# The temporal evolution of non-neutralized electric currents and the
complexity of solar active regions

Ioannis Kontogiannis Leibniz-Institut für Astrophysik Potsdam (AIP) An der
Sternwarte 16, 14482 Potsdam, Germany  Manolis K. Georgoulis Space Exploration
Sector, Johns Hopkins Applied Physics Laboratory, Laurel, MD 20723, USA
Research Center for Astronomy and Applied Mathematics of the Academy of
Athens, 11527 Athens, Greece

###### Abstract

We study the evolution of electric currents during the emergence of magnetic
flux in the solar photosphere and the differences exhibited between solar
active regions of different Hale complexity classes. A sample of 59 active
regions was analyzed using a method based on image segmentation and error
analysis to determine the total amount of non-neutralized electric current
along their magnetic polarity inversion lines. The time series of the total
unsigned non-neutralized electric current, $I_{NN,tot}$, exhibit intricate
structure in the form of distinct peaks and valleys. This information is
largely missing in the respective time series of the total unsigned vertical
electric current $I_{z}$. Active regions with $\delta$-spots stand out,
exhibiting 1.9 times higher flux emergence rate and 2.6 times higher
$I_{NN,tot}$ increase. The median value of their peak $I_{NN,tot}$ is equal to
$3.6\cdot 10^{12}$ A, which is more than three times higher than that of the
other regions of the sample. An automated detection algorithm was also
developed to pinpoint the injection events of non-neutralized electric
current. The injection rates and duration of these events were higher with
increasing complexity of active regions, with regions containing
$\delta$-spots exhibiting the strongest and longest events. These events do
not necessarily coincide with increasing magnetic flux, although they exhibit
moderate correlation. We conclude that net electric currents are injected
during flux emergence, but are also shaped drastically by the incurred
photospheric evolution, as active regions grow and evolve.

Delta sunspots, solar active regions magnetic fields, solar magnetic fields,
solar active regions, magnetohydrodynamics, solar activity

††journal: ApJ††facilities: SDO††software: SolarSoft IDL (Bentley & Freeland,
1998; Freeland & Handy, 1998), FLARECAST (Georgoulis et al., 2021)

##  1 Introduction

Active regions form when buoyant magnetic flux tubes rise from the interior of
the Sun (van Driel-Gesztelyi & Green, 2015), and appear in the photosphere,
forming pores, sunspots and/or groups of sunspots (Zwaan, 1985, 1987) of
varying size and complexity. Their white light morphology and distribution of
magnetic polarities can be used to categorize them in terms of magnetic
complexity (Hale et al., 1919). Active regions can be as simple as isolated
sunspots surrounded by plage regions ($\alpha$-type), bipolar or multipolar
sunspot groups consisting of well-, or not so well-, separated polarities
($\beta$\- and $\beta\gamma$-type), and in some cases they include opposite-
polarity umbrae sharing the same penumbra ($\delta$-type regions, Künzel,
1965). Different complexity categories can also reflect different evolutionary
states, as active regions emerge and evolve (McIntosh, 1990). Although most
regions are relatively simple and classified as $\beta$ (bipolar; Jaeggli &
Norton, 2016), the most complex active regions ($\delta$-spots) exhibit
sheared polarity inversion lines (PILs) characterized by exceptionally strong
electric currents and are the ones most often associated with strong flares
and coronal mass ejections (CMEs) (Sammis et al., 2000; Toriumi & Wang, 2019).

The existence of strong electric currents owing to complex magnetic fields was
noticed already several decades ago (Severnyi, 1965; Alfvén & Carlqvist,
1967). Although concerns were raised against the realism and feasibility of
calculation of these currents (Parker, 1996; Wilkinson et al., 1992), when
appropriate spectropolarimetric observations became commonplace, strong
indications occurred that the inferred currents from Ampere’s law are
reasonably realistic and reflect true physical characteristics (Leka et al.,
1996; Semel & Skumanich, 1998; Wheatland, 2000; Georgoulis et al., 2012).
These studies further showed that some active region electric currents
exhibited non-neutrality, meaning that their photospheric magnetic footprints
deviated from the archetypal isolated shielded flux tubes predicted by theory
(Parker, 1979) and discussed in studies of isolated magnetic elements
(Venkatakrishnan & Tiwari, 2009). A theoretical framework as well as merits
and caveats of the study of electric currents in solar active regions can be
found in Georgoulis (2018). In the nominal case of isolated flux tubes
embedded in a field-free plasma, a volume (direct) electric current is
shielded by an equal and oppositely directed (return) sheath current. This
symmetry possibly breaks in cases of very close proximity of opposite-polarity
footprints such as, prominently, intense magnetic polarity inversion lines. In
such exceptional cases, electric currents are non-neutralized and raw currents
are injected in the solar atmosphere, supplying the otherwise non-neutralized
currents of the overwhelmingly space-filling coronal magnetic fields (Longcope
& Welsch, 2000).

Recent studies have showcased the importance of the electric currents in the
context of the pre-eruption evolution of solar active regions (Georgoulis et
al., 2019), as well as the three-dimensional standard flare model (Aulanier et
al., 2012, 2013; Janvier et al., 2013, 2014). Overall, flaring active regions
exhibit at least an order of magnitude higher non-neutralized electric
currents than flare-quiet ones (Kontogiannis et al., 2017, 2018) and the same
quantity is highly correlated with the kinematic characteristics of CMEs
(Kontogiannis et al., 2019). Regions prone to eruptions are reported to
exhibit a higher degree of non-neutralization (Liu et al., 2017; Vemareddy,
2019; Avallone & Sun, 2020; Liu et al., 2024). Localized, intense changes of
the vertical electric current density have also been associated with pre-
flaring activity and free energy build-up (Mitra et al., 2020; Sahu et al.,
2023). Flux-ropes with non-neutralized electric currents at their footpoints
may well be kink-unstable (Tsap et al., 2022), which seems to be corroborated
by at least some observed flaring active regions (Barczynski et al., 2020;
Wang et al., 2023).

But what is the origin of non-neutralized electric currents in solar active
regions? Simulations show that the development of non-neutralized electric
currents is a consequence of flux emergence and the subsequent photospheric
motions, which lead to the formation of PIL and strong magnetic shear (Török
et al., 2014; Dalmasse et al., 2015). The coexistence of strong electric
currents and PILs is substantiated by observational studies (see e.g.,
Kontogiannis et al., 2019; Kazachenko et al., 2022). On the other hand,
Georgoulis et al. (2012) propose that the close interaction between opposite
magnetic polarities leads to the breaking of cylindrical symmetry and the
development of shear and strong electric currents through the Lorentz force.
In the same study, the coherence found in the electric current carried by the
magnetic polarities points to a sub-photospheric origin of electric currents.
What is, then, the evolution of the electric current in active regions and
what conclusions can be drawn from its scrutiny? This question is relevant to
how some active regions eventually become so complex. Although several studies
have focused on the evolution of the magnetic flux during flux emergence
(Otsuji et al., 2011; Norton et al., 2017; Kutsenko et al., 2019), no study so
far has been dedicated to the evolution of non-neutralized electric currents
in a sizeable sample of emerging active regions. Smaller samples of
regions/events have been scrutinized using the degree of current
neutralization (e.g., Liu et al., 2017; Avallone & Sun, 2020; He et al., 2020;
Wang et al., 2023; Liu et al., 2024), whose calculation is based on
assumptions regarding direct and return currents.

In the present study we examine a statistically significant sample of emerging
active regions of different magnetic complexity (in terms of Hale class),
following a methodology (Section 2) that calculates the non-neutralized
electric current using morphological image processing and taking into account
the systematic and numerical errors as in Georgoulis et al. (2012). In Section
3, we discuss a few case studies of specific active regions emphasizing the
evolution of electric current in different complexity classes. We then present
the characteristics of the non-neutralized electric currents time series in
comparison to the time series of the magnetic flux and as a function of Hale
class. Additionally, we analyze the time series in terms of injection events
and their properties. In Section 4, we summarize and discuss the results in
the context of the origin of electric currents in active regions and the
formation of $\delta$-spot regions.

Table 1: The sample of emerging active regions used in this study. The table contains information regarding their National Oceanic and Atmospheric Administration (NOAA) identifier, the start and end date of each HARP data set, the start and end heliographic longitude at which each region was observed, the resulting number of processed magnetograms (the number of points in the time series), the peak Hale class each region reached over the observation interval, and the number of flares associated with each region (see text).  Number | NOAA | Start Date | End Date | Start long. | End long. | Number of | Hale class | Flare class
---|---|---|---|---|---|---|---|---
|  |  |  | (Degrees) | (Degrees) | magnetograms |  |
1 | 11440 | 19-Mar-2012 00:46 | 26-Mar-2012 19:10 | -22.87 | 93.00 | 853 | $\beta\gamma\delta$ | 4 C
2 | 11267 | 4-Aug-2011 10:10 | 12-Aug-2011 11:22 | -46.62 | 58.76 | 963 | $\beta\gamma\delta$ | 1 B, 4 C
3 | 11465 | 19-Apr-2012 14:46 | 30-Apr-2012 13:10 | -57.92 | 90.97 | 1300 | $\beta\gamma\delta$ | 3 B, 16 C
4 | 11640 | 29-Dec-2012 12:10 | 8-Jan-2013 10:46 | -40.28 | 87.82 | 1191 | $\beta\gamma\delta$ | 7 B, 7 C
5 | 11726 | 19-Apr-2013 04:22 | 27-Apr-2013 09:10 | -19.41 | 88.68 | 924 | $\beta\gamma\delta$ | 11 B, 56 C, 1 M
6 | 11158 | 10-Feb-2011 21:58 | 21-Feb-2011 07:34 | -40.26 | 92.56 | 1247 | $\beta\gamma\delta$ | 1 B, 56 C, 5 M, 1 X
7 | 11750 | 15-May-2013 00:58 | 21-May-2013 18:58 | 0.4265 | 90.19 | 808 | $\beta\gamma\delta$ | 6 C
8 | 12673 | 28-Aug-2017 08:58 | 10-Sep-2017 11:10 | -81.77 | 89.12 | 1431 | $\beta\gamma\delta$ | 1 B, 54 C, 26 M, 4 X
9 | 11515 | 26-Jun-2012 03:58 | 10-Jul-2012 03:10 | -81.46 | 88.61 | 1668 | $\beta\gamma\delta$ | 2 B, 73 C, 30 M, 1 X
10 | 11884 | 26-Oct-2013 06:58 | 8-Nov-2013 10:46 | -83.82 | 88.84 | 1571 | $\beta\gamma\delta$ | 1 B, 15 C, 4 M
11 | 12257 | 4-Jan-2015 03:46 | 14-Jan-2015 14:46 | -50.51 | 89.60 | 1252 | $\beta\gamma\delta$ | 23 C, 3 M
12 | 11561 | 30-Aug-2012 00:22 | 5-Sep-2012 20:10 | -29.26 | 62.02 | 727 | $\beta\gamma\delta$ |
13 | 11645 | 2-Jan-2013 19:34 | 8-Jan-2013 23:10 | -12.42 | 75.48 | 736 | $\beta\gamma\delta$ | 2 C
14 | 11431 | 4-Mar-2012 11:10 | 10-Mar-2012 09:22 | 14.617 | 94.20 | 681 | $\beta\gamma\delta$ |
15 | 11273 | 16-Aug-2011 13:10 | 21-Aug-2011 16:34 | -18.69 | 52.27 | 618 | $\beta\gamma\delta$ |
16 | 11179 | 21-Mar-2011 09:58 | 26-Mar-2011 22:22 | -16.62 | 61.42 | 606 | $\beta\gamma$ |
17 | 11327 | 18-Oct-2011 23:58 | 28-Oct-2011 16:34 | -38.71 | 87.77 | 1107 | $\beta\gamma$ |
18 | 11813 | 6-Aug-2013 14:10 | 12-Aug-2013 11:34 | -13.30 | 69.01 | 699 | $\beta\gamma$ | 1 B
19 | 11776 | 18-Jun-2013 11:34 | 26-Jun-2013 00:58 | -11.47 | 90.45 | 907 | $\beta\gamma$ | 5 B, 5 C
20 | 11824 | 17-Aug-2013 06:10 | 22-Aug-2013 07:10 | 2.1014 | 75.38 | 597 | $\beta\gamma$ | 1 C
21 | 11922 | 10-Dec-2013 00:22 | 15-Dec-2013 19:22 | 4.4104 | 89.78 | 696 | $\beta\gamma$ |
22 | 12353 | 19-May-2015 11:10 | 27-May-2015 22:10 | -46.09 | 74.15 | 1008 | $\beta\gamma$ | 1 B, 3 C
23 | 11416 | 8-Feb-2012 13:58 | 18-Feb-2012 10:22 | -41.35 | 92.18 | 1060 | $\beta\gamma$ | 10 B, 1 C
24 | 11682 | 23-Feb-2013 19:58 | 6-Mar-2013 06:34 | -37.24 | 00.00 | 1135 | $\beta\gamma$ | 5 B
25 | 11781 | 27-Jun-2013 20:10 | 5-Jul-2013 01:46 | -11.43 | 82.70 | 859 | $\beta\gamma$ | 4 B, 1 C
26 | 11946 | 4-Jan-2014 04:58 | 14-Jan-2014 05:34 | -47.10 | 89.12 | 1196 | $\beta\gamma$ | 3 C, 1 M
27 | 12003 | 9-Mar-2014 15:34 | 16-Mar-2014 22:10 | -7.307 | 89.50 | 765 | $\beta\gamma$ | 9 C
28 | 12271 | 24-Jan-2015 06:22 | 2-Feb-2015 14:22 | -32.58 | 87.96 | 1087 | $\beta\gamma$ | 1 B, 4 C
29 | 12339 | 4-May-2015 16:34 | 18-May-2015 17:58 | -84.50 | 89.56 | 1636 | $\beta\gamma$ | 3 B, 56 C, 3 M, 1 X
30 | 12543 | 7-May-2016 16:58 | 16-May-2016 17:58 | -30.72 | 90.04 | 1043 | $\beta\gamma$ | 7 B, 5 C
31 | 11079 | 8-Jun-2010 03:34 | 13-Jun-2010 13:46 | 12.102 | 89.36 | 589 | $\beta\gamma$ | 1 B, 1 M
32 | 11326 | 20-Oct-2011 03:34 | 24-Oct-2011 15:46 | 23.325 | 86.04 | 534 | $\beta\gamma$ |
33 | 11385 | 22-Dec-2011 02:58 | 28-Dec-2011 22:22 | -22.34 | 63.62 | 811 | $\beta\gamma$ |
34 | 11472 | 29-Apr-2012 04:10 | 8-May-2012 00:34 | -51.84 | 60.73 | 915 | $\beta\gamma$ |
35 | 11807 | 28-Jul-2013 10:22 | 3-Aug-2013 12:58 | -2.951 | 81.79 | 729 | $\beta\gamma$ | 1 B
36 | 11081 | 11-Jun-2010 06:22 | 14-Jun-2010 23:22 | 35.170 | 90.15 | 439 | $\beta\gamma$ | 6 B, 7 C, 1 M
37 | 11148 | 17-Jan-2011 01:58 | 21-Jan-2011 09:46 | 21.677 | 79.32 | 509 | $\beta\gamma$ |
38 | 11116 | 16-Oct-2010 19:22 | 19-Oct-2010 02:58 | -2.952 | 30.11 | 259 | $\beta\gamma$ |
39 | 11510 | 18-Jun-2012 19:58 | 25-Jun-2012 04:22 | -18.74 | 71.90 | 713 | $\beta\gamma$ |
40 | 11141 | 30-Dec-2010 21:34 | 6-Jan-2011 01:34 | -1.415 | 81.40 | 736 | $\beta$ | 10 B, 1 C
41 | 11072 | 20-May-2010 16:22 | 29-May-2010 12:34 | -35.76 | 90.09 | 1050 | $\beta$ | 4 B
42 | 11076 | 31-May-2010 04:10 | 7-Jun-2010 21:34 | -16.68 | 89.09 | 925 | $\beta$ | 1 B
43 | 11096 | 3-Aug-2010 12:46 | 14-Aug-2010 02:10 | -63.10 | 74.95 | 1189 | $\beta$ | 2 B
44 | 11132 | 3-Dec-2010 23:10 | 11-Dec-2010 10:22 | -14.14 | 89.78 | 874 | $\beta$ | 4 B
45 | 11143 | 6-Jan-2011 00:46 | 13-Jan-2011 12:46 | -43.12 | 54.76 | 901 | $\beta$ |
46 | 11449 | 28-Mar-2012 09:10 | 3-Apr-2012 03:34 | -3.404 | 78.27 | 680 | $\beta$ |
47 | 11551 | 20-Aug-2012 03:46 | 25-Aug-2012 16:58 | -12.12 | 62.68 | 656 | $\beta$ |
48 | 11843 | 17-Sep-2013 07:46 | 23-Sep-2013 01:58 | -14.49 | 65.78 | 637 | $\beta$ |
49 | 11855 | 29-Sep-2013 18:10 | 7-Oct-2013 12:58 | -32.89 | 78.70 | 896 | $\beta$ |
50 | 12119 | 18-Jul-2014 09:34 | 26-Jul-2014 04:46 | -23.16 | 84.18 | 867 | $\beta$ | 1 B
51 | 11152 | 2-Feb-2011 15:22 | 8-Feb-2011 08:46 | -29.06 | 46.96 | 688 | $\beta$ | 8 B
52 | 11630 | 6-Dec-2012 17:10 | 15-Dec-2012 08:22 | -47.67 | 71.92 | 1028 | $\beta$ | 11 B, 2 C
53 | 11211 | 8-May-2011 15:10 | 11-May-2011 15:22 | 3.4807 | 45.80 | 356 | $\beta$ |
54 | 11446 | 22-Mar-2012 16:58 | 27-Mar-2012 19:58 | -17.52 | 52.96 | 570 | $\beta$ | 2 B
55 | 11105 | 2-Sep-2010 02:10 | 8-Sep-2010 12:46 | -0.349 | 92.62 | 772 | $\beta$ | 13 B, 2 C
56 | 11130 | 27-Nov-2010 15:10 | 5-Dec-2010 12:22 | -19.48 | 89.99 | 946 | $\beta$ | 26 B, 1 C
57 | 11242 | 28-Jun-2011 02:10 | 5-Jul-2011 07:58 | -9.091 | 90.76 | 860 | $\beta$ | 7 B,
58 | 11300 | 17-Sep-2011 00:34 | 22-Sep-2011 10:58 | 17.306 | 93.13 | 594 | $\beta$ |
59 | 11460 | 15-Apr-2012 11:34 | 26-Apr-2012 13:22 | -61.20 | 88.76 | 1290 | $\beta$ | 2 B, 5 C

##  2 Data and analysis

###  2.1 Data

The Helioseismic and Magnetic Imager (HMI; Scherrer et al., 2012; Schou et
al., 2012) onboard the Solar Dynamics Observatory (SDO; Pesnell et al., 2012)
provides uninterrupted, full-disk, spectropolarimetric observations of the
photosphere in the Fe I 6173 Å spectral line since 2010. The
spectropolarimetric observations are inverted to provide the magnetic field
vector at one level in the photosphere. This study utilizes the HMI Active
Region Patches (HARP; Hoeksema et al., 2014), which are cutouts that track
regions of interest (high-polarization regions such as active regions and
plages) and contain, among others, maps of the three components of the
magnetic field. The Cylindrical Equal Area (CEA) version of the definitive
data is used, that is, cutouts for which the 180°- ambiguity has been resolved
and the three components of the magnetic field vector $B_{r}$, $B_{p}$ and
$B_{t}$ (equivalent to the three Cartesian components $B_{z}$, $B_{x}$ and
$B_{y}$) have been deprojected and remapped to the solar disk center.

Figure 1: Illustration of the partitioning process, the calculation of the
non-neutralized electric current, and the automated injection event detection
method. a) Map of the $B_{r}$ component for NOAA 11158, at 01:58 UT on 13
February 2011. The magnetic field values are clipped between $\pm$500 G to
enhance visibility. The red (blue) contours indicate magnetic partitions with
positive (negative) total electric current. b) Map of the vertical electric
current density $J_{z}$, clipped between $\pm 7\cdot 10^{-6}$ A cm-2 to
enhance visibility. The red (blue) contours indicate magnetic partitions with
significant positive (negative) total electric current as per our selection
criteria. The contour indicated with “1” is the partition with the maximum (in
absolute value) non-neutralized electric current. c) Injection events (solid
red) detected for the $I_{NN,tot}$ time series of NOAA 11327, up to t = 6 d.

A sample of 59 emerging active regions was studied, consisting of 20 $\beta$-,
24 $\beta\gamma$\- and 15 $\beta\gamma\delta$-class regions (Table 1). These
Hale classes were the peak complexity classes reached by the active regions
while they traversed the Earth-facing solar disk. The magnetic complexity
information was derived from the Space Weather Prediction Center (SWPC) of the
National Oceanic and Atmospheric Administration
(NOAA)111https://www.swpc.noaa.gov/. Information regarding the flaring
activity of the regions was retrieved from the standardized multivariate data
set SWAN-SF (Angryk et al., 2020).

###  2.2 Derivation of non-neutralized electric currents

The method to derive the net electric currents is the one developed by
Georgoulis et al. (2012) and is illustrated schematically in Figure 1a and 1b.
The map of the vertical component of the magnetic field, $B_{r}$, is
partitioned using a flux tessellation method (Barnes et al., 2005), with
thresholds for $B_{r}$, minimum magnetic flux per partition and minimum
partition size equal to 100 G, 5$\times 10^{19}$ Mx and 40 pixels
($5.3\,Mm^{2}$), correspondingly. For each partition, the total current is
calculated via a surface integration of the electric current density, which is
itself calculated using the differential form of Ampere’s law, rather than
directly the integral form of Ampere’s law used by Georgoulis et al. (2012).
The reason for this was to speed up the calculations. In Kontogiannis et al.
(2017) it was shown that the difference between the two estimates were largely
insignificant, in comparison to the accuracy of the magnetic field
measurements. Then a baseline is needed to determine how much of the
calculated electric current could be due to numerical effects and/or errors.
First, the propagated error for the electric current $\delta I$ is calculated
using the corresponding errors $\delta B_{t}$ and $\delta B_{p}$. The
numerical effects are represented by the “current for the potential field”
$I_{pot}$. This is done by calculating the vector of the current-free
(potential) magnetic field via extrapolation (Alissandrakis, 1981) and
applying Ampere’s law on the potential magnetic field. The resulting electric
current should be equal to zero but this is not the case due to numerical
effects. Magnetic partitions are considered non-neutralized only if $I>5\times
I_{pot}$ and $I>3\times\delta I$.

For each HARP cutout the method essentially produces a set of electric-current
non-neutralized magnetic partitions, along with their corresponding net
electric currents $I_{NN}$. From this distribution, the total unsigned non-
neutralized electric current $I_{NN,tot}$ = $\Sigma$$|I_{NN}|$ and the maximum
unsigned non-neutralized electric current $I_{NN,max}=max(|I_{NN}|)$ of the
cut-out can be calculated (Kontogiannis et al., 2017). For a series of HARP
cutouts of a given active region, time series of $I_{NN,tot}$ and $I_{NN,max}$
were constructed, reflecting the evolution of the net electric currents in
this region. Additionally, from the maps of the non-neutralized partitions the
unsigned magnetic flux pertaining to those non-neutralized partitions
$\Phi_{NN,tot}$ can be calculated.

Non-neutralized partitions are found along PILs (Georgoulis et al., 2012), but
also during flux emergence events. These processes, i.e., emergence and PIL
formation will be further discussed in Section 3, through a comparative
analysis of $I_{NN,tot}$, $I_{NN,max}$, and $\Phi_{NN}$ time series, along
with those of the total electric current $I_{z}$ (namely, the surface integral
of the electric current density $J_{z}$) and the total unsigned magnetic flux,
$\Phi$, provided by the Space Weather HARP data (SHARP; Bobra et al., 2014),
for each HARP cut-out.

We note in passing that this method could potentially be used to study
individual partitions, too, although we do not attempt this here. Tracking
individual partitions should also account for potential merging or splitting
effects (see e.g., Švanda et al., 2021). Instead, our focus in this work is to
characterize active regions as a whole.

###  2.3 Automatic detection of injection events

An automated detection process was implemented to locate ascending branches of
conspicuous peaks in $I_{NN,tot}$, which signify events of net electric
current injection in the corresponding time series. To remove potential high
frequency noise, the time series were smoothed with a running 6-hour window,
corresponding to 31 points (as per the 12 minute cadence of the HMI vector
magnetograms). Then the local minima and maxima along with their positions
were located. The injection events are segments of notable and persistent
increase in $I_{NN,tot}$, and it is assumed that they are demarcated by a
local maximum and the local minimum that precedes it. The residual high-
frequency small-amplitude variability can cause the detection of several
consecutive, short segments along one injection event. A proximity criterion
was implemented to merge consecutive segments, i.e., for which the beginning
of the following segment is found less than 1 hour after the end of the
preceding. To avoid the merging of segments along decreasing parts of the time
series, we demanded that the following segment had a higher average
$I_{NN,tot}$ than the preceding. Finally, to avoid variations within the error
margin, segments resulting in an increase of less than 10% were discarded. An
example is illustrated in Figure 1c, for active region NOAA 11327. As per the
imposed criteria, fairly isolated conspicuous events are always detected but
consecutive weak events could be missed because they could be merged leading
either to decreasing or marginal, within the error, trend. This could produce
a slight underestimation of weak events, which does not, however, alter the
conclusions of our study.

Figure 2: a) Time series of $I_{NN,tot}$ (black solid), $I_{NN,max}$ (blue for
negative/red for positive) and $I_{z}$ (petrol) for active region NOAA 11551.
b) The total unsigned magnetic flux $\Phi$ (petrol solid) and the total
unsigned magnetic flux of the non-neutralized partitions $\Phi_{NN,tot}$
(black). The upper x-axis denotes central meridian distance in both a and b.
c-e) Maps of the vertical component $B_{r}$ of the photospheric magnetic field
at various instances. Blue and red contours indicate non-neutralized
partitions carrying negative and positive electric current, correspondingly.
The time is in decimal days since the first magnetogram of the series, taken
on 20 August 2012, at 03:46 UT ($t=0$).

For the detected time series segments we determined the duration and the
average injection rate using a linear fit. For the same segments we calculated
the corresponding rates of the total unsigned magnetic flux $\Phi$ and the
total unsigned flux of the non-neutralized partitions $\Phi_{NN}$.

##  3 Results

###  3.1 Example cases

####  3.1.1 Example of a $\beta$-class active region: NOAA 11551

Active region NOAA 11551 (Figure 2) started emerging on 20 August 2012, at
03:46 UT and evolved up to a $\beta$-type region. No flares were associated
with the region during its passage along the Earth-facing solar hemisphere.
Both $I_{z}$ and $\Phi$ (petrol lines in panels (a) and (b), respectively)
increased sharply within the first day of emergence and then started to decay.
The decrease of $I_{z}$ was sharp up to well into the second day, when the two
polarities became fully separated and then it started decreasing slowly until
the region decayed and rotated out of view. The magnetic flux exhibited an
increase after the third day, during which the decrease of the electric
current stagnated. After $t=4.5$ d both the magnetic flux and the electric
current decreased as the region moved past the 60° heliographic longitude. At
such high central meridian distances the measurement of the magnetic field
becomes less reliable due to projection and foreshortening effects, as well as
the increased noise pattern appearing towards the limb (Hoeksema et al.,
2014). The latter are known effects introducing an asymmetric bias in the
values of the total unsigned magnetic flux, leading to higher, on average,
values towards the western limb (see, e.g., Figure 6 of Kontogiannis et al.,
2018).

Figure 3: Same as Figure 2 for active region NOAA 11327. The first magnetogram
of the series was taken on 18 October 2011, at 23:58 UT ($t=0$).

The values of $I_{NN,tot}$ were around one order of magnitude lower than
$I_{z}$ (note the different y-axes in Figure 2a). Although the overall
evolution of $I_{NN,tot}$, $I_{NN,max}$ and $\Phi_{NN}$ followed the general
trend of the total unsigned magnetic flux and unsigned total current, there
were also notable differences. The total non-neutralized electric current
(black line in Figure 2a) also increased during the first day, but exhibited a
richer structure. The initial increase was followed by two sharp peaks,
reaching up to $5\times 10^{11}$ and $6\times 10^{11}$ A, as well as a lower
one seen during the decay. During the most pronounced peaks, $I_{NN,max}$ was
always significantly lower than $I_{NN,tot}$ revealing the existence of more
than one non-neutralized partitions (Kontogiannis et al., 2017), which
developed during the first day, as the magnetic field emerged and opposite
polarities started interacting. The sign of the $I_{NN,max}$ changed several
times indicating that during the evolution of the region different non-
neutralized partitions, with alternating signs of electric current, were
taking the lead. The photospheric magnetogram in Figure 2c shows three non-
neutralized partitions of neighboring opposite polarity magnetic flux. During
the third day, when $I_{NN,tot}$ dropped to almost zero, the polarities were
well separated (Figure 2d). After the end of the third day, $I_{NN,tot}$ and
$I_{NN,max}$ increased again, similarly to $I_{z}$, exhibiting a series of
peaks which reached up to $10^{11}$ A, until the region rotated out of view.
At all times the $\Phi_{NN}$ followed closely $I_{NN,tot}$; despite being
strongly correlated (Pearson correlation at 0.95), the two time series were
not identical as seen e.g., by the different relative size of the two peaks
around the end of the first day.

####  3.1.2 Example of a $\beta\gamma$-class active region: NOAA 11327

Active region NOAA 11327 (Figure 3) started emerging on 18 October 2011, at
23:58 UT, evolved up to a $\beta\gamma$-type region, but produced no flares
during its passage on the solar disk. The magnetic flux and the vertical
electric current increased gradually during the first day of emergence and
then increased precipitously during the following day reaching a maximum
around the end of the fourth day. Then, $I_{z}$ started to decrease slowly for
as long as the region was observed, while $\Phi$ remained relatively constant,
with a slight tendency to increase.

The $I_{NN,tot}$ was, overall, one order of magnitude lower than $I_{z}$ and
exhibited more structure, with discrete peaks and a more drastic decrease
after the maximum at the end of the fourth day. There was a broad, low peak
during the first day of emergence, when the two polarities were still close to
each other, but $I_{NN,tot}$ decreased briefly as the dipole separated. It
then increased sharply during the bulk of the emergence, during which more
non-neutralized partitions appeared in close proximity (see also Figure 3d).
In this case, however, the $I_{NN,max}$ did not change sign, which means that
the corresponding non-neutralized partitions always carried positive electric
current, as a result of one compact negative-flux, positive-current partition
having formed (Figure 3d). Sign changes were observed after the end of the
sixth day, when the net current in the active region had decreased
considerably and the magnetic flux was scattered into several smaller
partitions.

The total unsigned magnetic flux of the non-neutralized partitions,
$\Phi_{NN}$, followed a similar trend as the $I_{NN,tot}$, with one notable
difference: although $I_{NN,tot}$ exhibited sharp peaks at t = 2, 3 and 4 d,
as well as a drastic decrease after the fourth day, the corresponding peaks in
magnetic flux were not so pronounced and the overall variation of $\Phi_{NN}$
was relatively subtle until the end of the sixth day. This clearly implies
that interaction between magnetic partitions can change the injected electric
current significantly without notable changes in their magnetic flux.

Figure 4: Same as Figure 2 but for active region NOAA 11158. The first
magnetogram of the series was taken on 10 February 2011, at 21:58 UT ($t=0$).

####  3.1.3 Example of a $\beta\gamma\delta$-class active region: NOAA 11158

Active region NOAA 11158 (Figure 4) started emerging on 10 February 2011, at
21:58 UT and developed into a $\beta\gamma\delta$-type region. It produced the
first X-class flare of Solar Cycle 24 and, having been closely monitored by
the SDO, it was very well-studied (see, e.g., Tziotziou et al., 2013; Inoue et
al., 2014; Vemareddy et al., 2015, and references therein).

During the first two days of evolution two bipoles emerged, a smaller one at
the south-east (lower left) and a bigger one at the north-west (upper right).
This led to a slow increase of the magnetic flux (Figure 4b), during most of
which, there was also an increase in $I_{z}$ and $I_{NN,tot}$ (Figure 4a).
Initially, when the two polarities of each dipole started to separate, $I_{z}$
remained almost constant, following closely the evolution of $\Phi$, while
$I_{NN,tot}$ decreased until the end of the second day. Avallone & Sun (2020)
report a similar decrease in the degree of current neutralization. Our results
suggest that the observed decrease of $I_{NN,tot}$ is a consequence of the
separation of the dipoles.

Figure 5: Box-and-whiskers plots of time series characteristics for the
regions of Table 1. The lower and upper boundaries of the box indicate the
25th and 75th percentile, respectively, the horizontal line drawn inside the
box indicates the median and the whiskers indicate the minimum and maximum
value for each sample. All percentiles are calculated excluding outliers,
which are found 3$\sigma$ (the standard deviation of the sample) away from the
median and are marked with crosses. The top row contains the peak values of
$\Phi$ (left), $I_{NN,tot}$ (middle) and their ratio (right). The middle row
contains the time when the corresponding peak values occurred (left and
middle) as well as their time lag (right), with positive lag meaning that the
peak in $I_{NN,tot}$ occurred first. The bottom row contains the average
magnetic flux rate (left), the average $I_{NN,tot}$ rate (middle) and the
flux-normalized average $I_{NN,tot}$ rate. Green, blue and red indicate
$\beta$-, $\beta\gamma$\- and $\beta\gamma\delta$-regions, respectively. All
quantities were calculated for central meridian distances within $\pm$45°.

At the end of the second day the evolution of NOAA 11158 became dramatic, with
massive emergence of magnetic flux, accompanied by intense flaring activity,
comprising several C- and M-class flares, as well as one X-class flare, at t =
4.15 d. The total unsigned flux increased almost twice as fast in comparison
to the first two days, it reached a local maximum around t = 3.4 d and then
increased with a much lower rate until t = 5.3 d. The flux further increased
after t = 6 d. The increase of the electric current after the end of the
second day was sharper, with both $I_{z}$ and $I_{NN,tot}$ reaching local
maxima at t = 2.9 d. Then the two parameters followed a slightly different
evolution, $I_{z}$ reaching a maximum at t = 4 d and $I_{NN,tot}$ reaching a
maximum half a day later, at t = 4.5 d. As the active region evolved further
and entered its decay phase, both parameters decreased slowly, with
$I_{NN,tot}$ exhibiting a faster decrease. The $I_{NN,max}$ followed the
increase of $I_{NN,tot}$, but reached a plateau after the end of the third
day, switching sign three times, as indicated by the different colors in
Figure 4a. Since the flux was almost balanced during this phase (not shown)
and the magnetic flux carried by the non-neutralized partitions, $\Phi_{NN}$,
also varied slowly (Figure 4b), this is interpreted as different magnetic
partitions acquiring higher non-neutralized electric current as a result of
their interaction. In NOAA 11158, $I_{NN,tot}$ is not an order of magnitude
but only about 3-4 times lower than $I_{z}$, indicating that a significantly
higher part of the region is carrying non-neutralized electric currents.
Additionally, the $I_{NN,tot}$ is around five times higher than $I_{NN,max}$
indicating that many non-neutralized partitions along the PIL contribute to
the total net current (see also Figure 3 in Kontogiannis et al., 2017).
Strongly sheared and complicated PILs are typical characteristics of
$\delta$-spot regions, denoting a current-carrying magnetic structure in the
regions’ core.

Table 2: Table 2. Median values of the time series properties presented in Figure 5. Hale Class | Peak $\Phi$ | Peak $I_{NN,tot}$ | Peak $I_{NN,tot}$/$\Phi$ | Peak time $\Phi$ | Peak time $I_{NN,tot}$ |  $\Phi$ rate |  $I_{NN,tot}$ rate
---|---|---|---|---|---|---|---
| $10^{21}\,Mx$ | $10^{12}\,A$ | $10^{9}\,AMx^{-1}$ | days | days | $10^{21}\,Mxd^{-1}$ | $10^{12}\,Ad^{-1}$
$\beta$ | 6.0 | 1.0 | 0.17 | 2.5 | 2.0 | 2.4 | 0.5
$\beta\gamma$ | 5.9 | 1.0 | 0.17 | 2.7 | 2.2 | 2.6 | 0.6
$\beta\gamma\delta$ | 15.4 | 3.6 | 0.3 | 3.3 | 3.1 | 4.5 | 1.3

###  3.2 Temporal behavior of non-neutralized electric currents

####  3.2.1 The temporal evolution of $I_{NN,tot}$

To generalize the previous discussion on the full sample of active regions
contained in Table 1, we calculated some synoptic characteristics of the time
series, such as the peak values of $\Phi$ and $I_{NN,tot}$, their occurrence
times, as well as the corresponding average time lags between the peaks. These
are given in decimal days from the start of the emergence and provide a
measure of how fast the regions evolve up to their maximum magnetic flux and
electric current. Dividing the peak values by the occurrence times,
rudimentary $\Phi$ and $I_{NN,tot}$ variation rates were calculated. Figure 5
and Table 2 summarize these results.

The different magnetic complexity of active region classes $\beta$,
$\beta\gamma$ and $\beta\gamma\delta$ is reflected in their total unsigned
magnetic flux and electric current. As seen in the upper row of Figure 5,
$\delta$-regions clearly stand out in terms of maximum $\Phi$ and
$I_{NN,tot}$, with median values being more than three times higher than the
median $I_{NN,tot}$ values for $\beta$ and $\beta\gamma$-regions. We note in
passing that the four, non-outlier $\delta$-spot regions with the lowest
magnetic flux and non-neutralized electric current, found in the typical range
for $\beta$-regions, produced none (NOAA 11561, 11431 and 11273) or only two
C-class flares (NOAA 11645). From the $\beta\gamma$-regions (Figure 5, upper
row, blue boxes), the outlier is NOAA 12339, which in terms of flare
productivity is truly comparable to $\delta$-regions, having produced many C-
and M- as well as one X-class flare. Similarly, the two outliers from the
$\beta$-regions, namely NOAA 11130 and 11460, were also the most flare-
prolific $\beta$-regions of the sample. These findings agree with the results
of Kontogiannis et al. (2017), where a correlation was found between the flare
index (Abramenko, 2005) and the $I_{NN,tot}$, for a limited sample of regions.
In fact the Pearson correlation coefficient between the peak $I_{NN,tot}$,
$I_{NN,max}$ and the flare index for our sample is 0.82 and 0.87. A more
thorough investigation on the connection with imminent eruptive activity is
planned for the future.

The ratio of the peak $I_{NN,tot}$ over the peak $\Phi$ for $\delta$-spot
regions is almost twice as high as that of the rest of the sample. This is
strong indication that the excess net current in these regions is not due to
their larger size but due to their complexity. One may not completely rule out
the “big region syndrome” implying large values of non-potentiality metrics in
large active regions for extensive properties, albeit net currents are
inherently intensive properties, relying on internal structure and complexity.
It is also possible that these regions have intrinsically higher electric
current upon emergence.

The middle row of Figure 5 shows the absolute and relative timing of the
$\Phi$ and $I_{NN,tot}$ peak values, that is, the time it takes for active
regions to reach their maximum extent and net current content. For
$\delta$-spot regions, these times are longer, with NOAA 11515 and 12673
exhibiting the highest ones. These extreme active regions evolved over many
days, during which new flux emerged near already established flux systems
(Louis et al., 2014; Verma, 2018). From the $\beta$-regions sample, NOAA 11460
is an outlier in terms of peak $I_{NN,tot}$ time. Overall, the peak values in
$\Phi$ and $I_{NN,tot}$ are not synchronized. In most cases this difference is
in the order few hours and the time lags are predominantly positive, which
means that the peak in $I_{NN,tot}$ occurs earlier than the one in $\Phi$. In
some cases, magnetic field interactions or flux emergence in later stages of
evolution can lead to the peak $I_{NN,tot}$ value to occur later than the
$\Phi$ peak. This is also the case for the two outliers, NOAA 11273 and 11510,
a $\beta\gamma\delta$ and a $\beta\gamma$ region, respectively. Therefore, the
intrinsic details of flux emergence and subsequent photospheric motions
determine the relative timing of magnetic flux and non-neutralized electric
current.

Although it takes longer for the $\delta$-spot regions to reach their peak
$\Phi$ and $I_{NN,tot}$ values, these are much higher than those measured for
the other complexity classes. Hence, $\delta$-spot regions exhibit the highest
rates (Figure 5, bottom row). The median value for the magnetic flux rates
(see Table 2) is within the range of previous results by Otsuji et al. (2011),
Norton et al. (2017), and Kutsenko et al. (2019), if converted to $Mxh^{-1}$.
The flare-prolific $\beta\gamma$-region NOAA 12339 is an outlier also in terms
of magnetic flux rate. The median rate of $I_{NN,tot}$ for $\delta$ regions is
twice as high as the ones measured for $\beta$ and $\beta\gamma$ ones. The
flux-normalized increase rates of $I_{NN,tot}$ are still higher for the most
complex regions, which indicates that net current rates are not directly
proportional to the magnetic flux (and thus the size) of active regions.

####  3.2.2 The temporal evolution of $I_{NN,max}$

While the time series of $I_{NN,tot}$ show how the total amount of non-
neutralized electric currents evolves, the corresponding time series of
$I_{NN,max}$ show how the electric current carried by the most strongly non-
neutralized partition at each instance changes over time. As different parts
of the region emerge and evolve, the partition carrying $I_{NN,max}$ can
change as well, as does the associated magnetic flux $\Phi_{NN,max}$. This is
why, as shown in the descriptions of Figures 2, 3 and 4, the sign of the
$I_{NN,max}$ may change during its evolution. In Figure 6, the corresponding
time series of the sample are quantified, similarly to the preceding analysis,
in order to find differences between the different complexity classes.

As seen in the panels of the first row of Figure 6, $\delta$-spot regions are
associated with the highest $I_{NN,max}$ and $\Phi_{NN,max}$ values. This
means that as these regions evolve, their individual partitions develop the
highest electric currents in comparison to the simpler ones. The ratio of the
highest individual electric current $I_{NN,max}$ over the corresponding
magnetic flux $\Phi_{NN,max}$ is statistically higher as the complexity of the
region increases (NOAA 11682 is an outlier). These panels show that for the
most complex regions, the increase in the net current is not proportional to
the increase in the corresponding flux carried by the magnetic partitions.

The panels at the bottom row of Figure 6 show the time required for the peak
value of $I_{NN,max}$ and $\Phi_{NN,max}$ to develop in the examined sample of
active regions. Similarly to what was demonstrated in Figure 5, $\delta$-spot
regions take longer to evolve in comparison to $\beta\gamma$ and $\beta$
regions. However, here the distinction between the three classes in terms of
the time-lag between the peak values in $I_{NN,max}$ and $\Phi_{NN,max}$ is
clearer. Notably, active region NOAA 11460, a flare-prolific $\beta$-region is
again an outlier, demonstrating longer $I_{NN,max}$ peak time and shorter time
lag, as well as higher peak $\Phi_{NN,max}$. In fact, the majority of
$\delta$-spot regions exhibit negative lag, meaning that the peak value of
$\Phi_{NN,max}$ occurred earlier. This is another, clear indication that the
net electric current continues to increase in non-neutralized partitions after
they have fully emerged and reached their peak magnetic flux.

Figure 6: Same as Figure 5, but for the time series of $I_{NN,max}$ and the
associated magnetic flux $\Phi_{NN,max}$. The top row contains the peak values
of $\Phi_{NN,max}$ (left), $I_{NN,max}$ (middle) and their ratio (right). The
bottom row contains the time when the corresponding peak values occurred (left
and middle) as well as their time lag (right), with positive lag meaning that
the peak in $I_{NN,max}$ occurred first. All the characteristics were
calculated for central meridian distances lower than 45° EW. Figure 7: From
top to bottom: box-and-whiskers plots of the ratios $I_{NN,tot}/I_{z}$,
$\Phi_{NN}/\Phi$, $I_{NN,tot}/I_{NN,max}$, and $I_{NN,tot}/\Phi_{NN}$,
calculated at the time of peak $I_{NN,tot}$.

###  3.3 The relation between $I_{NN,tot}$, $I_{NN,max}$ and $I_{z}$

The three active regions presented in Figures 2, 3 and 4 differ not only in
the amount of non-neutralized electric currents they contained, but also in
the ratio of this electric current over $I_{z}$. The same is also true for
$\Phi_{NN}$, i.e., the magnetic flux associated with $I_{NN,tot}$. Given that
these quantities are the corresponding parts of $I_{z}$ and $\Phi$ that
pertain to the non-neutralized current part of the regions, it could be
surmised that the part of an active region that carries significant net
currents increases with magnetic complexity, which is highest for NOAA 11158
and lowest for NOAA 11551. For these regions the corresponding ratios were
even lower also when the two polarities were well separated and had increased
when magnetic flux was emerging or when opposite polarities were in close
proximity.

Figure 8: Electric-current injection event statistics for the active regions
of Table 1. a) Histograms of injection rates, b) $I_{NN,tot}$ rate versus
injection duration, c) $I_{NN,tot}$ rate vs. magnetic flux rate , and d)
$I_{NN,tot}$ rate vs. magnetic flux rate of the non-neutralized partitions. In
all panels, green, blue and red colors indicate statistics for $\beta$-,
$\beta\gamma$\- and $\beta\gamma\delta$-type regions, correspondingly.

Figure 7 extends these findings to the entire sample, showing the distribution
of the non-neutralized portion of the regions, in terms of electric current
and magnetic flux. These ratios were calculated at the times of maximum
$I_{NN,tot}$, for the smoothed time series and for central meridian distances
within $\pm$45°. As seen in the upper panel, $\delta$-spot regions exhibit
twice as high $I_{NN,tot}/I_{z}$ ratios compared to $\beta$ and $\beta\gamma$
regions, the bulk of which are close to 0.1. The ratio $\Phi_{NN}/\Phi$
exhibits smaller differences per complexity class, with a median around 0.65
for $\delta$-spot regions. The higher ratios and smaller separation between
the different classes could be a side effect of the calculation method,
because the entire flux contained in a non-neutralized partition is considered
non-neutralized although the net current may be occurring from a fraction of
this partition (see, e.g., Fig 1c). Thus, magnetic flux associated with the
current-carrying magnetic field could be overestimated. Nevertheless,
$\delta$-spot regions are located at the upper range of the $\Phi_{NN}/\Phi$
ratio, while $\beta$ and $\beta\gamma$ largely overlap each other. The ratio
of non-neutralized electric current is not significantly correlated with the
maximum magnetic flux of the region (Pearson correlation coefficient 0.2 for
$\delta$-regions). Therefore, the observed differences between different
complexity classes do not reflect their total magnetic flux content.

Another interesting finding pertains to the ratio $I_{NN,tot}/I_{NN,max}$. For
the bulk of $\beta$ and $\beta\gamma$ regions it is less than 3, meaning that
the $I_{NN,tot}$ is barely twice as high as $I_{NN,max}$, i.e., the electric
current carried by the most non-neutralized partition. Conversely, for
$\delta$ regions the corresponding ratio acquires higher values, indicating
the existence of many non-neutralized partitions in close proximity, forming
complicated PILs (Kontogiannis et al., 2017, 2019).

The ratio between the non-neutralized electric current over the current-
carrying magnetic flux is examined at the bottom panel of Figure 7. No notable
differences are seen for $\beta$\- and $\beta\gamma$-regions, but the current-
carrying flux in $\delta$-regions carries more electric current per unit non-
neutralized flux in comparison to regions belonging to the other two classes.
This result is in line with the observed highly-twisted structures (flux ropes
and/or sheared magnetic arcades) these regions often host (see e.g.,
Patsourakos et al., 2020, and references therein) and points to a systematic
characteristic exhibited by these active regions.

###  3.4 Net electric-current injection events

As seen in Figures 2, 3 and 4, peaks in $I_{NN,tot}$ are found in sequence or
superposition and indicate injection events and overall build-up of electric
currents as a result of both emergence and photospheric interactions. To study
these events in more detail, the detection algorithm was applied to the
$I_{NN,tot}$ time series of the sample. The statistics for the detected events
are summarized in Figure 8.

The relative frequency of the injection rates drops as the injection rate
increases (Figure 8a), but the highest injection rates are found in
$\delta$-spot regions, followed by $\beta\gamma$ ones. The former show
increased relative frequency of events with rates between 0.1 and $0.5\cdot
10^{12}\,Ah^{-1}$, whereas for $\beta$\- regions they reach up to $0.25\cdot
10^{12}\,Ah^{-1}$. Overall the injection rate is moderately correlated with
the event duration (0.45 Pearson correlation coefficient, Figure 8b). This
correlation is slightly stronger for $\delta$-regions (0.5) in comparison to
the rest (0.38) and the duration of the events is longer for the most complex
regions.

At the bottom row of Figure 8 the $I_{NN,tot}$ versus $\Phi$ (panel c) and
$\Phi_{NN}$ (panel d) rates of the events are plotted. There is an overall
positive correlation between these quantities (Pearson correlation coefficient
equal to 0.54 and 0.52, respectively), reflecting the fact that in many cases
the increase of the electric current is associated or happens during the
emergence of magnetic flux. This is the case seen, e.g., in NOAA 11158 (Figure
4), during the first day and, mainly, during the third day, where the bulk of
magnetic flux emerged and was accompanied by strong interaction between
magnetic footpoints. However, the $I_{NN,tot}$ increase is not always
associated with increasing magnetic flux since, as indicated by the two
scatter plots, there is a considerable number of events associated with very
small and even negative magnetic flux rates, indicating flux decay. The
correlation coefficients between $I_{NN,tot}$ and $\Phi$ rates for $\delta$\-
$\beta\gamma$\- and $\beta$-regions are 0.66, 0.30 and 0.47. The corresponding
correlation coefficient between the $I_{NN,tot}$ and $\Phi_{NN}$ rates (Figure
8d) are lower, that is, 0.52, 0.51 and 0.28. For the injection events with
rates higher than $0.12\cdot 10^{12}\,Ah^{-1}$ (which are mostly events
detected in $\delta$-regions, see also Figure 8a) the correlation for
$\delta$-regions, which exhibit most of these rates, drops to 0.47 and 0.20
for $\Phi$ and $\Phi_{NN}$, respectively. This correlation analysis for the
injection events corroborates the conclusions drawn from the preceding
analysis of time series properties. The current-carrying magnetic flux
$\Phi_{NN}$ peaks earlier than the $I_{NN,tot}$, which keeps increasing (see
Figure 6) and exhibiting injection events, even after $\Phi_{NN}$ is more or
less constant (see e.g., Figure 4).

##  4 Discussion and conclusions

This study presents, to the best of our knowledge, the first statistical study
of the evolution of non-neutralized electric currents in a statistically
significant sample of solar active regions. For the derivation of non-
neutralized electric currents the method of Georgoulis et al. (2012) was used,
that relies on the calculation of the vertical electric current density and
uses criteria on error propagation and numerical errors. The resulting time
series of $I_{NN,tot}$ are relevant to the non-neutralized current-carrying
part of an active region and exhibit a rich structure, not seen in the time
series of the surface integrated total unsigned current, $I_{z}$. Through a
comparative study of the $I_{NN,tot}$, $I_{NN,max}$, $\Phi_{NN}$ time series,
along with those of $I_{z}$ and $\Phi$, we reach a number of conclusions
regarding the evolution of net electric currents in active regions and the
differences between regions of different magnetic complexity.

In all types of regions, the unsigned magnetic flux and the electric current
exhibit similar (but far from identical) evolutionary trends, in terms of
overall increase and decay. However, the peak values of $\Phi$ and
$I_{NN,tot}$, representing their maximum size and complexity respectively, are
in principle not synchronized, instead, the relative timing of these peaks
depends not only on the emergence of magnetic flux but also on the subsequent
interactions between magnetic partitions. Thus the build-up of net electric
currents is a process driven by magnetic flux emergence but significantly
shaped by interactions between flux systems. The latter factor is more
pronounced in $\delta$-spot regions, where $I_{NN,tot}$ continues to increase
after the magnetic flux $\Phi_{NN}$, associated with the net currents has
peaked for most regions. The importance of polarity collision has been
highlighted for some cases of $\delta$ regions that were extremely flare-
prolific and exhibited various morphological configurations (Verma, 2018;
Toriumi et al., 2017; Chintzoglou et al., 2019). The results presented here
generalize these case studies, from the point of view of net electric current
injection.

During the flux decay phase, non-neutralized electric currents decrease faster
than the magnetic flux, implying the vital relation between net electric
currents and flux emergence along PILs, which leads to these currents’ fast
decrease during active region decay and their settling into more relaxed
configurations with scattered, well-separated polarities. This decrease is
more pronounced in $I_{NN,tot}$ than in $I_{z}$, corroborating the higher
relevance of $I_{NN,tot}$ to magnetic complexity. Even during decay, though,
incidental complexity due to interaction can still lead to injection of net
electric currents into the corona, as indicated by significant peaks often
seen in late stages of active region evolution.

To further study the injection of electric current in active regions, an
automated detection method was implemented and a statistical study was carried
out. The more complex regions, i.e., $\beta\gamma$ and $\beta\gamma\delta$
were characterized by injection events with correspondingly higher rate and
duration. The calculated correlations between the $I_{NN,tot}$ injection rates
with those of $\Phi$ and $\Phi_{NN}$ reflect again the combined role of
magnetic flux emergence and subsequent interactions in the formation of the
three complexity classes. A future step would involve studying the
characteristics of injection events on a larger sample of active regions, with
no limitation on evolutionary stage, and explore the association with imminent
flaring and eruptive activity. This might contribute clues toward prediction
of flares and eruptions (i.e., CMEs), which is a key challenge in solar
weather prediction (Kontogiannis, 2023).

In this work, we extended the findings of Kontogiannis et al. (2017) regarding
the relevance of non-neutralized electric currents to imminent flare
productivity using a statistically significant sample of active regions. In
several cases, $I_{NN,tot}$ evolution was unrelated to the size of active
regions, in terms of their magnetic flux. Such “intensive” behavior is
considered highly relevant to CME prediction (Bobra & Ilonidis, 2016). Our
results indicate that the relevance of the examined parameters to flare and
CME initiation is worth exploring in the future.

Regarding the relation between non-neutralized electric currents and
complexity, it was shown that active regions with $\delta$ spots differ both
in terms of peak values and evolution. They contain not only the highest
magnetic flux but also the highest net electric currents, even when the latter
are normalized to their peak magnetic flux content. The same is true for the
increase rate of $I_{NN,tot}$, as well as the ratio $I_{NN,tot}/I_{z}$, i.e.,
the part of the region that is current-carrying. For lower complexity cases,
$\beta$-regions are practically indistinguishable from $\beta\gamma$-regions
in terms of those properties. Since the non-neutralized electric currents are
directly associated with the presence of sheared PILs and, in our formulation,
the ratio of $I_{NN,tot}$ over $\Phi$ or $\Phi_{NN}$ is a proxy of twist, the
results presented here signify an inherently higher twist as well as different
formation processes for the regions hosting $\delta$-spots.

The study of Hale $\delta$-class active regions in terms of net currents is
extremely important both due to their high eruptive potential (Sammis et al.,
2000) and because of our incomplete knowledge on which mechanism drives their
formation (see e.g., López Fuentes et al., 2003; Knizhnik et al., 2018, and
references therein). The approach used in the present study can be used to
distinguish and probe the net current-carrying part of an active region.
Determining its properties can have implications on the formation mechanism,
even more so since typical photospheric proxies that characterize active
regions as a whole are not very efficient on distinguishing emerging flux
tubes of different sub-photospheric properties (Knizhnik et al., 2018). In
this context, Norton et al. (2022) found that the emergence rate of these
regions is almost twice as high (1.9 times) as those of $\beta$, which is also
corroborated by the present study (see Table 2). Here it was additionally
shown that the non-neutralized electric currents of $\delta$-spot regions
increase 2.6 times faster than in other active region types, with individual
injection events exhibiting even higher relative rates (Figure 8). The median
of the maximum ratio $\Phi_{NN}$/$\Phi$ was 0.65 (Figure 7), which is directly
comparable with the so called “degree of $\delta$”, which Norton et al. (2022)
defined as the magnetic flux fraction that participates in the $\delta$
configuration and amounts to 0.72 in their study. Recently, Levens et al.
(2023) examined a limited sample of regions and showed that the current
helicity, electric current and twist in magnetic knots, that is, the
constituents of $\delta$-spots, are all compatible with the kink instability.
Therefore, we conclude that the approach used in the present study could be
used in the future to contribute to the determination of $\delta$-spots
properties, providing more clues on the sub-surface condition in these
regions.

An additional benefit of our methodology is that the derivation of
$I_{NN,tot}$ is not based on assumptions concerning the chirality of the
active region (as e.g., in Avallone & Sun, 2020) and/or specific region
selection (Liu et al., 2017, 2024), which are required to determine the so-
called degree of current neutralization (Török et al., 2014). The presented
results are in line with those derived from the study of the degree of current
neutralization (Liu et al., 2017; Vemareddy, 2019; Avallone & Sun, 2020; Liu
et al., 2024; Barczynski et al., 2020; Wang et al., 2023; He et al., 2020).
Another future step would be a comparative analysis of the evolution of active
regions using both approaches, namely the degree of current neutralization
along with the ratio $I_{NN,tot}/I_{z}$ (or its complement
$1-I_{NN,tot}/I_{z}$) and the neutrality factor proposed by Georgoulis et al.
(2012).

We would like to thank the anonymous referee for providing useful comments,
which improved the content and presentation of this study. This work was
supported by grant KO 6283/2-1 of the Deutsche Forschungsgemeinschaft (DFG).
For this paper, data from SDO/AIA and SDO/HMI were used. These are courtesy of
NASA/SDO and the AIA, EVE, and HMI science teams and are publicly available
through the Joint Science Operations Center at the jsoc.stanford.edu. Some of
the calculations were performed using source codes made publicly available by
the H2020 project FLARECAST (https://dev.flarecast.eu/stash/projects/).

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