flare points minus the average of the five pre-flare points. For
eruptive events, both magnetic helicity and free energy decrease by
significant amounts after the flares. In contrast, only free energy is seen to
decrease after the flares for the confined events, and the magnitude of its
decrease from the pre-flare average is only significant at about the
1-$\sigma$ level. For the eruptive events, helicity decreases by about
$(0.45\pm 0.04)\times 10^{43}$ Mx2 (about 17% of the pre-event average), and
free energy decreases by about $(0.47\pm 0.07)\times 10^{32}$ ergs (about 10%
of the pre-flare average). The decrease in total magnetic energy for eruptive
events is only marginally statistically significant at the 1-$\sigma$ level,
and at roughly $(0.15\pm 0.10)\times 10^{32}$ ergs, it is only about 0.7% of
the pre-flare average. The fractional decrease in helicity is about twice as
large as that in free energy, implying that helicity is more sensitive to
changes in eruptive systems than free energy. For confined events, the
decrease in free energy is about $(0.59\pm 0.48)\times 10^{32}$ erg. In
confined events, the other quantities actually increase after flares, by 1.1%
for total helicity and 1.6% for total energy (Figure 11, discussed below in
our analysis of the ARs’ longer-term evolution, is also relevant here.)
However, this 1.1% increase in total helicity may not be significant because
computational errors can be larger than this change.

Figure 8: Superposed epoch plots of unsigned magnetic helicity (top), free
magnetic energy (middle), and total magnetic energy (bottom) in the active
regions for the eruptive (black) and confined (red) X-class flares. The
vertical green lines denote the flare start time. Five data points immediately
before the flares and five data points starting 30 minutes after the end of
the flares are chosen for these calculations. The time before the flares is
denoted as t0; the time 30 minutes after the end of the flares is denoted as
t1. Left column: Time series in physical units combined via SPE. Right column:
Normalized time series combined via SPE. Note that the horizontal axis shows
step number (not time) before / after each event; the time step in each set of
five points is 12 minutes.

We also find evidence that both the free energy and helicity released in
eruptive flares tend to be larger when more of these quantities were available
to be released, as shown in Figure 9. This is analytically suggested in linear
theory (Valori et al., 2015). Figure 9 may also suggest upper limits of
$\sim$20% of the free energy and $\sim$30% of the total helicity in the ARs to
be released by major eruptive flares. This is also consistent with estimates
from previous studies using both observational and synthetic data (see, e.g.,
Nindos et al., 2003; Moraitis et al., 2014).

The difference in free energy, $E_{\rm free,before}-E_{\rm free,after}$, is
expected to be positive if energy is released. The negative change estimated
for one event could arise from uncertainties in the NLFFF models, or from
other evolution in parallel, such as flux emergence, during the pre- to post-
flare interval. This relationship need not imply causality in the reverse
direction, with availability of more energy and helicity driving the release
of more, as opposed to simply enabling the release of more. Based on
observations of a nearly “universal” power-law distribution of flare number
frequency versus energy for solar ARs (e.g., Crosby et al. 1993; Aschwanden et
al. 2016 \- but see also Wheatland 2010 for a possible exception), a
reasonable prior expectation is that the free energy released in a given event
is a random variable drawn from a power-law parent distribution. The low free-
energy-loss and low helicity-loss outlier in the lower right corner of each
panel is consistent with this expectation of stochasticity, subject also to
the specific local conditions where the flare was triggered. An alternative
model is the build-up-release (BUR) scenario, described by Hudson (2020),
which predicts that available free energy in ARs is released as it is
accumulated. The BUR view appears inconsistent with our observations in that
released free energy is a small fraction of available free energy in all
cases. Whatever the underlying physical mechanisms governing free energy and
helicity release, the correlations here could have implications for space
weather prediction: when more energy is available, more tends to be released,
with more repercussions for associated or secondary phenomena, such as CMEs
and solar energetic particle events.

Figure 9:  Changes in free energy (left) and magnetic helicity (right) in
eruptive events, plotted against ARs’ available free energy and helicity,
respectively. Each flare is denoted by its flare ID listed in Table 1. Because
flares 5 and 6 were less than an hour apart, the two are treated as one event
here, plotted at the point labeled 5.

The SPE plots in Figures 3 and 8 show aspects of magnetic evolution close to
the times of X-class flares. What can we learn about such evolution from SPE
plots over longer time scales before and after these flares? Figure 10 shows
SPE plots from 48 hours pre-flare to 18 hours post-flare, for unsigned
helicity, $|H|$ (first row), unsigned helicity in the current-carrying field,
$|H_{j}|$ (second row), total energy, $E$ (third row), and free energy,
$E_{\rm free}$ (fourth row). As above, we combined ARs’ time series in both
physical units (left column) and after normalizing each AR’s time series
(right column) by that AR’s peak value over the interval of its observation.

In these longer duration SPE plots, it should be noted that, due to our small
sample of confined events, some observations are included more than once: for
the same AR, an observation $\Delta t_{1}$ hours before one flare can also be
$\Delta t_{2}$ hours before a second flare, and the synchronization of flare
times can include both when a long enough pre-flare interval is plotted. This
overlapping of multiple segments of an AR’s time series can, by summing
evolution at different phases, possibly obscure significant evolution that the
SPE approach is meant to highlight. Effects from shifting and combining
multiple observations from a single AR could even produce trends that are
spurious. Consequently, one expects stronger conclusions to be achieved by
applying SPE analysis to a larger sample of confined flares with no duplicate
epochs used in the superposition.

With this caveat in mind, few pre-flare trends in these quantities are
evident. Prior to confined events, most of these quantities increase
monotonically, but the small number of confined events in our sample must be
kept in mind. For eruptive events, no buildup of energy and helicity prior to
these X-class flares is obvious, but there is a clear tendency in all four
quantities for post-flare evolution toward pre-flare levels over the first 12
hours after each event. In particular, the helicity of the current-carrying
field and free energy appear to be essentially replenished on this time scale.
For smaller flares the replenishment might happen on shorter timescales, see
e.g., Dumbović et al. (2021); Thalmann et al. (2022a).

Because the emergence of new flux can affect all of the quantities plotted in
Figure 10, and might trigger flares/CMEs (e.g., Priest & Heyvaerts 1974), it
is worthwhile to also investigate the evolution of total unsigned (TUS) flux
in our AR ensemble. Flux emergence should cause increases in TUS flux, and
flux cancellation (another possible trigger of CMEs; e.g., Amari et al. 2010)
should cause decreases in TUS flux. We remark that flux cancellation might not
be discernible in ARs in which flux emergence is also occurring; but
superposing evolution of TUS flux in our sample regions might reveal it. For
each AR, we computed TUS flux in the [-48,+18]-hour interval around each flare
by summing the absolute value of radial field in pixels with total field
strength greater than 300 G (note that this is approximately twice the flux in
a single polarity.) We then superposed the TUS flux curves for all epochs. Our
results are shown in Figure 11. The top panel shows summed epochs for time
series in physical units; the bottom panel shows summed epochs for normalized
data. Error bars are smaller than the plotted circular symbols. For eruptive
events, no strong trends are visible, though a slight increase in flux tends
to occur during the 8 hr prior to event onset. The physical significance of
this minor increase is unclear. For confined events, a monotonic increase in
TUS flux is seen over the entire plotted interval. Hence, ongoing flux
emergence is a plausible explanation for the increases in total helicity, free
energy, and total energy for confined flares seen in Figure 10, as well as the
slight increases of total helicity and total energy near flare times seen in
Figure 8.

Figure 10: Same as Figure 8 but for longer time intervals. The temporal
profile of helicity of the current-carrying field ($|H_{j}|$), one component
of the helicity, is also plotted in the second row. Left column: Time series
in physical units, combined via SPE. Right column: Normalized time series,
combined via SPE. Figure 11: Superposed epoch plots of total unsigned (TUS)
magnetic flux (top) and normalized magnetic flux (bottom) in the active
regions for eruptive (black) and confined (red) X-class flares. The vertical
green lines denote the flare occurrence time. The absolute flux in all pixels
with field strength greater than 300 G was summed for these magnetic flux
estimates, i.e., flux in both polarities was included.

###  3.3 Flare-Induced Changes in Magnetic Twist

Non-potentiality of magnetic fields in ARs can be quantified by various
indices (e.g., Leka & Barnes, 2003). Here, we use magnetic twist as a measure
of non-potentiality to investigate magnetic changes in ARs associated with
X-class flares. Based on the force-free field assumption,

$\bf{\nabla}\times\bf{B}=\alpha\bf{B}~{},$ (9)

the force-free parameter, $\alpha$, is often treated as a measure of magnetic
twist in an AR (e.g., Seehafer, 1990; Pevtsov et al., 1995). A scalar,
$\alpha$ varies in space, and is often estimated for each pixel from vector
magnetograms via

$\alpha=[{\bf\nabla\times B}]_{z}/B_{z}~{},$ (10)

using finite-difference approximations. To characterize twist in an entire AR,
we choose a $B_{z}^{2}$-weighted, force-free $\alpha_{w}$, proposed by Hagino
& Sakurai (2004),

$\alpha_{w}=\frac{\displaystyle\int_{S}\alpha(x,y)\,B_{z}^{2}(x,y)\,dx\,dy}{\displaystyle\int_{S}\,B_{z}^{2}(x,y)\,dx\,dy},$
(11)

where $B_{z}$ is the vertical component of magnetic field. The integral is
done over the region of interest in the AR. One advantage of $\alpha_{w}$ over
a simple average of force-free $\alpha$ from Equation (10) is that
singularities at the polarity inversion lines (where $|{\bf B}|$ can be large
but $B_{z}$ is zero) are avoided (Tiwari et al., 2009). Because the $B_{z}$
weighting de-emphasizes values of $\alpha$ from noisy, weak-field pixels,
$\alpha_{w}$ is also believed to be one of the most robust approaches in
computing $\alpha$ (Leka & Skumanich, 1999). We note that $\alpha_{w}$ is an
intensive parameter – that is, it does not increase with an AR’s flux content
or size. This is in contrast to $E,dE/dt,H,$ and $dH/dt$, which are extensive
parameters, i.e., they tend to increase with system size. We remark that a
high-flux active region need not, in principle, also be highly non-potential.
In practice, however, Fisher et al. (1998) found that, in a sample of a few
hundred ARs, totals of unsigned vertical current in ARs were highly correlated
with their magnetic fluxes. Based on this observation, we expect integral
measures of non-potentiality to scale with AR size. We also note that some
parameters, or combinations of parameters, might not be readily classified as
either extensive or intensive. Leka & Barnes (2007); Welsch et al. (2009); Tan
et al. (2009), and Bobra & Couvidat (2015) found that extensive magnetic
parameters tended to be better predictors of flare productivity than intensive
parameters. In contrast, Bobra & Ilonidis (2016) found that intensive
parameters were better predictors of CME occurrence.

Figure 12:  Temporal profiles of normalized magnetic twist near the times of
X-class flares in our AR sample. Blue curves refer to eruptive flares; red
confined flares. The vertical black line denotes the occurrence time of the
flares.

Figure 12 shows normalized temporal profiles of unsigned magnetic twist for
the eruptive (blue) and confined (red) events. Sudden jumps occur at the times
of the flares for most events. This sudden increase appears to be permanent:
the twist stays at the increased level after the events. SPE plots of
$|\alpha|$ for our event sample are shown in Figure 13, computed from either
Equation (10) (left) or Equation (11) (right). The flare masks are applied for
both calculations, i.e., only the pixels in the flare masks are included for
the computations. The same flare mask is applied for all time steps for each
event.222 We have also tested the sensitivity of our results to variations in
flare-mask boundaries by dilating the flare masks. The time series
$\alpha_{ave}$ and $\alpha_{w}$ retain very similar shapes, though the
magnitudes of their changes in eruptive flares decrease with the level of
expansion of the mask, up to 46% with a half-width of 11 pixels for the
dilation. The magnitudes of both $\alpha_{ave}$ and $\alpha_{w}$ themselves
(cf., their changes) also become much smaller. We speculate that expanding the
masks adds pixels that are unrelated to the high-twist fields near the flaring
PILs, with the added $\alpha$ values randomly distributed in sign and
magnitude. Thus, the added pixels only increase the denominators in these
averages.  With either approach, eruptive (black) events show a significant
twist increase after the occurrence of the flares. We remark that twist in
confined events is about one order of magnitude smaller than in eruptive
events – a striking difference. Note that helicity flux in confined events in
Figure 3 is about 10 times larger than the helicity flux in eruptive events,
but here we find the twist content in confined events is much smaller for
confined events. The SPE helicity flux time series for the set of confined
events is dominated by AR 12192, which is a very high-magnetic-flux AR, making
its $|dH/dt|$ very large. This juxtaposition highlights the difference between
the scaling of extensive parameters, like $dH/dt$, and intensive parameters,
like $\alpha$. This accords with findings by Bobra & Ilonidis (2016), who
reported that high values of intensive twist measures were associated with CME
occurrence. The uncertainty in average twist is large relative to fluctuations
in twist for the confined flares, due both to our small sample of such events
and the relatively low value of twist in those ARs (lower signal-to-noise).
Consequently, changes in twist after confined flares (e.g., a small increase
in the right panel) are not significant.

The increase in ARs’ magnetic twist after flares must be caused by changes in
their photospheric magnetic fields, most notably the enhancement of horizontal
fields. Indeed, near flaring polarity inversion lines in strong flares,
horizontal field strengths are often observed to increase, while vertical
field strengths are not observed to change significantly (e.g., Wang et al.,
2012; Sun et al., 2012). Magnetic shear along the PIL also often increases
(Wang et al., 2012). The increases in average twist might reflect the
increased signal-to-noise in computing $\alpha$ values in stronger horizontal
fields: horizontal fields below our 300G noise-floor threshold are excluded
from our calculation, but the increase of horizontal field strengths after
flares implies more pixels are included in post-flare magnetograms. We note
that the fractional increase in twist is larger in the left panel of Figure
13. The $B_{z}^{2}$-weighting in Equation (11), for the right panel, possibly
acts to suppress the contributions from pixels near PILs, where $|\mathbf{B}|$
is large but $|B_{z}|$ is small. Hence, the difference between the fractional
changes in these plots suggests that the magnetic changes responsible for much
of the increase in twist occur in near-PIL fields.

The long-term trend of decreasing twist prior to events is significant, but we
do not have a definitive explanation for it. Decreasing pre-flare twist might
plausibly be an indication of expanding coronal magnetic structures (e.g.,
flux ropes) as coronal energy builds up. If the structure about to erupt
gradually ascends in a metastable evolutionary stage, this could conceivably
make the underlying photospheric fields less horizontal, with little or no
change in $B_{z}$. This scenario accords with observations of slow rising
motions seen before some eruptions (e.g., Schuck et al. 2004; Sterling & Moore
2005; Liu et al. 2012). This is, in essence, an evolutionary version of the
standard flare model in which the plasmoid and the X-point beneath it
gradually ascend a few hours before the eruption. Immediately after the
eruption, near-PIL horizontal fields would become stronger from the implosion
beneath coronal reconnection sites, and, overall, values of the predominant,
post-flare $\alpha$ would have higher magnitude. Unfortunately, the twist
measures that we study are only photospheric, so do not provide unambiguous
information about coronal magnetic structure. We note that the flares masks
over which these measures of average twist were derived post facto, and we
have not analyzed any flare-quiet epochs, so it is unclear whether these
trends could have any predictive capability. This finding may warrant futher
work.

Figure 13: Superposed epoch plots of unsigned averaged magnetic twist from
Equation (10) (left) and from Equation (11) (right). The pixels in the flare
masks with field strength greater than 300 G are included for the
calculations. The SPE for confined events is shifted upward along the y-axis
by an amount y_shift to aid comparison of the time evolution. y_shift =
$7.15\times 10^{-8}$ m-1 for averaged $\alpha_{ave}$ (left) and $10.99\times
10^{-8}$ m-1 for weighted averaged $\alpha_{w}$ (right). Blue vertical lines
denote flare start time.

An alternative interpretation of post-flare twist increases might involve the
Taylor relaxation hypothesis (Taylor, 1974), which predicts that magnetic
reconnection should tend to make $\alpha(x,y)$ more uniform across ARs.
Physically, the Taylor hypothesis is less applicable to coronal fields than to
laboratory plasmas, because it presumes that reconnection can operate on all
fields present, and, unlike a laboratory system, a given coronal magnetic
system is not truly isolated. As Antiochos (2003) and Longcope & Malanushenko
(2008) have noted, however, photospheric line-tying of coronal fields implies
that not all coronal fields can reconnect in a given flare. Instead,
reconnection can only act to make twist more uniform within the domain that
participates in a given flare. Also unlike laboratory systems, the corona is
an open volume: when a flare is eruptive, some helicity in the source region
is ejected into the heliosphere. Nonetheless, in general, helicity remains in
post-reconnection fields that remain closed: indeed, magnetic shear along PILs
is often observed to increase after major flares (e.g., Wang & Liu 2010). In
fact, this finding spurred a heated debate in the community when it was first
reported (Wang et al., 1994).

In a study of several dozen ARs, Nandy et al. (2003) reported evidence for
this effect in a subset of ARs whose twist was characterized for several
successive days. Their analysis included all GOES flares ascribed to each
region in their sample. It is plausible that multiple episodes of reconnection
occurred throughout the ARs that they studied, enabling twist to become more
uniform across these ARs over time. By following the variance in regions’
twist over several days, they estimated a characteristic timescale of circa
eight days over which regions’ twist tended to become more uniform.

To see whether twist within our AR sample became more spatially uniform after
our flares, we computed the difference (post-flare minus pre-flare) in the
standard deviation of $\alpha$ in our 20 events. For each event, we computed
$\alpha$ values by applying Equation 10 to pixels in our flare masks where the
field strength was greater than 300 G. These masks should delineate the fields
that participated in each flare’s reconnection, and that might therefore be
expected to exhibit more uniform post-flare twist. Two standard deviations
were separately calculated, one each from the sets of positive and negative
$\alpha$ values. The standard deviation for each event is chosen to be the
average of them. In Figure 14, we show the difference in standard deviations
in $\alpha$ versus the average of standard deviations. This difference is
negative for most of our flares (13 $<$ 0 versus 7 $>$ 0), meaning $\alpha$
tends to become more uniform in our flare sample. If positive and negative
changes were equally likely, the probability that 13 or more events among our
sample of 20 would have negative changes is about 13% – unlikely, but still a
reasonable possibility. The evolution of twist in our AR sample due to flares,
therefore, tends to accord with expectations based on the Taylor relaxation
hypothesis, but this effect is not strong. We also note that Equation 11 can
be also written, from Equation (11), as

$\alpha_{w}=\frac{\displaystyle\int_{S}J_{z}(x,y)\,B_{z}(x,y)\,dx\,dy}{\displaystyle\int_{S}\,B_{z}^{2}(x,y)\,dx\,dy},$
(12)

To the extent that $B_{z}^{2}$ does not change significantly due to a flare,
an increase of $\alpha_{w}$ indicates a larger $J_{z}$, albeit with an
inverse-$|B_{z}|$ weighting.

Figure 14:  Changes in the standard deviation of twist before and after 20
events, one difference per event, are shown for eruptive source regions
(black) and confined source regions (red). Twists from pixels enclosed in the
flare masks with field strengths greater than 300 G were included.

###  3.4 Decomposition of Relative Helicity and Eruptive Events

Figure 15:  Scatter plot between $|H_{total}|$ and $|H_{j}|$ for the events
listed in Table 1. Black dots represent eruptive events; red dots represent
confined events. Black line is a linear fit to the eruptive events (black
dots). The green line, with a slope of 0.1, has been suggested by Gupta et al.
(2021) as a threshold that divides eruptive from confined events. Both
$|H_{total}|$ and $|H_{j}|$ of AR 12192 for flares 13-17 are multiplied by a
factor of 0.1 in order to be included in the plots. The events are denoted by
their flare IDs.

The helicity of the current-carrying field, $H_{j}$, has been found to be
related to the eruptivity of events (e.g., Pariat et al., 2017). The ratio of
$|H_{j}|/|H|$ shows the potential to differentiate eruptive and non-eruptive
events (e.g., Thalmann et al., 2019b; Gupta et al., 2021). Gupta et al. (2021)
find a threshold value of 0.1 for this ratio. Above this threshold, flares
will tend to be eruptive. Our results do not contradict this finding, as shown
in Figure 15, albeit with statistical deviations. On the other hand, confined
events clearly do not obey this scaling. Thus, as discussed in Gupta et al.
(2021), the helicity ratio – alone or in combination with other quantities
such as the free energy ratio and flux-normalized current-carrying helicity –
may be useful to characterize the potential of an AR to flare.

##  4 Discussion: Summary, Conclusions, and Future Work

To understand how magnetic energy and magnetic helicity evolve in association
to major flares, we analyzed a dataset of photospheric vector magnetograms
observed by HMI around the times of 21 GOES X-class flares produced by 13
active regions. Six of these flares, from two active regions, were confined;
the remaining 15 flares, from 11 active regions, were eruptive.

We applied the DAVE4VM tracking method to these magnetogram sequences to
estimate the photospheric velocities that are needed to determine the fluxes
of magnetic energy (the Poynting flux) and magnetic helicity across the
photosphere. We also employed an optimization-based NLFFF code to extrapolate
to coronal magnetic fields. For these model coronal fields, we calculated the
magnetic energy, free magnetic energy, relative magnetic helicity, and
helicity of the current-carrying component of the field. From the magnetograms
themselves, we also calculated average twist over each AR (both unweighted and
with $B_{z}^{2}$-weighting), and total unsigned flux in strong-field ($|{\bf
B}|>300$ G) pixels.

To synthesize information from all events in order to characterize magnetic
evolution associated with these flares, we used superposed epoch (SPE)
analysis to combine time series from all regions. For each of the quantities
above, we extracted time series in a time interval around each event, and
averaged these series with the flares’ start times (the “key times” in SPE
terminology) co-aligned. We applied SPE to intervals of two durations: within
$\pm$ 5 hours of each flare, to identify typical rapid changes related to, or
resulting from, flares; and, separately, from 48 hours before to 18 hours
after each flare, to identify patterns in longer-time-scale evolution both
leading up to flares and in the hours following them.

Here, we itemize our most notable results:

  * •

Our main finding is that, on short time scales ($\sim$ 1h) after eruptive
flares we find decreases in both the coronal magnetic energy, $E$, and
helicity, $H$ (Figure 8); and their photospheric fluxes, $dE/dt$ and $dH/dt$
(Figure 3). Changes in free energy are statistically significant; changes in
total energy are less significant, given the roughly unchanged bulk energy of
the reference, potential magnetic field.

  * •

Using SPE to combine helicity estimates from our coronal extrapolations, we
find clear evidence of decreases in helicity in eruptive flares (Figures 8 and
9), consistent with the bodily removal of helicity from the corona by
jettisoned coronal ejecta. Low (2001) argued that the systematic removal of
magnetic helicity from the corona by CMEs might play an important role in the
solar dynamo. The insignificant decreases of total helicity that we find with
confined flares suggests these flares are an insignificant sink for helicity,
in accordance with the helicity conservation principle in confined systems.

  * •

Looking at the evolution of energy and helicity over longer time scales
(Figure 10) after eruptive events, we find that both quantities return to
near-pre-flare levels. The time scale for this replenishment is roughly 12
hours for the X-class flares analyzed in our work. If flares substantially
deplete the free energy in ARs, this would imply that there should be a
relative paucity of multiple X-class flares from the same ARs on time scales
shorter than this. We find, however, that eruptive flares only remove 10% of
free energy (Figure 8, middle row). Replenishment of helicity and energy after
large eruptive events is consistent with the idea, suggested by Longcope &
Welsch (2000), that the solar interior can act as a twist reservoir (or
“current driver”), to re-supply coronal helicity as it is removed by
ejections.

  * •

Consistent with the previous report of Bobra & Ilonidis (2016), we find that
the twist present in our small sample of ARs that produced confined flares is
roughly an order of magnitude smaller than the twist in our sample of ARs that
produced eruptions (see Figure 13).

  * •

Prior to eruptions, we find a tendency for average measures of twist within
flare masks to decrease. This might be explained by a toy-model with an
emerging flux rope with large internal twist, surrounded by lower-twist areas:
in the first phase of emergence, the photosphere cuts through the rope’s apex,
and a large area with high twist would be present in magnetograms; but as the
rope’s legs start to emerge, the area with strong, nonzero twist would
decrease; sufficient emergence could trigger eruption. We note that Figure 11
shows a small but significant, pre-eruption tendency for total unsigned flux
to increase, a signature of new flux emergence. Our averages of $\alpha$ might
also decrease if the pre-eruptive structure gradually ascends in a pre-flare,
slow-rise process, which might decrease horizontal field strengths at the
photosphere, lowering averaged values of $\alpha$.

  * •

Following eruptions, average magnetic twist in ARs is also observed to
increase significantly, as well as virtually instantly (i.e., within an hour
or so - Figure 13). The causes of this increase in twist remain unclear. This
might be explained by the strengthening of horizontal fields after eruptions,
due to implosion of fields beneath the coronal reconnection region: increases
in $|\alpha|$ would be expected if low-lying fields in eruptive regions have
more organized twist (i.e., flux ropes) than the structures hosting confined
flares. By studying the standard deviation of strong-field pixels’ twist
before and after individual flares, we also find some evidence for Taylor
relaxation (i.e., twist growing more uniform due to magnetic reconnection):
there is a marginally significant preponderance of ARs with decreases in the
standard deviation of twist after flares.

  * •

Flare-associated changes in $dE/dt$ and $dH/dt$ are shown in Figure 3. The
physical causes of the fluxes’ decrease is unclear, although we see clear
evidence that properties of the photospheric flows that drive these fluxes
change after flares. Average flow speeds in our event sample do not change
significantly. Delving deeper into flow changes, we computed the pre- and
post-flare distributions of pixels’ horizontal and vertical velocities as
functions of those pixels’ total field strength (Figure 6). We see increases
in speeds among weak-field pixels and decreases in speeds among strong-field
pixels. These changes might yield decreases in the magnitudes of $dE/dt$ and
$dH/dt$. It is known that strong flares often cause the magnetic field in some
areas to become “more horizontal” due to an increase in $B_{h}$, with the
normal field relatively unchanged (e.g., Hudson et al., 2008; Wang & Liu,
2010; Barczynski et al., 2019b). Given that stronger fields more effectively
suppress convection (e.g., Berger et al. 1998; Welsch et al. 2012), flare-
associated increases in $B_{h}$ might be partly responsible for decreases in
speeds in high-$|{\bf B}|$ pixels. Such changes, however, typically occur over
small areas, meaning that, by themselves, magnetic changes might be a minor
factor in altering the overall fluxes. We also computed the pre- and post-
flare distributions of pixels’ Poynting flux values as functions of those
pixels’ horizontal and vertical magnetic field, and, separately, as functions
of their horizontal and vertical velocities (Figure 7). Studying the
differences between the pre- and post-flare distributions, we see clear
increases in downward Poynting fluxes after flares. The nature of these post-
flare increases in the downward flow of magnetic energy warrants further
study.

  * •

For confined flares, only changes in the free magnetic energy are found to be
statistically significant. Given our small sample of confined flares, however,
we cannot be confident that a larger sample of such events would not exhibit
at least some of the changes we see in our eruptive sample.

In addition to the physical implications of our results for an improved
understanding of the roles of energy and helicity in strong flares, we briefly
refer to two methodology implications here: first, the SPE approach is useful
for finding patterns that could well be missed in studies of individual
events, because: (i) for an individual AR, time series of the variables that
we study exhibit strong fluctuations; and (ii) there is substantial variation
between ARs. Second, our methods of estimating both the photospheric fluxes of
energy and helicity and their coronal values are robust enough to enable
detection of systematic changes in these quantities, despite fluctuations in
each AR’s evolution and variability between ARs. The importance of combining
diagnostics from multiple, independent sources when magnetic energy and
helicity are studied was also highlighted by Thalmann et al. (2021).

Our results suggest several areas for future research:

  * •

Our analysis of magnetic evolution associated with confined flares was
severely hampered by our small sample size. A similar analysis on a larger
sample, particularly for confined events, would be both worthy and meaningful.

  * •

The nature of increases in downward Poynting flux is not yet understood. In
which magnetic structures do such changes occur? What effect(s) cause(s) these
changes? One approach on addressing these questions is to undertake case
studies, and in each case to identify areas with increases in downward
Poynting flux after flares. Magnetic structures in these areas, and their
relationships to flare morphology, such as ribbon emission, should be
investigated.

  * •

The nature of pre-flare decreases and post-flare increases in AR twist
associated with large, eruptive flares is also yet to be understood. From
differences in pre-flare and post-flare magnetograms, where do the primary
changes in twist occur? What is the magnetic structure in areas with
significant changes in twist? Are all such areas closely associated with the
flare site, or does twist sometimes change in significant ways in areas
peripheral to the core flare location? Again, case studies might be valuable.

We believe our results leave a lot to be understood and be learned about
flares and eruptions from studies of photospheric vector magnetograms, along
with physical quantities such as photospheric velocity fields and extrapolatd
coronal magnetic fields, to name a few, that are derived from them.

We wish to thank the anonymous referee for the valuable comments and
suggestions. BTW, SHP, and YL thank the US taxpayers for providing the funding
that made this research possible. We acknowledge support from NASA LWS
80NSSC19K0072 (YL and BTW). NASA’s SDO satellite and the HMI instrument were
joint efforts by many teams and individuals, whose efforts to produce the HMI
magnetograms that we analyzed here are greatly appreciated. YG is supported by
NSFC (11773016 and 11961131002) and 2020YFC2201201. GV acknowledges funding by
the Bundesministerium für Wirtschaft und Technologie through Deutsches Zentrum
für Luft- und Raumfahrt e.V. (DLR), Grants No. 50 OT 1001/1201/1901 as well as
50 OT 0801/1003/1203/1703, and by the President of the Max Planck Society
(MPG). EP acknowledges support from the French Programme National PNST of
CNRS/INSU co-funded by CNES and CEA. EP also acknowledges financial support
from the French national space agency (CNES) through the APR program. JT
acknowledges the Austrian Science Fund (FWF): P31413-N27. This article
profited from discussions during the meetings of the ISSI International Team
“Magnetic Helicity estimations in models and observations of the solar
magnetic field” (http://www.issibern.ch/teams/magnetichelicity/) and the ISSI
International Team “Magnetic Helicity in Astrophysical Plasmas”
(https://www.issibern.ch/teams/helicityastroplas/). BTW acknowledges support
from the Fundación Jesús Serra, which supported a visit to the IAC-Tenerife
during which a portion of this work was completed.

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