# Impact of Anomalous Active Regions on the Large-scale Magnetic Field of the
Sun

Shaonwita Pal Center of Excellence in Space Sciences India,
Indian Institute of Science Education and Research Kolkata, Mohanpur 741246,
West Bengal, India  Prantika Bhowmik Department of Mathematical Sciences,
Durham University, Durham, DH1 3LE, UK  Sushant S. Mahajan W.W. Hansen
Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, USA
Dibyendu Nandy<EMAIL_ADDRESS>Center of Excellence in Space Sciences
India,
Indian Institute of Science Education and Research Kolkata, Mohanpur 741246,
West Bengal, India  Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur 741246,
West Bengal, India

###### Abstract

One of the major sources of perturbation in the solar cycle amplitude is
believed to be the emergence of anomalous active regions which do not obey
Hale’s polarity law and Joy’s law of tilt angles. Anomalous regions containing
high magnetic flux that disproportionately impact the polar field are
sometimes referred to as “rogue regions”. In this study – utilizing a surface
flux transport model – we analyze the large-scale dipole moment build-up due
to the emergence of anomalous active regions on the solar surface. Although
these active regions comprise a small fraction of the total sunspot number,
they can substantially influence the magnetic dipole moment build-up and
subsequent solar cycle amplitude. Our numerical simulations demonstrate that
the impact of “Anti-Joy” regions on the solar cycle is similar to those of
“Anti-Hale” regions. We also find that the emergence time, emergence latitude,
relative number and flux distribution of anomalous regions influence the
large-scale magnetic field dynamics in diverse ways. We establish that the
results of our numerical study are consistent with the algebraic (analytic)
approach to explaining the Sun’s dipole moment evolution. Our results are
relevant for understanding how anomalous active regions modulate the Sun’s
large-scale dipole moment build-up and its reversal timing within the
framework of the Babcock-Leighton dynamo mechanism – now believed to be the
primary source of solar cycle variations.

##  1 Introduction

Active regions emerging on the solar surface have long been considered the
primary element through which the solar cycle manifests itself (Schwabe,
1844). Short and long-term evolution of the magnetic field associated with
active regions drives solar magnetic events such as flares, coronal mass
ejections, energetic particle releases, etc., besides controlling the Sun’s
total electromagnetic output. These activities subsequently affect the Earth
and its vicinity, specifically satellite operations, telecommunications, and
contribute to various aspects of space weather (Nandy, 2004; Nandy & Martens,
2007; Schrijver et al., 2015). Since surface magnetic field distribution and
its evolution play crucial roles in governing the short and long-term changes
in space weather and space climate, respectively, it is vital to study the
large-scale surface magnetic field during a solar cycle as well as its cycle
to cycle irregularities (Bhowmik & Nandy, 2018; Petrovay et al., 2020; Nandy,
2021; Bhowmik et al., 2023). The magnetic cycle of the Sun can be explained
using the Babcock-Leighton (B-L) Solar Dynamo theory (Leighton, 1964;
Piddington, 1972; Wang et al., 1991; Hazra & Nandy, 2016; Charbonneau, 2020;
Fan, 2021; Hazra et al., 2023). At the solar activity minimum, the global
magnetic field is primarily dominated by the poloidal field component, which
is predominantly dipolar. The solar differential rotation stretches this
poloidal field longitudinally to generate the toroidal field component in the
tachocline region. Amplified toroidal flux ropes eventually encounter magnetic
buoyancy in the convection zone and emerge primarily as Bipolar Magnetic
Regions (BMRs) (Nandy, 2002; Gilman, 2018; Fan, 2021). Near the Sun’s equator,
the leading sunspot polarities of two opposite hemispheres annihilate each
other, and the remaining trailing spot (mainly of uni-polar magnetic fields)
drifts towards the respective poles via large-scale meridional flow and
diffusion. During the first half of the solar cycle, many such trailing
polarities reaching the polar region eventually cancel the existing large-
scale polar field and create a new one with the opposite sign. This reversal
of the global dipolar field happens during the solar cycle maximum, and the
new polar field keeps growing till the end of the cycle. This whole process of
redistribution of active region-associated magnetic flux on the solar surface
is known as the B-L mechanism (Babcock, 1961; Leighton, 1964; Mackay & Yeates,
2012; Jiang et al., 2014). One of the existing hypotheses explaining the
dynamo cycle involves meridional circulation advecting the poloidal field down
to the tachocline region from the solar photosphere followed by the generation
and enhancement of toroidal field via Parker’s $\Omega$-effect (Parker, 1955)
\- thus completing the full solar cycle. Therefore, the B-L mechanism is the
primary means for evolving the polar field and also a crucial component
controlling variability in the amplitude of sunspot cycles over decadal to
century-scales (Nandy & Choudhuri, 2001; Dasi-Espuig et al., 2010; Muñoz-
Jaramillo et al., 2010; Bhowmik & Nandy, 2018; Kumar, Rohit et al., 2019; Dash
et al., 2020; Nandy et al., 2023).

Active regions associated with a solar cycle have their own characteristics
and largely follow specific patterns. They emerge closer to the equator as the
cycle progresses. This latitudinal variation of active regions in both
hemispheres primarily follows Sp$\ddot{o}$rer’s Law (Carrington, 1858). Hale’s
polarity law (Hale et al., 1919; Hale & Nicholson, 1925) determines the sign
of magnetic polarities (positive/negative) of active regions also known as
Bipolar Magnetic Regions (BMRs) in each hemisphere. According to Hale’s law,
the relative order of magnetic polarities of the leading and trailing spots
remains the same in a particular hemisphere, but it’s the opposite between the
northern and southern hemispheres. To elaborate, if, during a particular
cycle, the leading spots of BMRs emerging in the northern hemisphere have
positive polarity, the trailing spots will be of negative polarity. The
pattern would be opposite for the southern hemispheric BMRs with negative
leading spots and positive trailing spots. This polarity order also reverses
every solar cycle. Observations suggest that for most sunspot groups, their
leading and following spots are seated close to the equator and the pole,
respectively. Thus, the magnetic axis joining the centers of the two spots
(leader and follower) has a weak statistical tendency to have a slight
positive (negative) tilt angle with the axis parallel to the solar equator in
the northern (southern) hemisphere. The amplitude of tilt angles generally
increases with increasing latitude. This relation between active regions’
emergence latitudes and tilts is well known as Joy’s tilt Law (Hale et al.,
1919). Coriolis force, responsible for twisting the toroidal flux tube during
the time of emergence is believed to cause this tilt angle (Choudhuri &
Gilman, 1987; Fisher et al., 1995; Kleeorin et al., 2020).

Sp$\ddot{o}$rer’s law, Joy’s tilt law, and Hale’s polarity law - these three
well-observed laws that describe the location and orientation of sunspot pairs
on the solar surface, play a key role in understanding the solar magnetic
cycle and its variations through B-L mechanism. The emerging sunspots which do
not obey these laws can significantly impact the long-term behaviour of the
large-scale solar magnetic field. We have termed these spots ‘anomalous
regions’. Rogue region is also one category of anomalous regions whose flux
and tilt angle is very high (Nagy et al., 2017, 2019). In short, ‘anomalous’
classification introduced in this manuscript corresponds to all categories of
Anti-hale and Anti-joy regions (or combinations of both) irrespective of the
amount of flux and degree of tilt angle. We investigate the strength of impact
on the solar cycle for these different categories of anomalous regions due to
their diversity of tilt characteristics, flux content and spatio-temporal
distribution.

There are several theories on the generation of such anomalous regions. Anti-
Joy regions (with tilt angles opposite from what should be according to their
latitudinal positions) emerge due to two reasons primarily: (1) randomness in
Coriolis force (Schmidt, 1968), (2) convective buffeting of flux tubes (Weber
et al., 2013). For the Anti-Hale regions (regions with opposite polarity
orientation), studies suggest that they can be formed due to (1) kink
instability in highly twisted magnetic flux tube (Nandy, 2006; Knizhnik et
al., 2018), (2) oppositely oriented toroidal flux-tubes within the convection
zone (Stenflo & Kosovichev, 2012), (3) transport of Hale regions from opposite
hemisphere (McClintock et al., 2014), and (4) small-scale dynamo action
creating sunspots at the end of the cycle (Sokoloff et al., 2015).

Past studies suggest that tilt quenching and variation in the meridional
circulation are vital factors causing cycle irregularities (Wang et al., 2002;
Dasi-Espuig et al., 2010; Upton & Hathaway, 2014). Additionally, the
scattering in active regions’ tilt angles and Anti-Joy regions produce
significant changes in large-scale polar field build-up and open magnetic flux
dynamics during a solar cycle (Cameron et al., 2010; Jiang et al., 2014).
Besides, the appearance of a single large Anti-Joy and Anti-Hale region can
have a substantial effect on cycle amplitude, which was argued by past studies
(Yeates et al., 2015; Jiang et al., 2015) as a probable cause of weaker peaks
during solar cycles 23 and 24. A large rogue region emerging at different
cycle phases and latitudes with varying fluxes also affects the subsequent
sunspot cycle peak activity (Nagy et al., 2017, 2019). Previous observation-
based studies claimed that the Anti-Hale or Anti-Joy spots appearing on the
solar surface constitute a certain percentage of the total number of sunspots
in a cycle, varying from 3$\%$ to 10$\%$ (McClintock et al., 2014; Li, 2018;
Zhukova et al., 2020; Muñoz-Jaramillo et al., 2021). Thus, it is essential to
explore the effect of a group of these anomalous regions on the large-scale
surface magnetic field distribution and the overall polar magnetic field
variability - which we investigate in this present work. Our detailed
quantitative analyses are based on multiple numerical simulations using a
data-driven surface flux transport model (Bhowmik & Nandy, 2018). In these
simulations, we consider three classes of anomalous regions based on the
orientations and polarities of the two spots within the active regions and
study the consequences individually. We also investigate how their effects
vary depending on the following factors: their latitudinal position, emergence
timing relative to the cycle phase, associated magnetic flux and abundance
compared to remaining standard Hale-Joy regions.

The presented work is assembled as follows: in Section 2, we describe the
computational model used in our study. Section 3 summarizes the results we
obtain from our simulations. In section 4, we validate the results using
algebraic approximation and finally, we present our conclusions with relevant
discussion in Section 5.

##  2 Simulation Set-up

###  2.1 Surface Flux Transport Model

Surface Flux Transport (SFT) models are utilized to demonstrate the time
evolution of the large-scale photospheric magnetic field distribution
according to the Babcock-Leighton mechanism. This model includes the effects
of supergranular turbulent diffusion ($\eta$) and advective transport through
large-scale plasma flows like differential rotation [$\omega$($\theta$)] &
meridional circulation [$\mathrm{v}$($\theta$)] and particularly solves the
radial part of the magnetic induction equation which is given by,

$\frac{\partial B_{r}}{\partial t}=-\omega(\theta)\frac{\partial
B_{r}}{\partial\phi}-\frac{1}{\mathrm{R_{\odot}}\,\sin\theta}\frac{\partial}{\partial\theta}\Big{(}v(\theta)B_{r}\sin\theta\Big{)}+\frac{\eta}{\mathrm{R_{\odot}}^{2}}\Bigg{[}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial
B_{r}}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}B_{r}}{\partial\phi^{2}}\Bigg{]}+S(\theta,\phi,t).$
(1)

Where Br denotes the radial component of the magnetic field, $\theta$ and
$\phi$ are colatitude and longitude, respectively. We choose a constant
turbulent magnetic diffusivity of $\eta$ = 250 km2s-1 for our model (the same
value was used in calibrated century-scale observational data-driven SFT
simulations by Bhowmik & Nandy, 2018). The source term, S($\theta$,$\phi$,t),
mimics the emergence of new active regions on the solar surface. $R_{\odot}$
is the solar radius. Equation (1) is solved in a domain, $\theta:0$ to $\pi$
and $\phi:0$ to $2\pi$ radians, i.e., covering the whole photosphere. The same
SFT code based on spherical harmonics expansions (with degrees, $l$: 1 to 63)
has been used in many past studies (Nandy et al., 2018; Bhowmik & Nandy, 2018;
Bhowmik, 2019).

To include the observed large-scale axisymmetric plasma flow in the toroidal
direction, known as differential rotation, $\omega$($\theta$), we rely on the
following mathematical expression, provided by Snodgrass (1983),

$\omega(\theta)=13.38-2.30\>cos^{2}\theta-1.62\>cos^{4}\theta\,\,\>\textrm{degrees
${\mathrm{day}}^{-1}$.}$ (2)

This profile is in accordance with the helioseismic observation (Durney, 1974;
Schou et al., 1998). In addition, there is another observed weak large-scale
plasma flow named meridional circulation (Hathaway, 1993; Komm et al., 1993;
Mahajan et al., 2021; Hanasoge, 2022), which helps in transporting magnetized
plasma from the equatorial region to the poles in respective hemispheres. In
our simulation set-up, this flow is bounded within $0^{\circ}$ to $\pm$75∘ in
each hemisphere, attains the peak amplitude in the mid-latitudes and becomes
zero at the equator. We choose a time-independent meridional velocity profile
similar to van Ballegooijen et al. (1998),

$\displaystyle v(\lambda)$ $\displaystyle=-v_{0}\hskip 2.84544pt$
$\displaystyle\hskip 14.22636pt\text{if}|\lambda|<\lambda$ $\displaystyle
v(\lambda)$ $\displaystyle=0$ $\displaystyle\hskip
14.22636pt\text{if}|\lambda|>\lambda_{0}$ (4)

Where, $\lambda$ is latitude and $\lambda_{0}$ is the limiting latitude
($\lambda_{0}$ = 75∘) beyond which meridional flow vanishes. The amplitude of
the meridional flow (${v_{0}}$) is set at 15 m/s. We initialize our simulation
with a dipolar configuration of surface magnetic field concentrated primarily
near the 20∘ broad polar cap region ($\pm$70∘ to $\pm$90∘) in both
hemispheres. The unsigned strength of the initial polar flux in one hemisphere
is nearly about 2 x $10^{22}$ Mx which is consistent with observations.

###  2.2 Synthetic Sunspot Cycle Profile

Modeling the source term [$S(\theta,\phi,t)$] in equation (1) in SFT
simulations requires information on emergence latitude, longitude, tilt angle
and area of each region which can be constrained by observed properties of
active regions. For our study, we use the synthetic solar cycle data which is
prepared utilizing mathematical relations based on general properties of
observed solar cycles and their associated active regions. First, we consider
a time-dependent Gaussian curve to reproduce the sunspot number time series
following Hathaway (2010). The time-latitude distribution band of active
regions is inspired by Jiang et al. (2011). We use the following square root
relation, $\alpha$ = C $\sqrt{|\lambda|}$ to incorporate Joy’s tilt law in our
synthetic data set. Here, $\alpha$ is the tilt angle and $\lambda$ is the
latitudinal position of the centroid of the BMR. The tilt angle of BMRs can be
constrained with the variation of cycle strength and the localized inflows,
incorporated in the constant factor “C” (Jiang et al., 2014). We consider a
uniform distribution of active regions over the longitudinal domain of the
solar photosphere.

Since large sunspots rarely emerge during the initial and declining phase of a
cycle, it is preferable to use a polynomial distribution fitted to sunspot
observations to model their area distribution (Jiang et al., 2011; Muñoz-
Jaramillo et al., 2015). In order to prescribe the associated magnetic flux,
we follow the relationship: $\Phi$(A) = 7.0 $\times$ 1019 A (Maxwell), where A
represents the area of the whole active region in micro-hemispheres (Dikpati
et al., 2006). We assume that all active regions are typical $\beta$-spots
having equal flux in two polarities. We model each BMR with a Gaussian profile
prescribed by van Ballegooijen et al. (1998).

###  2.3 Preparation of Anomalous Active Regions

As discussed earlier, a few emerging active regions may have different
orientations and polarities in contrast to the standard Joy’s tilt law and
Hale’s polarity law. We categorize four possible combinations in which three
are considered (see the last three cases in Figure 1a) as candidates of
anomalous regions.

  * •

Configuration 1 (H-J): the standard Hale-Joy BMRs (non-anomalous) with
negative leading polarity and positive following polarity in the northern
hemisphere and the opposite pattern in the southern hemisphere (see first case
in Figure 1a). These strictly follow Joy’s tilt law and are the “ideal” spots
(denoted by H-J hereafter).

  * •

Configuration 2 (AH-J): BMR that does not follow Hale’s polarity law but
follow Joy’s tilt law. These configurations are also known as Anti Hale-Joy
BMRs (AH-J hereafter).

  * •

Configuration 3 (H-AJ): BMR that follows Hale’s polarity law but have an
opposite tilt, unlike H-J, thus, violating Joy’s tilt law. As an example,
consider the expected positive tilt angle of a BMR in the northern hemisphere
to be negative instead. Such a BMR is categorized as a Hale-Anti Joy BMR (H-AJ
hereafter).

  * •

Configuration 4 (AH-AJ): this is the rarest one where the BMR neither follows
Hale’s polarity law nor Joy’s tilt law. It is then an Anti Hale-Anti Joy BMR
(AH-AJ hereafter).

The middle and last columns of Figure 1 show observational evidence of the
above configurations of active regions appearing during Solar Cycle 24.

In most of our simulations, we consider 5$\%$ anomalous regions of any kind
(AH-J, H-AJ or AH-AJ) comprising 5$\%$ of the total unsigned flux associated
with the active regions of the whole cycle. We place anomalous regions over
the time-latitude domain using a random number generator (see the 1st row in
Figure 2; the black dots represent anomalous regions). In addition, we also
prepare six separate cases by varying emergence latitudes and phases
associated with AH-J BMRs. The butterfly diagrams corresponding to these cases
are shown in the middle and last panel of Figure 2. With each of these
different synthetic sunspot series, we perform disparate SFT simulations as
described in the following section 3.

##  3 Results

In order to understand the effect of anomalous active regions on the surface
magnetic field we primarily focus on large-scale magnetic field distributions.
Especially, it is well-established that the polar field and the axial dipole
moment at the end of the solar cycle, which closely reflect the large-scale
field distribution, strongly correlate with the succeeding cycle amplitude
(Yeates et al., 2008; Muñoz-Jaramillo et al., 2013; Bhowmik & Nandy, 2018). We
calculate the dipole moment in the following way,

$\mathrm{DM}(t)=\frac{3}{4\pi
R_{\odot}^{2}}\>\int_{\phi=0}^{2\pi}\>\int_{\lambda}\>\>B_{r}(\lambda,\phi,t)\>\sin\lambda\>\cos\lambda\>d\lambda\>d\phi,$
(6)

where $\lambda$ and $\phi$ represent latitude and longitude, respectively.
$R_{\odot}$ is the solar radius. DM(t) denotes the global axial dipole moment.
The dipole moment calculation involves the whole latitudinal and longitudinal
domain i.e. the entire photospheric radial magnetic field.

###  3.1 The Effect of Anomalous Regions on Axial Dipole Moment

We initially perform four SFT simulations (duration: 22 years each). One
simulation with all H-J spots and the other three simulations are with 5 $\%$
anomalous regions in configurations AH-J, H-AJ, and AH-AJ respectively, while
the remaining 95 $\%$ are H-J regions. It is noted that the spatio-temporal
distribution of anomalous regions is kept the same as sunspots in these cases.
Figure 3 depicts the evolution of the dipole moments associated with each of
these four simulations, where the anomalous regions were introduced beyond
year 11, i.e. in the second cycle. We notice that there is no significant
difference in the dipole moment variation for H-J and AH-AJ BMR distribution
while AH-J and H-AJ both behave similar to each other, yet significantly
different from H-J and AH-AJ. Quantitative details of the dipole moment
amplitude for these four configurations are provided in table 1 (see Sl No. 1
to 4).

In order to understand what leads to these similar outcomes from apparently
different BMR configurations, we investigate the temporal evolution of a
single BMR each having $\sim$ ${60}^{\circ}$ tilt angle corresponding to the
four configurations individually (see Figure 4).

Due to differential rotation, the large-scale surface plasma flow in the
azimuthal direction is faster near the equator than at higher latitudes.
Consequently, the following spot of an H-AJ region nearer to the equator will
advect more towards the right than the leading spot and eventually will attain
a configuration similar to that of AH-J. After a short period of evolution
($\sim$ 50 days in our simulations), the only difference remaining between an
H-AJ and an AH-J region is that H-AJ region has a higher tilt angle and lower
separation between its two polarities than AH-J (see the middle two rows in
Figure 4). The differential rotation has a similar effect on an AH-AJ region,
as it evolves to converge toward the standard H-J configuration with a lower
latitudinal separation and a higher tilt angle (see the top and bottom rows in
Figure 4). Now, the timescale of convergence between the two sets of
configurations (H-J & AH-AJ, AH-J & H-AJ) depends on the initial tilt and
emergence latitude, which primarily decide the angular separation between the
two spots of the BMR (given that their sizes are similar). In the case of a
typical observed active region, its tilt and, thus, the latitudinal separation
are much smaller. This makes the timescale of convergence for the two sets of
configurations longer. However, the time scale of empirical sunspot evolution
is short compared to long-term behaviour and can be considered transient to
the entire solar cycle progression. This explains the similar temporal
evolution and the final amplitude of the dipole moments at the end of the
22-year-long SFT simulations (H-J – AH-AJ and AH-J – AH-AJ), even with
apparently different classes of anomalous regions. Increasing the percentage
of anomalous regions to as high as 25 $\%$ does not alter this generic
behavior, indicating that the effect of the differential rotation on active
regions is quite robust and swift.

###  3.2 Effect on the Succeeding Cycle Amplitude and Time Period

The previous section demonstrates how the AH-J and H-AJ BMRs effectively cause
the same variation in the large-scale dipole moment build-up dynamics.
Hereafter we limit our analyses only to AH-J regions for simplicity. In the
context of solar cycle predictions using either dipole moment or polar flux,
we quantitatively evaluate the effect of AH-J regions on the final dipole
moment amplitude at the cycle minimum and speculate its implications on the
following cycle amplitude.

We utilize magnetic butterfly diagrams (Figure 5) depicting the spatio-
temporal evolution of the longitudinally averaged $B_{r}$ on the solar surface
to compare two SFT simulations: one with all H-J regions and another one with
5 $\%$ AH-J regions (remaining 95 $\%$ are H-J regions). We notice two
significant differences.

Firstly, in both hemispheres, magnetic fluxes predominantly from the following
spots migrate towards the poles as an effect of the B-L mechanism. The amount
of these unipolar surges, negative in the northern hemisphere (positive in the
southern hemisphere), is relatively higher in H-J configuration than in AH-J.

Secondly, in the second SFT simulation that contains AH-J regions, some
negative (in the northern hemisphere) polar surges are suppressed by the
magnetic flux associated with the positive following polarities of AH-J
regions (see within the blue rectangle in Figure 5). A similar scenario is
observed in the southern hemisphere with opposite polarity surges. The overall
effect on the evolution and build-up of the large-scale polar magnetic field
is manifested in two ways: delay in its polarity reversal and weaker final
amplitude at the end of the cycle in the second simulation with AH-J regions
compared to purely H-J simulation. Consequently, the succeeding solar cycle is
expected to be weaker and have a relatively extended cycle duration
(Waldmeier, 1935; Dikpati et al., 2008; Karak & Choudhuri, 2011).

Such changes should be more profound with more AH-J spots, which we verify by
performing several SFT simulations with increasing percentages of AH-J
regions. Statistical correlation analysis of the decrement of dipole moment
amplitude at the solar minimum with the increasing percentage of AH-J regions
shows a strong positive correlation with 99.9 $\%$ confidence (Spearman
coefficient is 0.963) in our SFT simulations. Subsequently, the delay in
dipole moment reversal compared to H-J regions bears a positive correlation
with the increasing number of AH-J regions (Spearman coefficient is 0.97 with
99.9 $\%$ confidence).

###  3.3 Relative Effectivity between the Number and Magnetic Flux of
Anomalous Regions

According to the basic features of the B-L mechanism, which dictates the
evolution of the large-scale magnetic field on the solar surface, the total
amount of magnetic flux associated with active regions should have more
significance than the total number of regions itself. Although past
observational studies (McClintock et al., 2014; Li, 2018; Muñoz-Jaramillo et
al., 2021) provide an estimate of the number of AH-J regions (3 $\%$ to 10
$\%$), there is scanty information about the total flux contained by these
anomalous regions. Thus we perform additional SFT simulations with the aim of
determining which one is more important: the number of AH-J regions (relative
to H-J regions) or the amount of AH-J region-associated magnetic flux
(relative to H-J flux). Thus we consider four distributions and compare the
corresponding dipole moment evolution (Figure 6).

  1. (a)

AH-J regions constituting 5 $\%$ of the total number of active regions and 5
$\%$ of the total active region-associated magnetic flux. Here the variation
in number and flux is mentioned as $\%$ number-$\%$ flux (thus, 5 $\%$ \- 5
$\%$ hereafter).

  2. (b)

AH-J regions constituting 10 $\%$ of the total active regions and 5 $\%$ of
the total flux (10 $\%$ \- 5 $\%$ hereafter).

  3. (c)

AH-J regions constituting 5 $\%$ of the total active regions but 10 $\%$ of
the total flux (5 $\%$ \- 10 $\%$ hereafter).

  4. (d)

AH-J regions constituting 10 $\%$ of the total active regions and 10 $\%$ of
the total flux (10 $\%$ \- 10 $\%$ hereafter).

In Figure 6a, we first observe a close resemblance between the dipole moment
evolution in cases with fixed flux but different numbers of AH-J regions: 5
$\%$ \- 5 $\%$ (orange curve) and 10 $\%$ \- 5 $\%$ (dark-orange curve).
Whereas changing the flux from 5 $\%$ to 10 $\%$ while keeping the number of
AH-J regions fixed to 5 $\%$ shows a more profound impact on the reversal
timing and final amplitude of the dipole moment at the end of the cycle (see
the orange and dark-orange curve in Figure 6a). This difference in the large-
scale magnetic field evolution becomes increasingly apparent in the later half
of the cycle. However, the importance of the number density of AJ-H regions
cannot be entirely disregarded. The same amount of magnetic flux (for example,
10 $\%$) can be distributed in one or many AH-J regions randomly positioned in
the activity belt. Depending on their emergence timing and latitude (and tilt
angle), additional variations will appear in the dipole moment evolution. We
test it by performing additional twenty SFT simulations with different
realizations of the time-latitudinal distribution of AH-J regions. The
corresponding dipole moment evolution is depicted in Figure 6b. The outer
curves encompassing the shaded area denote the dipole moment evolution with
maximum and minimum amplitudes for cases: 5 $\%$ \- 10 $\%$ (dark-magenta) and
10 $\%$ \- 10 $\%$ (magenta). The width and significant overlap of these two
regions indicate that the relative importance between the number and magnetic
flux is subjected to the particular distribution of the associated emergence
timing, location and latitude-based tilt angles of anomalous spots. The
details of these variations are discussed in the next section.

###  3.4 Effect of Spatio-temporal Variability of the Anomalous Active Region
Distribution

So far, in our SFT simulations, we randomly distribute the anomalous regions
over the activity belt of the entire solar cycle. However, results from the
previous section (3.3) indicate that emergence timing and latitude of such
regions could introduce more variability in the large-scale magnetic field
evolution. Moreover, earlier studies on observed sunspots (Sokoloff et al.,
2015; Li, 2018) claim preferential spatio-temporal distribution of anomalous
regions which differs from cycle to cycle. Thus, we perform six more SFT
simulations to address variations of AH-J region distribution as follows.

We consider that AH-J regions emerging during the: (1) initial phase of the
cycle, (2) middle phase i.e. around the cycle maximum, and (3) declining phase
of the cycle. Furthermore, we consider other cases with AH-J regions
concentrated (4) at higher latitudes, (5) at mid-latitudes and (6) near the
equatorial region (low latitudes). The corresponding activity belts are shown
in Figure 2 and the dipole moment evolution is depicted in Figure 7.

For the case with AH-J regions emerging predominantly in the initial phase
(simulation 1) of the cycle, we notice a slightly delayed reversal (Figure 7a)
of the dipole moment caused by unipolar surges from the following spots of
AH-J regions which have the same polarity as the existing polar field.
However, in the later stage of the cycle, the surface field is dictated by
mostly H-J regions, which facilitate a faster increase of the dipole moment,
thereby resulting in a final amplitude similar to the standard randomly
distributed AH-J regions all over the activity belt (see orange and violet
curve in Figure 7a). The impact of AH-J regions appearing around the cycle
maximum (simulation 2) is reflected in the significant delay in the reversal
of dipole moment (see the grey curve in Figure 7a). Note that the bulk of AH-J
regions are introduced from year 4 onwards, and the dipole moment almost
immediately starts deviating from the H-J case showing a rather sharp
deviation in year 5. The effect persists until the cycle minimum with a weaker
final dipole moment. Finally, in the case of AH-J regions emerging during the
declining phase of the cycle (simulation 3), the dipole moment reversal epoch
is similar to the H-J simulation (see the green and red curves in Figure 7a).
However, the final dipole moment strength is the weakest among all three cases
mentioned so far. It is caused by the magnetic flux contribution from the
following spots of AH-J regions (having opposite polarity compared to the
newly-built polar field) appearing mostly at the end of the activity cycle.

The dipole moment evolution for the three simulations with AH-J regions
preferentially emerging in the high (simulation 4), mid (simulation 5), and
low latitudes (simulation 6) are depicted in Figure 7b. We notice that the
high-latitude distribution has minimal effect on the dipole moment (pink curve
in Figure 7b), resulting in a similar evolution as in all H-J regions case,
also concurring with the findings of Yeates et al. (2015). However, the dipole
moment corresponding to the simulation with mid-latitude emergences of AH-J
regions is almost equivalent to the standard case (where AH-J regions are
distributed all over the cycle). The maximum impact on the dipole moment is
caused by the AH-J regions emerging at low latitudes (dark green curve in
Figure 7b), which is directly related to their negative contribution (higher
than the other cases) to the dipole moment build-up (explained in more detail
in the following sections). Quantitative details on how the dipole moment
evolution is affected in these six cases are provided in table 1 (see Sl No. 5
to 10).

##  4 Effectivity of Anomalous Regions on Global Dipole Moment: A Theoretical
Perspective

In our analyses so far, we have considered the large-scale magnetic field
evolution on the solar surface, primarily the dipole moment and polar flux,
both of which are longitudinally invariant by definition. While focusing only
on these two measures, our spatially two-dimensional SFT simulation (function
of latitude, longitude and time) can be reduced to only one spatial dimension
with surface magnetic field dependent on latitude and time (Petrovay et al.,
2020). Under such simplification, each tilted BMR, after averaging
azimuthally, will appear as a pair of flux rings of opposite polarity with a
bipole source having a finite latitudinal separation. This methodology has
been utilized earlier (Petrovay et al., 2020; Nagy et al., 2020; Yeates, 2020)
to propose a mathematical perspective of calculating the individual
contribution of each active region emerging during a solar cycle to the final
global dipole moment generation at the cycle minimum.

As demonstrated by Petrovay et al. (2020) and Nagy et al. (2020), the change
in the global dipole moment during the nth cycle can be expressed as the sum
of the contributions from individual active region dipole moments as follows,

$\mathrm{DM}_{n+1}-\mathrm{DM}_{n}=\Delta\mathrm{DM}=\sum_{i=1}^{N_{\mathrm{total}}}\delta
D_{\mathrm{U},i}$ (7)

where $\delta D_{\mathrm{U},i}$ corresponds to the ‘ultimate’ dipole moment
contribution from the ith active region during nth cycle and assuming there
are $N_{\mathrm{total}}$ regions in that cycle. Now, the final contribution,
$\delta D_{\mathrm{U},i}=f_{\infty,i}\,\delta D_{1,i}$, where $f_{\infty,i}$
is the asymptotic factor (Petrovay et al., 2020). The initial unsigned dipole
moment of any active region can be expressed as $\delta D_{1}=\frac{3}{4\pi
R^{2}}\,\Phi\,d_{\lambda}\cos\lambda$, where $\lambda$ is the emergence
latitude and $d_{\lambda}$ is the latitudinal separation of the two polarities
of the sunspot. $\Phi$ represents the magnetic flux in the BMR’s leading (or
following) spot. The polarity of the spots closest to the equator determines
whether $\delta D_{1}$ will positively or negatively contribute to the global
dipole moment.

The asymptotic factor is a Gaussian function of latitude,
$f_{\infty}=\mathrm{C}\,\mathrm{exp}(-\lambda^{2}/2\,\lambda_{R}^{2})$, such
that its amplitude decreases with increasing latitude. $\lambda_{R}$ and C are
dependent only on the transport parameters used in the SFT model, which are
fixed in all our simulations. Thus, in summary, the latitude dependency of the
ultimate dipole moment contribution from an individual region can be expressed
as,

$\delta D_{\mathrm{U}}\propto
d_{\lambda}\,\cos\lambda\,\mathrm{e}^{-\lambda^{2}}.$ (8)

The function, $\delta D_{\mathrm{U}}$ has decaying amplitude with increasing
latitude, provided $d_{\lambda}$ is constant for a 1D system (see Figure 8a).
This theoretical aspect delineated above can be utilized to explain the
variations seen in the dipole moment evolution in multiple SFT simulations
presented in section 3. Firstly, under this one-dimensional formulation,
active regions (in Figure 2) are reduced to the bipole sources with finite
latitudinal separation, whereas the longitudinal separation between the
leading and following spots becomes irrelevant. Thus we find pair-wise
similarities of dipole moment evolution for the cases: H-J and AH-AJ, and AH-J
and H-AJ (provided all emerge at the same latitude, $\lambda$ with the same
separation, $d_{\lambda}$). Secondly, any AH-J (or H-AJ) region will
contribute negatively to the dipole moment as its relative polarity
configuration is the exact opposite of an H-J (or AH-AJ) region, thus
diminishing the global dipole moment as found in sections 3.1 and 3.2. These
findings are reflected in Figure 8b where we carry out a qualitative
comparison between the net change in solar dipole moment in SFT simulations
with the mathematical approximations.

The latitude dependency shown in equation 8 also supports our findings in
section 3.4. In the first three SFT simulations with the varying spatio-
temporal distribution of AH-J regions on the activity belts, we see notable
changes in global dipole moment in cases with the anomalous regions emerging
during either middle or declining phases of the cycle (see Figure 7a). A
careful inspection of the latitudinal distribution of these spots on the
activity belt (see the second row in Figure 4) suggests that more spots
emerging in mid-to-low latitudes (in contrast to more high latitude
emergences) are responsible for this. Similarly, for cases with AH-J regions
appearing in different latitudinal belts (see the third row in Figure 4),
preferential emergence in low latitudes (smaller $\lambda$) will increase the
negative contribution according to equation 8. Therefore, it will cause the
maximum reduction of the final global dipole moment (see the pink curve in
Figure 7b). Figure 8b shows the qualitative comparison between these results
(high/mid/low latitudinal emergence impacts) from our SFT simulations and
aforementioned algebraic prescription.

##  5 Concluding Discussions

Emergence of active regions and transport of the associated magnetic flux on
the solar surface primarily determines the Sun’s polar field and dipole moment
build-up. The global dipolar field acts as the seed which modulate the
amplitude of the future sunspot cycle underscoring its importance. In this
study, we focus on the orientation of the active regions and investigate their
impacts on the large-scale magnetic field of the Sun, especially, the dipole
moment and hemispheric polar field. We keep the flux transport parameters and
some BMR properties like the separation, area, latitudinal and longitudinal
position fixed, eliminating variations originating from these properties. This
enables us to segregate contributions from “anomalous sunspot regions” in the
creation of irregularities in the solar cycle. It is to be noted that “rogue”
regions (Nagy et al., 2017) with very high tilt angle or high magnetic flux
(or a combination of both) are a subset of anomalous sunspots according to our
classification. How a few rogue or highly tilted active regions can
significantly impact the dipole moment evolution has been studied earlier
(Jiang et al., 2014; Nagy et al., 2017). In comparison, our focus is not on
rogue regions specifically but on a more comprehensive study of the impact of
diverse classes of anomalous regions with varying numbers, flux content
emergence timing and location.

We consider all active regions as bipolar sunspots (i.e., $\beta$-spots) and
do not consider any regions with complex morphology (e.g., $\delta$-spots). By
performing multiple SFT simulations we find that a fraction of the total
sunspot number emerging as anomalous regions can significantly influence the
large-scale magnetic field evolution during the solar cycle. The appearance of
AH-J/H-AJ regions changes the polarity reversal timing and eventually
suppresses the axial dipole moment at the end of the solar cycle – in
conformity with the findings of (Jiang et al., 2015; Nagy et al., 2017).

In this work, we establish a crucial point through SFT simulations: the AH-J
and H-AJ regions contribute almost identically in the dipole moment evolution,
polar field build-up and their reversal epoch; this is because over a few
rotational timescales the Sun’s differential rotation makes their orientation
similar. In addition, AH-AJ active regions behave similarly to H-J regions in
their long-term evolution. We perform an analysis which shows that our
simulation results are consistent with the algebraic formulation motivated
from mathematical theory (see section 4).

Based on our findings we conclude that in the context of the the long-term
evolution of the Sun’s large-scale magnetic field, the sunspots’ orientation
can be grouped into two classes: 1) Those that contribute positively to the
dipole moment – Hale-Joy sunspots (H-J) and 2) Which contribute negatively –
Anti Hale-Joy sunspots (AH-J). These two distinct classes of active region
configurations also exhibit characteristically distinct dynamics over shorter
time scales leading to disparate interactions with the local magnetic field
distribution.

Our analyses demonstrate that the amount of flux and the number density of
anomalous regions play an essential role in the dynamics of surface magnetic
field evolution. Although not always, we notice that in some cases a large
population of anomalous sunspots can impact the ultimate dipole moment more
than a small population carrying the same amount of flux. Therefore we
speculate that sometimes a larger group of anomalous spots can cumulatively be
more influential than a single large rogue region due to their latitudinal and
phase distribution diversity.

It is known that the total amount of unsigned polar flux is of the order of
the unsigned flux contained in a single but very large active region. A
similar amount of net flux distributed over a fraction of sunspots of the
anomalous class of regions appearing at distinct phases and latitudes will
affect the cycle differently. Our simulations show that AH-J regions near the
equator have the highest impact on the ultimate dipole moment strength,
whereas their appearance during the mid-phase of the solar cycle has the
highest impact on the reversal timing. Near-equatorial emergence is understood
to be more influential in general. This carries over to the case of anomalous
regions. The leading polarity of AH-J/H-AJ regions is opposite to the sign of
the new polar field (that is being imparted) in the opposite hemisphere. Thus
upon appear near the equator and diffusing to the other hemisphere, it may
reduce the strength of the new polar field of the opposite hemisphere (and the
ultimate dipole moment) during the descending phase of the solar cycle. In
sharp contrast, at high latitudes, the initial phase population of anomalous
regions hardly influences the overall magnetic cycle evolution and its
inherent dynamics. We believe that along with the diverse phase and
latitudinal distributions, the hemispheric asymmetry in the distribution of
AH-J/H-AJ regions may translate to the asymmetry seen in the polar field
weakening.

The result from different scenarios based on different synthetic sunspot
distributions is described in section 3 and tabulated in table 1. Note that
the statistics reported correspond to one random realization of each scenario.
Although these values may differ from one random realization to the other, the
qualitative behaviour of the global magnetic field evolution under different
possible scenarios still have a resemblance to the observed Sun.

In summary, our simulations provide insight on how the interplay of anomalous
and regular active regions modulate the solar dipole moment evolution. Given
that the dipole moment at the minima of a cycle is the most important factor
determining the strength of the future sunspot cycle, these insights are
important in the context of solar cycle predictions. In order to constrain
polar field evolution and dynamics it would be important to extract the
combined data of AH-J, H-AJ and AH-AJ regions (with their precise number, flux
content and the emergence phase-latitude information) along with the H-J
regions from observation. Such an exercise will provide the necessary
observational constraints for driving more precise predictive models of
surface flux transport evolution and illuminate the fine subtleties of the
Babcock-Leighton mechanism for solar polar field production.

##  6 Acknowledgement

The Center of Excellence in Space Sciences India (CESSI) – where this work was
carried out – is funded by IISER Kolkata, Ministry of Education, Government of
India. S.P. acknowledges funding from the University Grants Commission,
Government of India. P.B. acknowledges support from the project ST/W00108X/1
funded by the Science and Technology Facilities Council (STFC), United Kingdom
and S.S.M. acknowledges support from NASA contract NAS5-02139 (HMI) to
Stanford University. The authors acknowledge useful discussions at ISSI Team
Meetings on “What Determines The Dynamo Effectivity Of Solar Active Regions?”
supported by the International Space Science Institute, Bern, Switzerland. The
authors thank the referee for useful discussions.

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Table 1: A list of quantitative results: Anomalous active regions (AARs) coming from distinct distributions Sl No. |  Type of AAR distribution |  Decrease in Dipole Moment |  Delay in time reversal
---|---|---|---
1) |  5 $\%$ AH-J all over the cycle (5$\%$ flux) |  13.7 $\%$ |  2.61 $\%$
2) |  10 $\%$ AH-J all over the cycle (10$\%$ flux) |  48.17 $\%$ |  7.85 $\%$
3) |  5 $\%$ AH-J appearing more at the initial phase |  3.99 $\%$ |  3.14 $\%$
4) |  5 $\%$ AH-J appearing more at the middle phase |  11.04 $\%$ |  5.75 $\%$
5) |  5 $\%$ AH-J appearing more at the end phase |  26.17 $\%$ |  1.57 $\%$
6) |  5 $\%$ AH-J appearing more at the upper latitude |  3.73 $\%$ |  3.14 $\%$
7) |  5 $\%$ AH-J appearing more at the middle latitude |  9.75 $\%$ |  3.14 $\%$
8) |  5 $\%$ AH-J appearing more near the equator |  40.64 $\%$ |  4.71 $\%$
Figure 1: Panel (a): cartoon of bipolar magnetic region (BMR) with different
orientations and polarities in each hemisphere. Here BMR of configuration 1
follows both Hale’s and Joy’s law (H-J region); configuration 2 represents
regions which follow Joy’s law but violate Hale’s polarity law (AH-J region);
configuration 3 does not follow Joy’s tilt law but obeys Hale’s law (H-AJ
region) and lastly, configuration 4 disobeys both Hale’s and Joy’s law (AH-AJ
region). Panel (b): surface magnetic field distribution observed in line-of-
sight magnetogram from the Helioseismic Magnetic Imager (HMI) instrument on
board the Solar Dynamics Observatory (SDO). It denotes four full disk HMI
magnetograms observed in Solar Cycle 24. Panel (c): cut outs of the green box
(in panel b) that serve as examples of the four configurations (H-J, AH-J,
H-AJ and AH-AJ regions).

Figure 2: Butterfly diagrams reflecting the spatio-temporal variability of the
distribution of anomalous regions. Histograms of phase distributions and
latitude distributions are plotted on the top axis and right axis respectively
in each subplot. Top panel reflects the distribution where 5$\%$ anomalous
regions are spread all over the cycle. The next panel constitutes the
diversity in emergence phase of the anomalous active regions – (1), (2) and
(3) dictate population dispersed at initial phase, middle phase and the
declining phase, respectively. Third panel denotes the high, mid and low
latitude distributions (4, 5 and 6, respectively).

Figure 3: Time variation of axial dipole moment over two solar cycles
including 5 $\%$ of four differently configured BMRs. Green and pink curves
denote the axial dipole moment evolution of configuration 1 and configuration
4 (AH-AJ), respectively. Orange and blue curves represent the time evolution
of dipole moment for configuration 2 (AH-J) and 3 (H-AJ) respectively.

Figure 4: Time evolution of the surface magnetic field ($B_{r}$) for different
configurations. Here configuration 1 stands for H-J bipolar magnetic region
(BMR), and configurations 2, 3, and 4 indicates the anomalous regions
(mentioned in Figure 1). SFT simulations are performed for these four initial
configurations, and the evolved magnetic field configuration on day 1, 25 and
50 are shown in the left, middle and right panels, respectively.

Figure 5: Butterfly diagram of the surface magnetic field ($B_{r}$) for
configuration 1 and configuration 2. Panel (a): represents the butterfly
diagram of H-J regions. Panel (b): shows the butterfly diagram of 95 $\%$ H-J
regions with 5 $\%$ AH-J regions. Cyan circles represents the AH-J regions
distributed all over the solar cycle. Yellow line in both plots depict the
reversal timing of the polar field. Black solid and black dashed lines
represent positive and negative polarities respectively. Blue arrow in Panel
(a) denotes the newly built up polar field region, whereas blue box indicates
the opposite polarity surges for H-J BMRs. On the other hand shorter blue
arrow in Panel (b): reflects the accumulation of lesser amount of negative
flux during the end phase of the cycle. Blue box indicates the missing
opposite polarity surges in the same panel.

Figure 6: Time evolution of axial dipole moment over 22 years incorporating a
variation in the number and flux content of the AH-J regions only in the
second cycle. Here the variation in number and flux is mentioned as
%number-%flux. Orange (5 $\%$-5 $\%$) and dark-orange (10 $\%$-5 $\%$) curves
depict the time evolution of dipole moment with AH-J regions having 5 $\%$
flux but 5 $\%$ and 10 $\%$ in number respectively. Violet (10 $\%$-10 $\%$)
and dark-violet (5 $\%$-10 $\%$) curves represent AH-J regions having 10$\%$
flux but 5 $\%$ and 10 $\%$ in number respectively. Green curve denotes the
corresponding result for $100\%$ H-J regions. Panel (a): represents the
simulation with single realization. Panel (b): depicts the simulations with 20
random realizations for 10 $\%$-10 $\%$ and 5 $\%$-10 $\%$ cases.

Figure 7: Variation in the time series of the dipole moment (only for the
second cycle in the simulation) with different distribution of anomalous
(AH-J) regions shown in 2. Panel (a): represents the dipole moment evolution
of different phase distributions. The effect of AH-J regions having more
density near the starting of the cycle, middle of the cycle and the end of the
cycle on the dipole moment is denoted by violet, grey and red curves,
respectively. Panel (b): pink, light blue and dark green curves depict the
dipole moment evolution for AH-J regions emerging in high, mid and low
latitudinal positions, respectively. In all of these plots, green and orange
curves represent H-J and AH-J, respectively. Panel a1, b1 and a2, b2 are the
cut out regions of panels a, b highlighted in cyan and violet, respectively.
These sub-panels portray the delay in time-reversal near the cycle maximum and
the decrement of dipole moment/polar flux at the minimum of the same cycle.

Figure 8: Qualitative comparison of net change in solar dipole moment in our
SFT simulation with the mathematical approximations. Panel (a): depicts the
latitudinal ($\lambda$) dependency of different analytical functions
contributing to the ultimate dipole moment. High, mid and low latitudinal
regions are denoted by cyan, violet and yellow. Panel (b): change in dipole
moment ($\Delta$ DM) of our simulations(violet traingles) are compared with
the mathematically derived approximated values (orange circles) for six
different cases labelled on the horizontal axis.