# Flux emergence and generation of flare-productive active regions

Shin Toriumi<EMAIL_ADDRESS>Institute of Space and Astronautical
Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku,
Sagamihara, Kanagawa 252-5210, Japan

###### Abstract

Solar flares and coronal mass ejections are among the most prominent
manifestations of the magnetic activity of the Sun. The strongest events of
them tend to occur in active regions (ARs) that are large, complex, and
dynamically evolving. However, it is not clear what the key observational
features of such ARs are, and how these features are produced. This article
answers these fundamental questions based on morphological and magnetic
characteristics of flare-productive ARs and their evolutionary processes,
i.e., large-scale flux emergence and subsequent AR formation, which have been
revealed in observational and theoretical studies. We also present the latest
modeling of flare-productive ARs achieved using the most realistic flux
emergence simulations in a very deep computational domain. Finally, this
review discusses the future perspective pertaining to relationships of flaring
solar ARs with the global-scale dynamo and stellar superflares.

###### keywords:

Solar flares, Coronal mass ejections, Sunspots, Active regions,
Magnetohydrodynamics

††journal: Advances in Space Research

##  1 Introduction

The Sun ceaselessly exhibits activity events of various scales over the whole
spectrum of electromagnetic waves. Solar flares and coronal mass ejections
(CMEs) are among the most prominent energy-releasing phenomena in the current
solar system (Fletcher et al., 2011; Shibata & Magara, 2011). Most powerful
events often emanate from active regions (ARs) that harbor substantial
magnetic non-potentiality.

ARs are created owing to the emergence of dynamo-generated toroidal magnetic
flux from the subsurface layer (flux emergence; Parker 1955). In the
convection zone, where the gas pressure surpasses the magnetic pressure (i.e.,
plasma-$\beta$ is greater than unity), magnetic flux is buffeted by convective
turbulence and accumulates twist during its ascent. However, as the flux
reaches the photosphere and moves further into the atmosphere, where the
plasma density drastically decreases and the magnetic field is dominant
($\beta<1$), the flux begins to build up ARs. Owing to the drastic reduction
of the external gas pressure, the flux expands into the atmosphere, and the
accumulated twist can be released. If an AR has a sufficient amount of
magnetic flux, sunspots are gradually produced, and flares and CMEs may
occasionally occur if the free magnetic energy ($\Delta E_{\rm mag}$) stored
in the corona is suddenly released, where

$\displaystyle\Delta E_{\rm mag}=\int\frac{B^{2}}{8\pi}\,dV-\int\frac{B_{\rm
pot}^{2}}{8\pi}\,dV$ (1)

and $B$ and $\mbox{\boldmath$B$}_{\rm pot}$ denote the actual and potential
magnetic fields, respectively.

Fig. 1 presents the observation of an M-class flare (occurred in February
2011) in NOAA AR 11158. The chromospheric image (top panel) displays two
bright patches, known as flare ribbons, that extend at both sides of the
polarity inversion line (PIL), where the magnetic polarity ($B_{z}$, or the
Stokes-I/V signal in the middle panel) alters its sign.

In the flare theories such as the CSHKP model (after Carmichael, 1964;
Sturrock, 1966; Hirayama, 1974; Kopp & Pneuman, 1976) and its variations, the
flare ribbons brighten in response to the injection of energy that
precipitates along the loops, sourcing from the site of magnetic reconnection
in the corona. The heated chromospheric plasma evaporates and fills the
coronal loops, which leads to the stark enhancement of the soft X-ray flux, as
shown in the bottom panel of Fig. 1. The loops gradually cool down with time
through radiation and thermal conduction, and eventually, the X-ray light
curve returns to the background level.

Fig. 1: M6.6-class flare in NOAA AR 11158 as observed by Hinode/SOT (Kosugi et
al., 2007; Tsuneta et al., 2008) with the GOES soft X-ray light curve. The
detailed analysis suggests that the hook-shaped magnetic field on the PIL
triggered the event (Kusano et al., 2012; Bamba et al., 2013). Figure
reproduced from Toriumi et al. (2013) by permission of the AAS.

As a result of reconnection, a helical flux rope is ejected upward, probably
with the support of plasma instabilities, and, if successful, it develops into
a CME that travels through the interplanetary space. The flare event in Fig. 1
was reportedly accompanied by the CME eruption (e.g., Liu et al., 2012; Inoue
et al., 2018).

The overall description above, i.e., from the dynamo, flux emergence, AR
formation, to the subsequent occurrence of flares and CMEs, is accepted in the
research community. However, in this article, we address the following two key
questions:

  * 1.

What are the important observational features of the flare-productive ARs?

  * 2.

If there exist any, how are these features created?

As the genesis of flaring ARs is related to the prediction and forecasting of
flare eruptions, it is recognized as one of the primary targets in solar
physics to be explored in the next decade (see, e.g., Schrijver et al., 2015).

In the rest of this article, we focus on the flux emergence and formation of
ARs that are prone to major solar flares, and thus, for the understanding of
flares and CMEs, readers are advised to refer to other excellent reviews
(e.g., Priest & Forbes, 2002; Chen, 2011; Fletcher et al., 2011; Shibata &
Magara, 2011; Janvier et al., 2015; Benz, 2017, and references therein).

Section 2 provides an overview of flare-productive ARs by reviewing the
observational features of the ARs. Section 3 presents theoretical studies to
numerically reproduce the formation processes of flaring ARs. Section 4
introduces the latest modeling attempts of flaring ARs with the most realistic
setup. Section 5 summarizes the paper and discusses some remaining problems.

##  2 Observational aspects

This section deals with the observational aspects of flare-productive ARs.
Since the discovery of solar flares (Carrington, 1859; Hodgson, 1859),
significant effort has been made to scrutinize the mechanism and origin of
eruptions. Considering that strong flares mostly occur in ARs, we can directly
examine the relationship between the flares and morphological and magnetic
characteristics of their hosting sunspots, which we elaborate in the following
subsections.

###  2.1 $\delta$-sunspots and flare activity

Observers have noticed that the structure of sunspots is one of the key
factors that determine the flare activity. For instance, in the following Mt.
Wilson sunspot classification, $\delta$-spots are known to be very flare-
active (Künzel, 1960):

  * 1.

$\alpha$, a unipolar spot group

  * 2.

$\beta$, a single bipolar spot group of both positive and negative polarities

  * 3.

$\gamma$, a complex spot group in which spots of both polarities are
distributed so irregularly as to prevent classification as a $\beta$ group

  * 4.

$\delta$, a spot group in which umbrae of opposite polarities are separated by
less than 2∘ and situated within the common penumbra

Sammis et al. (2000) showed that the flare magnitude increases with the size
and complexity of the spot groups. According to these authors, all
$\geq$X4-class flares occur in spot groups of areas greater than 1000 MSH and
classified as the most complex $\beta\gamma\delta$.

Major spaceweather events in history were mostly caused by $\delta$-spots. For
instance, the great geomagnetic storm in March 1989, which caused a severe
power outage in Quebec, Canada, and damaged devices at power plants in New
Jersey, USA, is related to NOAA AR 5395 and its series of strong flares (e.g.,
Allen et al., 1989). The images and magnetograms of this AR, as shown in Wang
et al. (1991), reveal the existence of large complex-shaped $\delta$-spots.

Zirin & Liggett (1987) categorized the formation of $\delta$-spots into three
types:

  * 1.

Type 1: spots emerging all at once, intertwined

  * 2.

Type 2: satellite spots near larger older spots

  * 3.

Type 3: collision of two bipoles

The representative spot group of Type 1 is NOAA 5395 (March 1989), but other
events, such as NOAA 5629 (August 1989), 5747 (October 1989), and 6659 (June
1991), have been extensively analyzed (Tang & Wang, 1993; Schmieder et al.,
1994). A clear example of Type 2 is NOAA 10930 (December 2006). An X3.4-class
flare occurred between sunspots of positive and negative polarities, but the
positive spot was a part of a newly-emerging bipole, whose counterpart was
quickly disappearing (Kubo et al., 2007; Wang et al., 2008): Observations of
NOAA 10930 are summarized in Section 8 of Hinode Review Team et al. (2019).
NOAA 11158 (Fig. 1: February 2011) is the typical Type 3 $\delta$-spot. In
this event, two bipoles emerged and underwent fly-by motion (near head-on
collision). A series of strong flares occurred between the positive and
negative spots, originated from different emerging bipoles, at the center
(Chintzoglou & Zhang, 2013; Toriumi et al., 2014).

Fig. 2: Four representative categories of flaring ARs. From top to bottom, the
polarity distribution, possible 3D structures of magnetic fields, and the
sample events. In the magnetograms (bottom panels), temporally stacked flare
ribbons are indicated by orange and turquoise contours. Figures reproduced
from Toriumi et al. (2017) by permission of the AAS.

Fig. 2 displays four categories of flaring ARs (Toriumi et al., 2017).
Surveying 51 $\geq$M5-class events that occurred within 45∘ from the disk
center over six years, these authors classified ARs based on their
evolutionary patterns: (1) spot-spot, a complex, compact $\delta$-spots, in
which an elongated PIL extends across the entire AR (probably equivalent to
Type 1 $\delta$-spot); (2) spot-satellite, where a new emerging flux collides
with an existing spot (Type 2); and (3) quadrupolar, the collision of two
bipoles (Type 3). It was found that 83% of the analyzed ARs displayed
$\delta$-structures. However, two flares (X1.2 and M5.1) occurred at the
boundaries among separated independent ARs (inter-AR cases), indicating that
X-class events do not require $\delta$-spots.

Therefore, it is possible to categorize the formation of flaring ARs based on
the apparent motions of individual polarities. However, we cannot investigate
the subsurface flux emergence from direct optical observations. This is why
the series of flux emergence simulations have been conducted.

###  2.2 Observational characteristics

In flaring ARs, PILs show a strong horizontal field ($>$4000 G), strong
$B_{z}$ gradient ($\sim$100 G Mm-1), and strong magnetic shear (80∘–90∘ from
the direction of the potential field) (Severnyj, 1958; Hagyard et al., 1984;
Wang et al., 1996; Schrijver, 2007). Recently, Okamoto & Sakurai (2018), Wang
et al. (2018), and Castellanos Durán et al. (2020) detected superstrong
magnetic fields of $\gtrsim$6000 G in the transverse fields of $\delta$-spots
light bridges (i.e., PILs). These values are probably the strongest ever
measured on the Sun, including the sunspot umbrae. These strong-field, strong-
gradient, highly-sheared PILs can be direct manifestation of the non-
potentiality of magnetic fields. Therefore, observational parameters that
characterize flaring PILs have been extensively used to predict flares and
CMEs (e.g., Falconer et al., 2002; Leka & Barnes, 2003; Schrijver, 2007;
Kusano et al., 2020).

Such flaring PILs are often produced in the course of spot rotations and
counter-streaming flows at PILs (Harvey & Harvey, 1976). For instance, the
flares in NOAA 10930 occurred along the PIL between the two spots, and Min &
Chae (2009) reported a fast spot rotation with an angular speed of $\sim
8^{\circ}$ hr-1 before the X-class event, which sheared the magnetic fields
around the PIL and supplied free energy. Chintzoglou et al. (2019) pointed out
that flaring PILs are created by the relative movement of the spots,
especially when the shearing and flux cancellation develop owing to the spot
collision (collisional shearing). These PILs sometimes exhibit the “magnetic
channel,” an alternating pattern of elongated positive and negative polarities
(Zirin & Wang, 1993a, b).

The complexity of ARs are often quantified by the magnetic helicity, a measure
of magnetic topology such as twists, kinks, and internal linkage (Elsasser,
1956). The helicity of the magnetic field $B$ fully contained in volume $V$ is
defined as

$\displaystyle H=\int_{V}\mbox{\boldmath$A$}\cdot\mbox{\boldmath$B$}\,dV,$ (2)

where $A$ is the vector potential of $B$, i.e.,
$\mbox{\boldmath$B$}=\nabla\times\mbox{\boldmath$A$}$. In practical situations
in solar physics, the magnetic field lines cross the boundary (e.g., the
photosphere); therefore, it is more convenient to use the relative helicity,

$\displaystyle H_{\rm
R}=\int_{V}(\mbox{\boldmath$A$}+\mbox{\boldmath$A$}_{0})\cdot(\mbox{\boldmath$B$}-\mbox{\boldmath$B$}_{0})\,dV,$
(3)

where $\mbox{\boldmath$A$}_{0}$ and $\mbox{\boldmath$B$}_{0}$ are the
reference values of $A$ and $B$, respectively (Berger & Field, 1984; Finn &
Antonsen Jr, 1985). $H_{R}$ has the advantage of being gauge invariant, and,
in many cases, the potential field $\mbox{\boldmath$B$}_{\rm
pot}\,(=\nabla\times\mbox{\boldmath$A$}_{\rm pot})$ is chosen as the reference
field. Numerous observations have claimed the relationship between helicity
injection and flare occurrence (see, e.g., Démoulin & Pariat, 2009; Valori et
al., 2016, and references therein).

The “magnetic tongues,” the two extended polarities on magnetograms at both
sides of the PILs, especially when they have a yin-yang pattern, are assumed
to be the vertical projection of the poloidal component of the twisted
emerging magnetic flux (López Fuentes et al., 2000). Therefore, the
arrangement of the tongues and PILs represent the sign of helicity in ARs, and
observations reveal that flaring ARs show such ying-yang tongues (e.g.,
Mandrini et al., 2014; Poisson et al., 2015).

In the atmosphere above the PIL, magnetic flux ropes (bundles of twisted field
lines) are sometimes observed as “sigmoids” in soft X-rays (Rust & Kumar,
1996; Canfield et al., 1999) and “filaments” in H$\alpha$ images (Pevtsov,
2002). The observations indicate that the filament formation is in favor of
the model proposed by van Ballegooijen & Martens (1989), in which converging
and shearing flows around the PIL cause cancellation and twisting of the field
lines to eventually produce a flux rope (filaments/prominences) above the PIL.

Other characteristics, such as non-neutralized electric currents, structural
complexity of photospheric magnetograms, non-thermal broadening of coronal
emission lines, and helioseismic (subsurface) signatures, exist. Moreover, the
triggering of flare eruptions are often associated with the rapid development
of photospheric magnetic fields. For more comprehensive discussion, readers
may refer to Section 3 of Toriumi & Wang (2019).

##  3 Theoretical aspects

As ARs are created through flux emergence from below the solar surface, it is
essential to trace the evolution of magnetic flux from the deeper convection
zone into the atmosphere. To realize this, the magnetohydrodynamic (MHD)
simulations of flux emergence have been performed (Archontis, 2008; Fan, 2009;
Cheung & Isobe, 2014; Schmieder et al., 2014). In typical flux emergence
simulations, in the stratified atmospheric layers representing the solar
convection zone, photosphere, chromosphere, and corona, a flux tube is
initially embedded into the convection zone. The flux tube starts rising by,
for instance, perturbing velocity fields inside the tube or partially reducing
the density from the tube to trigger the buoyant emergence (Parker
instability: Parker 1979). As it ascents to the atmosphere, the flux tube
builds up bipolar spots in the photosphere and magnetic loops in the corona.

To explain the generation of flare-productive ARs, several scenarios have been
suggested and tested by performing such flux emergence simulations. In this
section, we review the past attempts to theoretically model the observed
characteristics of flaring ARs.

###  3.1 Idealized models

Many of the flux emergence simulations conducted so far fall into the category
of idealized models, where the subsurface layer is given as an adiabatically
stratified, plane parallel atmosphere; i.e., the interior is not convectively
unstable. This treatment sets aside the complex effect of turbulent convection
and allows modelers to focus on the essential physical processes.

Fig. 3: Four representative flux emergence models of flare-productive ARs. The
green arrows in the second column point to the satellite spots, which
originate from the parasitic flux tube. Figures reproduced from Toriumi &
Takasao (2017) by permission of the AAS.

One of the representative scenarios of $\delta$-spot formation is the “kinked
tube” model, in which a strongly twisted tube rises owing to the kink
instability (Tanaka, 1991). Using the anelastic MHD code, Fan et al. (1999)
calculated the emergence of a kink-unstable flux tube in the interior and
discovered that the horizontal slice of the emerging flux tube displays a
compact bipolar pair with sheared PIL. Although this morphology is reminiscent
of $\delta$-spots, these simulations were limited to the convection zone due
to the anelastic approximation. Takasao et al. (2015) coped with this problem
by solving fully compressible MHD equations and extending the computational
box to the corona. Interestingly, their modeled spots present, in spite of the
simple bipolar configuration, a quadrupolar structure comprising main bipolar
spots with an elongated pair in between (left column of Fig. 3). Owing to the
release of the helicity, the main spots show strong rotational motions, which
is one of the key characteristics of the flaring ARs. The complex morphology
in the magnetogram suggests that the kinked tube model may explain Type 1
$\delta$-spots (see also Knizhnik et al., 2018).

The second group is the “interacting tube” model, in which multiple flux
bundles interfere with each other to build a single, perhaps complex, AR
(e.g., Murray & Hood, 2007). As shown in Fig. 3, second column from the left,
a parasitic flux tube is placed right above the main tube along the orthogonal
direction. Lifted by the main tube, the parasitic tube produces satellite
spots in the proximity to the main spot (indicated by green arrows).
Therefore, this scenario may be one of the mechanisms of Type 2
$\delta$-spots, in which newly emerging flux appears within the penumbra of
the existing spot (see also Jouve et al., 2018).

In the “multi-buoyant segment” scenario, a single flux tube becomes buoyant at
two segments along the tube, probably caused by the convective pinning at the
middle, and produces a pair of emerging bipoles on the surface, i.e., a
quadrupolar AR (Toriumi et al., 2014). In this case, the two bipoles cause
head-on collision in between, and as a result, strongly packed positive and
negative polarities are created with a highly sheared PIL in the center of the
quadrupolar AR (Fig. 3, third column). The overall evolution resembles Type 3
$\delta$-spots (Fang & Fan, 2015; Syntelis et al., 2019).

These models succeeded in reproducing $\delta$-configurations, sheared PILs,
flux ropes, and others, all of which are the essential ingredients of the
flaring ARs. Toriumi & Takasao (2017) performed a systematic survey of the
aforementioned models using similar numerical conditions with as little
difference as possible (Fig. 3). Key findings are summarized below.

  * 1.

Highly sheared PILs are formed owing to the combined effects of advection,
stretching, and compression of magnetic fields, exerted by the spot motion.

  * 2.

Flux ropes are created above the PIL. However, access to the outer atmosphere
(interplanetary space) depends on the magnetic structure of the entire AR, and
this may determine the CME productivity.

  * 3.

Free energy is accumulated in the corona in the form of current sheets.
Magnetic parameters that predict solar flares with high accuracy (e.g., Bobra
& Couvidat, 2015) reflect the free magnetic energy stored in the corona well.

###  3.2 Realistic models

Contrary to idealized models, flux emergence models that incorporate the
effect of thermal convection enable us to understand complex realistic aspects
of the evolution, in particular, the coupling of magnetic field and
turbulence. For instance, Cheung et al. (2010) modeled the emergence of a
twisted flux tube from the convective convection zone and eventual formation
of proto-sunspots. By kinematically inserting the flux tube from the bottom
boundary, placed at 7.5 Mm below the photosphere, as a sequentially updated
bottom boundary, the authors succeeded in reproducing spots with umbral dots,
light bridges, and penumbral filaments, which are the manifestation of
magnetoconvection (see also Rempel & Cheung, 2014).

Using the updated version of the same code, which is now extended to the solar
corona, Cheung et al. (2019) mimicked the emergence of small satellite spots
within the penumbra of the existing spot to reproduce X-class flares in NOAA
12017 (April 2014). Their numerical results explain the properties of solar
flares, such as temporal evolution of X-ray light curve, chromospheric
evaporation, and flare ribbon expansion.

The interplay between the global-scale convection and magnetic flux has been
strenuously investigated by using the 3D spherical shell models. For example,
Nelson et al. (2013) simulated the self-consistent generation of
$\Omega$-loops from a bundle of toroidal flux, whereas Jouve et al. (2013)
modeled the emergence of a magnetic flux tube that is placed at the depth of
the convection zone. Jouve et al. (2018) also investigated the interaction of
multiple emerging flux tubes and pointed out the possibility that such
interactions introduce complexity to ARs. Although these computations are
limited to the convection zone due to the anelastic approximation and, thus,
cannot simulate the emergence into the photosphere, they revealed the
importance of the turbulent background convection and the mutual interaction
of flux tubes.

##  4 New twist

Flux emergence simulations have succeeded in explaining various aspects of the
mechanisms behind the formation of flare-producing ARs. However, most
simulations are idealized: many models lack the realistic convection zone, or
the emergence is triggered by artificially imposed buoyancy. Even in
convective flux emergence simulations, the bottom boundary is not deep enough,
at most $-30$ Mm (Chen et al., 2017), and flux tubes are kinematically
injected from the bottom boundary. However, in reality, ARs are formed as the
outcome of interaction between magnetic flux and background turbulent
convection. To elucidate the effect and importance of deep large cells on the
flux emergence, there has been a demand for detailed calculations.

Fig. 4: Volume rendering of entropy perturbation in the R2D2 simulation that
is normalized by the averaged entropy at each depth level. Figure courtesy of
H. Hotta.

To overcome these issues, Toriumi & Hotta (2019) leveraged the newly developed
radiative MHD code R2D2 (Hotta et al., 2019; Hotta & Iijima, 2020). In brief,
this code reproduces the realistic solar convection in a very deep domain by
employing the reduced sound-speed technique (Rempel, 2005; Hotta et al., 2012)
and allows for the exploration of magnetoconvective evolution in the whole
solar convection zone. First, the 3D computation box that spanned from
$-139.2$ Mm to $+700$ km in height with a horizontal width of $98.3\ {\rm
Mm}\,\times 98.3\ {\rm Mm}$ was prepared, and the box was numerically resolved
by a $1024\times 1024\times 256$ grid (see Fig. 4). Considering that the
previous convective emergence models reached down to approximately $-30\ {\rm
Mm}$ and that the solar convection zone is a strongly stratified medium, the
simulation with the bottom boundary at $-140\ {\rm Mm}$ is a significant
advancement: The gas pressure at $-140\ {\rm Mm}$ is eight orders of magnitude
greater than that in the photosphere. The periodic boundary condition was
applied to both horizontal directions.

After the development of thermal convection in the box, a twisted force-free
flux tube was placed at the depth of $16.7\ {\rm Mm}$, given as the Lundquist
field (Lundquist, 1951),

$\displaystyle B_{x}=B_{\rm tb}J_{0}(\alpha r),B_{\phi}=B_{\rm tb}J_{1}(\alpha
r),$ (4)

where $r$ is the radial distance from the tube axis, $B_{\rm tb}=10\ {\rm kG}$
the axial field strength, $J_{0}$ and $J_{1}$ the Bessel functions,
$\alpha=a_{0}/R_{\rm tb}$, $a_{0}=2.404825$, and $R_{\rm tb}=7\ {\rm Mm}$ the
tube’s radius. Unlike previous models, the flux tube was neither endowed with
an artificial buoyancy nor forcibly injected into the domain. The tube was
simply advected by the turbulent background convection.

Note that using $16.7\ {\rm Mm}$ as the initial depth allows us to investigate
the emergence from the critical layer, where the approximations assumed in the
previous simulations become invalid (around $-20\ {\rm Mm}$ for the anelastic
and thin flux tube approximations; Fan 2009). The artificial influences caused
by adding a flux tube into the domain at one time step where there was none at
the previous step (e.g., Cheung et al., 2008), such as a lack of the perfect
mechanical balance, can be circumvented using the flux injection method.
However, these influences would be minor because vigorous turbulence
drastically reforms the magnetic configuration (see the following paragraphs).

Fig. 5: Generation of $\delta$-spots in the R2D2 simulation. From left to
right, normalized emergent intensity, magnetic field strength $B_{z}$
(magnetogram), and normalized field strength
$|\mbox{\boldmath$B$}|/(4\pi\rho_{0})^{1/2}$ (thus in units of velocity) on
the vertical cross-section at 32.0 hr (upper row) and 42.3 hr (lower row)
after the flux tube is initially placed. A pair of bipoles (P1-N1 and P2-N2)
collide to produce $\delta$-spots N1-P2 and N2-P1. Figure reproduced from
Toriumi & Hotta (2019) by permission of the AAS.

Fig. 5 shows the evolutions of intensity, surface magnetogram, and magnetic
field in the vertical cross-section. First, a large convection upflow elevates
the flux tube at the side edge of the box at $x=0$ and $98.3\ {\rm Mm}$
(because of the periodic boundary condition), and a bipolar pair P1-N1 appears
in the photosphere. As time progresses, the secondary pair P2-N2 emerges in
the middle of the domain owing to another upflowing region and, consequently,
a quadrupolar AR is formed. The right column demonstrates that two strong
downdrafts (approximately $x=30$ and $70\ {\rm Mm}$) drag down the flux tube
to make a W-shaped configuration. On the surface, the two emerging bipoles
approach each other as they expand, and eventually, two $\delta$-spots, N1-P2
and N2-P1, are formed. The visibility in the intensity map shows that in each
spot, the umbrae of opposite polarities reside in a common penumbra,
satisfying the definition of the $\delta$-spot (Section 2.1).

The PIL between the two spots within each $\delta$-spot shows a narrow lane of
elongated granular cells (light bridge), and the polarity pattern resembles
the magnetic channel structure. The previous simulations revealed that the
light bridges in regular spots comprise granules, which, especially in the
case of narrow light bridges, resemble the penumbra with an elongated magnetic
field (Cheung et al., 2010; Toriumi et al., 2015a, b). The $\delta$-spot light
bridge reproduced here shares a similar magnetoconvective structure.

In the photosphere, all four spots show counter-clockwise rotations owing to
the releasing of the twist of the tube, which was imposed as the initial
condition. As a result, a sheared (or counter-streaming) flow is excited
around the PIL between the two neighboring spots of each $\delta$-spot, and
this velocity field arranges the transverse component of the magnetic field to
make the PIL highly sheared. The field strength pertaining to the PIL area far
exceeds the equipartition value, which is of the order of 1 kG. Hotta &
Toriumi (2020) performed high-resolution calculations, where the grid spacing
is up to four times finer, and discovered that a very strong horizontal field
of $\gtrsim$6000 G is formed at the edge of the $\delta$-spot light bridge.
The field strength matches the recent observations made by Okamoto & Sakurai
(2018) and others.

Fig. 6: Bird’s eye view and top-down view of magnetic field lines above the
$\delta$-spot N2-P1 in Fig. 5. A helical flux rope structure is formed above
the PIL. It should be noted that this flux rope is created as a result of
magnetic reconnection between the field lines originating from the two
independent but neighboring sunspots N2 and P1, rather than the direct
emergence of the original flux tube. Figure reproduced from Toriumi & Hotta
(2019) by permission of the AAS.

Fig. 6 shows the field line structure above one of the $\delta$-spots in Fig.
5. Driven by the counter-streaming flow in the photosphere, the overlying
arcade field is sheared and forms a flux rope structure. The 6-kG horizontal
field (Hotta & Toriumi, 2020) is created by the strong shear at the
$\delta$-spot PIL, which is sustained owing to the downward magnetic force
caused by the overlying arcade.

The above simulations demonstrate that the flux emergence in the deep-enough
computational box self-consistently reproduces $\delta$-spots and other key
features. The entire evolution of $\delta$-spots follows the multi-buoyant
segment scenario, in which a single flux system in the interior rises at two
locations to produce colliding bipoles. It should be emphasized that the
$\delta$-spot formation is a result of coupling between the magnetic flux and
turbulent background convection.

##  5 Summary and discussion

In this article, we discussed important observational features of flare-
productive ARs, including $\delta$-spots, sheared PILs, magnetic channels,
rotating spots, and flux ropes. These structures are reproduced by numerical
models that have been performed over the past decades. In particular, state-
of-the-art realistic flux emergence simulations that contain large convective
cells in deeper layers are now able to reproduce all these features in a self
consistent manner, and we understand that the interaction between magnetic
flux and turbulent convection introduces such complexity in the ARs (see Fig.
7).

Fig. 7: Selected key observational features (mainly photospheric) discussed in
this article and their relationships as revealed by the series of
observational and theoretical studies. The morphological and magnetic
complexities of flare-productive ARs are derived ultimately from the close
interplay between magnetic flux and turbulent convection, i.e., the
magnetoconvective nature of dynamo mechanisms in the deep interior.

Although most sophisticated models reveal the critical aspects of the
formation of flare-productive ARs, there still remains a scale gap between the
localized flux-emergence simulations and global-scale dynamo models. For
instance, almost none of the initial flux tubes in the flaring AR models
introduced in Sections 3 and 4 is given as a dynamo-generated toroidal flux,
and it is not obvious if these initial conditions are truly natural.
Therefore, the next step of the R2D2 calculation would be to test the
realistic flux emergence from much deeper layers. Ultimately, it is necessary
to simultaneously solve the dynamo action in the deep convection zone of a
global scale and the development of ARs on the surface to gain comprehensive
understanding of the entire history from the interior to the surface.

From the observational aspect, it has been pointed out that the flare
reconnection is triggered by small-scale, perhaps subarcsec-sized,
photospheric fields around the PIL (Wang et al., 2017, and references
therein). Therefore, to understand the origin and development of flare
triggers, unprecedented high-resolution polarimetry as achieved by DKIST
(Rimmele et al., 2020) is demanded. Further, UV observations suggest that the
charging of flare energy in the atmosphere may be quantified by the non-
thermal broadening of coronal emission lines (Harra et al., 2009; Imada et
al., 2014). Such spectroscopic signatures of energy accumulation over the
course of AR development would be one of the optimal targets of the next-
generation space UV observatories Solar-C (EUVST) (Shimizu et al., 2019) and
MUSE (De Pontieu et al., 2020). Nevertheless, these efforts would be
facilitated more by the support of advanced modeling capabilities.

Recent progress in observing stellar flares indicates the occurrence of
superflares on solar-type stars, which are 10–1000 times stronger than those
on the Sun (e.g., Maehara et al., 2012), and this leads to the discussion on
the possibility of superflares on the Sun (e.g., Aulanier et al., 2013;
Shibata et al., 2013). Then, how do such superflaring starspots occur? As
described in Section 2, on the Sun, we know that large, complex, rapidly-
evolving ARs tend to produce massive flares. Therefore, it is required to
obtain the information of starspot evolution and its relation with stellar
flares (Namekata et al., 2019). Recently, Toriumi et al. (2020) focused on the
Sun-as-a-star spectral irradiance variations of transiting sunspots and
revealed that the multi-wavelength monitoring of the stars may help us
understand the structures and evolutions of starspots. In future, probably by
combining with the numerical models, we may have a more detailed approach on
finding the cause of stellar flares.

## Acknowledgments

We are grateful to the Special Issue Guest Editors (B. Schmieder and C. H.
Mandrini) and the anonymous referees for improving the quality of the
manuscript. This work was supported by JSPS KAKENHI Grant Numbers JP18H03723
(PI: Y. Katsukawa), JP20KK0072 (PI: S. Toriumi), JP21H01124 (PI: T. Yokoyama),
and JP21H04492 (PI: K. Kusano), and by the NINS program for cross-disciplinary
study (grant Nos. 01321802 and 01311904) on Turbulence, Transport, and Heating
Dynamics in Laboratory and Astrophysical Plasmas: “SoLaBo-X.”

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