11institutetext: Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104
Freiburg, Germany
11email<EMAIL_ADDRESS>
# Evolution of the flow field in decaying active regions

Transition from a moat flow to a supergranular flow

H. Strecker  N. Bello González

(Received October 24, 2017; accepted September, 21 2018)

###### Abstract

Context. Fully fledged sunspots are known to be surrounded by a radial outflow
called the moat flow.

Aims. We investigate the evolution of the horizontal flow field around
sunspots during their decay, that is, from fully fledged to naked spots, after
they loose the penumbra, to the remnant region after the spot has fully
dissolved.

Methods. We analysed the extension and horizontal velocity of the flow field
around eight sunspots using SDO/HMI Doppler maps. By assuming a radially
symmetrical flow field, the applied analysis method determines the radial
dependence of the azimuthally averaged flow field. For comparison, we studied
the flow in supergranules using the same technique.

Results. All investigated, fully fledged sunspots are surrounded by a flow
field whose horizontal velocity profile decreases continuously from 881 m s-1
at 1.1 Mm off the spot boundary, down to 199 m s-1 at a mean distance of 11.9
Mm to that boundary, in agreement with other studies. Once the penumbra is
fully dissolved, however, the velocity profile of the flow changes: The
horizontal velocity increases with increasing distance to the spot boundary
until a maximum value of about 398 m s-1 is reached. Then, the horizontal
velocity decreases for farther distances to the spot boundary. In
supergranules, the horizontal velocity increases with increasing distance to
their centre up to a mean maximum velocity of 355 m s-1. For larger distances,
the horizontal velocity decreases. We thus find that the velocity profile of
naked sunspots resembles that of supergranular flows. The evolution of the
flow field around individual sunspots is influenced by the way the sunspot
decays and by the interplay with the surrounding flow areas.

Conclusions. Observations of the flow around eight decaying sunspots suggest
that as long as penumbrae are present, sunspots with their moat cell are
embedded in network cells. The disappearance of the penumbra (and consequently
the moat flow) and the competing surrounding supergranular cells, both have a
significant role in the evolution of the flow field: The moat cell transforms
into a supergranule, which hosts the remaining naked spot.

###### Key Words.:

sunspots – Sun: evolution – Sun: photosphere

##  1 Introduction

A fully evolved sunspot with a penumbra is usually surrounded by an annular
cell, the so-called moat cell. This region shows a horizontal-radial flow of
gas away from the spot. Sheeley (1972) first analysed the flow around sunspots
and described its annular form. The extension of the flow, as measured from
the outer spot boundary, is stated to be in a range of 10–22 Mm (Sheeley 1972;
Muller & Mena 1987; Shine et al. 1987; Brickhouse & Labonte 1988; Balthasar &
Muglach 2010; Verma et al. 2018). Löhner-Böttcher & Schlichenmaier (2013)
studied the flow field of 31 H-class sunspots in HMI Doppler maps and obtained
extension values in the range of 5–15 Mm with a mean value of 9.2 Mm. Time-
distant helioseismic inversions by Švanda et al. (2014) lead to moat flow
extensions of 12 Mm for an average over 104 H-class spots. Rempel (2015)
simulated two sunspots, one with a fully developed penumbra and one naked
sunspot that had just lost its penumbra. He found an extension of 10 Mm of the
flow field around both sunspots.

Sheeley (1972) measured horizontal flow velocities of 0.5–1 km s-1. The
comparison of several analysis methods (e.g. tracking granules and Doppler
shifts) by Brickhouse & Labonte 1988 led to flow velocities in the range of
0.2–1.4 km s-1. Muller & Mena (1987) described the velocity of the outflow
motion as a function of the distance to the sunspot, with the velocity slowing
down near the outer edge of the flow region. Using local correlation tracking,
Balthasar et al. (2013) obtained a decreasing velocity from the spot boundary
with 1 km s-1 down to 0.2 km s-1 at the outer edge of the moat flow region.
Löhner-Böttcher & Schlichenmaier (2013) measured velocities of 0.8–1.2 km s-1
just outside the sunspot with a monotonic decrease to a mean value of 0.18 km
s-1 at the outer edge of the moat cell where the noise level is reached. The
moat flow around the fully developed sunspot simulated by Rempel (2015) shows
a similar velocity profile. The horizontal velocity drops from 1.5 km s-1 at
the boundary of the sunspot down to 0.2 km s-1 at a distance of
$R\,=\,R_{\mathrm{spot}}$ \+ 10 Mm from the spot centre. The flow field
enclosing the simulated decaying sunspot displays a different profile for the
horizontal velocity. At a distance of 1 Mm to the spot boundary, the naked
spot is surrounded by an outflow whose horizontal velocity increases up to 0.5
km s-1 to later decrease until it drops below 0.2 km s-1 at a distance of 10
Mm from the outer edge of the spot.

The mechanism driving the moat flow is a controversial topic in solar physics.
Vargas Domínguez et al. (2008) described the unmagnetised moat flow as an
extension of the magnetised Evershed flow. This flow is observed as a radially
outward directed horizontal flow in the photospheric layers of the penumbra
(Evershed 1909) of magnetised plasma. The main argument for a connection of
both flows is the non-existence of the moat flow when a sunspot does not have
a penumbra. Vargas Domínguez et al. (2007) analysed spots with different
penumbral configurations and found that the moat only appears when a penumbra
is present. The flow can only be measured in the direction of the penumbra
filaments and is not detectable in directions transverse to them.

The idea of a connection between the Evershed flow and the moat flow is
rejected by the observation of an outflow after the penumbra has dissolved and
because the Evershed flow is magnetised, whereas the moat flow is a mainly
unmagnetised flow (Deng et al. 2007; Verma et al. 2012; Balthasar et al.
2013). Pardon et al. (1979) found a persisting flow in the remnant region two
days after the spots had disappeared. They described the sunspot in its final
stage to be surrounded by a stable cell that became unstable and collapsed
when the spot was gone. According to Deng et al. (2007), the symmetry of the
flow pattern becomes lost when the sunspot decays. They observed a decrease in
motion when the penumbra had dissolved. In addition, the outward motion is not
in the immediate surroundings of the naked spot, but is separated by an inward
motion similar to that reported by Rempel (2015) in his simulated naked spot.
Balthasar et al. (2013) measured higher velocities of the outflow on the
penumbral side of a decaying sunspot compared to the flow velocity on the side
where the penumbra had already dissolved.

Meyer et al. (1974) described the moat flow as a horizontal outflow resulting
from the rising of gas beneath the penumbra that bends horizontally as a
result of the shape of the penumbra. Nye et al. (1988) developed a model based
on this idea: The flow of energy to the surface is blocked, causing a heating
of gas beneath the sunspot. The rising gas cannot be efficiently transported
because the strong magnetic field of the sunspot prevents this, which leads to
an increasing gas pressure below the spot. Consequently, the gas rises, but is
blocked by the funnel shape of the penumbra and develops a strong horizontal
component. In this model, the velocity of the flow depends on the depth of the
penumbra. Sobotka & Roudier (2007) found the moat size to be mostly
independent of the spot radius. Instead, it seems to depend on the
evolutionary stage of the sunspot. The investigation of Löhner-Böttcher &
Schlichenmaier (2013) also showed no dependence between the spot radius and
the extension of the moat flow and supported the model by Nye et al. (1988).
Rempel (2015) found clear evidence in his simulations for the decoupled origin
of the moat and Evershed flows. The outflow around the sunspots is caused by
the perturbation of the up- and downflow balance that is due to the presence
of the sunspot: Because around spots, cooler downflows are reduced, the
average temperature rises, causing a vertical upflow close to the spot that
becomes visible as the moat flow when it reaches the photosphere.

The lifetimes of sunspots can vary from several days to several weeks. In a
group of sunspots, the follower spot seldom achieves a stable configuration
and is normally destroyed within days, while the leader spot can evolve into a
round and stable (H-class) sunspot whose decaying process can last up to
several weeks (Bumba 1963). The slow decay of a sunspot is a diffusive
process, which means a gradually shrinking by erosion at its boundary (Stix
2004). Nevertheless, Sheeley et al. (2017) observed decaying sunspots that
lost their round shape and became more elongated. Then, part of the flux
separated from the spot, and the spot is reshaped into a smaller roundish
spot. The authors concluded that sunspot decay occurs in a series of episodic
bursts. Bellot Rubio et al. (2008) observed that the loss of the penumbra
takes three days, in which the umbra splits and only three distinct finger-
like structures remain, which extend from the umbra into the quiet Sun. Deng
et al. (2007) described the decay of the penumbra in relation with a change in
and around flux concentrations of the local flow field. The photospheric
plasma flow takes action and advects the flux, which contributes to the decay
process of the sunspot. The spreading of magnetic flux over a larger area
leads to the disappearance of the dark area in intensity maps, but does not
remove magnetic flux from the photosphere (Martínez Pillet 2002). Convective
flows shape the field and destroy the annular pattern around the spot.
Surrounding supergranular flows are able to carve out field-free regions and
sweep away the flux to the cell edges. This leads to a fragmentation of the
magnetic field, and in connection with magnetic flux of opposite polarity, it
leads to flux cancellation (van Driel-Gesztelyi & Green 2015).

Simon & Leighton (1964) described supergranulation as a system of atmospheric
currents in the photosphere. The currents form a cellular network, which is
visible in Doppler maps after the reduction of other larger scale flows (e.g.
differential rotation and convective blueshift). Inside the cells, the flow is
radially directed from the cell centre to its boundary. The diameters of
supergranules are in the range of 10 Mm up to 45 Mm (Simon & Leighton 1964;
Roudier et al. 2014; Orozco Suárez et al. 2012). Depending on the method,
different values are measured (Hirzberger et al. 2008). A mean horizontal
velocity of 0.4 km s-1 in supergranules was measured by Simon & Leighton
(1964). Orozco Suárez et al. (2012) investigated supergranular convective
flows using Fourier local correlation tracking and intergranular magnetic
elements. The flow velocity in supergranules increases outwards for larger
distances from the centre. After reaching a maximum of 0.35 km s-1, the
velocity decreases monotonically. By averaging over 222 976 supergranular
cells and applying time-distant helioseismic inversions, Švanda et al. (2014)
found a symmetrical flow in supergranules, which is directed radially away
from the centre of the cells to its periphery, with horizontal velocities in
the range of 0.3 to 0.6 km s-1. Their velocity profile is found to increase
with increasing distances to the supergranule centre up to a maximum value
from where on a continuous decrease for larger distances starts. According to
Simon & Weiss (1968), the non-stationary cells only survive their turnover
time, then the flow lapses into disorder. Supergranular cells have a mean
lifetime of 1.5 days, which can extend up to 4 days (Roudier et al. 2014;
Hirzberger et al. 2008). De Rosa & Toomre (2004) observed the creation of
supergranules due to fragmentation or merging of older cells and stated that
each supergranular cell takes part in a minimum of one merging or splitting
event during its lifetime. This interaction seems to be the preferred mode of
evolution of the observed supergranular pattern. The mechanism of advecting
magnetic field elements to the network and the similar appearance suggests a
relation between the supergranular pattern and the quiet-Sun magnetic network,
as was recognised by Simon & Leighton (1964). De Rosa & Toomre (2004) observed
a strengthening or weakening of the network lanes due to splitting or merging
of supergranules. Although there is no definite evidence of a one-to-one
relation (see review by Rieutord & Rincon (2010) and references therein), in
the large-scale picture, the supergranular pattern can be equated with the
magnetic network. The horizontal, radially outward directed gas motion of the
moat flow, which is visible in Doppler maps, resembles the characteristics of
supergranular flows. In addition, magnetic lanes can form around the moat cell
due to magnetic features (MMFs), which cross the moat and conglomerate,
resembling the magnetic network. This relation has been described by Simon &
Leighton (1964), Sheeley (1972), and Vrabec (1974), for example, who proposed
the sunspot to be sitting in the centre of a supergranule. Meyer et al. (1974)
concluded from the observations that sunspots are related to supergranular
convection, but these two flows should be distinguished because of their
difference in size and occurrence by taking into account the unique relation
of the moat flow to its sunspot. Švanda et al. (2014) found the moat flow to
be asymmetric, while flows in supergranules are symmetric, but the
investigated cells approximately show the same size. In addition, the authors
described the moat cell as a downflow region, while the motion in
supergranules shows an upflow$-$downflow behaviour. Various investigations
have been carried out to study the differences and similarities between the
moat flow and supergranules (see e.g. Sobotka & Roudier (2007) and the reviews
by Solanki (2003) and Rieutord & Rincon (2010) and references therein). An
overview of the characteristics of the moat flow and supergranules is given in
Table 1.

Table 1: Characteristics of the moat flow and supergranules summarised as an
overview.

| Supergranule | Moat flow
---|---|---
Extension |  5–22.5 Mm(5,8) |  5–20 Mm (including spot)(2,4,7,10)
(radius) | mean: 12.5 Mm | mean: 9 Mm (radius w/o spot)
Velocity |  0.3–0.6 km s-1 (5,9,10) |  1.4–0.18 km s-1(1,4,7)
|  mean: 0.4 km s-1 |  mean: 0.55 km s-1
Lifetime |  mean: 1.5 days(3,8) |  several days(6)
| max: 4 days | as long as an H-class sunspot

(1) Balthasar et al. (2013); (2) Brickhouse & Labonte (1988); (3) Hirzberger
et al. (2008); (4) Löhner-Böttcher & Schlichenmaier (2013); (5) Orozco Suárez
et al. (2012); (6) Pardon et al. (1979); (7) Rempel (2015); (8) Roudier et al.
(2014); (9) Simon & Leighton (1964); (10) Švanda et al. (2014)

In our analysis, we study the characteristics of the flow field in which a
sunspot is embedded for different stages of the sunspot evolution. We start
the analysis for fully fledged H-class sunspots and follow their evolution
until they dissolve. The paper is organised as follows: In Sect. 2 we describe
the selection criteria and analyse the data. We apply the method not only to
the flow region around sunspots (Sect. 3.1), but also to study the flow in
supergranules (Sect. 3.2). In Sect. 4 we discuss the results, and we conclude
in Sect. 5.

##  2 Data selection and data analysis

We analyse the evolution of the flow field around fully fledged sunspots
throughout their decay as well as the flow field in supergranules. For this
purpose, we used Doppler maps from the Solar Dynamics Observatory (SDO)
(Pesnell et al. 2012). SDO data allow for continuous observations over long
time ranges, which is necessary to follow the evolution of a sunspot and its
surrounding area. We selected eight sunspots according to two criteria:

  1. 1.

The sunspots should be fully fledged and roundish (H-class sunspots) when they
appear on the eastern limb of the solar disc.

  2. 2.

The sunspots should decay before reaching the western limb.

We used the sub-frame data provided by the HMI/AIA Joint Science Operations
Center (JSOC)111http://jsoc.stanford.edu/. The provided patches are generated
by taking into account the NOAA number. In addition, the sub-frames are
already rotated to switch the north-south orientation of the original
data222Full disc data of the Helioseismic and Magnetic Imager (HMI) (Schou et
al. (2012)) show a rotation angle of 180∘. . Full-disc data of the
Helioseismic and Magnetic Imager (HMI) (Schou et al. (2012)) show a rotation
angle of 180∘. To ensure that we have the whole flow field surrounding the
spot in the field of view (FOV), we chose a data size of $500\times 500$ pix2,
that is, $252\times 252$ arcsec2.

The analysis method of our investigation is based on the work by Löhner-
Böttcher & Schlichenmaier (2013). They analysed the moat flow around 31 stable
sunspots at a given heliocentric angle. We present here a brief summary of the
analysis and refer to Löhner-Böttcher & Schlichenmaier (2013), Strecker
(2015), and Strecker & Bello González (2016) for a more detailed description
of the method. In the latter, the method by Löhner-Böttcher & Schlichenmaier
(2013) is extended to include the variation of the heliocentric angle during
the passage of sunspots across the solar disc.

###  2.1 Reduction of Doppler maps

We used the HMI 720 s Doppler maps provided by
JSOC333http://jsoc.stanford.edu/ajax/exportdata.html. These are low-noise data
generated by combining filtergrams recorded over a time interval of 1260 s.
Several velocity patterns, which are superimposed in one Doppler map, have to
be taken into account and were removed in order to properly study the flow
field around sunspots. These velocity components are

  1. 1.

the radial and tangential velocity components due to the orbital motion of the
satellite, as given in the fits headers of the HMI files;

  2. 2.

the differential rotation of the Sun for the (time-dependent) B0 inclination
angle;

  3. 3.

the centre-to-limb variation of the convective blueshift; and

  4. 4.

a residual pattern caused by instrumental effects (generated with data from
2013 and 2014).

To generate the models of these four velocity components, we followed Löhner-
Böttcher & Schlichenmaier (2013). We obtained a different residual pattern
(Strecker 2015), as our models are generated with data from 2013 and 2014. For
a further reduction of the noise level and the effect of p-modes, we followed
Löhner-Böttcher & Schlichenmaier (2013) and created 3 h time averages (out of
15 successive Doppler maps).

###  2.2 Localisation of the region of interest

For consistency, the simultaneous intensity maps and LOS-magnetograms were
also averaged over three hours. The fomer were used to localise the sunspot as
the centre of gravity and to determine the outer boundary of the sunspot by an
intensity threshold of the spot compared to the surrounding region. After the
disappearance of the spot in the intensity maps due to sunspot decay, an
approximate localisation of the centre of the flow field by eye by taking into
account the remaining flux in the LOS magnetograms. The obtained coordinates
were used to crop out a field of view of $200\times 200$ pix2 in the Doppler
maps. The centre of the flow-field measurements is thus defined as the centre
of the spot (as long as it is present) and located in the centre of the image.
Animations available as online material show the evolution of each spot as
seen in intensity, magnetograms, and Doppler maps (see Appendix A and Figs. 7
\- 14 therein for temporal snapshots for each spot). The spot boundary is
outlined by a red contour at each time. The position of the maximum horizontal
velocity and extension of the horizontal flow, that is, the cell size as
determined with our analysis method, are marked by green and pink contours,
respectively. These allow following the spot evolution from fully fledged to
vanished. To study supergranules, we defined their centre by hand as the
centre of the magnetic network cells, as seen in the LOS magnetograms (see
Fig. 15 in Appendix B) .

###  2.3 Analysis of the flow field

Owing to the spherical surface of the Sun, the velocities of horizontal flows
on the solar surface such as the Evershed flow or the moat flow are redshifted
for flows drifting away from disc centre and blueshifted for flows directed
towards disc centre. We follow Löhner-Böttcher & Schlichenmaier (2013) in
their analysis and assumed circular sunspots surrounded by an axially
symmetrical flow field, which should also hold for supergranules. This allowed
us to determine azimuthally averaged flow properties (Schlichenmaier & Schmidt
2000) by reading out the LOS velocity along circles with different radii
around the centre of the sunspot or supergranule. Sunspots and so the
surrounding flow field seem to have an elliptical shape off disc centre
because of foreshortening effects. Therefore, the circles to read out the LOS
velocity become elliptical according to the heliocentric angle, which defines
the position of the region of interest on the solar disc. Based on the work by
Löhner-Böttcher & Schlichenmaier (2013) (their Eq. 5), the LOS horizontal
velocity component $\upsilon^{\mathrm{LOS}}_{\mathrm{h}}$ for radial pixel
step is obtained as the amplitude of the LOS velocity along the ellipse. For
the LOS vertical component, which can be obtained as the offset of the fit, an
accurate value cannot be obtained as it is strongly influenced by the residual
velocities in the HMI Doppler maps. To compare the horizontal flow velocities
for different heliocentric angles ($\theta$),
$\upsilon^{\mathrm{LOS}}_{\mathrm{h}}(R,\theta)$, with $R$ the radial
distance, they have to be independent of LOS projection effects. This is
reached by

$\upsilon_{\mathrm{h}}(R)=\frac{{\upsilon^{\mathrm{LOS}}_{\mathrm{h}}(R,\theta)}}{\sin(\theta)}.$
(1)

Following Löhner-Böttcher & Schlichenmaier (2013), we defined the size of the
flow region at the point where the root mean square (rms) exceeds the LOS
horizontal velocity component
$\upsilon^{\mathrm{LOS}}_{\mathrm{h}}\leq\sigma_{\mathrm{rms}}$. The rms gives
the azimuthal velocity fluctuation around the sine fit $f(\phi_{i})$ and was
computed for all distances $R\,=\,R_{\mathrm{spot}}\,+\,r_{\mathrm{F}}$ with
$r_{\mathrm{F}}\geq 3\,$pixel from the spot boundary:

$\sigma_{\mathrm{rms}}(R)=\sqrt{\frac{1}{n}\cdot\sum_{i=1}^{n}(\upsilon^{LOS}(R,\phi_{i})-f(\phi_{i}))^{2}}.$
(2)

Figure 1 displays a snapshot of AR11641 (2 January 2013 at 07:30 UT) as an
example of the output of the analysis method: In the left panel, a Doppler map
is overplotted, with the green contour outlining the spot boundary as seen in
intensity, and the various ellipses for different radii ($r_{\mathrm{F}}$)
along which the flow field around the spot is measured. The right panel
displays the variation of the LOS horizontal velocity with the radial distance
to the spot boundary. The green curve displays LOS horizontal velocities
higher than the rms value (error bars). The red curve displays LOS horizontal
velocities that can no longer be detected above the noise (rms) level. This
transition (from green to red) defines the outer boundary of the flow field
(Löhner-Böttcher & Schlichenmaier 2013). The blue curve displays the
horizontal velocity corrected for projection effects (Eq. 1). The information
on the maximum horizontal velocity and the extension of the flow were then
extracted from $\upsilon_{\mathrm{h}}$ (independent of heliocentric angle) and
$\upsilon^{\mathrm{LOS}}_{\mathrm{h}}$, respectively, for all studied sunspots
and at different stages during their evolution across the disc.

Figure 1: Doppler map (left panel) of AR11641 on 2 January 2013 at 07:30 UT.
The green line displays the outline of the spot, and exemplary ellipses, along
which the velocity values are read out, are plotted in black. The extension of
the flow field is delimited by velocities higher than the $rms$ values (green
curve in the right panel). The $rms$ values are plotted as error bars. The
blue curve represents the velocity profile corrected for projection effects.

The sunspots we studied do not keep their roundish shape during their decaying
process. We defined the radius of the flow field to be $r_{\mathrm{F}}=0$ pix
at the spot boundary in intensity maps. To take into account the varying shape
of the spot during its decaying process, which can lead to an underestimation
of the radius by the automatic procedure, we started our analysis at a
distance of $r_{\mathrm{F}}=3\,\mathrm{pix}\approx 1.1$ Mm from the sunspot
boundary and took 51 pixel steps to measure the velocity across the whole flow
field. As the spot morphology changes, it also influences the morphology of
the surrounding flow field. We took this aspect into account by examining the
sunspot’s magnetic structure and its surroundings in the corresponding LOS
magnetograms, as we describe in the next section. This is also illustrated in
the animations provided in Appendix A.

##  3 Results

In the first part of this section, we describe the results of the analysis of
the flow field around decaying sunspots. The second part contains the results
of the analysis of the flow in supergranules.

###  3.1 Flow field around sunspots

We analysed eight sunspots in different time spans, covering their evolution
from fully fledged to naked sunspots to their eventual disappearance on the
disc, when observable. We generated 3 h data averages, which provided us with
a set of eight data points per day. As required, all sunspots had fully
fledged penumbrae at the beginning of the analysis. Table 4 lists the main
results for the analysed active regions. Columns. 3 to 5 contain maximum
horizontal velocity values for the three decaying stages of each spot. Columns
6 to 8 contain the extension of the flow field also for the three decaying
stages of each spot. The values in Cols. 3 and 6 display averages over the
first day in the analysis, that is, over eight samples in the fully fledged
sunspot phase. Columns 4 and 7 are time averages over the naked-spot phase,
while Cols. 5 and 8 are averages over the remaining analysis period, that is,
after sunspot disappearance. We note that values in Cols. 4 and 7 and 5 and 8
have different averaging periods depending on the evolution of each spot.

Table 2: NOAA number of the investigated active regions with the extension of
the flow field from the sunspot boundary $\bar{r}_{\mathrm{F_{end}}}$ and
selected velocity values.

AR | Period |  $\bar{\upsilon}_{\mathrm{h}}$(3 pixel) [m s-1] |  $\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})$ [m s-1] |  $\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})$[m s-1] |  $\bar{r}_{\mathrm{F_{end}}}$ [Mm] |  $\bar{r}_{\mathrm{F_{end}}}$ [Mm] |  $\bar{r}_{\mathrm{F_{end}}}$ [Mm]
---|---|---|---|---|---|---|---
|  | (1st day) | (w/o penumbra) | (w/o spot) | (1st day) | (w/o penumbra) | (w/o spot)
11676 | 19.–27.02.2013 | 677 | 373 | 237 | 14.5 | 10.43 | 6.6
11641 | 02.–08.01.2013 | 838 | 488 | 342 | 14.8 | 12.9 | 13.1
11646 | 05.–10.01.2013 | 957 | 400 | 221 | 9.4 | 8.0 | 4.3
11841 | 13.–21.09.2013 | 711 | 519 | \- - | 9.8 | 8.4 | \- -
12013 | 21.–26.03.2014 | 929 | 289 | \- - | 8.8 | 3.8 | \- -
12163 | 09.–16.09.2014 | 968 | 443 | 130 | 9.9 | 10.3 | 4.1
12169 | 18.–26.09.2014 | 1051 | 318 | 266 | 15.2 | 6.8 | 6.5
12170 | 19.–26.09.2014 | 916 | 356 | 309 | 12.6 | 6.8 | 6.7
mean |  | 881 | 398 | 251 | 11.9 | 8.4 | 6.9

444$\bar{\upsilon}_{\mathrm{h}}$(3 pixel) is the mean maximum horizontal
velocity (average over the first day of analysis), which is found at the
sunspot boundary. $\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})$
is the mean maximal velocity after the penumbra/the spot has fully dissolved
for different locations in the flow field.

####  3.1.1 Horizontal flow velocity

At their initial stage in the analysis, that is, during their fully fledged
evolutionary stage, all sunspots show a velocity profile that decreases
continuously from a maximum value at 1.1 Mm ($r_{\mathrm{F}}=3$ pix) off the
spot boundary with increasing distance to the spot boundary. See an example
for active region (AR) 11641 in Fig. 1. On the first day of the analysis, the
measured velocities range from a mean maximum velocity (average over the
eight-sunspot sample) of 881 m s-1 at a distance of 3 pixels from the boundary
of the sunspot (see Table 4, third column) down to 199 m s-1 at the outer edge
of the flow region. During the decay of the sunspots, the maximum velocity
values show individual changes due to the distinct evolution of each sunspot.
Nevertheless, all studied sunspots show a decrease in velocities at the inner
boundary of the flow field during sunspot decay. This evolution of the
velocity measured closest (1.1 Mm off) to the sunspot is illustrated in Fig. 2
(black), with individual symbols for each sunspot. All times are shifted to
show the loss of penumbra as a common point in time (t = 0).

Figure 2: Evolution of the maximum velocity
$\upsilon^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})$ (black) and velocity at
the spot boundary $\upsilon_{\mathrm{h}}$(1.1 Mm) (red) of the flow field for
all analysed active regions. If the maximum velocity is localised at the
sunspot boundary, these values are equal to $\upsilon_{\mathrm{h}}$(1.1 Mm)
and are not visible in the diagram. The abscissa values are given in days with
the loss of the penumbra being set as a common point in time.

Studying the velocity profiles for the different evolutionary stages of
sunspots decay unveils a change in the profile behaviour. The case for AR11641
is illustrated in Fig. 3: The horizontal velocity decreases with sunspot
decay, especially close to the spot boundary. The continuously decreasing
profile, characteristic for the moat flows, disappears. The profile becomes
flatter. After the penumbra dissolves (see Fig. 3e), the velocity profile
changes dramatically: the velocity at the (naked) spot boundary is now lower
than the velocity measured 5$-$6 Mm apart. We estimate a mean maximum
horizontal flow of 398 m s-1 in the surroundings for the eight naked sunspots
(see Table 4, fourth column, for the individual values of the single spots).
We note that when determining the average maximum horizontal velocity
$\upsilon^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})$ in the flow field, the
distance to the spot boundary is not a fixed value, but rather,
$r_{\mathrm{F}}$ varies for each (naked) spot. This is outlined by the green
contour in the animations (see Appendix A) on the spot evolution.

Figure 3: Velocity profiles of AR11641 for different times in the spot
evolution. Transition from green to red in the profiles outlines the outer
edge of flow field. A change in the shape of the velocity profile becomes
apparent as the sunspot decays and the penumbra dissolves on 7 January 2013 at
01:30 UT. The vertical lines mark the distance to the spot for which the
horizontal velocity is maximum.

The change in the shape of the velocity profile becomes even more pronounced
as the remaining naked spot dissolves: A velocity profile develops whose
amplitude first increases with distance to the (dissolved-)spot centre. A
maximum horizontal velocity of 251 m s-1 (averaged over the eight ARs) is
reached. Farther away from the (dissolved-)spot centre, the velocity decreases
again (see e.g. Fig. 3f). This mean maximum velocity of 251 m s-1 of the flow
field measured after the sunspots disappearance is lower than the velocity
measured in the surroundings of the naked spots (398 m s-1) described above.
The two blank rows in Table 4, fifth column, are due to measuring maximum
velocities after the sunspots disappearance in only six of the eight studied
regions. AR11841 does not completely dissolve before reaching the western
limb. In the surrounding region of AR12013, no consistent outflow pattern can
be measured after the spot has fully dissolved. The behaviour of AR12013 and
its surrounding flow field is discussed further in Sect. 3.1.3 (where we
explain the absence of such an outflow).

To visualise the change in velocity profiles for all active regions, the
horizontal velocity by the spot boundary (black) is plotted together with the
maximum horizontal velocity (red) measured over the flow field in Fig. 2. As
long as the two values coincide, only the values closest to the sunspot are
visualised (black). Thus, the detachment of the red and black values reveals
the change in the profile of the horizontal velocity component of the radial
flow. The onset of the profile change is coincident with the full
disappearance of penumbrae, indicated by the blue vertical line in Fig. 2,
which shows the intimate coupling between penumbrae and outflows in the
surroundings of sunspots.

It is important to note here that the process of decay, distinct for each
sunspot, also plays a role in the measured outflows. Thus, a change in the
location of the maximum velocity in the flow field before the penumbra has
fully dissolved can be ascribed to the separation of part of the penumbra
(flux expulsion) from the sunspot, for instance: the evolution of AR11641
(Appendix A, from 04.01.2013 00:00 UT – 09:00 UT) three days before the
penumbra dissolves (red crosses in Fig. 2 at t = -3 days) shows that the flux
expelled into the moat region influences the measurement of the flow velocity
until it vanishes. Similarly, the early onset of the change in velocity
profile in AR11841 (red triangles) is due to an asymmetric disappearance of
the penumbra (see animation in Appendix A). The process of the final loss of
the penumbra leads to asymmetries in the shape of the other spots as well and
therefore also in the surrounding flow fields. This causes the scattering of
the shift of the measured locations for the maximum velocity. In addition, the
final dissolving of the penumbra is a mostly continuous process, and this
point in time is selected by eye in intensity maps, which can also lead to
discrepancies and to the spreading around the defined point in time. The
reshaping of the radial profile is observed in all cases, however.

####  3.1.2 Flow extension and projection effects

The analysis method of the flow extension as described in Sect. 2.3 relies on
the fact that off disc centre, and due to projection effects, a horizontal
flow shows a component along the LOS with measurable velocities. Therefore,
the measured flow extensions depend on the LOS horizontal velocity and on the
$rms$ values. Closer to disc centre, the projection effect decreases, and
lower velocity values are measured. The lower values vanish easily in the
noise and lead to smaller measured flow extensions close to disc centre. This
effect is shown in Fig. 4 for the individual sunspots, as the extension of the
flow field seems to decrease and at some point in time starts to increase
again when sunspots cross over the zero-meridian. Velocities of sunspots in
the eastern and western hemisphere are outlined by black and red symbols. The
values are again shifted to have the loss of the penumbra as a common point in
time. We note that the analysed time range is different for each sunspot and
does not start at the same point in time in the sunspot evolution. Sunspot
decay, especially the loss of the penumbra, leads to a large scattering in the
values of the flow extension. The erosion and detachment of magnetic flux from
the decaying sunspot also affects the measurement of the extension of the
outflow.

For fully fledged sunspots, a mean flow extension of 11.9 Mm is measured (see
Table 4, sixth column, for the mean value for each spot). Figure 4 displays a
decrease in the flow extension as evolution progresses when comparing similar
heliocentric angles on the eastern (black) and western (red) hemispheres of
the solar disc. After the penumbra has dissolved, the extension of the outflow
also decreases to a mean value of 8.4 Mm. The values for each sunspot before
and after losing the penumbra, as given in Table 4 (sixth and seventh column)
display an individual behaviour of the flow extension for the spots. Around
some active regions, for example, AR11641 and AR11841, only a slight decrease
is found. For AR12163, even a higher value is measured. This can be ascribed
to the influence of a part of the penumbra detaching from the spot. However,
the extension of the flow field around other active regions, such as AR12169
and AR12170, decreases by more than 5 Mm.

We also measure the extension of the flow after the naked sunspot has
dissolved. We find a mean extension of 6.9 Mm for the remaining flow cell
without spot. Nevertheless, a distinct behaviour for each flow cell is again
obtained (see Table 4, last column). The flow cell size of AR11641 remains
mostly stable when the values of the different steps of the sunspots evolution
are taken into account. The flow extension around AR12169 and AR12170, for
instance, also remains constant from the stage of naked spot to the eventual
spot extinction. The flow cell around some other active regions, such as
AR11646 and AR12163, decreases within half a day after the spot disappears,
however. The flow field around AR12013 even disappears immediately with the
disappearance of the spot, as we discussed earlier.

####  3.1.3 Interaction between the flow field around sunspots and surrounding
network (supergranular) cells

We find a main common behaviour for the horizontal flow velocity profile of
the different sunspots, but the extension of the flow field surrounding a
sunspot shows a more individual behaviour. Zwaan (1992) observed breaks in the
flow field around a sunspot and suggested a dependence of the shape of the
cell on the magnetic environment. To study whether the surrounding of sunspots
influences the extension of the flow field, we used HMI LOS magnetograms. When
sunspots have a penumbra, the LOS magnetograms show the well-known scenario of
a sunspot as a strong accumulation of magnetic flux with moving magnetic
features streaming away from it, and with the spot centred in a (network)
cell, as shown in Fig. 5a. The magnetic flux accumulating around the spot
seems to form a firm boundary against the surrounding network cells. The
network (supergranular) cells evolve in continuous competition with each
other, including the sunspot cell. At some locations, the surrounding network
cells are able to interrupt the sunspot’s flow cell, as is shown in the
network cell centred at around x = 56 Mm, y = 26 Mm, in Fig. 5a, for instance.
In other cases, such as Fig. 5c, the sunspot cell implodes, squeezed by the
surrounding cells. Nevertheless, as long as the penumbra is present, the cell
formed around the sunspot is only partially distorted and can re-establish
itself again and push the surrounding network cells back. The evolution of
each sunspot as seen in the magnetograms can be followed in the central panel
of the animations (see Appendix A and Figs 7 \- 14 for snapshots).

Figure 4: Extension of the flow field around all analysed active regions.
Black symbols indicate the active regions to be localised on the eastern side
on the meridian, while red symbols show values for the active region in the
western hemisphere. The black vertical lines and symbols display the
approximate crossing of the meridian by the respective sunspot in time. The
loss of the penumbra is set as a common point in time. Figure 5: LOS
magnetograms of AR11676 (upper panels) and AR11646 (lower panels) for
different stages in their evolution (left columns: sunspots with penumbra, and
right columns: dissolved spots). The area of the spot is outlined by the green
ellipses. The extension of the flow field is outlined by the red ellipses.

As the penumbra dissolves, the morphology of the spot changes and the strength
of the flow seems to weaken. This influences the measured extension of the
flow field (see previous Sect. 3.1.2). We observe two different scenarios: (1)
For some cases, in regions of the spot cell boundary in which magnetic flux
accumulates, the influence of surrounding (supergranular/network) cells
appears to be weak and a cell can form around the spot. In such cases, the
cell remains stable, even after the spot vanishes, and the flow extension can
be measured. An example is illustrated in Fig. 5b for AR11676. With time, the
cell size slowly decreases as the surrounding network cells push against it,
leading to a long-lasting (¿ 1 day) dissolving process. The full evolution
process can be monitored in the animations in Appendix A. (2) In contrast, if
there is weak magnetic flux in the spot cell boundary, the surrounding cells
are able to distort the flow field by squeezing the spot cell. This leads to a
decrease in flow extension. The effect of squeezing the spot cell becomes even
more pronounced as the spot vanishes in the intensity maps: Eventually, the
spot cell implodes owing to the action of the surrounding cells, as shown in
Fig. 5c and Fig. 5d for AR11646, for instance. We observe a similar behaviour
for AR12013 and AR12163. For the former, the flow even disappears with the
disappearance of the spot.

These observations suggest that the variety of measured extension values of
the flow field surrounding a spot (Sect. 3.1.2) can be explained by boundary
conditions, namely, the interaction with surrounding network cells. In
addition, the accumulation of magnetic flux, forming part of the network,
seems to influence the stability of the cell.

###  3.2 Flow field of supergranules

To analyse the flow field in supergranules, we applied the same method as for
the study of the flow field around sunspots. To localise the centre of
supergranules, we used LOS magnetograms and assumed the magnetic network as a
suitable outliner for the supergranular pattern manifest in the averaged
Doppler maps. The centre of such a cell was then manually localised by eye. In
this way, we selected 18 supergranules in the magnetograms and read out the
LOS velocity values from the Doppler maps along circles and ellipses around
the defined barycentre point. Examples of the contouring analysis of three
supergranules of the sample as seen in intensity, magnetograms, and Doppler
maps can be found in the figures provided in Appendix B.

####  3.2.1 Horizontal flow velocity

The mean horizontal velocity profiles are displayed in Fig. 6d for all
selected supergranules. On average, the velocity increases from a mean value
of 215 m s-1 for increasing distance to the cells centre, up to a maximum of
355 m s-1 to later decrease for larger distances. The maximum horizontal
velocity, in the range of 216–511 m s-1, is always localised farther out in
the flow field and not at the centre of the cell.

####  3.2.2 Flow extension

According to our analysis method, the cell size is determined as the distance
to the cell centre where the measured LOS horizontal velocity is lower than
the noise level. We measure a mean distance of 8.8 Mm, with values in a range
from 6.1 Mm to 12.9 Mm, which translates into an estimate of the diameter of
the cells of 12.2 Mm to 25.8 Mm.

Table 3: Summary of results.

Spot with penumbra | Naked spot | Dissolved spot | Supergranule
---|---|---|---
$\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})=881\mathrm{m\,s}^{-1}$ | $\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})=398\,\mathrm{m\,s}^{-1}$ | $\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})=250\,\mathrm{m\,s}^{-1}$ | $\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})=355\,\mathrm{m\,s}^{-1}$
($\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})=1000\,\mathrm{m\,s}^{-1}$)(1) | ($\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})=500\,\mathrm{m\,s}^{-1}$)(3) |  | ($\bar{\upsilon}^{\mathrm{max}}_{\mathrm{h}}(r_{\mathrm{F}})=300-500\,\mathrm{m\,s}^{-1}$)(2,5)
$\bar{r}_{\mathrm{F_{end}}}=11.9\,\mathrm{Mm}$ | $\bar{r}_{\mathrm{F_{end}}}=8.4\,\mathrm{Mm}$ | $\bar{r}_{\mathrm{F_{end}}}=6.9\,\mathrm{Mm}$ | $\bar{r}_{\mathrm{F_{end}}}=8.8\,\mathrm{Mm}$
($\bar{r}_{\mathrm{F_{end}}}=9.2\,\mathrm{Mm}$)(1) | ($\bar{r}_{\mathrm{F_{end}}}=10\,\mathrm{Mm}$)(3) |  |  $(\bar{r}_{\mathrm{F_{end}}}=5-22.5\,\mathrm{Mm}$)(2,4)

555 The first three columns display the mean values obtained for the different
stages of the sunspots evolution (sunspot with penumbra, naked spot, dissolved
spot). The fourth column depicts the mean values of the supergranular flow.

(1) Löhner-Böttcher & Schlichenmaier (2013), (2) Orozco Suárez et al. (2012),
(3) Rempel (2015), (4) Roudier et al. (2014), (5) Švanda et al. (2014)

##  4 Discussion

We have studied how the radial outflow around sunspots, the moat flow, evolves
with sunspot decay. For comparison, we have measured the flow in supergranules
by similar means. In the following, we discuss our results in the context of
the findings reported by other authors, and we investigate the possible
relation between these two large-scale plasma flows.

###  4.1 Flow around fully fledged sunspots

The measured extension of the moat flow around fully fledged sunspots (day one
in our evolutionary study) are in agreement with the results obtained by
Löhner-Böttcher & Schlichenmaier (2013), who applied the same analysis method.
Our mean extension value of 11.9 Mm (see Table 5, first column and Fig. 6a,
change from green to red) is somewhat higher than their mean flow extension of
9.2 Mm. This can be ascribed to our reduced (eight sunspots) sample compared
to theirs (31 sunspots). It is well within their range of values of 5–15 Mm,
however. Other investigations by Sheeley (1972), Brickhouse & Labonte (1988),
Balthasar & Muglach (2010), Švanda et al. (2014) and Verma et al. (2018), for
example, led to a larger flow extension ranging from 10–22 Mm. The measured
profile shape of the horizontal velocity, showing a continuous decrease with
increasing distance from the spot boundary (see Fig. 6a), agrees with the
results by Balthasar et al. (2013), Löhner-Böttcher & Schlichenmaier (2013),
and Rempel (2015). The maximum velocity values, ranging between 677 m s-1 and
1051 m s-1 and measured close to the sunspot boundary (grey shaded area in
Fig. 6a at 1.1 Mm), are comparable to those reported by Löhner-Böttcher &
Schlichenmaier (2013) (0.8 to 1.2 km s-1).

###  4.2 Flow around naked sunspots

As the penumbra dissolves, we measure a decrease in the horizontal velocity
around the spot. This is in agreement with the findings by Balthasar et al.
(2013), who reported on lower flow velocity on the side of the sunspot where
its penumbra has dissolved compared to the outflow velocity where the penumbra
is still present. The measured evolution of the velocity profile (see Fig. 6b)
is also in agreement with the results of the simulations by Rempel (2015). He
reported a horizontal velocity at a distance of 1 Mm to the spot boundary that
increased with distance up to 500 m s-1 to decrease later. We measure a mean
maximum velocity value of 398 m s-1 (see Table 5, second column and Fig. 6b,)
with individual values ranging from 289 m s-1 to 519 m s-1. In Fig. 6b, the
grey shaded area reaches an even higher value because this area displays the
dispersion of all individual profiles for all times during the naked-spot
phase and for all the spots in the sample. In addition, the maximum horizontal
velocity is not found to be at the same distance to the spot boundary for the
different time steps and sunspots. The extension of the flow field reported by
Rempel (2015) does not change in his simulated naked spot. This disagrees with
our measured mean value, which decreases from 11.9 Mm to 8.4 Mm (compare Table
4, Cols. 6 and 7 and Table 5, Cols. 1 and 2), although we also have active
regions that show a more constant flow extension (e.g. AR11646, AR11841, and
AR12163, see Table 4) during sunspot decay. According to our study, the
interaction with surrounding cells (see Sect. 3.1.3) causes the measured
decrease of the flow extension. The discrepancy of the simulations by Rempel
(2015) with our results can be ascribed to the missing influence of
supergranular flows in the surrounding of the naked-sunspot cell in the
simulations. This was also noted by Sheeley et al. (2017), whose observations
and suggestion of supergranules compressing the outflow on one side agree with
our observations and interpretations. Sheeley (1972) and Zwaan (1992) have
also reported about irregularities in the flow cell around sunspots due to
sections of the network penetrating it. In addition, Meyer et al. (1974)
explained discrepancies in the measurement values of decaying sunspots as the
lack of symmetry and incompleteness of the flow field. In our analysis, the
horizontal velocity is obtained as the amplitude of the LOS velocity read out
along circles around the sunspot (see Sect. 2.3). The penetration of a
neighbouring supergranule into the flow region of the sunspot in the direction
to its axis of symmetry will have a stronger influence on the measured value
than the penetration of a supergranule perpendicular to the spot axis of
symmetry. Therefore, we stress that for an irregular cell shape, the
measurement of the flow extension around a naked sunspot, with our analysis
method, will depend on the direction of the penetration of the neighbouring
supergranules.

###  4.3 Flow in the remnant region of dissolved spots

For the six analysed cases in which a flow can be measured after the spot has
dissolved, we measure a mean extension of 6.9 Mm (see Table 5, third column
and Fig. 6c, change from green to red) and a mean maximum horizontal velocity
of 250 m s-1. This means that after sunspot decay, the remaining flow cells
have a mean diameter of 13.8 Mm. These cells are mostly outlined by the
magnetic network as seen in magnetograms (see animations in Appendix A and
Figs. 7 \- 14, centre panels). Directly after the disappearance of the spot as
seen in the intensity maps, it can be distinguished from a regular network
cell (supergranule) because of the enhanced magnetic flux remaining from the
spot. This flux is then spread in time, possibly advected by convection as
described by van Driel-Gesztelyi & Green (2015). To interpret the remaining
velocity profile (Fig. 6c), we recall that the grey shaded area takes into
account all individual profiles obtained for the different remnant cells and
the maximum values are not localised at the same distance to the cells’
centre. Therefore, the individual profiles would show intersections, that is,
some might reach maximum velocity closer to the centre and show a faster
decreasing radial velocity profile than others.

Pardon et al. (1979) reported on 2 (out of 18) moat flows that persisted for
two days after the spot disappeared, as seen in magnetograms. We are not aware
of any other study on the remnant flow cell of a decaying sunspot. Because of
the evident similarities between supergranules and these remnant cells as seen
in magnetograms and Doppler maps, we compare the results we obtained for both
of them in Sect. 4.5.

Figure 6: Mean horizontal velocity of the flow field of (a) fully fledged
sunspots, (b) naked spots, (c) dissolved spots, and (d) supergranules.

###  4.4 Supergranular flow

Our study of the flow in 14 supergranules led to a mean diameter of 17.6 Mm
for supergranular cells (see Table 5 last column). This is smaller than the
mean cell diameter of 25 Mm reported by Roudier et al. (2014). Nevertheless,
our individual values, in the range of 12.2 Mm to 25.8 Mm, are within the
diameter of supergranular cells measured by Simon & Leighton (1964) and
Roudier et al. (2014), which vary from 10 Mm to 45 Mm. We surmise that our
lower mean value is due to our limited sample and the manual selection of the
single cells. This was done by eye, and therefore cells with approximately the
same size have been selected. Hirzberger et al. (2008) also ascribes the wide
range of cell sizes to the different analysis methods reported by different
authors. The horizontal velocity profiles measured for the supergranular flows
resemble each other. They all increase with the distance to the cell centre,
reach a (averaged over all samples) mean maximum velocity of 355 m s-1 , and
decrease for larger distances (see Fig. 6d). Our supergranular velocity
profiles share a similar shape and maximum velocity with those by Orozco
Suárez et al. (2012), who fitted their profiles to the supergranular
convection kinematic model by Simon & Weiss (1989). Yet, our flow cells are
somehow smaller. Our maximum velocities also agree with those measured by
Švanda et al. (2014).

###  4.5 Supergranular flow and flow in decaying active regions

By comparing the radial velocity profile of the supergranular flow and the
flow field around sunspots after the penumbra has fully dissolved, the similar
curve progression becomes obvious (see Fig. 6b and d). The mean maximum
velocity of the flow around the remaining naked spot of 398 m s-1 is somewhat
higher than the mean maximum horizontal flow velocity in supergranules (355 m
s-1) (see Table 5, Cols. 2 and 4), but still within the range of single
measured values. The mean value of the flow extension around the naked spot of
8.4 Mm is of the same size as the measured extension of a mean supergranule
(8.8 Mm).

Once the decaying spot dissolves, the remaining cell certainly resembles a
network cell in magnetograms. Comparison of the individual cases shows that
the remaining flow in the active regions squeezed by the surrounding
supergranules do not share similarities with standard supergranular flows (see
e.g. AR11646 and AR12013, Table 4). For spots whose cell persists after the
spot has dissolved, the remaining flow values (see Table 5, Col. 3) agree with
the lower range values of supergranular flows, however. This also becomes
apparent by comparing the profiles of the horizontal velocity (see Fig. 6c).
The profile for the remnant flow region is flatter, but the maximum velocity
is not localised at the spot boundary.

###  4.6 Moat flow decay: interaction with the surroundings

Our results do not allow us to rule out at this point any of the scenarios
propsed by Sheeley (1972) or Vargas Domínguez et al. (2008), as described in
Sect. 1. We clearly corroborate the intimate relation between the decay of the
moat flow around sunspots and the disappearance of the penumbra during active
region decay.

The close agreement of the properties measured in the flow around naked
sunspots and the flow in supergranules leads us to conjecture that a
supergranule takes over the remaining flow cell in a decaying active region.
This remaining flow appears to be more stable as long as the naked spot is
still present, and it weakens when the spot fully dissolves. The interaction
with the surrounding supergranules is observed to result in the expansion and
eventual fragmentation of the flowing cell or its merging as a result of
implosion, as also described by Simon & Weiss (1968) and De Rosa & Toomre
(2004). In this process, the remaining magnetic flux would either be spread
over a larger area (Harvey & Harvey 1973) by advection or be concentrated,
hence becoming part of the network, respectively (see Fig. 5).

Our results lead us to conclude that the presence of a flow cell enclosing a
spot and how it evolves strongly depends on the behaviour of the surrounding
competing supergranules and related magnetic network, that is to say, it
strongly depends on the surrounding environment and not only on the presence
of the sunspot itself. Our findings are in line with those by Simon & Weiss
(1968), who concluded from their observations that the shape of the moat
depends on the magnetic surrounding and ascribed breaks in the moat to
substantial magnetic structure nearby.

##  5 Summary and conclusions

From the detailed analysis of the flow velocity, especially the variation in
radial velocity profile and the flow extension, the evolution of the flow
field around sunspots can be described as follows: For fully fledged sunspots,
the flow is characterised by a continuous outwards plasma motion, the moat
flow. The horizontal velocity component is maximum close to the sunspot as
long as the penumbra is present, and decreases while the sunspot decays. The
moat flow generates a moat cell around the spot, which is generally outlined
by a magnetic boundary. Network cells (supergranules) outlined by the magnetic
network in the surrounding area interact with the moat flow. Expanding network
cells push the moat flow backwards, towards the sunspot, thus destroying the
annular form of the moat cell. The moat flow is mostly strong enough to
reinstall and also repel the competing network (supergranular) cells. This
leads to the asymmetric shape of most moat cells.

When the penumbra dissolves, the flow profile changes. As the naked spot
remains, the surrounding outflow resembles the flow in supergranules, that is,
the maximum velocity is now found at larger distance from the spot boundary.
The interaction between this flow field and the surrounding network cells
remains. Without penumbra and decreasing flow velocities, however, the
neighbouring cells are able to further push back the outward-moving plasma
towards the decaying sunspot. This can lead to the deformation of the cell. In
some cases, the network cells are even squeezed until they reach the boundary
of the naked spot. If the remaining outflow is still strong enough, a cell
smaller than the original moat cell prevails, which then dissolves within
supergranular timescales.

Early studies by Meyer et al. (1974), Simon & Leighton (1964), and Sheeley
(1972) have proposed the possible coupling between the supergranular and moat
flows. Our study does not allow for a definite proof of this relation. Our
findings, making use of HMI data, give indications of the moat flow
transitioning into a supergranular flow when the penumbra disappears or
sunspot decays, however, which supports the findings and scenarios proposed by
those authors.

The supergranular convection kinematic model by Simon & Weiss (1989) also
agrees with our findings for the flow in supergranules. This also holds for
the results of Orozco Suárez et al. (2012), although our flow cells are
somehow smaller.

In conclusion, sunspots embedded in a moat cell decay into naked spots
embedded in a supergranule. The supergranule can remain after the naked spot
has dissolved or becomes squeezed by surrounding supergranular cells, in the
very same manner in which standard supergranular cells compete against each
other.

###### Acknowledgements.

HS likes to thank J. Löhner-Böttcher for sharing his codes and for the helpful
discussions. The authors thank R. Schlichenmaier for the careful reading of
the manuscript and discussions. HS has been funded by SOLARNET, an EU-FP7
integrated activity project and the DFG project RE 328 2/1-2. NBG acknowledges
financial support by the Senatsausschuss of the Leibniz-Gemeinschaft, Ref.-No.
SAW-2012-KIS-5 within the CASSDA project. SDO is a mission for NASA’s Living
With a Star (LWS) Program. This research has made use of NASA’s Astrophysics
Data System.

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##  Appendix A Snapshots and animations of sunspot evolution

The snapshots (see Fig. 7 \- 14) show the first time step of the analysed data
set for each of the eight sunspots in the intensity map (left), LOS
magnetograms (middle), and LOS Doppler maps (right), as listed in Table 4. For
each of the spots, an animation with the full temporal evolution of the active
regions

is provided as online material. The spot boundary is outlined by a red contour
at each time. The position of the maximum horizontal velocity and extension of
the horizontal flow, that is, the cell size as determined with our analysis
method, are marked by green and pink contours, respectively. The white arrow
in the intensity map points in direction of disc centre and therein visualises
the movement of the spot across the solar disc.

Figure 7: Snapshots of the evolution AR11641 in intensity maps (left), LOS
magnetograms (middle), and LOS Doppler maps (right). The position of the
maximum horizontal velocity and extension of the horizontal flow are marked by
green and pink contours, respectively. The white arrow in the intensity map
points in the direction of disc centre. Figure 8: Same as Fig. 7, except for
AR11646. Figure 9: Same as Fig. 7, except for AR11676. Figure 10: Same as Fig.
7, except for AR11841. Figure 11: Same as Fig. 7, except for AR12013. Figure
12: Same as Fig. 7, except for AR12163. Figure 13: Same as Fig. 7, except for
AR12169. Figure 14: Same as Fig. 7, except for AR12170.

##  Appendix B Figures of supergranules

Figure 15: Snapshots of three supergranules in intensity maps (left), LOS
magnetograms (middle), and LOS Doppler maps (right). The position of the
maximum horizontal velocity and extension of the horizontal flow, i.e., the
cell size as determined with our analysis method, is marked by green and pink
contours, respectively. The black arrow in the intensity map points in the
direction of disc centre.


]https://www.aovgun.com

# Cosmology with variable $G$ and nonlinear electrodynamics

Gabriel W. Joseph<EMAIL_ADDRESS>Physics Department, Eastern
Mediterranean University, Famagusta, North Cyprus, 99628 via Mersin 10, Turkey
Ali Övgün<EMAIL_ADDRESS>[ Physics Department, Eastern Mediterranean
University, Famagusta, North Cyprus, 99628 via Mersin 10, Turkey

###### Abstract

In a bid to resolve lingering problems in cosmology, more focus is being
tilted towards cosmological models in which physical constants of nature are
not necessarily real constants but vary with cosmic time. In this paper, we
study a cosmological model in nonlinear electrodynamics with Newton’s
gravitational constant $G$, which is not a constant but varies in terms of a
power law of the scale factor of the universe. Hence, the evolution of the
scale factor $a(t)$ is studied in this model, which depends on a fine-tuning
term of nonlinear electrodynamics, $\alpha$. Finally, we verify the stability
of the model using the speed of sound.

Cosmology; Nonlinear electrodynamics; Inflation; Acceleration of the universe;
Causality; Classical stability; Variable $G$

###### pacs:

95.36.+x, 98.80.-k

##  I Introduction

Remarkable achievements recorded in the field of cosmology in the past decades
can be attributed to observations of cosmic microwave background (CMB)
radiation and type-Ia supernovae, suggesting that cosmic expansion is
accelerating supernova1 . It is well-known that Maxwell’s electrodynamics, as
a source of the Friedmann-Robertson-Walker spacetime, which is the central to
the standard cosmological model (SCM), causes an initial singularity: the
breakdown in geometrical structure of space and time at a finite time in the
past. This initial singularity around the very early universe has revealed an
unpredictable large value of the energy density and curvature of spacetime,
thus departing from the domain of the SCM. This setback has given rise to
other secondary cosmological problems, such as the horizon, flatness, and
monopole problems. The secondary problems have been resolved by using scalar
fields. Moreover, magnetic universe models, which portray no initial
singularity due to strong electromagnetic field in the modified nonlinear
electrodynamics (NED) in the early universe, have been used in the literature
to resolve this puzzle DeLorenci:2002mi ; vol1 ; Kruglov:2015fbl ;
Ovgun:2016oit ; Ovgun:2017iwg ; Otalora:2018bso ; Sarkar:2020fhp ;
Golovnev:2008cf ; novello1 ; Novello:2006ng ; Kruglov:2017vca ;
Kruglov:2014hpa ; Kruglov:2016cdm ; Kruglov:2016lqd ; Kruglov:2020aje ;
Sharif:2017pdd ; Campanelli:2007cg . The introduction of cosmological constant
$\Lambda$, or scalar field with a type of potential function or modifying the
theory of gravity with the aid of some function $f(r)$, can effectively
explain the acceleration of the universe born1 .

All these models are not without their own setbacks Poplawski:2012ab ;
Poplawski:2014dea ; Oztas:2018jsu ; Dil:2019qos . For instance, the smallness
of the cosmological constant when compared with the vacuum energy is very
difficult to explain. Furthermore, the choice of $f(r)$ function in the
modified gravity is not unique. Moreover, inflationary cosmological models
provide convincing solutions to the horizon problem, flatness problem, and
small value of cosmological constant, no model of inflation has been
experimentally confirmed.

In scalar-tensor theory, much potential leads to cosmic inflation and
acceleration of the universe. When NED is coupled to a gravitational field, it
can give the necessary negative pressure and enhance cosmic inflation.
Therefore, it is interesting to explore a new model of NED with variable
gravitational constant where stochastic magnetic field drives the inflation of
the universe.

Magnetic fields at different levels in the past have effects on various
cosmological and astrophysical phenomena observed in the universe. Throughout
the universe, physical processes over a large range of scales are influenced
by magnetic fields in the universe. The present and associate magnetic fields
could therefore arise from the flow of materials in the high-temperature
vicinity of the big bang. This corresponds to the primordial magnetic fields
(PMF). Adequate knowledge of the PMF offers information about the early
universe between the inflationary and recombination eras. Such magnetic fields
originate from cosmic phase transition with the bluer spectrum and vector
potential produced during the inflation era. Stochastic inconsistencies of EM
field in relativistic electron-positron plasma produce PMF from thermal
instabilities in the pre-recombination plasma. The plasma sustains the
magnetic fluctuations in the early universe before the era of Big Bang
nucleosynthesis during the radiation-dominated era.

Using the new type of NED containing a stochastic magnetic background with
nonvanishing $B^{2}$ gives cosmological models void of initial singularities.
Other cosmological models lacking a primordial singularity, such as non-
minimal coupling, quantum gravity effect, Lagrangian with quadratic term,
inflation by scalar fields, and NED without modification of general
relativity, have been introduced in the literature to solve the puzzle of
cosmology and mystery of inflation muk ; cos1 ; cos2 ; born1 ; camara ; eliz ;
coupling ; durrer ; kunze ; Azri:2017uor ; Azri:2018qux ; Azri:2014apa ;
Hernandez-Almada:2020ulm ; Garcia-Aspeitia:2019yni ; Vazquez:2018qdg ;
Sert:2019qfl ; Adak:2016led ; Okcu:2020ybv ; Cruz:2020hkh ; Cruz:2018xzn ;
Cruz:2017bcv ; Cordero:2013pua .

Max Born and Leopold Infeld used the idea of Gustav who in 1909 began
constructing a pure electromagnetic theory of charged particles to propose a
new theory in 1934: fully relativistic and gauge-invariant nonlinear
electrodynamics Born:1934gh . Born-Infeld proposed a nonlinear fields
Lagrangian with an interesting attribute of transforming to Maxwell’s theory
for low electromagnetic fields. Since there are no new degrees of freedoms,
such as scalar fields or branes, works of cosmology described by NED should
have some interesting features of cosmic importance. The sources of cosmic
inflation can be traced to nonlinear electromagnetic radiation which is
explained by modified Maxwell’s equations. When coupled with the gravitational
fields, NED may give negative pressure and also can lead to cosmic inflation
born1 . The evolution of the universe, when explored with a new NED model such
that electromagnetic field coupled to gravitational field prevents cosmic
singularity at the big bang. The electromagnetic and gravitational fields were
very strong during the evolution of the early universe, thereby leading to
quantum correction and giving birth to NED camara ; durrer ; kunze . One can
present a model of the universe filled with a Born-Infeld type of nonlinear
electromagnetic field, which is inhomogeneous and anisotropic without
singularity.

Recently, interest is being geared towards cosmological models in which
physical constants of nature are varying with time Barrow:1998df ;
Albrecht:1998ir ; Cohen:1998zx . For instance, in the varying speed of light
(VSL) theories, where the speed of light is spatial, temporal and wavelength-
dependent, most pending problems of standard cosmological models are being
resolved without considering inflation Moffat:2014poa ; Leszczynska:2017bnj ;
Camara:2007zz . The VSL theories, which deal with the shortcomings of
inflation, have not address issues related to the initial big bang
singularity. The Newtonian gravitational constant $G$ occurs in the source
term of Einstein’s field equation of the general theory of relativity, which
is a fundamental equation for developing every model of cosmology. In
Einstein’s field equation, $G$ acts as a coupling constant between the
geometry of spacetime and matter. In quantum mechanics, $G$ is essential in
the definition of the Planck constant Kantha:2016ylw . While in SCM, $G$ is an
invariant quantity. It has been noted that there is significant evidence that
the gravitational constant $G$ can vary in time Gv1 . Dirac, in 1937, argued
that variations in $G$ of about 5 parts in $10^{11}$ per annum could explain
the relatively small strength of the gravitational force compared to other
fundamental forces of nature. In order to unify gravitation and elementary
particle physics, Einstein’s theory with time-varying $G$ is already in the
literature weyl ; dirac ; Leszczynska:2017bnj ; Singha:2007ey ; ggtjp ;
Malekjani:2015yzp ; Sheykhi:2012zzb ; Lu:2014rza ; Sharif:2012dr ; Lu:2009iv ;
Sheykhi:2010jn ; Dungan:2009fp ; Borah:2014gca . Singh ggtjp  has presented a
cosmological model with $G\sim a^{2/\alpha}$, where $a$ is the scale factor of
the universe and $\alpha$ a constant. When either $G$ or the cosmological
constant $\Lambda$ is varied with time, the Einstein field equation is still
preserved. It has been shown that, the variable Newtonian gravitation constant
can account for dark energy and most of its effects, and current dynamical
dark energy models using time-dependent cosmological constant terms are being
considered Dungan:2009fp .

In this paper, we study a model involving non-linear electrodynamics coupled
to general relativity, additionally assuming a time-dependent gravitational
constant, and study its cosmological dynamics, in order to unify the different
epochs during the evolution of the universe. We show that variable $G$ in the
study of nonlinear electromagnetic radiation is a source of inflation in the
early universe.

The structure of the paper is as follows: in section II, we briefly introduce
the cosmology of a universe filled with nonlinear magnetic monopole fields. In
section III, we obtain the evolution of the universe filled with nonlinear
magnetic monopole fields and variable gravitational constant. In section IV,
we check the stability of the model and give our conclusion in section V.

##  II Non-linear magnetic monopole fields and cosmology

In nonlinear electrodynamics, we define the Lagrangian density by novello1

$\mathcal{L}_{NED}=-\frac{\mathcal{F}^{\alpha}}{4},$ (1)

where $\alpha$ is the electromagnetic fine structure constant and
$\mathcal{F}$ denotes an invariant quantity known as the Maxwell invariant.
Since the matter part of the Lagrangian is independent of the metric’s
derivatives, in tensorial language the matter energy-momentum definition using
(1) is given as Kruglov:2017vca

$T_{\mu\nu}=-K_{\mu\lambda}F_{\nu}^{~{}\lambda}+g_{\mu\nu}\mathcal{L}_{NED},$
(2)

with

$K_{\mu\lambda}=\frac{\partial\mathcal{L}_{NED}}{\partial\mathcal{F}}F_{\mu\lambda},$
(3)

where $g_{\mu\nu}$ is the metric of spacetime and the indices $\mu$ and $\nu$
run from 0 to 3. Here, it is assumed that on the cosmic background, there
exists a dominant stochastic magnetic field whose wavelengths are less than
its curvature. Hence, the mean electromagnetic fields then become the source
of Einstein equations tolman . The averaged electromagnetic fields are given
by DeLorenci:2002mi :

$\langle E\rangle=\langle B\rangle=0,\langle E_{i}B_{j}\rangle=0,$ (4)
$\langle E_{i}E_{j}\rangle=\frac{1}{3}E^{2}g_{ij},\langle
B_{i}B_{j}\rangle=\frac{1}{3}B^{2}g_{ij},$

where $\langle$ $\rangle$ denotes averaging brackets used for taking mean of
volume and the indices $i$ and $j$ run from 1 to 3. The wavelength of
radiation is considered to be lower than the volume and the volume smaller
than the curvature.

However, the case of real nonlinear magnetic monopole is when $E^{2}=0$.
Therefore, as obtained from equation (1), the energy density
$\rho=-T^{0}_{~{}0}$ and the pressure $p=T^{i}_{~{}i}/3$ of the nonlinear
monopole magnetic field is Ovgun:2016oit

$\displaystyle\rho_{NED}=-\mathcal{L}_{NED},$ (5) $\displaystyle
p_{NED}=\mathcal{L}_{NED}-\frac{2B^{2}}{3}\frac{\partial\mathcal{L}_{NED}}{\partial\mathcal{F}},$
(6)

where the definition of $\mathcal{L}_{NED}$ is given in Eq. (1) with
$\mathcal{F}=B^{2}/2$.

From the above equations, we obtain the energy density equation $\rho$ and
pressure $p$ as thus:

$\begin{array}[]{l}\rho=\rho_{NED}=\frac{2^{-\alpha}(B^{2})^{\alpha}}{4}\\\
p=p_{NED}=\frac{2^{-\alpha}}{12}(B^{2})^{\alpha}\left(4\hskip
1.69998pt\alpha-3\right).\end{array}$ (7)

##  III Cosmology with variable $G$ and nonlinear electrodynamics

In varying $G$ theories, the action is still

$S=\int dx^{4}\left(\sqrt{-g}\left(\frac{R}{16\pi
G}+\mathcal{L}_{NED}\right)\right).$ (8)

Taking the variation of the action with respect to the metric and ignoring
surface terms leads to

$G_{\mu\nu}-g_{\mu\nu}\Lambda=\frac{8\pi G}{c^{4}}T_{\mu\nu}.$ (9)

In cosmological context, the Friedmann Robertson Walker metric is written as

$ds^{2}=-c^{2}dt^{2}+a(t)^{2}\left[\frac{dr^{2}}{1-Kr^{2}}+r^{2}d\Omega\right],$
(10)

where $a(t)$ is the scale factor, $t$ is the comoving time and $K=0,1,-1$
represent a flat, closed and open FRW universe, respectively.

For the case of flat FRW $(K=0)$ and $c=1$, the Einsteins equations are camara
; durrer ; kunze ,

$H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G(t)}{3}\rho,$ (11)
$\frac{\ddot{a}}{a}=-\frac{4\pi G(t)}{3}(\rho+3p).$ (12)

where $H$ represents the Hubble parameter and dot is the differentiation with
respect to time.

However, the conservation equation that follows from (11)-(12) is for time
variation in $G(t)$ is now Barrow:1998df ; Albrecht:1998ir :

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\rho\frac{\dot{G}}{G},$ (13)

where $\omega=\frac{p}{\rho}$ denotes the equation of state parameter for the
dark energy.

Without diving into the dynamics of variable, we shall use Barrow’s ansatz in
which the gravitational constant $G$ varies in form of the power-law of the
scale factor as Barrow:1998df ; Albrecht:1998ir :

$G=G_{0}a^{m},$ (14)

where $G_{0}$ is a positive constant. Since we know that $G$ increases with
time, $m$ must be positive. Furthermore, since $G$ depends on the scale factor
of the universe, its time derivatives $\dot{G}$ must be greater than zero.

From the conservation equation (13), we obtain:

$-\frac{\partial\mathcal{L}_{NED}}{2\partial\mathcal{F}}\cdot\left(\frac{\mathrm{d}}{\mathrm{d}t}((B(t))^{2})+4\hskip
1.69998pt\frac{B(t)^{2}\dot{a}}{a}\right)-\frac{\mathcal{L}_{NED}\dot{G}}{G}=0.$
(15)

The solution to the above equation gives an important relation between $B(t)$
and $a(t)$ as follows:

$B(t)=a(t)^{-1/2\hskip 1.18999pt\frac{4\hskip
0.84999pt\alpha+m}{\alpha}}B_{0}.$ (16)

Conveniently, when written in terms of the scale factor, the evolution of
energy density and pressure are given by:

$\rho=\frac{2^{-\alpha}}{4}\left(\mathit{B_{0}}^{2}a(t)^{\frac{-4\hskip
0.84999pt\alpha-m}{\alpha}}\right)^{\alpha},$ (17)
$p=\frac{2^{-\alpha}}{12}\left(\mathit{B_{0}}^{2}a(t)^{\frac{-4\hskip
0.84999pt\alpha-m}{\alpha}}\right)^{\alpha}\left(4\hskip
1.69998pt\alpha-3\right).$ (18)

Then we have:

$\rho+p=\frac{2^{-\alpha}}{3}\left(\mathit{B_{0}}^{2}a(t)^{\frac{-4\hskip
0.84999pt\alpha-m}{\alpha}}\right)^{\alpha}\alpha,$ (19)
$\rho+3p=2^{-1-\alpha}\left(\mathit{B_{0}}^{2}a(t)^{\frac{-4\hskip
0.84999pt\alpha-m}{\alpha}}\right)^{\alpha}\left(2\hskip
1.69998pt\alpha-1\right),$ (20)

and the EoS parameter $\omega$ is

$\omega=\frac{4}{3}\alpha-1.$ (21)

Eq. (21) shows some interesting cases of the universe. It follows that at
$\alpha=0$, $\omega=-1$, is a cosmological constant case, at

$\alpha=0.5$, $\omega=-1$/3 indicates case of dark energy and at $\alpha=1$,
$\omega=1/3$ denotes the case of ultra-relativistic. The matter content of the
universe is related to its acceleration equation:

$\frac{\ddot{a}}{a}=-\frac{4\pi G(t)}{3}(\rho+3p).$ (22)

On checking the singularity in energy density and pressure at $a(t)\rightarrow
0$ and $a(t)\rightarrow\infty$, we obtain,

$\lim_{a(t)\rightarrow 0}\rho(t)=\lim_{a(t)\rightarrow 0}p(t)=0,$ (23)
$~{}~{}\lim_{a(t)\rightarrow\infty}\rho(t)=\lim_{a(t)\rightarrow\infty}p(t)=0.$
(24)

Using the Einstein’s field equation and energy momentum tensor, the Ricci
Scalar $R$ which gives the curvature of spacetime is calculated as follows:

$R=8\hskip 1.69998pt\pi\hskip 1.69998pt\mathit{G_{0}}\hskip
1.69998pta(t)^{m}(\rho-3p).$ (25)

The Ricci tensor squared $R_{\mu\nu}R^{\mu\nu}$and the Kretschmann scalar
$R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$ are also obtained Ovgun:2016oit
as:

$R_{\mu\nu}R^{\mu\nu}=(8\hskip 1.69998pt\pi\hskip
1.69998pt\mathit{G_{0}}\hskip 1.69998pta(t)^{m})^{2}(\rho^{2}+3p^{2}),$ (26)
$R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}=(8\hskip 1.69998pt\pi\hskip
1.69998pt\mathit{G_{0}}\hskip
1.69998pta(t)^{m})^{2}\left(\frac{5}{3}\rho^{2}+2\rho p+3p^{2}\right),$ (27)
$\lim_{a(t)\rightarrow 0}R(t)=\lim_{a(t)\rightarrow
0}R_{\mu\nu}R^{\mu\nu}=\lim_{a(t)\rightarrow
0}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}=0.$ (28)

The nature of the scale factor gives the behaviour of the curvature scalar.
Taking the limit of the above equations as the universe accelerates at
$a(t)\rightarrow 0$, we obtain no singularities in the curvature scalar, Ricci
tensor and the Kretschmann scalar.

##  IV The Evolution of the Scale Factor of the Universe

The first Friedmann equation with variable $G(t)$ for the flat universe is
given by

$H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G(t)}{3}\rho.$ (29)

When a particle moves in one dimension in a potential $V(a)$, the equation of
motion is

$\dot{a}^{2}+V(a)=0.$ (30)

The potential function

$V(a)=-\frac{1}{3}\hskip 1.69998pt2^{1-\alpha}\pi\hskip 1.69998ptG_{0}\hskip
1.69998pt(a(t))^{m+2}\left(B_{0}^{2}(a(t))^{\frac{-4\hskip
0.84999pt\alpha-m}{\alpha}}\right)^{\alpha},$ (31)

is negative and possesses a maximum at $a=a_{c}=-C_{1}$.

Using the (14) and (5), it becomes

$\hskip 1.69998pt\frac{-2^{1-\alpha}\pi\hskip 1.69998ptG_{0}\hskip
1.69998pta^{-4\hskip 1.18999pt\alpha+2}B_{0}^{2\hskip 1.18999pt\alpha}+3\hskip
1.69998pt\dot{a}^{2}}{3a^{2}}=0,$ (32)

then we find the scale factor $a(t)$ is equal to

$\displaystyle a(t)\approx\frac{1}{2}\hskip
1.69998pt\sqrt{B_{0}}2^{\frac{3}{4}}2^{\frac{3}{4\alpha}}\left(G_{0}\hskip
1.69998pt\alpha^{2}(C_{2}-t)^{2}\right)^{\frac{1}{4\alpha}}3^{-\frac{1}{4\alpha}}\pi^{\frac{1}{4\alpha}}.$
(33)

Equation (33) shows that at $t{\rightarrow}0$, the scale factor
$a(t){\rightarrow}$

$\displaystyle\frac{1}{2}\hskip
1.69998pt\sqrt{B_{0}}2^{\frac{3}{4}}2^{\frac{3}{4\alpha}}\left(G_{0}\hskip
1.69998pt\alpha^{2}C_{2}^{2}\right)^{\frac{1}{4\alpha}}3^{-\frac{1}{4\alpha}}\pi^{\frac{1}{4\alpha}}.$
(34)

This implies that the size of the universe $a(t)$ was never zero.

Figure 1: Plot of the scale factor $a$ versus the time $t$ (for
$B_{0}=G_{0}=C_{2}=m=1$).

In the early universe, there is a de Sitter phase because of the nonlinear
corrections to Maxwell’s theory, as shown in Fig.1. Thus, for $\alpha<1$, the
universe enters into its accelerating expansion phase which indicates the
present dark energy dominated epoch. When $\alpha\geq 1$ present a radiation-
dominated era with the universe in the phase of decelerating expansion
Carroll:1997ar .

Introducing the quantity $q$ (the deceleration parameter) Carroll:1997ar , we
described the expansion of the universe by:

$q=-\frac{\ddot{a}a}{(\dot{a})^{2}}=9/2\hskip 1.69998pt\frac{\rho+3\hskip
1.69998ptp}{a\rho}.$ (35)

There is a inflation phase for $q<0$ for $\alpha<1$ and deceleration phase for
$q>0$ for $\alpha>1$ shown in Fig.2.

Figure 2: Plot of the deceleration parameter $q$ versus the scale factor $a$
(for $B_{0}=G_{0}=C_{2}=m=1$).

To estimate the amount of the inflation, we use the definition of e-foldings

$N=\ln\frac{a(t_{end})}{a(t_{in})}$ (36)

where $t_{end}$ is the time inflation ends while $t_{in}$ is the time it
begins. For $N\simeq 70$ e-folding, the cosmic flatness and horizon problems
can be resolved. Hence, we obtain the scale factor for beginning time of
inflation (for $m=1,\alpha=1,G_{0}=1,B_{0}=1$):

$a(t_{in})=3.46\times 10^{-31}.$ (37)

Using the second Friedmann equation Eq. (22) which is known as the
acceleration equation for the universe:

$\frac{\ddot{a}}{a}=\frac{(1-2\hskip 1.69998pt\alpha)}{4\alpha^{2}t^{2}}.$
(38)

It is clear that acceleration stops at $\alpha=0.5$.

##  V Test of Causality

For any cosmological model to survive, it is an established that the speed of
sound can not exceed the local speed of light, $c_{s}\leq 1$. The second
requirement for stability is that the square of the speed of sound must be
positive, i.e., $c^{2}_{s}>0$ . In case the model is a classically stable one,
sound  at $E=0$, we obtain:

$c_{s}^{2}=\frac{dp}{d\rho}=\frac{dp/d\mathcal{F}}{d\rho/d\mathcal{F}}=-\frac{7}{3}+\frac{4}{3}\alpha.$
(39)

A requirement of the classical stability $c^{2}_{s}>0$ is $\alpha>\frac{7}{4}$
and the causality $c_{s}\leq 1$ is $\alpha\leq\frac{5}{2}$. Hence, the best
value of $\alpha$ for both stability conditions are
$\frac{5}{2}\geq\alpha>\frac{7}{4}$.

##  VI Conclusion

This work studied cosmology with a varying gravitational constant $G$ and NED
in a flat FRW universe. Under changes to the scale factor, the evolution of
the magnetic field reduced to $B(t)=a(t)^{-1/2}B_{0}$, as obtained in
Kruglov:2017vca  where $m=0$. The evolution of the scale factor shows that the
models give an accelerating, expanding universe with

$\displaystyle a(t)\approx\frac{1}{2}\hskip
1.69998pt\sqrt{B_{0}}2^{\frac{3}{4}}2^{\frac{3}{4\alpha}}\left(G_{0}\hskip
1.69998pt\alpha^{2}(C_{2}-t)^{2}\right)^{\frac{1}{4\alpha}}3^{-\frac{1}{4\alpha}}\pi^{\frac{1}{4\alpha}},$
(40)

where $B_{0}$ represents the magnetic induction field at present time $t_{0}$,
and $\alpha$ is a free parameter presented in Fig. 1. The value of the scale
factor obtained indicates that the size of the universe was never zero. By
incorporating NED and variable gravitational constant, we present a model free
from the initial big bang singularity and inflation. As observed in equations
(23), (24), and (28), the models portray no singularity in the energy density,
pressure, and curvature terms, respectively. For $\alpha\geq 1$, the universe
is in a radiation-dominated era with the phase of decelerating expansion.
However, when $\alpha=0.5$, $\omega=-1$/3 indicates a transition into the
present epoch of dark energy with accelerating expansion. Furthermore, we
studied the stability of this model and observed that it depends of the
constant $\alpha$ and is classically stable for
$\frac{5}{2}\geq\alpha>\frac{7}{4}$. In future studies, we intend to
investigate the evolution of the universe with both varying $G$ and $c$ in
NED.

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