##  Chapter 1 Solar Flares and Magnetic Helicity

Sung-Hong Park

[1]Shin Toriumi

###### Abstract

Solar flares and coronal mass ejections are the largest energy release
phenomena in the current solar system. They cause drastic enhancements of
electromagnetic waves of various wavelengths and sometimes eject coronal
material into the interplanetary space, disturbing the magnetic surroundings
of orbiting planets including the Earth. It is generally accepted that solar
flares are a phenomenon in which magnetic energy stored in the solar
atmosphere above an active region is suddenly released through magnetic
reconnection. Therefore, to elucidate the nature of solar flares, it is
critical to estimate the complexity of the magnetic field and track its
evolution. Magnetic helicity, a measure of the twist of coronal magnetic
structures, is thus used to quantify and characterize the complexity of flare-
productive active regions. This chapter provides an overview of solar flares
and discusses how the different concepts of magnetic helicity are used to
understand and predict solar flares.

Keywords: Solar flares, Coronal mass ejections, Active regions, Magnetic
fields, Magnetic helicity

###  1.1 Introduction

Figure 1.1: Solar flare and CME. (Left) An X2.2-class flare in NOAA AR 11158,
as observed by the Hinode satellite. The top panel shows the intensity of the
Ca II H line, in which bright “flare ribbons” expand across the AR, whereas
the bottom panel displays the corresponding photospheric magnetogram
(white/black corresponds to positive/negative polarity), both detected by
Hinode/SOT (Kosugi et al., 2007; Tsuneta et al., 2008). (Right) Lightbulb-
shaped CME captured by SOHO/LASCO on 2000 February 27. Courtesy of SOHO (ESA &
NASA).

In astronomy, the term “flare” generally refers to the sudden brightening of
electromagnetic waves at a variety of wavelengths. In the Sun, the flares
(Figure 1.1) are occasionally observed as brightenings across the spectrum
from X-rays to radio waves with time scales of minutes to hours (Fletcher et
al., 2011; Benz, 2017). The typical amount of energy released during the
flares ranges from $10^{28}$ to $10^{32}$ erg, making flares the largest
energy-releasing phenomena in the current solar system.111The magnitude of
solar flares is characterized by the maximum soft X-ray flux of 1-8 Å measured
by the Geostationary Operational Environmental Satellite (GOES). The GOES
classes are A, B, C, M, and X, corresponding to $10^{-8}$, $10^{-7}$,
$10^{-6}$, $10^{-5}$, and $10^{-4}$ W m-2 at Earth. Solar flares affect
orbiting planets in different ways. For instance, X-ray and ultraviolet
emissions enhance the degree of ionization of planetary atmospheres,
accelerating the escape of atmospheres from the planets. Coronal mass
ejections (CMEs), often accompanying large-scale flares (Chen, 2011), may hit
the magnetosphere of the Earth, causing magnetic storms and malfunctions in
telecommunication systems. Therefore, in modern civilization with highly
advanced information technology, the prediction and forecasting of flares and
CMEs is an urgent issue (Barnes et al., 2016; Kusano et al., 2021).

Observations have shown that particularly strong flares are typically
associated with active regions (ARs), conglomerates of magnetic flux with
sunspots, which strongly suggests that flares are a release of the free
magnetic energy ($\Delta E_{\rm mag}$) that is accumulated in the corona
during the evolution of ARs:

$\Delta E_{\rm mag}=E_{\rm mag}-E_{\rm
p}=\int_{V}\frac{\boldsymbol{B}^{2}}{8\pi}\,dV-\int_{V}\frac{\boldsymbol{B}_{\rm
p}^{2}}{8\pi}\,dV,$ (1.1)

where $\boldsymbol{B}$ is the actual magnetic field and $\boldsymbol{B}_{\rm
p}$ is the potential (current-free) field, $\nabla\times\boldsymbol{B}_{\rm
p}=0$. Here, the potential field can be proven to be the minimum possible
magnetic energy for a given distribution of the normal magnetic field $B_{\rm
n}$ on the photospheric surface.222This property is known as Thomson’s
theorem, which requires the assumption that the magnetic field is solenoidal.
In a modern view, flares occur in association with magnetic reconnection, a
magnetohydrodynamic (MHD) process that converts (free) magnetic energy into
kinetic energy and thermal energy with nonthermal particle acceleration
(Priest and Forbes, 2002; Shibata and Magara, 2011).

Statistically, stronger flares occur in sunspot groups of not only larger spot
areas, or larger amounts of magnetic flux in the photospheric surface (${\rm
S}_{\rm ph}$),

$\Phi=\int_{\rm S_{ph}}|B_{\rm n}|\,dS,$ (1.2)

but also with more complex shapes (Sammis et al., 2000; Toriumi et al., 2017).
Most of the sunspot groups are categorized as $\beta$-spots, in which umbrae
of the positive and negative polarities have their own penumbrae. However, it
is known that the strongest flares tend to occur in sunspot groups called
$\delta$-spots, in which the positive and negative polarities are close to
each other and surrounded by a single penumbra. In such ARs, magnetic
structures often display signs of helical configurations, such as sheared
polarity inversion lines (PILs), rotating sunspots, and magnetic flux ropes
(Toriumi and Wang, 2019). Therefore, magnetic helicity is widely used as an
index to quantitatively measure the degree of twist and complexity of ARs. It
is expected that elucidating the relationship between the magnetic helicity
supplied to the solar surface and accumulated magnetic energy will lead to an
understanding of flares and CMEs (Park et al., 2010b, 2012; Tziotziou et al.,
2012; Kim et al., 2017), and give rise to better prediction and forecasting
capabilities (Bobra et al., 2014; Bobra and Couvidat, 2015; Bobra and
Ilonidis, 2016; Leka et al., 2019).

This chapter discusses the concept of magnetic helicity in the context of AR
evolution and its relationship with flare eruptions. For general accounts of
flares and CMEs, readers may refer to other reviews by, e.g., Priest and
Forbes (2002), Fletcher et al. (2011), Shibata and Magara (2011), and Benz
(2017). Regarding the observational characteristics of the flare-productive
ARs and the attempts to theoretically reproduce them, refer to Toriumi and
Wang (2019).

The remainder of this chapter is organized as follows. In Section 1.2, we
provide an overview of the basic concepts and formulations of magnetic
helicity. In Section 1.3, we provide methods for measuring the volume helicity
and helicity flux. Then, in Section 1.4, we review some practical applications
of these methods to actual AR data. Section 1.5 describes the numerical
reconstruction and modeling of coronal magnetic fields, which are important
for accurately measuring the helicity and validating the techniques. Finally,
in Section 1.6, we summarize this chapter and discuss future perspectives. We
note that, although other forms of helicity (e.g., current helicity and
kinetic helicity) are also discussed in the context of flare activity in ARs,
in this chapter, we keep our primary focus on magnetic helicity.

###  1.2 Basic concepts and formulations

Magnetic helicity is a well-conserved quantity that represents the topology of
a magnetic field, e.g., twists, kinks, and internal linkages (Elsasser, 1956;
Woltjer, 1958; Moffatt, 1969). The helicity $H$ of a magnetic field
$\boldsymbol{B}$ contained within a given volume $V$ is defined as

$H=\int_{V}\boldsymbol{A}\cdot\boldsymbol{B}\,dV,$ (1.3)

where $\boldsymbol{A}$ denotes the corresponding vector potential of
$\boldsymbol{B}$, i.e., $\boldsymbol{B}=\nabla\times\boldsymbol{A}$. It is
strictly conserved in ideal MHD (Woltjer, 1958). The dissipation is relatively
weak even in resistive MHD (Taylor, 1974; Berger and Field, 1984), including
typical flares (Berger, 1984), which is one of the reasons why magnetic
helicity has been used for studying the eruptivity.

Equation (1.3) is gauge-invariant only for magnetic fields fully contained in
a closed volume. To circumvent this limitation, for open systems such as the
solar corona, Berger and Field (1984) and Finn and Antonsen (1985) introduced
the relative helicity,

$H_{\rm
R}=\int_{V}(\boldsymbol{A}+\boldsymbol{A}_{0})\cdot(\boldsymbol{B}-\boldsymbol{B}_{0})\,dV,$
(1.4)

where $\boldsymbol{B}_{0}$ is the reference field and $\boldsymbol{A}_{0}$ is
its vector potential, $\boldsymbol{B}_{0}=\nabla\times\boldsymbol{A}_{0}$. For
practical purposes, the potential field $\boldsymbol{B}_{\rm p}$ is often
chosen as the reference field,

$H_{\rm R}=\int_{V}(\boldsymbol{A}+\boldsymbol{A}_{\rm
p})\cdot(\boldsymbol{B}-\boldsymbol{B}_{\rm p})\,dV.$ (1.5)

The relative helicity (1.5) can be decomposed into two separately gauge-
invariant components (Berger, 2003): $H_{\rm R}=H_{\rm j}+2H_{\rm pj}$ with

$H_{\rm j}=\int_{V}(\boldsymbol{A}-\boldsymbol{A}_{\rm
p})\cdot(\boldsymbol{B}-\boldsymbol{B}_{\rm p})\,dV,$ (1.6) $H_{\rm
pj}=\int_{V}\boldsymbol{A}_{\rm p}\cdot(\boldsymbol{B}-\boldsymbol{B}_{\rm
p})\,dV,$ (1.7)

where $H_{\rm j}$ is the helicity of the current-carrying, or non-potential,
component of the magnetic field, $\boldsymbol{B}_{\rm
j}=\boldsymbol{B}-\boldsymbol{B}_{\rm
p}=\nabla\times(\boldsymbol{A}-\boldsymbol{A}_{\rm p})$, and $H_{\rm pj}$ is
the mutual helicity between $\boldsymbol{B}_{\rm j}$ and $\boldsymbol{B}_{\rm
p}$.

However, care must be taken when considering Equation (1.5) because the
integrand $(\boldsymbol{A}+\boldsymbol{A}_{\rm
p})\cdot(\boldsymbol{B}-\boldsymbol{B}_{\rm p})$ is only meaningful if
integrated over the whole domain, and the simple sum of the relative helicity
in contiguous subvolumes is not equal to the relative helicity of the entire
volume (Berger and Field, 1984; Longcope and Malanushenko, 2008). To overcome
this additivity problem, Longcope and Malanushenko (2008) proposed dividing
the volume into a set of subvolumes whose boundaries between them are magnetic
flux surfaces with no magnetic field penetrating them. The additive self
helicity is thus defined as the relative helicity integrated only over the
corresponding subvolume with respect to the reference potential field
restricted to the subvolume:

$H_{{\rm R}i}=\int_{V_{i}}(\boldsymbol{A}+\boldsymbol{A}_{{\rm
p}i})\cdot(\boldsymbol{B}-\boldsymbol{B}_{{\rm p}i})\,dV,$ (1.8)

where $V_{i}$ is the subvolume bounded by the flux surface, and
$\boldsymbol{B}_{{\rm p}i}$ and $\boldsymbol{A}_{{\rm p}i}$ are its potential
field and the vector potential, respectively. The additive self helicity is
integrated to obtain the relative helicity:

$H_{\rm R}=\sum_{i}H_{{\rm R}i},$ (1.9)

where the $i$ indexes denote the subvolumes.

However, in the definition by Longcope and Malanushenko (2008), it is
difficult to find a suitable set of subvolumes in practical situations, and
the finest possible decomposition may be to take an infinitesimally thin tube
around each field line. In this limit, the reference potential field would be
the magnetic field itself, so that the additive self helicity would vanish.
Yeates and Page (2018) followed a different approach, decomposing the relative
helicity into a relative field-line helicity, which vanishes along any field
line where the original field matches the reference (potential) field. The
integral of the relative field-line helicity gives the relative helicity.
Although this is also true for the additive self helicity of Equations (1.8)
and (1.9), it is only in relative to the sum of the local reference fields
rather than the global reference field used in the relative helicity.

As described in detail in Pariat et al. (2015), the temporal variation in the
relative magnetic helicity $H_{\rm R}$ in a closed volume $V$ can be expressed
as

$\frac{dH_{\rm
R}}{dt}=2\int_{V}\boldsymbol{A}\cdot\frac{\partial\boldsymbol{B}}{\partial
t}\,dV+\int_{\partial V}\left[(\boldsymbol{A}-\boldsymbol{A}_{\rm
p})\times\frac{\partial(\boldsymbol{A}-\boldsymbol{A}_{\rm p})}{\partial
t}\right]\cdot d\boldsymbol{S}-2\int_{V}\boldsymbol{A}_{\rm
p}\cdot\frac{\partial\boldsymbol{B}_{\rm p}}{\partial t}\,dV.$ (1.10)

It should be noted that only the sum of the three integrals on the right-hand
side is gauge-invariant, whereas each integral is not. By using the Faraday’s
law and the Gauss divergence theorem and assuming an ideal MHD condition at
the boundary of the volume (i.e., $\boldsymbol{E}|_{\partial
V}=(-\boldsymbol{v}\times\boldsymbol{B})|_{\partial V}$), $dH_{\rm R}/dt$ of
Equation (1.10) can be decomposed into two volume integrals and four flux
terms on the surface of $V$ as

$\frac{dH_{\rm R}}{dt}=\frac{dH_{\rm R}}{dt}\bigg{|}_{\rm diss}+\frac{dH_{\rm
R}}{dt}\bigg{|}_{\rm B_{p},var}+F_{\rm v_{n}}+F_{\rm B_{n}}+F_{\rm
AA_{p}}+F_{\rm\phi},$ (1.11)

where

$\displaystyle\frac{dH_{\rm R}}{dt}\bigg{|}_{\rm
diss}=-2\int_{V}\boldsymbol{E}\cdot\boldsymbol{B}\,dV$ (1.12)
$\displaystyle\frac{dH_{\rm R}}{dt}\bigg{|}_{\rm
B_{p},var}=2\int_{V}\frac{\partial\boldsymbol{\phi}}{\partial
t}\nabla\cdot\boldsymbol{A_{\rm p}}\,dV$ (1.13) $\displaystyle F_{\rm
v_{n}}=-2\int_{\partial
V}(\boldsymbol{B}\cdot\boldsymbol{A})\boldsymbol{v}\cdot d\boldsymbol{S}$
(1.14) $\displaystyle F_{\rm B_{n}}=2\int_{\partial
V}(\boldsymbol{v}\cdot\boldsymbol{A})\boldsymbol{B}\cdot d\boldsymbol{S}$
(1.15) $\displaystyle F_{\rm AA_{p}}=\int_{\partial
V}\left[(\boldsymbol{A}-\boldsymbol{A}_{\rm
p})\times\frac{\partial(\boldsymbol{A}-\boldsymbol{A}_{\rm p})}{\partial
t}\right]\cdot d\boldsymbol{S}$ (1.16) $\displaystyle
F_{\rm\phi}=-2\int_{\partial V}\frac{\partial\boldsymbol{\phi}}{\partial
t}\boldsymbol{A}_{\rm p}\cdot d\boldsymbol{S},$ (1.17)

and $\phi$ is the solution of the Laplace equation $\nabla^{2}\phi=0$ (i.e.,
satisfying $\boldsymbol{B}_{\rm p}=\nabla\phi$). Choosing the Coulomb gauge
(CG; i.e., $\nabla\cdot\boldsymbol{A}_{\rm p}=0$) for the potential field
$\boldsymbol{B}_{\rm p}$, as well as having the following two additional
constraints for the boundary condition:

$\displaystyle\boldsymbol{A}|_{\partial V}=\boldsymbol{A}_{\rm p}|_{\partial
V},$ (1.18) $\displaystyle\boldsymbol{A}_{\rm p}\cdot{\rm
d}\boldsymbol{S}|_{\partial V}=0,$ (1.19)

we can obtain a simplified expression of $dH_{\rm R}/dt$ as

$\frac{dH_{\rm R}}{dt}\bigg{|}_{\rm CG}=\frac{dH_{\rm R}}{dt}\bigg{|}_{\rm
diss}+F_{\rm v_{n}}+F_{\rm B_{n}}.$ (1.20)

Figure 1.2: Schematic diagram that demonstrates relevant examples of $F_{\rm
tot}>0$ (top) and $F_{\rm tot}<0$ (bottom) across the solar surface (shown in
gray), in the context of the two distinct helicity flux components $F_{\rm
v_{n}}$ (panels a1–a4) and $F_{\rm B_{n}}$ (panels b1–b2). The magnetic flux
tubes are described by field lines with right-handed (positive; red) and left-
handed (negative; blue) helicity. Here, $t_{0}$ and $t_{1}$ are arbitrarily
chosen only to demonstrate the time progress ($t_{0}<t_{1}$).

Now in the context of investigating the transport rate of the relative
magnetic helicity across ${\rm S}_{\rm ph}$ of a given AR into the corona, the
concept of the relative magnetic helicity flux $F_{\rm tot}$ has been used in
many previous studies of flaring ARs with photospheric magnetic field
observations (e.g., Berger and Field, 1984),

$\displaystyle F_{\rm tot}$ $\displaystyle=-(F_{\rm v_{n}}+F_{\rm B_{n}})$
(1.21) $\displaystyle=2\int_{\rm S_{ph}}\left[(\boldsymbol{A}_{\rm
p}\cdot\boldsymbol{B}_{\rm h})v_{\rm n}-(\boldsymbol{A}_{\rm
p}\cdot\boldsymbol{v}_{\rm h})B_{\rm n}\right]\,dS,$

where the subscripts “n” and “h” indicate the normal and horizontal
components, respectively. In Equation (1.21), the $F_{\rm v_{n}}$ term is
associated with the emergence or submergence of a twisted magnetic structure
through ${\rm S}_{\rm ph}$ (often called the emergence term). The $F_{\rm
B_{n}}$ term is related to the twisting or untwisting of magnetic field lines
by horizontal footpoint motions on ${\rm S}_{\rm ph}$ (the so-called shear
term). The negative sign in front of the sum of $F_{\rm v_{n}}$ and $F_{\rm
B_{n}}$ is because the surface unit vector $d\boldsymbol{S}$ is chosen
outwardly from the volume $V$ in this particular case of helicity injection
into the corona. It should be noted, however, that $F_{\rm tot}$ can be
attributed to the transport of helicity into either the coronal domain or the
subsurface across ${\rm S}_{\rm ph}$. In Figure 1.2, the general concept of
$F_{\rm tot}$ is demonstrated with notable examples of $F_{\rm tot}>0$ (top)
and $F_{\rm tot}<0$ (bottom) by the two distinct helicity flux terms of
$F_{\rm v_{n}}$ (panels a1–a4) and $F_{\rm B_{n}}$ (panels b1–b2),
respectively.

Once $F_{\rm tot}$ is determined as a function of time, the change in $H_{\rm
R}$ due to helicity transport across ${\rm S}_{\rm ph}$ over the time interval
$\Delta t=t-t_{0}$ can be obtained:

$\Delta H_{\rm R}(t)=\int_{t_{0}}^{t}F_{\rm tot}\,dt,$ (1.22)

where $t_{0}$ is the reference time (often considered as the start time of
observations), and $t$ is the time of interest. In the literature, $\Delta
H_{\rm R}$ has often been referred to as the injection or accumulation of
helicity into the AR coronal volume through ${\rm S}_{\rm ph}$. Nevertheless,
strictly speaking, a more appropriate term may be helicity change as a result
of the helicity flux transport. Again, as described in Figure 1.2, the
transport can be achieved not only by injection into the coronal volume but
also by removal from the corona into the subsurface.

###  1.3 Methods of helicity estimation

####  1.3.1 Volume helicity

If the coronal magnetic field is obtained as a result of extrapolation based
on photospheric magnetic field measurement or numerical simulation (Section
1.5), it is possible to directly calculate the helicity in a coronal volume.
As mentioned in Section 1.2, the relative helicity, $H_{\rm
R}=\int(\boldsymbol{A}+\boldsymbol{A}_{\rm
p})\cdot(\boldsymbol{B}-\boldsymbol{B}_{\rm p})\,dV,$ has a degree of freedom
in the choice of gauge. In many practical applications, the relative helicity
is calculated for the magnetic field
$\boldsymbol{B}=\nabla\times\boldsymbol{A}$ in a finite volume, often of the
3D computational box with the Cartesian coordinates $(x,y,z)$ above a vector
magnetic field on the photospheric surface $(z=0)$. In the past, the CG (i.e.,
$\nabla\cdot\boldsymbol{A}=0$) has been used to calculate the volume helicity
(e.g., Rudenko and Myshyakov, 2011; Thalmann et al., 2011). However, the CG,
when implemented, could be computationally expensive.

To overcome this issue, Valori et al. (2012) adopted a gauge by DeVore (2000),
in which one component of the vector potential is 0, for instance, $A_{z}=0$,
which reduces the calculation cost. For a Cartesian box
$V=[x_{1},x_{2}]\times[y_{1},y_{2}]\times[z_{1},z_{2}]$, we are free to select
a gauge $\hat{\boldsymbol{z}}\cdot\boldsymbol{A}=0$ that satisfies

$\boldsymbol{A}=\boldsymbol{A}_{0}-\hat{\boldsymbol{z}}\times\int_{z_{1}}^{z_{2}}\boldsymbol{B}(x,y,z^{\prime})\,dz^{\prime},$
(1.23)

where the vector potential $\boldsymbol{A}_{0}$ is a solution to the
$z$-component of $\boldsymbol{B}=\nabla\times\boldsymbol{A}$ and is described
as $\boldsymbol{A}_{0}=\boldsymbol{A}(x,y,z_{1})=(A_{0x},A_{0y},0)$. We take a
simple solution to this:

$\displaystyle
A_{0x}=-\frac{1}{2}\int_{y_{1}}^{y}B_{z}(x,y^{\prime},z=z_{1})\,dy^{\prime},$
(1.24) $\displaystyle
A_{0y}=\frac{1}{2}\int_{x_{1}}^{x}B_{z}(x^{\prime},y,z=z_{1})\,dx^{\prime}.$
(1.25)

Similarly, the vector potential for the reference potential field,
$\boldsymbol{A}_{\rm p}$, is calculated using the same $\boldsymbol{A}_{0}$ as

$\boldsymbol{A}_{\rm
p}=\boldsymbol{A}_{0}-\hat{\boldsymbol{z}}\times\int_{z_{1}}^{z_{2}}\boldsymbol{B}_{\rm
p}(x,y,z^{\prime})\,dz^{\prime}.$ (1.26)

The relative helicity for the finite volume, $H_{\rm R}$, can be computed
using the set of $\boldsymbol{B}$, $\boldsymbol{A}$, $\boldsymbol{B}_{\rm p}$,
and $\boldsymbol{A}_{\rm p}$.

It should be noted here that the calculation results of the relative helicity
may differ depending not only on the choice of gauge (CG or DeVore gauge) but
also on the implementation method. Valori et al. (2016) systematically
compared the relative helicity with multiple calculation methodologies and
investigated the difference in calculation results and the dependence on
spatial resolution. As a result, it was found, for instance, that the accuracy
in computing $H_{\rm R}$ is found to vary by different methods, especially for
the CG methods, and that the spread in $H_{\rm R}$ estimates between different
methods converge in general at higher spatial resolutions.

####  1.3.2 Magnetic helicity flux

To determine the relative magnetic helicity in a coronal volume, it is
essential to have a 3D coronal magnetic field, which requires complicated
modeling efforts based on either the photospheric magnetic field observation
or the polarization of coronal lines. On the other hand, it is straightforward
to determine the time series of the magnetic helicity flux $F_{\rm tot}$ and
examine the temporal variation and total amount of helicity transported across
the AR photosphere.

One of the first practical methods for estimating $F_{\rm tot}$ in ARs was
developed by Chae (2001, hereafter $F_{\rm tot}^{\rm JC}$), using line-of-
sight magnetograms obtained by the Michelson Doppler Imager (MDI; Scherrer et
al., 1995) onboard the Solar and Heliospheric Observatory (SOHO),

$F_{\rm tot}^{\rm JC}=-2\int_{\rm S_{ph}}(\boldsymbol{A}_{\rm
p}\cdot\boldsymbol{v}_{\rm LCT})B_{\rm n}\,dS,$ (1.27)

where $\boldsymbol{v}_{\rm LCT}$ is the horizontal velocity of the apparent
motions of magnetic elements in the photosphere, which is determined using the
technique of local correlation tracking (LCT; November and Simon, 1988).
$\boldsymbol{A}_{\rm p}$ is calculated from $B_{\rm n}$ using the fast Fourier
transform (FFT) as

$\displaystyle\boldsymbol{A}_{\rm p,x}={\rm FFT^{-1}}\left(\frac{jk_{\rm
y}}{k_{\rm x}^{2}+k_{\rm y}^{2}}{\rm FFT}(B_{\rm n})\right),$ (1.28)
$\displaystyle\boldsymbol{A}_{\rm p,y}={\rm FFT^{-1}}\left(-\frac{jk_{\rm
x}}{k_{\rm x}^{2}+k_{\rm y}^{2}}{\rm FFT}(B_{\rm n})\right).$

$F_{\rm tot}^{\rm JC}$ is a good proxy of $F_{\rm tot}$ under the condition
that $\boldsymbol{v}_{\rm LCT}$ reasonably represents the magnetic field-line
footpoint velocity (sometimes called the flux transport velocity)
$\boldsymbol{u}\equiv\boldsymbol{v}_{\rm t}-(v_{\rm n}/B_{\rm
n})\boldsymbol{B}_{\rm t}$. However, we caution here that $\boldsymbol{v}_{\rm
LCT}$ is not valid in places where newly emerging or completely submerging
flux patches appear. Figure 1.3 shows an example of the implementation of the
$F_{\rm tot}^{\rm JC}$ method for NOAA AR 8011 observed by SOHO/MDI on 1997
January 17.

Figure 1.3: An example of estimating $F_{\rm tot}$ with the $F_{\rm tot}^{\rm
JC}$ method for NOAA AR 8011. The line-of-sight magnetic field $B_{\rm LOS}$
was obtained by SOHO/MDI when the AR was located near the center of the solar
disk at 17:00 UT on 1997 January 17, therefore, it can be well approximated to
the normal component $B_{\rm n}$. The grayscale image of $B_{\rm LOS}$ is
overplotted with $\boldsymbol{A}_{\rm p}$ (arrows in panel a) and
$\boldsymbol{v}_{\rm LCT}$ (arrows in panel b). $F_{\rm tot}$ is estimated by
integrating proxy values, i.e., $-2(\boldsymbol{A}_{\rm
p}\cdot\boldsymbol{v}_{\rm LCT})B_{\rm n}$ (grayscale image in panel c), for
the relative magnetic helicity flux density over the entire AR photospheric
surface under consideration. Figure reproduced from Chae (2001) by permission
of the AAS.

Another method of estimating $F_{\rm tot}$ was developed by Kusano et al.
(2002, hereafter $F_{\rm tot}^{\rm KK}$), applying the LCT technique and an
inversion from the normal component of the magnetic induction equation,
respectively, to infer $\boldsymbol{v}_{\rm h}$ and $v_{\rm n}$, and using the
helicity flux formula of Equation (1.21). Several follow-up studies were
carried out using the same $F_{\rm tot}^{\rm KK}$ method, in particular,
considering the temporal variations of $F_{\rm v_{n}}$ and $F_{\rm B_{n}}$
separately (e.g., Kusano et al., 2003; Yokoyama et al., 2003). Since then,
$F_{\rm tot}$ has been estimated using several different methods to infer the
plasma flow velocity from a sequence of (vector) magnetic field data (e.g.,
Welsch et al., 2004; Longcope, 2004; Schuck, 2006, 2008; Chae and Sakurai,
2008).

In addition, there was an approach by Pariat et al. (2005, hereafter $F_{\rm
tot}^{\rm EP}$) to determine $F_{\rm tot}$ explicitly using a specific form of
$\boldsymbol{A}_{\rm p}$ that satisfies the CG (i.e.,
$\nabla\cdot\boldsymbol{A}_{\rm p}=0$),

$\boldsymbol{A}_{\rm p}=\frac{1}{2\pi}\boldsymbol{\hat{n}}\times\int_{\rm
S_{ph}^{\prime}}B_{\rm
n}(x^{\prime})\frac{\boldsymbol{r}}{r^{2}}\,dS^{\prime}.$ (1.29)

By inserting $\boldsymbol{A}_{\rm p}$ of Equation (1.29) into Equation (1.21),
$F_{\rm tot}^{\rm EP}$ can be expressed as

$F_{\rm tot}^{\rm EP}=-\frac{1}{2\pi}\int_{\rm S_{ph}}\int_{\rm
S_{ph}^{\prime}}\frac{\boldsymbol{r}\times\left[\boldsymbol{u}-\boldsymbol{u^{\prime}}\right]}{r^{2}}\bigg{|}_{\rm
n}B_{\rm n}\,B_{\rm n}^{\prime}\,dS\,dS^{\prime}.$ (1.30)

As discussed in Pariat et al. (2005), the $F_{\rm tot}^{\rm EP}$ method was
shown to be more efficient in reducing spurious signals in the map of the
relative magnetic helicity flux density (i.e., the integrand in Equation
(1.30)) compared to that from the $F_{\rm tot}^{\rm JC}$ method (i.e., the
integrand in Equation (1.27)).

###  1.4 Practical applications

####  1.4.1 Temporal variation

Since several practical methods were developed in the 2000s to estimate
$F_{\rm tot}$ across the AR photosphere from solar magnetic field observations
(refer to Section 1.3.2), there have been many studies to find out any
temporal variations in the magnetic helicity over various timescales in
relation to flaring activity of the source ARs, particularly for comparing
differences of flaring versus non-flaring ARs. In the context of timescales
under consideration, time series analyses of $F_{\rm tot}$ can be generally
classified into two categories: (1) short-term (tens of minutes to a few
hours) and (2) long-term (a few hours to several days). In the case of short-
term variation, for example, there were reports of a large, impulsive
variation in the time series of $F_{\rm tot}$ over the course of the flare
(e.g., Moon et al., 2002a, b; Hartkorn and Wang, 2004; Romano et al., 2005; Bi
et al., 2018). The impulsive variation in $F_{\rm tot}$ typically occurs
during the rise phase of the flare X-ray flux, and the magnitude of the
impulsive helicity variation (i.e., $\Delta H=\int_{t_{0}}^{t_{1}}F_{\rm
tot}\,dt$ with $t_{0}$ and $t_{1}$ as the flare start and end times)
correlates with the flare X-ray peak flux. Although this impulsive variation
in $F_{\rm tot}$ was found for multiple flare events, we should be cautious
because photospheric magnetograms derived from spectropolarimetric
observations (in particular, pixels where white-light flare kernels are
located) can be significantly affected by flare emission, as mentioned by
Hartkorn and Wang (2004).333An example of such disturbed polarimetric signals
can be found in the magnetogram of Figure 1.1 of this chapter, in which the
core of the positive (negative) polarity sunspot shows a negative (positive)
signal. We note that a detailed investigation of short-term variations in
$F_{\rm tot}$ may help us to understand the flare-trigger process (e.g.,
tether-cutting magnetic reconnection, rapidly growing MHD instabilities), and
the relevant flare-associated evolution of the photospheric magnetic field.

Figure 1.4: Long-term trend of increase in $\Delta H_{\rm R}$ across the
photospheric surface of NOAA AR 9672 before the onset of an X1.3 flare on 2001
October 25. The MDI line-of-sight magnetic field and EIT 195 Å intensity
images of the AR, taken at $\sim$16:00 UT, are presented in panels (a) and
(b), respectively. In panel (c), the temporal variations in $\Delta H_{\rm R}$
(cross-solid line), the total unsigned flux (diamond-solid line), and GOES
soft X-ray flux (dotted line) are shown, together with the two distinct (i.e.,
increase and steady) phases in the observed profile of $\Delta H_{\rm R}$.
Figure reproduced from Park et al. (2008) by permission of the AAS.

With respect to flare-associated variations in $F_{\rm tot}$ on a long-term
scale, one of the most remarkable observational findings is that flaring ARs
show a large, continuous increase in $\Delta H_{\rm R}$ (i.e., the time
integral of $F_{\rm tot}$) over an interval of several hours to a few days
before the flare onset. Figure 1.4 presents an example of this long-term
increasing trend of $\Delta H_{\rm R}$ (cross-solid line in panel c) in NOAA
AR 9672, which produced an X1.3 flare (dotted line in panel c) on 2001 October
25. Interestingly, $\Delta H_{\rm R}$ increased monotonically during a pre-
flare phase of $\sim$24 hour (marked by the dark gray bar and labeled I) and
then remained almost constant for the post-flare phase (labeled II). On the
other hand, AR’s total unsigned magnetic flux (diamond-solid line in panel c)
changed little over the entire investigation period, making it difficult to
find any flare-associated temporal variations. This trend of increase in
$\Delta H_{\rm R}$ was found in many other large flaring ARs (e.g., Park et
al., 2008, 2010a, 2010b, 2012, 2013; Kusano et al., 2002; Magara and Tsuneta,
2008; Vemareddy et al., 2012; Romano et al., 2014; Vemareddy, 2015; Romano et
al., 2015), whereas no characteristic evolution of $\Delta H_{\rm R}$ was
observed in flare-quiet ARs. Some statistical studies of this increase in
helicity in relation to the flaring activity are discussed in Section 1.4.2.

Figure 1.5: Long-term increasing trend of $H_{\rm R}$ in the coronal volume of
NOAA AR 11158 prior to an X2.2 flare on 2011 February 15. (Left) A selected
set of coronal magnetic field lines extrapolated from the photospheric
vertical magnetic field (grayscale image), is displayed with different colors
according to their maximum altitude. (Right) Temporal variations of $H_{\rm
R}$ (red line), $\Delta H_{\rm R}$ (blue line), the total unsigned magnetic
flux (black line), and the GOES soft X-ray flux (gray line). Figure reproduced
from Jing et al. (2012) by permission of the AAS.

In the same context, as shown in Figure 1.5, a long-term increase in helicity
was also found in the time profile of the relative magnetic helicity $H_{\rm
R}$ in the coronal volume of ARs that produced large flares (e.g., Park et
al., 2010a; Jing et al., 2012, 2015; Thalmann et al., 2019). The observed
increasing patterns of both $\Delta H_{\rm R}$ and $H_{\rm R}$ prior to flares
support the idea that $H_{\rm R}$ in the AR corona is mainly supplied by
$F_{\rm tot}$ through the bottom surface (i.e., the AR photosphere). It is
also thought that $H_{\rm R}$ is well preserved over timescales of a few hours
to several days, for example, as shown by Park et al. (2010a), based on the
comparison of approximately five-day profiles between $H_{\rm R}$ and $\Delta
H_{\rm R}$ for NOAA AR 10930. Moreover, over the course of AR evolution, the
pre-flare increasing trend of the helicity may reflect a complicated process
of building up the free magnetic energy in the corona by the emergence of
twisted magnetic flux tubes and/or footpoint shear motions of the magnetic
field line.

Figure 1.6: Maps of the relative magnetic helicity flux density for NOAA AR
8100, observed at 00:49 UT (panel a) and 23:11 UT (panel b) on 1997 November
3, are presented in the blue and red color scale. The AR produced an X2.1
flare at 05:52 UT on November 4. The positive and negative vertical magnetic
fields are indicated by pink and black contours, respectively. Some regions of
interest are marked by the rectangular boxes with the following labels. A: the
main PIL; B: a southern part of the main positive polarity sunspot; C and D:
satellite patches in which the magnetic polarity is opposite to that of the
nearby main spot; F: a region with strong upflows on the photospheric surface;
G: a region in which the helicity flux density is predominantly negative in
sign. Figure reproduced from Kusano et al. (2002) by permission of the AAS.
Figure 1.7: Time profiles of $\Delta H_{\rm R}$ (black line) and $\Phi$ (blue
line) for four ARs that produced multiple eruptive flares with CMEs. The red
vertical lines indicate the times when the CMEs first appeared in the LASCO/C2
field of view. Figure reproduced from Park et al. (2012) by permission of the
AAS.

One of the most important aspects of studying the magnetic helicity is that it
measures the handedness of twisted magnetic field lines: i.e., positive
helicity in the case of right-handed twist and negative helicity for the left-
handed case. As shown in Figure 1.6, Kusano et al. (2002) examined the spatial
distribution and evolution of the relative magnetic helicity flux density
across the entire photospheric surface of NOAA AR 8100 during a five-day
interval from 1997 November 1 to 5. They found that negative helicity
transport was predominantly caused by $F_{\rm v_{n}}$ over the first 2.5 days
of the investigated interval, while helicities of positive and negative signs
were transported by $F_{\rm v_{n}}$ and $F_{\rm B_{n}}$, respectively, for the
rest of the interval. In the latter phase, multiple flares, consisting of
three M-class flares and one X2.1 flare, occurred in the region (labeled G in
panel b of Figure 1.6), where negative helicity was consistently transported
primarily owing to the shear motion along the main PIL. Such a characteristic
follow-up transport of helicity via a localized area (typically around PILs)
with a sign opposite to the preloaded helicity in the AR corona has been
reported in many other observations of flaring ARs (e.g., Yokoyama et al.,
2003; Wang et al., 2004; Park et al., 2010a; Vemareddy et al., 2012; Jing et
al., 2015). When the subsequent opposite-sign helicity flux in the localized
areas of the AR becomes larger than the helicity flux in the rest of the
areas, the sign of $F_{\rm tot}$ is actually reversed over the course of its
measurement, which is called the helicity sign reversal.

Park et al. (2012) performed time series analysis of $F_{\rm tot}$ for a set
of different AR samples, each of which produced eruptive flares and
accompanying CMEs. They found that all the investigated eruptive flares
occurred following a remarkable increase in $\Delta H_{\rm R}$ first and the
subsequent reversal of the helicity sign (see Figure 1.7). Here the helicity
sign reversal is shown as the slope of the time profile of $\Delta H_{\rm R}$
changes its sign. In contrast, examining temporal variations of $F_{\rm tot}$
for NOAA AR 12257, Vemareddy (2021) reported that there were only small
C-class flares without CMEs even when the helicity sign reversal proceeded.
However, it should be noted that in the case of NOAA AR 12257, there was no
significant transport of helicity before the sign reversal, while a large
amount of helicity transport was made for the ARs under study in Park et al.
(2012). A careful investigation is needed when interpreting the helicity flux
density map; for example, artificial signals may be generated owing to various
limitations in observational data and practical methods, as indicated by
Pariat et al. (2005, 2006). Meanwhile, Jing et al. (2015) conducted an
interesting comparison study of eruptive versus confined flares by analyzing
the evolution of $H_{\rm R}$ in the AR corona. They found that $H_{\rm R}$
remarkably decreased $\sim$4 hour prior to the eruptive X2.2 flare on 2011
February 15, whereas no such decrease in $H_{\rm R}$ was found in the case of
the confined X3.1 flare on 2014 October 24.

It is noteworthy that the observed signatures of (1) the helicity sign
reversal and (2) the sudden decrease in $H_{\rm R}$ prior to eruptive flares
can be explained by the flare-trigger MHD simulation model of Kusano et al.
(2012), in which, as a flare trigger, a small-scale bipolar magnetic structure
supplies helicity of opposite sign into the pre-existing sheared arcade
system.

Figure 1.8: (Left) Coronal magnetic field lines extrapolated from the vertical
magnetic field (grayscale image) at the photospheric surface of (a) AR 11158
at 21:00 UT on 2011 February 14 and (b) AR 12192 at 19:00 UT on 2014 October
24. (Right) Time profiles of $|H_{\rm j}|/|H_{\rm R}|$ for (c) AR 11158 and
(d) AR 12192. In panels (c) and (d), the shaded areas represent the spread of
the values of $|H_{\rm j}|/|H_{\rm R}|$ from the three different methods used
to estimate the volume helicity. The vertical dashed and solid lines indicate
the GOES peak times of the M- and X-class flares, respectively. The horizontal
dashed line marks the value of $|H_{\rm j}|/|H_{\rm R}|=0.17$ as a reference.
Figure reproduced from Thalmann et al. (2019) by permission of the AAS.

There have been longstanding discussions on the role of magnetic helicity as a
sufficient indicator of solar eruptions. Based on practical methods of
estimating the magnetic helicity (refer to Section 1.3), many trials have been
conducted so far to determine the magic parameter that has the capability of
providing a threshold above which flares and CMEs occur. In this respect,
Pariat et al. (2017) proposed a new parameter for the solar eruptivity
criterion, which is the ratio of the magnetic helicity of the current-carrying
magnetic field to the total relative helicity, $|H_{\rm j}|/|H_{\rm R}|$. For
NOAA AR 12673, which produced the largest X-class flare of solar cycle 24,
Moraitis et al. (2019) found that the helicity ratio increased and then
relaxed to lower values before and after, respectively, each of the two major
X2.2 and X9.3 flares on 2017 September 6. Through the analysis of observations
and coronal field extrapolations of two flaring ARs with eruptive versus
confined flares, as shown in Figure 1.8, Thalmann et al. (2019) found a
similar increasing/decreasing trend of $|H_{\rm j}|/|H_{\rm R}|$ before and
after the major eruptive flares in NOAA AR 11158. However, this was not
observed in NOAA AR 12192, which produced only confined flares without CMEs.
Another follow-up study of Gupta et al. (2021), with a set of 10 different
NOAA-numbered AR samples, shows results supporting that the helicity ratio has
a strong ability to indicate the eruptive potential of an AR, albeit with a
few exceptions. A further statistical study with more samples is needed.

Then, why does the helicity ratio, $|H_{\rm j}|/|H_{\rm R}|$, seem to predict
flare eruptions well? There have been intensive studies on critical conditions
for a current-carrying structure (e.g., twisted flux ropes) to successfully
erupt against its surrounding confinement field, as represented by the torus
instability (Kliem and Török, 2006; Démoulin and Aulanier, 2010) and the
double-arc instability (Ishiguro and Kusano, 2017). In fact, various
“relative” quantities have been proposed in the framework of instabilities for
a current system in the corona and tested with observations of the eruptive
and confined flares (Sun et al., 2015; Toriumi et al., 2017; Lin et al., 2020;
Li et al., 2020; Kusano et al., 2020; Kazachenko et al., 2022). In this
respect, $|H_{\rm j}|/|H_{\rm R}|$ may reflect the magnetic relationship
between the current-carrying structure (the flux rope) and the entire system
(the whole AR), which is one of the key factors that determine these
instabilities.

Recently, using wavelet analysis, attempts have been made to determine any
characteristic periodicities in the time series of $F_{\rm tot}$ for flaring
ARs. For example, Korsós et al. (2020) and Soós et al. (2021) reported that
there are some differences in the identified periodicities between flaring ARs
with large M- and X-class flares and ARs with smaller B- and C-class flares.
In this context, more detailed studies in the future may help us to better
understand whether the evolution of the magnetic field on the photosphere is
inherently different between large flaring ARs and the others, as well as to
predict the likelihood of flare occurrence for a target AR.

####  1.4.2 Statistical trends

Examination of the magnetic helicity on many sample ARs has revealed the
statistical tendency that flare-rich ARs harbor a larger amount of magnetic
helicity or helicity flux. For instance, LaBonte et al. (2007) surveyed the
helicity injections for 48 X-flaring ARs and 345 reference regions without
X-flares. They derived an empirical threshold for the occurrence of an X-class
flare that the peak helicity flux, $\max{\lvert\langle F_{\rm
tot}\rangle\rvert}$ given by Equation (1.27), should exceed a magnitude of
$6\times 10^{36}\ {\rm Mx}^{2}\ {\rm s}^{-1}$.

Park et al. (2010b) expanded the analysis to 378 ARs and investigated the
correspondence between the 24 hour-averaged helicity flux, $\lvert\langle
F_{\rm tot}\rangle\rvert$ based on Equation (1.27), and the next 24-hour flare
index (i.e., the sum of GOES soft X-ray peak intensities of flares that
occurred in the next 24 hour after the helicity measurement). They
demonstrated that, for 91 subsamples with an unsigned magnetic flux of
$(3$–$5)\times 10^{22}\ {\rm Mx}$, the helicity flux for the flaring ARs was
approximately twice that of the quiescent ARs. On the other hand, 118 ARs with
large helicity rates did not show a significant difference in magnetic flux
between the flaring and quiescent groups. As shown in Figure 1.9, these
authors also found that the larger value of $\lvert\langle F_{\rm
tot}\rangle\rvert$ an AR has, the higher the flaring activity of the AR is in
all cases of C-, M-, and X-class flares.

Figure 1.9: Flare-productive probability as a function of $\lvert\langle
F_{\rm tot}\rangle\rvert$, respectively, for C-class (diamonds), M-class
(triangles), and X-class (squares). The gray bars indicate the total number of
AR samples in the defined bins. Figure reproduced from Park et al. (2010b) by
permission of the AAS.

Regarding the differences between CME-eruptive and non-eruptive flares, Nindos
and Andrews (2004) examined the coronal magnetic helicity, derived under the
assumption that the coronal field of each AR has a constant force-free
$\alpha$ (Section 1.5.1) for 133 M-class events from 78 ARs. They revealed the
statistical tendency that the pre-flare $\alpha$ values and coronal helicity
of the non-eruptive flares are smaller than those of the eruptive flares. Park
et al. (2012) examined 47 CMEs in 28 ARs and showed that there was also good
correlation between $\lvert\langle F_{\rm tot}\rangle\rvert$ and the CME
speed. Interestingly, by tracing the helicity evolution, the CMEs analyzed
were divided into two groups: a group in which helicity increased
monotonically with one sign of helicity, and a group in which the sign of
helicity reverses after a significant helicity injection (reversed shear:
Kusano et al. 2004, 2012).

Figure 1.10: Scatter plots of ARs as functions of free magnetic energy
(denoted as $E_{\rm c}$ in this diagram) and absolute relative magnetic
helicity ($|H_{\rm m}|$ here). The blue diamonds, red squares, and red
asterisks denote the non-flaring, M-class flaring, and X-class flaring ARs,
respectively. The dashed horizontal and vertical lines indicate the thresholds
for M-class events ($2\times 10^{42}\ {\rm Mx}^{2}$ and $4\times 10^{31}\ {\rm
erg}$). The dotted and dash-dotted lines are the least-square best fit and
least-square best logarithmic fit for the data points, respectively. Figure
reproduced from Tziotziou et al. (2012) by permission of the AAS.

Tziotziou et al. (2012) investigated the relationship between the free
magnetic energy ($\Delta E_{\rm mag}$) and relative helicity ($H_{\rm R}$) on
42 ARs, finding that the flaring ARs possess higher free energy and relative
helicity. As demonstrated in Figure 1.10, there exists a monotonic scaling
between the two variables with a threshold for M-class events, which is
$4\times 10^{31}\ {\rm erg}$ and $2\times 10^{42}\ {\rm Mx}^{2}$ for the
energy and helicity, respectively. This energy-helicity diagram was further
examined by Tziotziou et al. (2014). By using 3D numerical simulations of
eruptive and non-eruptive cases and two observed, eruptive and non-eruptive
ARs, they reconstructed the energy-helicity diagram and found a consistent
monotonic scaling law.

However, care must be taken when considering such statistical results because
magnetic helicity is homogeneous to the square of the magnetic flux. As shown
in Section 1.1, ARs with larger areas or magnetic fluxes are more prone to
flares, which may be included in the statistical results using extensive
parameters (those scaling with AR size) including magnetic helicity (Welsch et
al., 2009; Bobra and Couvidat, 2015; Sun et al., 2015; Toriumi and Takasao,
2017). Therefore, to assess the eruptivity of ARs purely owing to the
structural complexity and not to the AR area, it is worth considering the
helicity normalized by the flux squared, $H_{\rm R}/\Phi^{2}$ (Démoulin and
Pariat, 2009). On the other hand, the results of statistical studies clearly
demonstrate that there exist quantitative differences between flare-productive
and quiescent ARs. This suggests that helicity-related variables may be useful
for predicting and forecasting imminent flare occurrences.

####  1.4.3 Magnetic tongues: A proxy for the helicity sign

It has been suggested that magnetic tongues in ARs, the elongated yin-yang
shaped patches of positive and negative magnetic polarities on the solar
surface, can be used as a proxy for the magnetic helicity (López Fuentes et
al., 2000). It is thought that the positive and negative polarities extended
on both sides of the PIL in longitudinal (or line-of-sight) magnetograms
reflect the vertical projection of the poloidal (azimuthal) component of the
emerging magnetic flux tube. When an arched flux tube rises and penetrates the
photospheric surface, the horizontal cross section of the tube displays a pair
of opposite polarity patches (magnetic tongues) with the PIL in between. If
the flux tube is twisted or writhed, that is, if the emerging flux has a non-
zero magnetic helicity (the situation in panels (a1) or (a4) in Figure 1.2),
the tongue structure has an axisymmetric pattern, and the PIL deviates from an
orthogonal angle to the main bipolar axis. Therefore, the layout of the
tongues and the direction of the PIL were used to determine the sign of the
magnetic helicity of the ARs. In fact, flux emergence simulations that assume
twisted flux tubes often display magnetic tongues in the photosphere (e.g.,
Archontis and Hood, 2010; Toriumi et al., 2011).

Chandra et al. (2009) analyzed the evolution of the magnetic field in NOAA AR
10365 in May 2003 and found that it had a positive magnetic helicity. In fact,
this AR produced an M1.6-class flare leaving a pair of J-shaped flare ribbons,
another indication of twisting in ARs. Mandrini et al. (2014) analyzed the
force-free parameter, $\alpha$ (Section 1.5.1), for NOAA ARs 11121 and 11123
and confirmed that the positive value for $\alpha$ agrees with the observed
magnetic tongue.

Statistical analysis of magnetic tongues by Luoni et al. (2011) on 40 ARs
showed that the helicity signs determined from the tongues were consistent
with those derived from the photospheric helicity fluxes. Poisson et al.
(2015a, b) systematically surveyed 41 bipolar ARs and demonstrated that the
twist estimated from the tongues in general matched the force-free $\alpha$
derived from the coronal field extrapolations. By analyzing 187 ARs, Poisson
et al. (2016) further investigated the dependence of tongues on the activity
cycle. They found that the angle of the PIL relative to the bipolar axis (AR
tilt) has only a weak sign dominance in each solar hemisphere ($\gtrsim 50$%).
In addition, the characteristics of the tongues, not only the PIL angle but
also the amount of magnetic flux, polarity size, latitude, emergence rate,
etc., are not dependent on the periods of the cycle.

###  1.5 Numerical models

####  1.5.1 Coronal field reconstructions

Figure 1.11: NLFFF extrapolation on NOAA AR 10953. The best-fit NLFFF result
is shown in panel (b) as white lines. The stereoscopically reconstructed field
lines are displayed in panels (d) and (e), in which the solid black cubes
outline the NLFFF computational domain. The loops inside the box are colored
according to their misalignment angle from the NLFFF solution (yellow to
orange corresponds to 5∘ to 45∘). Figure reproduced from De Rosa et al. (2009)
by permission of the AAS.

Owing to the limitations of magnetic field measurements in the solar corona,
we rely on coronal field models to estimate the magnetic field energy and
helicity. The simplest approach is to extrapolate the coronal field based on
the observed photospheric magnetic field under the assumption that non-
magnetic forces (e.g., pressure gradient and gravity) are negligible and that
the Lorentz force vanishes, i.e.,

$\boldsymbol{j}\times\boldsymbol{B}=0,$ (1.31)

where $\boldsymbol{j}=c/(4\pi)(\nabla\times\boldsymbol{B})$ is the electric
current. This force-free condition can also be written as

$\nabla\times\boldsymbol{B}=\alpha\boldsymbol{B},$ (1.32)

where $\alpha$ is the force-free parameter. If $\alpha=0$, or equivalently
$\boldsymbol{j}=0$ (current-free), the coronal field is the potential field
($\boldsymbol{B}_{\rm p}$), which is often used as the reference field when
estimating the relative magnetic helicity (as in Equation (1.5)). If $\alpha$
is non-zero and constant everywhere in the coronal volume under consideration,
the magnetic field is called a linear force-free field (LFFF). If $\alpha$ is
non-uniform, the field is a non-linear force-free field (NLFFF: Figures 1.5,
1.8, and 1.11).

Here, only the vertical component of the photospheric magnetic field ($B_{\rm
n}$) is required as the bottom boundary condition to obtain the potential
field, whereas all components of the vector magnetogram ($\boldsymbol{B}$) are
used for the LFFF and NLFFF. Extrapolation models, particularly NLFFF methods,
have been extensively used for the direct determination of volume helicity in
the AR corona (see discussions in the previous sections). The readers are
referred to, e.g., Inoue (2016) and Wiegelmann and Sakurai (2021) for detailed
accounts of extrapolation methods.

These reconstruction models, however, only provide steady-state coronal fields
and may not be applicable to flare-productive ARs, which are highly dynamic in
nature. To overcome this issue, temporally evolving models have been used. One
approach of such coronal field reconstructions is data-constrained models,
where the initial coronal field is prepared by extrapolations based on a
snapshot magnetogram, and the subsequent dynamical evolution is obtained by
solving the MHD equations.

Figure 1.12: (Top and middle) Magnetic fields for ground-truth flux-emergence
simulation (GT), potential field extrapolation (PF), and four data-driven
coronal field reconstructions (MF, MHD1, MHD2, and MHD3). The bottom boundary
shows the vertical GT magnetic field (i.e., the photospheric magnetogram). The
colored tubes indicate the field lines, where the tubes with reddish (bluish)
colors are integrated from the seeds placed in the positive (negative)
polarity. The seeds were identical in all six cases. The time for MHD1 is 1000
s after the photospheric appearance of the emerging flux, while all other
times are 4000 s. (Bottom) Temporal evolution of the apex height $z_{\rm
apex}$, magnetic energy $E_{\rm mag}$, and relative magnetic helicity $H_{\rm
R}$ for GT and the four coronal field models, MF, MHD1, MHD2, and MHD3. The
normalization factor for time $\tau_{0}$ corresponds to 25 s. Figure
reproduced from Toriumi et al. (2020) by permission of the AAS.

Data-driven models, in which the coronal field evolves in response to
sequentially updated bottom boundary magnetograms, may provide even more
realistic coronal reconstructions. To investigate how accurately different
data-driven models can reproduce a coronal field and quantities such as the
magnetic helicity, Toriumi et al. (2020) used a flux emergence simulation as
the ground truth (GT) dataset (Toriumi and Takasao, 2017). In this GT
simulation, a twisted flux tube initially placed in the convection zone rises
into the atmosphere and eventually builds up coronal loops. These authors
compared the GT coronal field with those reconstructed using multiple data-
driven models based on sequential GT magnetograms. As illustrated in Figure
1.12, the helical flux rope structure was reproduced in all coronal field
models. On the other hand, model-dependent results were obtained for
quantitative comparison, with the magnetic energies ($E_{\rm mag}$ and $\Delta
E_{\rm mag}$) and relative magnetic helicity ($H_{\rm R}$) varying from the
cases almost comparable to GT to those differed by orders of magnitude. The
magnetic helicity was derived by the method in DeVore (2000), following Valori
et al. (2012). The observed model discrepancies were attributed to a highly
non-force-free input bottom boundary (GT magnetograms) and to modeling
treatments of the background atmosphere, bottom boundary, and spatial
resolution.

####  1.5.2 Idealized simulations

Numerical models of flux rope eruptions (i.e., CMEs) have also been employed
to test the concept of magnetic helicity. For instance, Malanushenko et al.
(2009) used eruption models to show that the additive self helicity ($H_{{\rm
R}i}$: Equation (1.8)) is useful for determining the threshold beyond which a
flux rope becomes unstable. The additive self helicity divided by the square
of the magnetic flux through a cross section of the tube, $H_{{\rm
R}i}/\Phi^{2}$, can be considered as a generalization of the twist number
(Berger and Field, 1984), which has been used to discuss the threshold of kink
instability. The authors investigated the evolution of $H_{{\rm R}i}/\Phi^{2}$
in the simulations, in which a twisted flux rope kinematically inserted into
the computational domain interacted with a pre-existing coronal arcade and
deformed into a writhed configuration owing to the kink instability (Fan and
Gibson, 2003). Over time, the (absolute) magnitude of $H_{{\rm R}i}/\Phi^{2}$
increased until it reached a threshold value of $\sim$1.5 for the instability
to occur.

Figure 1.13: Examination of helicity ratio ($|H_{\rm j}|/|H_{\rm R}|$) by
Pariat et al. (2017). (Left) Snapshots of two simulation cases (non-eruptive
and eruptive) with medium arcade strength. (Right) Time evolution of the
helicity ratio for the seven parameter cases. The eruptive cases show higher
values above 0.45. Note that the relative helicity $H_{\rm R}$ is denoted in
the figure as $H_{\rm V}$. Reproduced with permission from Astronomy &
Astrophysics, ©ESO.

Another simulation model that was employed to validate the usage of helicity
was given by Leake et al. (2013, 2014), where a magnetic flux emerges into a
magnetized corona, which may or may not erupt depending on the direction of
the coronal arcade (Figure 1.13). Pariat et al. (2017) investigated the
temporal evolution of several global scalar quantities to determine whether
they could discriminate between eruptive and non-eruptive cases. It was found
that the helicity ratio $|H_{\rm j}|/|H_{\rm R}|$ can determine the
eruptivity, whereas other quantities such as the magnetic flux ($\Phi$),
magnetic energies ($\Delta E_{\rm mag}$, $\Delta E_{\rm mag}/E_{\rm mag}$,
etc.), and total relative helicity ($|H_{\rm R}|$) cannot (see Section 1.4.1
for the details).

###  1.6 Summary and Discussion

In this chapter, we have reviewed the following: (1) solar flares and CMEs
tend to occur in ARs with complex structures; (2) therefore, various concepts
of magnetic helicity, which is the physical quantity that measures the
complexity of magnetic fields, help to understand the occurrence of flares and
CMEs; and (3) based on the statistical findings, these concepts may be useful
for the prediction and forecasting of such eruptive events. Magnetic helicity
is not limited to mere theoretical research, but its practicality has been
proven by a variety of studies that exploit actual observational data.
However, there remain important topics that cannot be addressed in detail in
this chapter.

For instance, there is an issue that the absolute value of magnetic helicity
may vary, depending on which gauge is employed for both $\boldsymbol{A}$ and
$\boldsymbol{A}_{\rm p}$, as discussed in Pariat et al. (2015) and Schuck and
Antiochos (2019). This gauge issue can be problematic, particularly when
comparing results obtained from different methods of estimating helicity and
investigating a critical value of helicity for flare- or CME-trigger
instabilities. In addition, as in Equation (1.11), the definition of the
temporal variation of the relative magnetic helicity $dH_{\rm R}/dt$ and its
flux components on the boundary surface may not be fully consistent between
previous studies with somewhat different gauge choices. We also note that each
flux component may not be gauge-invariant, but only their sum is. On the other
hand, different ways of implementing helicity formulas, as well as in
preprocessing observational data, can result in inconsistent results due to
numerical errors and artificial noise. Future community-wide efforts, such as
“Helicity Thinkshops” and “Helicity2020”, may need to be made to discuss and
achieve a general consensus on the issues addressed here.

It is important to note that we still need to resolve various sources of
uncertainty in determining the evolution of magnetic helicity for ARs. First,
as a problem on the observational side, there are inaccuracies in photospheric
magnetograms. For instance, if there are magnetic field structures with the
sizes smaller than the spatial resolution, magnetic helicity may be
underestimated. Other causes of inaccuracies include the 180 degree ambiguity
of the transverse field, general difficulties in calibration, improperly
assumed atmospheres in the Stokes inversions, corrugation of line formation
heights, and inadequate sampling of spectral profiles (see, e.g., del Toro
Iniesta 2003; some of these issues will be discussed later in this section).
Second, there is a difficulty of using the photospheric data in estimating the
relative helicity and/or helicity flux for the corona. The dynamic
chromosphere lying between the photosphere and the corona may have an effect
on the coronal energetics that cannot be perceived by photosphere observation
alone, and this may also be one of the causes of misassessing magnetic
helicity of the corona. Third, as mentioned in the previous paragraph, because
the relative helicity is not invariant to gauge selection, it is critically
important to confirm that the applied gauges are the same when comparing
different relative helicity estimates. Otherwise, the comparison will lead to
a wrong conclusion about the AR evolution and its relationship with eruptivity
(see Sections 1.2 and 1.3.1 for the details).

To understand the flaring activity of ARs, thus far, most magnetic helicity
studies have focused on the temporal evolution of magnetic helicity with a set
of individual ARs over a span (typically, a few days to two weeks) of their
(partial) passages on the solar disk. On the other hand, very recently, a
novel approach was designed by Park et al. (2021) to examine whether there are
any flare-associated differences in heliographic regions on a larger scale
with respect to the degree of complying with the hemispheric sign preference
(HSP) of magnetic helicity. The HSP of magnetic helicity has a unique cycle-
independent tendency, in which various features in the northern hemisphere of
the Sun tend to exhibit negative or left-handed helicity, while they are
positive in the southern hemisphere. In the study of Park et al. (2021),
heliographic regions were defined in the Carrington longitude-latitude plane
with a grid spacing of 45∘ in longitude and 15∘ in latitude. The authors
examined the relative magnetic helicity flux $F_{\rm tot}$ across the
photospheric surface for 4,802 samples of 1,105 unique ARs observed during
solar cycle 24. They found that heliographic regions with lower degrees of HSP
compliance are likely to show higher levels of flaring activity as well as
larger values of the average magnetic flux. What does this observational
result tell us? Based on simulations of the rise of magnetic flux tubes deep
in the convection zone by Manek and Brummell (2021), we can deduce that if the
helicity sign of a rising flux tube is against the HSP, then the tube’s field
strength is relatively larger for its successful rise compared to the other
case of a tube with its helicity sign following the HSP. The assumption behind
this speculation is that the background poloidal field strength is quasi-
isotropic in the tachocline. The association of the lower HSP with the higher
flaring activity may also suggest that there are localized regions in the
convection zone where large-scale downflows and strong turbulence are present.
Rising flux tubes therein may have an opposite sign of magnetic helicity as
compared to the one expected from the HSP, may possess greater magnetic
complexity (such as $\delta$-spots), and may produce large flares. Such HSP
studies may offer an indirect understanding of the elusive magnetic nature of
the solar interior as well as the interaction between flux tubes and
convection flows, which may be the origin of bearing flare-productive ARs
(Toriumi and Wang, 2019). Moreover, realistic dynamo simulations and
helioseismology studies will help us better understand the formation and
evolution of flaring ARs even before their emergence on the solar surface.

In the last decades, especially since the sequential (vector) magnetograms
became available from space, we have deepened our understanding of the
usefulness of magnetic helicity. While there still remain difficulties in
accurately estimating the magnetic helicity, such as those caused by the
aforementioned ambiguity in deriving the transverse magnetic field in the
photosphere, we expect that such difficulties can be overcome or at least
mitigated in the near future. For example, methods that employ stereoscopic
disambiguation may be helpful (e.g., with Solar Orbiter: Rouillard et al.
2020; Valori et al. 2022). The photospheric magnetic field, from which the
coronal field is reconstructed, deviates significantly from the force-free
state. However, it may be possible to improve the coronal field reconstruction
by incorporating the chromospheric field, which yields a better force-freeness
(e.g., with DKIST and Sunrise-3: Wiegelmann et al. 2008; Fleishman et al.
2019). Numerical simulations have been used to validate the helicity
estimation methods, but by utilizing more realistic flux-emergence simulations
that have appeared in recent years, it will be possible to examine these
methods with even higher accuracy (Cheung et al., 2019; Toriumi and Hotta,
2019; Hotta and Iijima, 2020). These advanced studies are expected to lead to
further understanding and better prediction of flares and CMEs in the near
future.

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