11institutetext: Yunnan Observatories, Chinese Academy of Sciences, Kunming
Yunnan 650216, PR China
11email<EMAIL_ADDRESS>22institutetext: CAS Key Laboratory of Solar
Activity, National Astronomical Observatories, Beijing 100012, PR China
33institutetext: Center for Astronomical Mega-Science, Chinese Academy of
Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, PR China
44institutetext: University of Chinese Academy of Sciences, Yuquan Road,
Shijingshan Block Beijing 100049, PR China 55institutetext: Instituto de
Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain 66institutetext:
Universidad de La Laguna, Dpto. Astrofisica, E-38206 La Laguna, Tenerife,
Spain

# Naked emergence of an anti-Hale active region

I. Overall evolution and magnetic properties

Jincheng Wang 112233 Xiaoli Yan 1133 Defang Kong 1133 Zhike Xue 1133 Liheng
Yang 1133 Qiaoling Li 1144 Yan Zhang 1144 Hao Li 5566

(Received March 1, 2021; accepted **** **, ****)

###### Abstract

Context.

Aims. In order to understand the emergence of the active region, we
investigate the emerging process and magnetic properties of a naked anti-Hale
active region during the period between August 24 to 25, 2018.

Methods. Using the data from Helioseismic and Magnetic Imager on board the
Soar Dynamic Observatory and the New Vacuum Solar Telescope, we calculated
different evolving parameters (such as pole separation, tilt angle) and
magnetic parameters (such as vertical electric current, force-free parameter,
relative magnetic helicity) during the emergence of the active region. With
these calculated parameters and some reasonable assumptions, we use two
different methods to estimate the twist of the active region.

Results. The magnetic flux and pole separation continue increasing while the
tilt angle exhibits a decreasing pattern during the emergence of the active
region. The increase of the pole separation is mainly contributed as a result
of the enhancement in the longitude direction. A power-law relationship
between pole separation and total flux is found during the emergence of the
active region. On the other hand, it is found that both the positive and
negative electric currents increased equivalently and the average flux-
weighted force-free parameter $\tilde{\alpha}$ remains almost consistently
positive, on the order of $\sim$ 10-8 m-1. The relative magnetic helicity is
mainly contributed by the shear term, while the relative magnetic helicity
injection flux of the shear term changes its sign at the latter stage of the
emergence. The twist number of the whole active region remains on the order of
10-1 turns during the emergence of the active region.

Conclusions. We find that the magnetic flux tube with low twist also could
emerge into the solar atmosphere.

###### Key Words.:

Sun: sunspot – Sun: evolution – Sun: magnetic fields

##  1 Introduction

It is widely known that the solar atmosphere is dominated by magnetic fields
and almost all the activities taking place in the atmosphere are associated
with the evolution of magnetic fields (e.g.,Thornton & Parnell, 2011;
Wiegelmann et al., 2014; van Driel-Gesztelyi & Green, 2015; Cai et al., 2019).
Therefore, the question of the origin of magnetic fields is the most
significant aspect in understanding the nature of solar activities. The
observational sunspots and active regions (ARs) are thought to be born of the
emerging magnetic flux, which is the concentration of the magnetic fields (van
Driel-Gesztelyi & Green, 2015). On the other hand, many solar activities are
triggered by newly emerging magnetic flux, such as coronal mass ejections
(CMEs), flares, formations or eruptions of the filaments, jets, and small-
scale bursts (e.g., Feynman & Martin, 1995; Chen & Shibata, 2000; Isobe et
al., 2007; Yan et al., 2017; Chen et al., 2018; Wang et al., 2018, 2019; Young
et al., 2018; Tian et al., 2018). Such an emerging magnetic field not only
plays a vital role in understanding active regions and sunspots, but it also
has a great influence on many of the activities in taking place in the solar
atmosphere.

Flux-emerging regions (FERs), or general ARs, are thought to be formed by the
rise of flux tubes from the convection zone to the solar surface (van Driel-
Gesztelyi & Green, 2015). Fu & Welsch (2016) found that about half of the
investigated emerging active regions show a two-step emergence pattern that
indicates that the flux emergence rate was relatively low at the early phase,
undergoing a significant enhancement in the next phase. According to the
numerical simulations, a twisted flux tube can emerge from the convection zone
into the corona, passing through the photosphere and chromosphere as a result
of the magnetic buoyancy or Parker instability, which leads to the formation
of an $\Omega$-shaped flux tube (Leka et al., 1996; Fan, 2001; Magara, &
Longcope, 2003). However, some studies have suggested that not all the twisted
flux tubes embedded in the subsurface could emerge into the upper atmosphere.
Murray et al. (2006) simulated different emerging flux tubes with different
twist values. They found that the highly twisted flux tube could easily emerge
into the upper atmosphere, while the weakly twisted flux tube fails to do so.
Toriumi et al. (2011) also obtained a similar result that only the flux tube
with high twist (q$H_{0}$ $>$ 0.1, $q=B_{\theta}/rB_{l}$, $H_{0}$=170 km) can
be launched into the corona and the low twisted flux tube could hardly emerge
into the photosphere due to the influence of the horizontal expansion.
However, it is rare to observe a high twisted flux tube emerging from the
subsurface. Several researchers have provided some observational evidence to
ascertain the emergence of a small twisted flux rope (Okamoto et al., 2009;
Yan et al., 2017). Poisson et al. (2015) analyzed the “magnetic tongue ” of 41
emerging bipolar active regions with a new systematic and user-independent
method, deducing that flux ropes embedded in the subsurface before the
emergence have a low amount of twist and that highly twisted flux tubes are
relatively rare. The discrepancy between the numerical simulations and the
observations raises a significant question regarding how much twist the flux
tube needs for its emergence.

Magnetic fields emerging from the Sun’s interior would carry intrinsic
information about the magnetic field in the subsurface to the upper
atmosphere. Emerging magnetic fields play a key role in the transportation of
the magnetic energy and helicity. When a flux tube emerges into the solar
atmosphere from the convective zone (CZ), it not only brings magnetic flux
that contributes to the formations of active regions or sunspots but that also
injects magnetic energy and magnetic helicity into the solar atmosphere
(Démoulin & Pariat, 2009). Using a three-dimensional MHD numerical simulation,
Magara, & Longcope (2003) investigated the injection process of magnetic
energy and relative magnetic helicity into the upper atmosphere. They found
that both the magnetic energy and relative magnetic helicity are contributed
by the emergence term during the early phase and the shear term during the
latter phase. Liu & Schuck (2012) studied the injections of magnetic energy
and helicity in two emerging active regions, and they found that rates of
energy injection from both the emergence term and the shear term have a
consistent evolution in phase during the entire process of flux emergence.
They also derived that the injection of magnetic helicity and magnetic energy
in the corona are mainly contributed by the shear term and the emergence term,
respectively. A similar result was also obtained by Vemareddy (2015). However,
with the analysis of 28 emerging active regions, Liu et al. (2014) certified
again that the shear term is dominant in the helicity injection, however, both
terms, namely, the shear and emergence terms contribute approximately
equivalent energy to the solar atmosphere. On the other hand, Vemareddy &
Démoulin (2017) even found that opposite magnetic helicity was injected into
the corona during the emergence of the active region and suggested that it may
be caused by the emergence of a flux rope with different sign of twist between
its apex and its legs. Investigations of the magnetic helicity or energy
injection during the emergence of magnetic flux tube provide a good window for
us to improve the understanding of the non-potential origination of emerging
magnetic flux and a potential way to explore the nature of the magnetic flux
embedded in the subsurface.

In general, the leading sunspots of a bipolar AR in the northern hemisphere
have the opposite polarity with regard to leading spots in the southern
hemisphere and the leading magnetic polarities alternate in successive sunspot
cycle, known as Hale’s law (Hale & Nicholson, 1925). This feature can be
interpreted well in the models of “dynamo theory” (e.g., Parker, 1955;
Babcock, 1961; Leighton, 1964; Wang & Sheeley, 1991; Choudhuri et al., 1995).
Shearing motion by differential rotation, called the $\Omega$-effect, plays a
key role in these models (Krause & Raedler, 1980; Dikpati & Gilman, 2001).
According to the Hale’s law, the leading spots of bipolar ARs should be
negative and the following spots should be positive in the northern hemisphere
in the solar cycle 24. The ARs with the reverse polarities in leading spots
and following spots are considered as anti-Hale ARs. People found that a few
ARs could violate Hale’s law and the percentage of anti-Hale ranges from a few
to 10% (Wang & Sheeley, 1989; Li & Ulrich, 2012; McClintock et al., 2014; Li,
2018; Zhukova et al., 2020). Li (2018) found that the distribution of tilt
angles and magnetic fluxes between Hale and anti-Hale is different while the
pole separation of anti-Hale ARs is smaller than Hale ones. The anti-Hale and
Hale ARs exhibit similar latitudinal distributions (McClintock et al., 2014;
Li, 2018). On the other hand, anti-Hale ARs may hide the imprint of
fundamental processes in solar interior. With the clue of the increase of the
percentage of anti-Hale regions during the solar minimum, Sokoloff et al.
(2015) concluded that small-scale dynamo is active in the solar interior.
Based on a surface flux transport simulation and observational data, Jiang et
al. (2015) proposed that bigger anti-Hale ARs emerging around low latitudes
are responsible for the weak polar fields that affect the activity of the
subsequent cycle. Thus, studies of anti-Hale ARs may shed light on our
understanding of solar magnetic field evolution and supply a clue to probe the
performing mechanism of magnetic field.

In this paper, we focus on the evolution and the magnetic characteristics in a
strictly naked emerging anti-Hale active region NOAA 12720. The evolution of
the active region and variations of different related-magnetic parameters
during its emergence are studied in detail. The sections of this paper are
organized as follows. The observations and methods are described in Section 2,
the results are given in Section 3, and the summary and discussions are
presented in Section 4.

##  2 Observations and methods

###  2.1 Observations

In this study, data from Helioseismic and Magnetic Imager (HMI: Schou et al.,
2012) on board the Solar Dynamic Observatory111https://sdo.gsfc.nasa.gov (SDO:
Pesnell et al., 2012) are used to analyze the magnetic properties of an
emerging Active Region NOAA 12720 during the period from 16:12 UT on August 24
to 12:00 UT on August 25, 2018. The SDO/HMI provides continuous full-disk
measurement of the Stokes vector of the photospheric Fe $\rm{I}$ 6173
$\rm{\AA}$ line about every 135 s. The HMI science team had pipelined the
process of retrieving vector field information from filtergrams and derived
the available data (hmi.sharp.cea.720s) for solar researchers (Hoeksema et
al., 2014). The pipeline procedure involves inversion of stokes vectors using
the Very Fast Inversion of the Stokes Vector algorithm (VFSIV: Borrero et al.,
2011; Centeno et al., 2014) based on the Milne-Eddington atmospheric model and
removing the 180$\degr$ azimuthal ambiguity using the minimum energy method
(Metcalf, 1994; Leka et al., 2009). Space Weather HMI Active Region Patch
(SHARP) products provide the data of AR’s line-of-sight and vector magnetic
field, continuum intensity, Doppler velocity, with a pixel scale of about 0.5
$\arcsec$ and a cadence of 12 minutes (Hoeksema et al., 2014). These data are
remapped using Lambert (cylindrical equal area method (Calabretta, & Greisen,
2002)) projection centered on the midpoint of the active region, which is
tracked at the Carrington rotation rate. The continuum intensity images from
SHARP permit us to investigate the evolution of the active region, whereas the
vector magnetograms (VMs) from SHARP allow us to analyze the magnetic
properties of the active region. Moreover, the high-resolution TiO and
H$\alpha$ images from the New Vacuum Solar Telescope222http://fso.ynao.ac.cn
(NVST:Liu et al., 2014; Yan et al., 2020) are also utilized to exhibit the
photospheric and chromospheric details. The TiO images have a pixel size of
0.″052 and a cadence of 30 s. The field of view (FOV) of TiO images is
120″$\times$ 100″. The field of view (FOV) of H$\alpha$ images is 150″$\times$
150″, with a 45s cadence and a spatial resolution of 0″165 per pixel. These
data are calibrated from Level 0 to Level 1 with dark current subtracted and
flat field corrected, and then speckle masking method was used to reconstruct
the calibrated images from Level 1 to Level 1+ (Xiang et al., 2016).

###  2.2 Methods

####  2.2.1 Pole separation and tilt angle

In order to determine the pole separation of the active region, we first need
to determine the locations of each polarity. For each moment, the centroids of
positive and negative polarities is determined by using the flux-weighted
method as follows:

$(\tilde{x}_{\pm},\tilde{y}_{\pm})=(\frac{\Sigma xB_{\pm}}{\Sigma
B_{\pm}},\frac{\Sigma yB_{\pm}}{\Sigma B_{\pm}}),$ (1)

in which $B_{\pm}$ is the magnetic field strength in each pixel and ($x$, $y$)
is the position coordinate of each pixel. The subscript signs “+” and “-”
denote positive and negative polarities, respectively. As the centroids
$(\tilde{x}_{\pm},\tilde{y}_{\pm})$ of the positive and negative polarities
have been derived, the pole separation is calculated by using the following
equation:

$d=\sqrt{(\tilde{x}_{+}-\tilde{x}_{-})^{2}+(\tilde{y}_{+}-\tilde{y}_{-})^{2}}.$
(2)

On the other hand, we define the tilt angle as the angle between the pole
separation vector and the solar east. Thus, the tilt angle can be calculated
by using the following equation:

$\vartheta=-\arctan(\frac{\tilde{y}_{+}-\tilde{y}_{-}}{\tilde{x}_{+}-\tilde{x}_{-}}).$
(3)

####  2.2.2 Magnetic flux, vertical electric current, and force-free parameter
$\alpha$

Magnetic fluxes are integrated the positive and negative polarity magnetic
fields by their area, respectively. Positive and negative magnetic fluxes are
calculated by using the following equations:

$\displaystyle\varphi_{zp}=\int_{S_{ph}}B_{z}(x,y)_{+}dS,$
$\displaystyle\varphi_{zn}=\int_{S_{ph}}|B_{z}(x,y)_{-}|dS,$ (4)

where $B_{z}(x,y)_{+}$/$B_{z}(x,y)_{-}$ is the vertical positive/negative
magnetic field and the S denotes the integrated area with positive or negative
magnetic field.

According to Ampere’s law, the electric current density in the corona can be
derived by the following equation:

$\textbf{\emph{J}}=\frac{c}{4\pi}(\triangledown\times\textbf{\emph{B}}),$ (5)

in which $c$ is the speed of the light and B denotes the magnetic field vector
in the corona. According to Eq. 5, by neglecting the effect of the electric
displacement current, the electric current density component perpendicular to
the solar surface ($J_{z}$) can be calculated by the following equation:

$J_{z}(x,y)=\frac{c}{4\pi}(\triangledown\times\emph{{B}})_{z}=\frac{c}{4\pi}(\frac{\partial
B_{y}(x,y)}{\partial x}-\frac{\partial B_{x}(x,y)}{\partial y}),$ (6)

where $B_{x}$ and $B_{y}$ are the two perpendicular components of the
transverse magnetic fields. With the vector magnetic fields from SDO/HMI and
Eq. 6, the distribution of vertical electric current density in the
photosphere can be approximated by using a five-point stencil method.
Therefore, the vertical current can be integrated by the vertical current
density in the active region. The positive and negative currents calculated by
using the following equations:

$\displaystyle I_{zp}$ $\displaystyle=$
$\displaystyle\int_{S_{ph}}J_{z}(x,y)_{+}dS,$ $\displaystyle I_{zn}$
$\displaystyle=$ $\displaystyle\int_{S_{ph}}|J_{z}(x,y)_{-}|dS,$ (7)

where $J_{z}(x,y)_{+}$/$J_{z}(x,y)_{-}$ is the vertical positive or negative
current density and S denotes the integrated area. On the other hand, we also
calculate the current neutralization ratio ($\Re$)(Török et al., 2014; Liu et
al., 2017):

$\Re=\frac{I_{ret}}{I_{dir}},$ (8)

where $I_{ret}$ and $I_{dir}$ are the return and direct currents,
respectively. They are express as

$\displaystyle I_{ret}=\int_{S_{ph}}\kappa_{ret}(x,y)J_{z}(x,y)dS,$
$\displaystyle I_{dir}=\int_{S_{ph}}\kappa_{dir}(x,y)J_{z}(x,y)dS,$ (9)

with

$\displaystyle\kappa_{ret}=\left\\{\begin{array}[]{cl}0&$ $\text{if}$
$J_{z}(x,y)B_{z}(x,y)<0\\\ 1&$ $\text{otherwise,}\end{array}\right.$ (12)
$\displaystyle\kappa_{dir}=\left\\{\begin{array}[]{cl}0&$ $\text{if}$
$J_{z}(x,y)B_{z}(x,y)>0\\\ 1&$ $\text{otherwise.}\end{array}\right.$ (15)

The current neutralization ratio ($\Re$) is an important parameter in
evaluating the equilibrium of the AR system, which is related to flares and
CMEs (Melrose, 1991; Liu et al., 2017; Avallone & Sun, 2020).

Under the assumption of force-free field, electric currents are parallel or
anti-parallel to the magnetic field lines. The electric current could be
expressed as:

$J=\alpha B,$ (16)

in which $\alpha$ is called force-free parameter, and it is constant along
each field line. The parameter $\alpha$ could usually be used to represent a
measure of magnetic twist in an AR (Seehafer, 1990; Pevtsov et al., 1995;
Hagino, & Sakurai, 2004). A calculable quantity
($\alpha_{z}=\frac{J_{z}(x,y)}{B_{z}(x,y)}$) is a commonly used proxy of the
twist at the photospheric level (Pevtsov et al., 1994). Here, we calculate an
averaged flux-weighted force-free parameter $\tilde{\alpha}$ over the whole AR
in the photosphere, based on the following equation (Hagino, & Sakurai, 2004;
Kutsenko et al., 2019):

$\tilde{\alpha}=\frac{\int j_{z}(x,y)B_{z}(x,y)dS}{\int B_{z}(x,y)^{2}dS},$
(17)

in which $B_{z}$ is the vertical magnetic field. S denotes the area of the
whole active region.

####  2.2.3 Magnetic helicity injection flux

Relative magnetic helicity (for simplicity, we use magnetic helicity to refer
to relative magnetic helicity in this paper) can be transported from the solar
interior to the corona by the new emerging fluxes and the various motions of
magnetic flux in the photosphere. The magnetic helicity injection flux across
a surface S is expressed as (Berger, & Field, 1984; Démoulin, & Berger, 2003):

$\dot{H}=2\int_{S}(\textbf{\emph{A}}_{p}\cdot\textbf{\emph{B}}_{t})V_{\bot
n}dS-2\int_{S}(\textbf{\emph{A}}_{p}\cdot\textbf{\emph{V}}_{\bot t})B_{n}dS,$
(18)

where $\textbf{\emph{A}}_{p}$ is the vector potential of the potential field
$\text{\emph{B}}_{p}$, $\textbf{\emph{B}}_{t}$, and $B_{n}$ are the tangential
and normal components of the magnetic field, respectively.
$\textbf{\emph{V}}_{\bot t}$ and $\textbf{\emph{V}}_{\bot n}$ are the
tangential and normal components of the velocity perpendicular to the magnetic
field lines, respectively. The velocity ($\textbf{\emph{V}}_{\bot}$) is
further corrected by removing the irrelevant field aligned plasma flow as
$\textbf{\emph{V}}_{\bot}=\textbf{\emph{V}}-\frac{\textbf{\emph{V}}\cdot\textbf{\emph{B}}}{\textbf{\emph{B}}^{2}}\textbf{\emph{B}}$.
The velocity vector field (V) is derived by using the Differential Affine
Velocity Estimator for Vector Magnetograms (DAVE4VM) method (Schuck, 2008) in
our study. The DAVE4VM method implements a variational principle with the
magnetic induction equation to track the velocity of magnetic footpoints. We
adopt 19 $\times$ 19 pixels as the window size for DAVE4VM, which is
determined by examining the slope, Pearson linear correlation coefficient, and
Spearman rank order between $\bigtriangledown_{n}\cdot(\textbf{V}B_{n})$ and
$\Delta B_{n}/\Delta t$ (Schuck, 2006, 2008). The first term in the right of
the equation is named emergence term ($\dot{H}_{e}$), which is associated with
the emergence of magnetic flux tube from the solar subsurface. The second term
is named shear term ($\dot{H}_{s}$), which is generated by shearing magnetic
field lines due to tangential motions on the surface (Berger, & Field, 1984;
Kusano et al., 2002; Pariat et al., 2005; Liu et al., 2014). As the target
active region is small enough, the curvature effect could be ignored. The
solar photosphere S can be assumed to be a planar. Therefore, the helicity
ejection flux can be expressed as follows (Pariat et al., 2005; Liu & Schuck,
2012):

$\displaystyle\dot{H}$ $\displaystyle=$ $\displaystyle\dot{H}_{e}+\dot{H}_{s}$
(19) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{s}\int_{s\arcmin}\textbf{\emph{n}}\cdot\frac{\textbf{\emph{x}}-\textbf{\emph{x}}\arcmin}{|\textbf{\emph{x}}-\textbf{\emph{x}}{\arcmin}|^{2}}\times\\{\textbf{\emph{B}}_{t}(\textbf{\emph{x}})V_{\bot
n}(\textbf{\emph{x}})B_{n}(\textbf{\emph{x}}\arcmin)$
$\displaystyle-\textbf{\emph{B}}_{t}(\textbf{\emph{x}}\arcmin)V_{\bot
n}(\textbf{\emph{x}}\arcmin)B_{n}(\textbf{\emph{x}})\\}dS\arcmin dS$
$\displaystyle-\frac{1}{2\pi}\int_{s}\int_{s\arcmin}\textbf{\emph{n}}\cdot\frac{\textbf{\emph{x}}-\textbf{\emph{x}}\arcmin}{|\textbf{\emph{x}}-\textbf{\emph{x}}{\arcmin}|^{2}}\times\\{[\textbf{\emph{V}}_{\bot
t}(\textbf{\emph{x}})$ $\displaystyle-\textbf{\emph{V}}_{\bot
t}(\textbf{\emph{x}}\arcmin)]B_{n}(\textbf{\emph{x}})B_{n}(\textbf{\emph{x}}\arcmin)\\}dS\arcmin
dS$ $\displaystyle=$ $\displaystyle\int_{s}G_{\theta}(\textbf{\emph{x}})dS,$

in which x and $\textbf{\emph{x}}\arcmin$ represent two photospheric positions
and n is the surface normal vector pointing into the corona.
$G_{\theta}(\textbf{\emph{x}})$ could be used to present the approximate proxy
of the magnetic helicity injection flux density (Pariat et al., 2005). We use
Eq.19 to calculate the total helicity flux in the whole active region.
Accumulated helicity can be derived by the integration of the helicity
injection rate with time.

To minimize the influence of measurement error in the magnetic field of HMI,
we only consider pixels with a magnetic field strength exceeding 300 G
(roughly 3$\sigma$ noise level of the vector magnetic field) to calculated
above magnetic parameters (Hoeksema et al., 2014). The errors of positive and
negative fluxes are estimated by the uncertainties from HMI observations of
SHARP data. On the other hand, a Monte-Carlo experiment was used to estimate
the errors in Eqs.1, 2, 3, 7, 8, 17, and 19. In the Monte Carlo experiment, we
randomly added noises to three components of the vector magnetic field, and
repeated the computations of above equations. The added noises have a Gaussian
distribution, and the widths ($\sigma$) of Gaussian distributions are set to
be the uncertainties from HMI observations in each components of the vector
magnetic field. This test was repeated 200 times. The two-hour running average
of the root mean square (rms; $\sigma$) of these 200 experiments is used to be
the representative error as the method estimated by Liu & Schuck (2012).

##  3 Results

###  3.1 Emerging process of the active region

During the period from August 24 to 25 2018, an active region NOAA 12720
gradually emerged into the solar surface in the northern hemisphere, and
appeared near the center of the solar disk (N07, W16). Figure 1 shows the
overview of this emerging active region at around 09:24 UT on August 24, 2018.
Panel (a) displays the full disk line-of-sight magnetic field observed by
SDO/HMI. This active region occurred in a purely clean environment, which is a
perfect isolated AR. Panel (b) shows the TiO image observed by NVST. A lot of
elongated granules and stretched dark lanes marked by black arrows in panels
(b) could be identified between two main pores or sunspots, which are
considered to be the signatures of the emerging flux through the photosphere
(Strous et al., 1996; Schlichenmaier et al., 2010). The panels of the right
column exhibit the images from SHARP data at the corresponding moment. Panel
(c) is the continuum intensity image. The elongated granules and stretched
dark lanes marked by the black arrows also could be found in the flux-emerging
region. Panel (d) shows the vector magnetogram at 09:24 UT on August 24. The
horizontal magnetic field ($\textbf{B}_{t}$) is depicted by blue and red
arrows indicating the direction and strength, while the background is the map
of the vertical magnetic field component ($B_{z}$). The relatively
concentrated leading positive polarity and the dispersed following negative
polarity indicate the fact that this active region does not obey with the
hemisphere pattern of Hale’s law (Hale & Nicholson, 1925) for the active
region in northern hemisphere in the solar cycle 24. Thus, the emerging active
region is an anti-Hale one. Panel (e) shows the Doppler velocity map in the
photosphere. Blue and red colors indicate upflows and downflows, respectively.
The fascinating feature is that the regions with strong magnetic flux show
strong red shift in the photosphere. This may be associated with the falling
plasma. As the magnetic fields ascend into higher altitude, the plasma
residing in the emerging magnetic field would fall down through the footpoints
along the magnetic field lines due to the solar gravity, and returns to the
solar surface. These falling plasma shows the red shifted signatures at the
footpoints of the magnetic field lines.

Figure 1: Overview of the emerging active region. (a) The global line-of-sight
magnetic field observed by SDO/HMI. The yellow dashed line marks the solar
equator. The yellow and black boxes outline the fields of view of panel (b)
and panels (c)-(e), respectively; (b) TiO image observed by NVST; (c)
continuum intensity image from SHARP data; (d) vector magnetograms from SHARP
data; blue and red arrows indicate the transverse field with positive and
negative flux, respectively; (e) Doppler velocity from SHARP data.

Figure 2 exhibits the emerging process of the active region NOAA 12720. The
panels of left column show vector magnetograms at different moments, while the
ones in the mid and right columns are corresponding continuum intensity and
horizontal velocity, respectively. The active region initially appeared as a
simple small bipolar pore (see panels (a1) & (b1)), which eventually developed
into a large active region including a leading positive sunspot, a following
negative sunspot and some mixed polarities between two main sunspots (see
panels (a4) & (b4)). The detailed evolution of the active region is displayed
by the animation of the Fig. 2. Most of the newly emerging flux initially
appeared between two main polarities and gradually accumulated toward each
polarity. Some moving dipolar features (Bernasconi et al., 2002) could be
found between two main polarities, which is marked by the yellow circle in the
panel (a3). These types of structures are important to form the long coronal
loop connecting two main polarities and release the mass from rising magnetic
field at the later stage of emergence (Centeno, 2012). Panels (b1)-(b4) show
the evolution of two main sunspots during the emergence of the active region
in continuum intensity images. The leading sunspot tends to be more
concentrated than the following one. On the other hand, the leading sunspot
drifted westward while the following sunspot drifted eastward. The leading
sunspot moved faster than the following one, which is demonstrated by the
inclinations of two dashed blue lines in the mid column. This feature is
reasonable for the leading (or following) sunspot during emergence. In
general, the leading sunspot is close to the solar equator and drift westward
while the following sunspot is far away from the solar equator and drift
opposite direction. With the influence of solar rotation and differential
rotation in latitude, the leading sunspot is expected to drift faster than the
following one during their emergence. Based on the velocity map derived by
DAVE4VM in the panels of the right column, the positive polarity exhibited a
strong motion toward the southwest at the early phase of emergence, while the
negative polarity moved toward the northeast (panel (c1)-(c3)). At the later
phase of emergence, these motions became weak and inconsistent (panel (c4)).

Figure 2: Evolution of the emerging active region NOAA 12740. (a1)-(a4) The
vector magnetograms. (b1)-(b4) The continuum intensity images. The white and
black contours indicate the magnetic field strength (800G) with positive and
negative polarities, respectively. The white asterisks in panel (b2) denote
the centroids of positive and negative polarities; (c1)-(c2) The transverse
velocity maps derived with DAVE4VM. The blue and green arrows indicate the
transverse velocity with positive and negative polarities, respectively.

Figure 3 shows the changes in photosphere and chromosphere between the early
and latter stages of emergence. The images in top row are TiO observations
from NVST, while the ones in bottom row are corresponding H$\alpha$
observations from NVST. At early stage of emergence, a lot of elongated
granules and stretched dark lanes can be found in the middle of active region
in the photosphere, which are marked by black circle in panel (a) of Fig. 3.
Meanwhile, the sheared-arcade structures appeared in the chromosphere (see in
panel (c)). At the latter stage of emergence, the active region was
approaching mature, which consists of leading, following sunspots and some
fragments. It is interesting to find that the leading sunspot has a complete
penumbra while the following sunspot does not (see panel (b)). Meanwhile, more
complicated structures (such as filaments or small-scale filamentary
structures) could be seen in the chromosphere (see panel (d)).

Figure 3: Photospheric and chromospheric observations during the emergence.
(a) & (b) TiO images observed by NVST; (c) & (d) H$\alpha$ images observed by
NVST. Black circle in panel (a) points out some elongated granules and
stretched dark lanes.

###  3.2 Magnetic flux, pole separation and tilt angle

Based on the equations described in Sect. 2.2, we calculate the magnetic flux,
pole separation, and tilt angle of the emerging active region during its
emergence. Figure 4 (a) exhibits the temporal variations of positive and
negative magnetic fluxes. The error bars are derived by integrating the
uncertainties from HMI observation with integrated area. Both positive and
negative magnetic fluxes show a equivalent increase from about 0.2 $\times
10^{21}$ Mx at 16:12 UT on August 23 to 3.6 $\times 10^{21}$ Mx at 12:00 UT on
August 25. The mean increasing rate of magnetic flux is about 0.78 $\times
10^{20}$ Mx/hr in each polarity. We can find that both positive and negative
magnetic fluxes maintained a slow increase during the period before 05:00 UT
on August 24. The mean increasing rate of each polarity was only about 0.17
$\times 10^{20}$ Mx/hr during this period. We define this period as the early
phase of emergence. Both positive and negative magnetic fluxes experienced a
rapid increase during the period from 05:00 UT on August 24 to 9:00 UT on
August 25. The mean increasing rate of each polarity was 1.16 $\times 10^{20}$
Mx/hr. We define this period as the main phase of emergence. Therefore, the
emergence of this naked anti-Hale AR is also consistent with the two-step
emergence pattern (Fu & Welsch, 2016). Unfortunately, the data during the
periods from 06:24 to 08:48 UT on August 24 and 25 were unavailable because of
the eclipse in the SDO orbit (Pesnell et al., 2012). Both positive and
negative fluxes exhibit little change during the period from 09:00 UT to 12:00
UT on August 25. This indicates that the active region became mature at around
12:00 on August 25. Panel (b) shows the time variations of tilt angle and pole
separation. The blue line denotes the tilt angle of the active region, while
the black line denotes the pole separation. The uncertainties are depicted by
the black and blue representative error bars. They are estimated by conducting
a Monte Carlo experiment, as described in Sect. 2.2. There are several
features on the profile of tilt angle. Firstly, at the early phase of the flux
emergence, the active region maintained a high value for the tilt angle (about
23$\degr$) and began to decrease. Secondly, the main decrease of the tilt
angle took place during the phase from 03:36 UT to 18:00 UT on August 24.
Thirdly, the active region kept at low value of tilt angle (about 12$\degr$)
after 18:00 UT, when the magnetic flux continued to increase. The pole
separation shows a continuously increasing pattern with a constant rate during
the entire time, which was from about 18 Mm at the beginning to about 70 Mm in
the end. The mean increasing rate of pole separation was about 2.92 Mm/hr. We
divide the pole separation into two components: longitude component, $d_{x}$,
and latitude component, $d_{y}$. Panel (c) shows the evolutions of $d_{x}$ and
$d_{y}$. The longitude component $d_{x}$ increased significantly all the time
from about 16 Mm to 68 Mm. The enhancement of pole separation in latitude
direction ($d_{y}$) is much smaller than that in the longitude direction.
Therefore, the increase of the polar separation is mainly contributed from the
enhancement in the longitude direction ($d_{x}$). This means that the two
poles are separated from each other in longitude as the active region emerges.

Figure 4: Variations of magnetic flux, pole separation, and tilt angle of the
emerging active region during the period from 16:12 UT on August 23 to 12:00
UT on August 25, 2018. (a) The evolution of positive and negative magnetic
flux. The red and blue lines indicate the positive and negative magnetic flux.
The errors in the positive and negative magnetic flux are estimated by the
uncertainties from HMI observations of SHARP data; (b) The evolutions of pole
separation and tilt angle. The black line denotes pole separation and the blue
one denotes tilt angle; (c) The evolution of pole separation in longitude
direction ($d_{x}$) and latitude direction ($d_{y}$). The black and blue lines
denote the pole separation in longitude direction and latitude direction,
respectively. Error bars, reported at several representative times in panels
(b) & (c), are estimated by using the Monte-Carlo method.

In order to investigate the relationships between the magnetic flux and pole
separation or tilt angle during the emergence of the active region, we plot
the total magnetic flux as a function of pole separation and tilt angle in
Fig. 5. Figure 5 (a) shows the total magnetic flux ($\varphi$) as a function
of the pole separation ($d$). A positive correlation between these two
parameters is found. By using a linear fitting for the logarithm of these
variables, a correlation of $log_{10}(\varphi)=2.72*log_{10}(d)+16.94$ plotted
by blue dotted line can be derived. This correlation is equivalent to $d$
$\sim$ $\varphi^{0.37}$. In detail, when the pole separation is smaller than
$\sim$ 36.5 Mm ($log_{10}d$ $<$ $10^{1.56}$ Mm), we get that
$log_{10}(\varphi)=1.17*log_{10}(d)+19.11$ shown by the yellow line and is
equivalent to $d$ $\sim$ $\varphi^{0.85}$. When the pole separation is between
36.5 Mm and 58.4 Mm ($10^{1.56}<log_{10}d$ $\leqslant 10^{1.77}$ Mm), the
logarithm of magnetic flux equals to 3.00$*log_{10}(d)+16.53$ shown by the red
line, which is equivalent to $d$ $\sim$ $\varphi^{0.33}$. When the pole
separation is larger than 58.4 Mm ($log_{10}d$ $>$ $10^{1.77}$ Mm), the
logarithm of magnetic flux equals to 0.55$*log_{10}(d)+20.85$ plotted by the
pink line, which is equivalent to $d$ $\sim$ $\varphi^{1.82}$. In other words,
during the emergence of active region, the magnetic flux has a relatively
small increasing rate when the pole separation is shorter than 36.5 Mm and
longer than 58.4 Mm. The magnetic flux experiences a significant increase at
the moderate pole separation of the range from 36.5 Mm to 58.4 Mm. This fact
manifests that the magnetic flux of naked AR main increases in the moderate
pole separation. Fig. 5 (b) exhibits the total magnetic flux as a function of
the tilt angle. We also use a linear function to fit the scatter plot. A
correlation of $\vartheta$ $=$ $-10.66*log_{10}(\varphi)+224.10$ can be
derived. This means that there is a negative correlation between the tilt
angle and magnetic flux during the dipolar AR emergence. In detail, the active
region with small magnetic flux ($<$ 1020.8 Mx) kept a high and almost
constant tilt angle of $\sim$ 23$\degr$-24$\degr$. The active region with
large magnetic flux ($>$ 1021.6 Mx) kept in a small and almost constant tilt
of $\sim$ 12$\degr$-13$\degr$. The tilt angle decreased gradually from about
23$\degr$ to 13$\degr$, when the total magnetic flux increased from 1020.8 to
1021.6 Mx. This also manifests that the newly emerging magnetic flux with a
high tilt angle emerges from subsurface at first. And then two polarities of
emerging magnetic flux separate mainly from each other in longitude direction,
which results in the decrease in tilt angle.

Figure 5: Correlation between the magnetic flux and pole separation or tilt
angle; (a) profile of total magnetic flux as a function of the pole
separation. The different color lines denote the fitting lines with different
slopes. The different color functions depict the results of different fitting
lines by using the linear fit; (b) profile of total magnetic flux as a
function of the tilt angle. The fitting line by linear function is indicated
by the blue dotted line.

###  3.3 Magnetic properties

Figure 6 shows the distributions of the force-free parameter $\alpha_{z}$,
vertical electric current ($J_{z}$), and proxy of heilicty flux density
($G_{\theta}$) in the active region. Panels (a1)-(a4) exhibit the
distributions of the force-free parameter $\alpha_{z}$ in the photosphere at
different moments. The force-free parameter shows a disordered pattern in each
polarity. It is interesting to find that the strong force-free parameter
mainly located in the periphery of main polarities. Panels (b1)-(b4) show the
distributions of the vertical electric current density ($J_{z}$) in the
photosphere at the corresponding moments. We also find that the vertical
electric current shows a disordered in each polarity and the strong vertical
electric current also located in the periphery of main polarities. Panels
(c1)-(c4) display the distributions of magnetic helicity ejection flux density
($G_{\theta}$) in the photosphere at different moments. The density of the
helicity ejection flux shows a relative uniform pattern unlike the force-free
parameter and vertical electric current (positive and negative helicity
ejection fluxes are dominant in each polarity). At around 04:54 UT and 14:42
UT on August 24, the helicity ejection flux was small and dominated by
positive value (see panels (c1)-(c2)). At around 00:30 UT on August 25, it was
also dominated by positive helicity ejection flux for the whole active region.
However, the leading sunspot (positive polarity) was dominated by negative
helicity ejection flux (see panel (c3)). At around 10:18 UT on August 25, the
negative helicity ejection flux was dominant in the active region, especially
in the leading sunspot (positive polarity).

Figure 6: Distributions of force-free parameter ($\alpha_{z}$), vertical
electric current density, proxy of heilicty flux density ($G_{\theta}$);
(a1)-(a4) the force-free parameter ($\alpha_{z}$); (b1)-(b4) the vertical
electric current density ($J_{z}$). The pink and black contours denote the
positive and negative magnetic field with the levels of 500 G, 1500 G,
respectively; (c1)-(c4) the proxy of helicity flux density (Gθ). It only plots
the pixels with the total magnetic field strength bigger than 300 G.

Figure 7 shows the evolution of different magnetic parameters in the whole
active region during the emergence of the active region. Panel (a) exhibits
the variation of the average flux-weighted force-free parameter
$\tilde{\alpha}$ calculated by the Eq.17. The average force-free parameter is
almost positive during the emergence of the active region, which does not
follow the hemisphere rule. The magnitude of the average force-free parameter
is the order of $\sim$ $10^{-8}$ m-1, which is consistent with results from
previous studies (Wang & Abramenko, 2000; Liu et al., 2014; Kutsenko et al.,
2019). It rises up to 6 $\times$ $10^{-8}$ m-1 at 10:00 UT on Aug 24. At the
end of the emergence, the average force-free parameter is only $\sim$ 0.5
$\times$ $10^{-8}$ m-1. In contrast to the result from Wang & Abramenko
(2000), it is variant instead of constant during the emergence of the active
region. Panel (b) exhibits the evolutions of the vertical electric current and
the current neutralization ratio ($\Re$). The red line indicates variation of
the positive electric current while the blue line indicates variation of the
negative electric current. The positive and negative electric currents are
almost equivalent during the entire emergence of the AR. This fact manifests
that no any net current is carried into the atmosphere during the emergence of
this isolated AR. It is reasonable to assume that the total current in the
isolated AR stays neutralized during its emergence; this is because the total
current $I_{total}$ over the whole photospheric AR is equal to
$\frac{c}{4\pi}\oint\bf{B_{t}}\cdot\it d\bf{l,}$ and $\bf{B_{t}}$ would be
balanced along the path outside the boundary of the isolated AR (Parker,
1996). At the early phase of emergence, they remained in low magnitude of less
than 1 $\times$ $10^{12}$ A. At the main phase of emergence, they experienced
a rapid increase at first and then decreased at the latter phase. During the
period from around 03:48 UT on Aug 24 to 03:24 UT on Aug 25, both positive and
negative electric currents increased mainly from about 0.5 $\times$ $10^{12}$
A to about 7.5 $\times$ $10^{12}$ A. After around 04:00 UT on Aug 25, they
experienced a decrease until the end time of investigation. The peak time of
electric current (about 03:24 UT on August 25) was earlier than that of the
magnetic flux (about 10:00 UT on August 25) (see Figs. 4 & 7). The black line
shows the variation of the current neutralization ratio ($\Re$). The $\Re$
values are in the range from about 0.8 to 1.1, which suggests that the AR kept
neutral current during its emergence. This result is consistent with the
previous observational studies, which also supports the premise that currents
become more neutralized during magnetic flux emergence in flare-quiet ARs (Liu
et al., 2017; Avallone & Sun, 2020).

Figure 7: Evolution of different magnetic parameters in the active region
during the period from 16:12 UT on August 23 to 12:00 UT on August 25, 2018:
(a) The flux-weighted force-free parameter ($\tilde{\alpha}$); (b) Vertical
electric currents and the current neutralization ratio. Red and blue lines
denote the positive and negative vertical current, respectively. The black
line plots the profile of current neutralization ratio; (c) The magnetic
helicity injection flux. Red and blue lines denote the emergence term and
shear term, respectively; (d) Accumulative helicity. Red and blue lines
indicate the accumulative helicity contributed from emergence term and shear
term. The total accumulative helicity is shown by the black line. The green
line denotes the flux of GOES soft X-ray 1-8 $\rm\AA$. The two vertical dashed
lines denote the times of 05:00 UT on August 24 and 09:00 UT on August 25.
Some B-class flares are marked by blue arrows. Error bars estimated by the
Monte-Carlo experiment are plotted at representative times.

With the methods described in Sect. 2.2, we also calculated the magnetic
helicity injection flux of the active region during the emergence of the
active region. According to Eq.19, we could obtain the magnetic helicity
injection flux from the emergence term and the shear term. Figure 7 (c)
exhibits the evolution of magnetic helicity injection flux. The magnetic
helicity flux contributed by the emergence term is plotted by the red line
while the magnetic helicity flux contributed by the shear term is plotted by
the blue line. There are several features in the magnetic helicity flux.
Firstly, both terms of the helicity injection flux kept a low value at the
early phase of the emergence. Secondly, the shear term changed its sign from
positive to negative at about 04:00 UT on Aug 25. Thirdly, the emergence term
stayed in positive during the whole period. On the other hand, we calculate
the accumulated helicity by the time integral of the helicity injection flux
with setting the accumulated helicity to be zero at the beginning of
investigation. Figure 7 (d) shows the time variation of accumulative helicity.
At the early phase, the accumulative helicity was very low. At the main phase
of the emergence, the total accumulative helicity also showed the same pattern
as the vertical current, which was increasing at first and then began to
decrease. Overall, this active region is mainly dominated by the positive
helicity. This feature does not obey the hemisphere rule (being positive in
the northern hemisphere and negative in the southern hemisphere at the 24
solar cycle). Furthermore, during the most of time, the total magnetic
helicity in the active region was contributed mainly by the shear term, which
is also consistent with previous findings (Liu & Schuck, 2012; Liu et al.,
2014; Wang et al., 2019).

The green line in panel (d) denotes the variation of soft X-ray 1-8 $\rm\AA$
flux during the emergence of the active region. We can find that several
B-class flares took place in this active region. These flares marked by blue
arrows in panel (d) mainly appeared at the latter phase of the emergence. As
is well known, both the electric current and magnetic helicity are the proxies
of non-potential magnetic field. In general, the flares would transport the
magnetic energy to other forms of energy (such as thermal energy and kinetic
energy of plasma) by magnetic reconnection. It implies that these flares may
be associated with the decrease of electric current and the magnetic helicity
at the latter phase.

###  3.4 Magnetic twist

Figure 8: Evolution of twist numbers ($\bar{n}$ and $T_{n\alpha}$) in the
entire active region during the emergence of the active region. The asterisk
denotes the profile of $\bar{n}$ while the solid line denotes the one of
$T_{n\alpha}$. The horizontal dotted line denotes zero of twist number. The
two vertical dashed lines denote the times of 05:00 UT on August 24 and 09:00
UT on August 25. The errors are transferred from the parameter’s errors
estimated by the Monte-Carlo experiment.

On one hand, the magnetic helicity inside a uniformly twisted flux tube with n
turns, can be estimated by $\rm H=n\varphi^{2}$ (Berger, & Field, 1984), where
$\varphi$ is axial magnetic flux. Therefore, if we assume that the magnetic
field of active region is a uniformly twisted flux tube, the twist number of
active region can be derived by the magnetic helicity and magnetic flux. The
black line of asteriskes in Figure 8 plots the twist number,
$\bar{n}\approx\rm H(t)/\bar{\varphi}^{2}$, where $\bar{\varphi}$ is the
average magnetic flux between two polarities
($\bar{\varphi}=(\varphi_{zp}+\varphi_{zn})/2$) and H(t) is total accumulated
helicity. This quantity ($\bar{n}$) indicates how much twist is in the the
magnetic configuration of the entire active region. At the early phase, the
twist number had changed its sign several times. This feature may be related
to the small and mixed helicity injection flux from the subsurface at the
early phase. At the main phase, the twist number increased to about 0.05 at
first and then it lowered slightly and remained at about 0.04. After that, it
experienced a sharp decrease until the end of our investigation time frame.
Except for the early phase, the twist number almost kept in positive during
the emergence of the active region. Due to the small magnetic helicity and
magnetic flux at the early phase, we only focus on the twist number in the
main phase of emergence. We were able to find that the twist number was on the
order of 10-1 during the emergence. The positive sign of the twist number is
consistent with the positive sign of force-free parameter, $\tilde{\alpha}$,
which does not follow the hemisphere rule either. On the other hand, we assume
that the emerged magnetic field is a linear force-free field. The force-free
parameter is set to be $\tilde{\alpha}$. Therefore, the twist number (Tnα) of
magnetic field could be estimated by the function of
T${}_{n\alpha}=\frac{1}{4\pi}\int\tilde{\alpha}dL$, in which the L is the
length of the magnetic field and simply assume to be $\frac{\pi d}{2}$. Based
on the above equation and reasonable assumptions, the Tnα could be calculated
by using the evolutions of $\tilde{\alpha}$ and pole separation (d). The
profile of Tnα shows by the solid blue line as the function of time in Fig. 8.
Interestingly, Tnα was also on the order of 10-1 turns, which is consistent
with the twist number ($\bar{n}$) derived by the helicity calculation. In
detail, the Tnα is slightly bigger than $\bar{n}$ at the early stage of main
emergence. At the end of emergence, the Tnα is also about 0.04 turns, which is
consistent with $\bar{n}$. This might be related to the continuous increasing
twist in the emerged magnetic field at the early stage of main emergence.
Nevertheless, both Tnα and $\bar{n}$ show almost the same value at the end of
emergence. These results manifest that the twist of this naked anti-Hale
active region is very low and in the order of 10-1 turns. The errors of
$\bar{n}$ mainly come from the uncertainty of accumulated helicity related to
photospheric velocity and magnetic field, while the errors of Tnα mainly come
from the uncertainty of force-free parameters related to magnetic field. We
use a Monte Carlo experiment method to estimate the uncertainty of these
parameters. Therefore, the errors of Tnα and $\bar{n}$ can be estimated by the
propagation from these parameters’ errors, which are shown by using error bars
in Fig. 8. On the other hand, the lack of data during periods from 06:24 to
08:48 UT on August 24 and 25 might lead to an underestimation of the
accumulated helicity, which results in the underestimation of $\bar{n}$.
Furthermore, we use a photospheric averaged flux-weighted force-free parameter
$\tilde{\alpha}$ to represent the force-free parameter of magnetic field,
which might bring on some uncertainties in estimating Tnα. Therefore, we take
a conservative estimate that the twist number of this emerging active region
is on the order of 10-1 turns. Nevertheless, we use two different methods to
estimate the twist number of the active region and obtain the consistent
result, which demonstrates that the result is reliable.

##  4 Summary and discussion

This paper presents a detailed study on the evolution and magnetic properties
of an anti-Hale emerging active region NOAA 12720. Our main results are as
follows.

(1) During the emergence of the anti-Hale active region, the leading sunspot
with positive magnetic field drifted faster than the following one with
negative magnetic field. Most of the newly emerging magnetic fluxes initially
occurred between the two main polarities.

(2) A positive correlation between magnetic flux and pole separation is found
during the emergence. The increase of the pole separation is mainly
contributed from the enhancement in the longitudinal direction. The tilt angle
of the active region maintains a high value (about 23$\degr$) at the early
phase and a low value (about 12$\degr$) at the latter phase. In contrast to
the pole separation, the tilt angle shows a negative correlation with total
magnetic flux. On the other hand, a power-law relationship between pole
separation and total flux is found during the emergence of the active region.

(3) The force-free parameter ($\alpha_{z}$) and vertical electric current
density ($J_{z}$) exhibit a disorderly distribution in each polarity. The
average flux-weighted force-free parameter and accumulative helicity in the
entire active region was positive, a result that does not obey the hemispheric
rule. The electric current and magnetic helicity in the active region are
mainly built up at the main phase of the emergence. In addition, the magnetic
helicity injection in the active region is mainly contributed by the shear
term.

(4) Through the calculation of the helicity and the force-free parameter to
estimate the twist number, we infer that the twist number of the whole active
region is on the order of 10-1 turns.

According to the statistical result of 4385 sunspot groups determined by Li
(2018), only 8.1 $\pm$ 0.4 % of these active regions are the anti-Hale sunspot
groups. Therefore, anti-Hale active region is relatively rare. This anti-Hale
active region NOAA 12720 emerged in a clean environment, which provides a
unique opportunity for us to investigate the nature of emerging magnetic flux
and anti-Hale active region. We find that the active region emerged with a
high tilt angle at the early phase. The tilt angle of the active region
decreased gradually when the active region continued to emerge. The tilt angle
of the active region came up to a low value at the mid-phase of emergence and
kept invariant until the end of the emergence. If we hypothesize that the
anti-Hale active action before emergence is a flux tube embedded in the
subsurface, the evoution of the tilt angle in this anti-Hale active region is
consistent with the emergence of the flux tube with low twist. As a flux tube
emerges with low twist from the convective zone into the solar atmosphere, the
tilt angle of two main polarities exhibits high value at the initial stage and
then decreases to a lower value. As the bottom of the lifting flux tube
escapes, the tilt angle remains invariant at a low value (Knizhnik et al.,
2018). A negative correlation between tilt angle and magnetic flux could be
found during this bipolar AR emergence. However, Kosovichev & Stenflo (2008)
studied 715 bipolar ARs and deduced that the tilt angle does not dependent on
the amount of magnetic flux. These facts manifest the finding that the
evolution of tilt angle depends on the magnetic flux during the emergence of
one active region but the tilt angle does not depend on the amount of magnetic
flux in various active regions.

During the emergence of this naked active region, the pole separation has a
power-law dependence on the magnetic flux. The power-law index in our study is
0.37. Interestingly, Kutsenko et al. (2019) also found a similar relationship
between pole separation and maximum of magnetic flux using 423 ARs. Their
power-law index is about 0.36 which is a similar value to ours. It should be
noted that our study just focuses on the evolution of one active region, while
the statistical study of Kutsenko et al. (2019) includes the properties of
many different emerging active regions. Therefore, the dependence of
polarities separation on magnetic flux is a fundamental relationship – either
in the evolution of one emerging active region or between different emerging
active regions. On the other hand, there are three states on the relation
between pole separation and magnetic flux (see Fig. 5(a)). When the pole
separation is shorter than 36.5 Mm and longer than 58.4 Mm, the magnetic flux
increases with a relatively small increasing rate. When the pole separation is
moderate in the range from 36.5 Mm to 58.4 Mm, the magnetic flux increases
with a large increasing rate. The similar relationship between pole separation
and magnetic flux is also seen in the emergence phase of NOAA active region
8167 (see Fig. 1 (f) in Kosovichev & Stenflo (2008)). Combining (a) & (b) in
Fig. 3, we see that these three states are mainly associated with the
evolution of magnetic field while the pole separation increase with an almost
constant rate. The first two states may be related to a two-step emergence
pattern in the active region (Fu & Welsch, 2016), with a relatively gradual
emergence followed by a relatively sudden change to more rapid emergence. This
feature has been seen in simulations of “two-step” model (Matsumoto et al.,
1993; Magara, 2001; Toriumi & Yokoyama, 2010). In models of two-step
emergence, the rate of emergence in the emerging flux tube becomes rapidly
increasing from a relatively small as the emerging flux tube reaches around
the photosphere or chromosphere. In addition, Chintzoglou & Zhang (2013)
suggested that the emerging magnetic structure with a weaker-field structure
preceding a stronger-field structure is responsible for the two-step emergence
of AR 11158. The third presumes that the magnetic flux has a low increasing
rate with pole separation at the later stage, which might be related to
emergence of the serpentine magnetic field at the latter stage of emergence.
These serpentine magnetic fields would release mass through magnetic
reconnection (Cheung et al., 2010; Centeno, 2012), resulting in the
cancellation of magnetic flux on the photosphere. The cancellation of magnetic
flux would slow down the growth of magnetic flux in the whole active region.
Otherwise, these cancelled magnetic fluxes were often seen as the signatures
of moving dipolar features (see Fig. 2 (a3)) on the photosphere before
cancellation.

The injection of magnetic helicity in this anti-Hale active region was
contributed mainly from the shear term, which is consistent with previous
results (Liu & Schuck, 2012; Liu et al., 2014). According to the 3D numerical
simulation of a twisted flux tube, Magara, & Longcope (2003) found that
relative magnetic helicity is dominated by the emergence term at the early
stage of the emergence. However, we does not find this feature in our study.
This may be related to the low twist in this emerging active region. We obtain
that the twist number of the emerging active region is in the order of 10-1
turns. This means that the emerging active region carried the low twist into
the atmosphere. The low twist in the magnetic flux leads to a little injection
of magnetic helicity contributed by the emergence term at the early stage. In
other words, the low magnetic helicity injection flux contributed by the
emergence term at the early stage manifests that the twist of the emerging
magnetic flux is low.

According to the definition of the twist number in the axial symmetric
cylindrical flux tube, we can get $B_{\theta}/B_{l}$ = $n*\frac{2\pi r}{L}$,
where $B_{\theta}$ and $B_{l}$ are, respectively, the magnetic field strength
of azimuthal and longitudinal components, while r is the radius of flux tube
and L is the length of the flux tube. According to the definition of the twist
parameter ($q=\frac{B_{\theta}}{rB_{l}}$), we can obtain that q$H_{0}$ =
$n*\frac{2\pi H_{0}}{L}$, where $H_{0}$ is the pressure scale height
($H_{0}$=170 km) (Toriumi et al., 2011). With the assumption that the whole
emerging active region is a flux tube, the length L equals $\frac{\pi d}{2}$.
At 04:00 UT on August 25, the pole separation (d) of the active region equals
to about 60 Mm. Combinated with the above values of the parameters, the twist
parameter (q$H_{0}$) in this active region is estimated to be about 1 $\times$
10-3. Comparing with the critical value (q$H_{0}$ = 0.1) for the emergence of
flux tube derived by Toriumi et al. (2011), the twist of this emerging active
region is relatively weak. Therefore, this result is opposed to the conclusion
in which the flux tube with weak twist would fails to emerge into the upper
atmosphere (Murray et al., 2006; Toriumi et al., 2011).

While the magnetic twist plays an important role in the activity and evolution
of the active region, the studies of twist in the active region are needed to
reveal the intrinsic properties of magnetic field (Nandy, 2006; Luoni et al.,
2011; Poisson et al., 2015). Our result indicates that the low twist in the
sub-photospheric emerging part of a flux rope is coherent with previous
studies (Liu & Zhang, 2006; Poisson et al., 2015). Yang et al. (2009) studied
58 emerging Hale active regions and obtained that the average twist number in
these Hale active region is about 0.039 turns, a value that is comparable to
our result. This implies that either Hale or anti-Hale active regions operate
under the same rising mechanism in the solar interior. Otherwise, this can
also demonstrate that this anti-Hale active region is likely to be generated
by rising of toroidal flux tube with opposite orientation in the interior
(Stenflo & Kosovichev, 2012), instead of causing by the kink instability
mechanism of highly twisted flux tube (Nandy, 2006; Knizhnik et al., 2018).
However, the positive sign of twist including positive twist number and
average force-free parameter ($\tilde{\alpha}$) is opposite the sign of
structural chirality of solar active region in northern hemisphere (Seehafer,
1990). The active region violating Hale rule might be responsible for this
phenomenon.

Here, we consider the role played by the twist in the emergence of the
magnetic flux tube. In our observational study, we use two different methods
which are based on magnetic helicity calculation and force-free parameter to
estimate the twist in this anti-Hale active region. We find that the emerged
magnetic field covering the whole active region exhibits a lowly twisted
structure. The twist number is in the order of 10-1 turns. More recently,
Knizhnik et al. (2018) simulated the emergence of different twisted flux tubes
from the convection zone into the corona. They found that the behaviours of
low twisted flux tube emergence matched the observed properties of the single
active region (such as $\beta$ style), while the ones of high twisted flux
tube matched the complex active region ($\delta$ style). Moreover, some
authors found that an untwisted flux tube could also be able to rise through
the upper convection zone and emerge into the photosphere to form spots (Stein
et al., 2011; Rempel & Cheung, 2014; Knizhnik et al., 2021). Knizhnik et al.
(2021) explained that the rise of untwisted toroidal flux rope is triggered by
the undular instability instead of buoyant instability. On the other hand,
most studies of numerical simulation often set a mature magnetic flux rope
embedding in the convection zone bottom as the initial condition. The reality
is that the origin of the magnetic field of the active region is still unclear
at present. One possibility is that the magnetic field of the flux rope or
flux tube is forming while they are rising under the subsurface. Thus, a
greater number of episodes ought to be considered in the numerical simulation.
After all, we conclude that the simple active region does not need too much
initial twist in the intrinsic magnetic field for its emergence.

###### Acknowledgements.

We appreciate the referee’s careful reading of the manuscript and many
constructive comments, which helped greatly in improving the paper. SDO is a
mission of NASA’s Living With a Star Program. The authors are indebted to the
SDO, NVST teams for providing the data. This work is supported by the National
Key R&D Program of China (2019YFA0405000), the National Science Foundation of
China (NSFC) under grant numbers 12003064, 11873087, 11803085, Yunnan Science
Foundation of China (2019FD085), the CAS “Light of West China” Program under
number Y9XB018001, the Open Research Program of the Key Laboratory of Solar
Activity of Chinese Academy of Sciences (grant No.KLSA202014), the Yunnan
Talent Science Foundation of China (2018FA001), the Yunnan Science Foundation
for Distinguished Young Scholars under No. 202001AV070004, and the Key
Research and Development Project of Yunnan Province under number
202003AD150019..

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