11institutetext: Dept. d’Astrophysique/AIM, CEA/IRFU, CNRS/INSU, Université
Paris et Paris-Saclay,
91191 Gif-sur-Yvette Cedex, France 22institutetext: IRAP, Université Toulouse
III - Paul Sabatier, CNRS, CNES, Toulouse, France 33institutetext: Université
Paris-Saclay, CNRS, Institut d’Astrophysique Spatiale, 91405, Orsay, France

# Impact of far-side structures observed by Solar Orbiter on coronal and
heliospheric wind simulations

B. Perri 11 A. Finley 11 V. Réville 22 S. Parenti 33 A. S. Brun 11 A.
Strugarek 11 É. Buchlin 33

(Received ??; accepted ??)

###### Abstract

Context. Solar Orbiter is a new space observatory that provides unique
capabilities to understand the heliosphere. In particular, it has made several
observations of the far-side of the Sun and therefore provides unique
information that can greatly improve space weather monitoring.

Aims. In this study, we aim to quantify how the far-side data will affect
simulations of the corona and the interplanetary medium, especially in the
context of space weather forecasting.

Methods. To do so, we focused on a time period with a single sunspot emerging
on the far-side in February 2021. We used two different input magnetic maps
for our models: one that includes the far-side active region and one that does
not. We used three different coronal models typical of space weather modeling:
a semi-empirical model (potential field source surface or PFSS) and two
different magnetohydrodynamic models (Wind Predict and Wind Predict-AW). We
compared all the models with both remote sensing and in situ observations in
order to quantify the impact of the far-side active region on each solution.

Results. We find that the inclusion of the far-side active region in the
various models has a small local impact due to the limited amount of flux of
the sunspot (at most 8% of the total map flux), which leads, for example, to
coronal hole changes of around 7% for all models. Interestingly, there is a
more global impact on the magnetic structure seen in the current sheet, with
clear changes, for example, in the coronal hole boundaries visible in extreme
ultra-violet (EUV) on the western limb, which is opposite to the active region
and the limb most likely to be connected to Earth. For the Wind Predict-AW
model, we demonstrate that the inclusion of the far-side data improves both
the structure of the streamers and the connectivity to the spacecraft.

Conclusions. In conclusion, the inclusion of a single far-side active region
may have a small local effect with respect to the total magnetic flux, but it
has global effects on the magnetic structure, and thus it must be taken into
account to accurately describe the Sun-Earth connection. The flattening of the
heliospheric current sheet for all models reveals that it causes an increase
of the source surface height, which in return affects the open and closed
magnetic field line distributions.

###### Key Words.:

Magnetohydrodynamical simulations – Solar physics – Solar coronal streamers –
Solar corona – Solar extreme ultraviolet emission – Solar atmosphere

##  1 Introduction

Space weather forecasting is usually performed using a chain of models that
connect the solar surface up to the near-Earth space environment, such as in
the Virtual Space Weather Modeling Center (VSWMC) for Europe (Poedts et al.,
2020), the Solar Wind Modeling Framework (SWMF) for the United States
(Odstrcil, 2003; Tóth et al., 2012), or the SUSANOO model in Japan (Shiota et
al., 2014). The first element of this chain is the coronal model, which
typically takes as input some measurements between 1 and 3 solar radii
(magnetic field in most cases but sometimes density or white-light images are
used, as in Réville et al. (2023)), and it provides as output the state of the
solar corona near 20 solar radii. This output is then used to drive
heliospheric models that propagate the solar wind along with transients, such
as coronal mass ejections (CMEs), out to the Earth’s orbit at one astronomical
unit. The coronal part is one of the most crucial elements of this chain of
modeling, not only because it is the source of the solar wind structures that
will influence the rest of the chain but also because it is one of the regions
that contains the most challenging physics. Different modeling can thus
drastically change the propagation of transients and the description of the
space weather situation at Earth (Samara et al., 2021).

To achieve an accurate description of the solar wind and magnetic field in the
solar corona, there are many different possible approaches. The least
resource-intensive technique is to use empirical extrapolations such as the
WSA method (Wang-Sheeley-Arge, see Wang & Sheeley, 1990; Arge & Pizzo, 2000),
which computes the wind speed based on an inverse relationship with the flux
tube expansion from a static force-free extrapolation of the coronal magnetic
field. Capturing the non-force free features in the corona requires the use of
magneto-frictional models (Pomoell et al., 2019) or even full-on
magnetohydrodynamic (MHD) models. Even for MHD modelling, there is a wide
range of approximations being used for the heating and acceleration of the
wind. One-dimensional wind models have proved to be interesting for their
capacity of including realistic physics at reasonable computational costs
(Lionello et al., 2001; Suzuki & Inutsuka, 2005; Grappin et al., 2010; Pinto &
Rouillard, 2017). Three-dimensional MHD models have been developed and
improved over the years with systematic comparisons to both in situ and
remote-sensing data for validation, such as AWSoM (van der Holst et al., 2010,
2014; Sachdeva et al., 2019), Wind Predict (Réville et al., 2015a; Perri et
al., 2018; Réville et al., 2020; Parenti et al., 2022), or COCONUT (Perri et
al., 2022, 2023; Kuźma et al., 2023). Other models can be even more
sophisticated, but this comes at a computational cost such that they can no
longer be used in operational space weather forecasting (Mikić et al., 2018).
Some models even go beyond the single fluid approximation by taking into
account the multi-species nature of the solar wind (Usmanov et al., 2014,
2016, 2018; Chhiber et al., 2021). Yet other models have focused on boundary
conditions to provide the most physical conditions for time-dependent models
(Wu et al., 2006), such as in Yalim et al. (2017) and Singh et al. (2018).
(For a somewhat exhaustive list of solar wind models used for space weather
forecasting, please see Reiss et al. (2022).)

In all the aforementioned approaches, coronal and heliospheric simulations are
data driven and therefore suffer when given incomplete data as an input, the
most systematic of which is a lack of far-side observations of the Sun. As a
result, synoptic maps based on solar data used to initialize numerical
simulations are never synchronic, that is, they need to include data from
different time periods in order to cover the full 360 degrees of the Sun in
longitudes. Using asynchronous maps is often a reasonable approximation when
solar activity is close to minimum, as there is little evolution during a
solar rotation. However, at maximum solar activity, this begins to cause
serious issues in the input data, as active regions (ARs) can emerge on the
far-side without being taken into account into the synoptic maps. New
developments in helioseismology are currently trying to address this issue,
but synoptic products based on this method are still in development (Heinemann
et al., 2023). The synchronicity of solar observations was solved briefly by
the two STEREO spacecraft that were spread around the Sun in order to have
full coverage of solar activity. However, due to the loss of STEREO-B in 2014
and the migration of STEREO-A back towards the Earth, there were no far-side
observations until recently with the launch of Solar Orbiter in 2020 (Müller
et al., 2020). Solar Orbiter is currently the only observatory that observes
the Sun with multiple multi-wavelength remote sensing instruments far outside
the Sun-Earth line, which means that there are moments when Solar Orbiter is
observing the far-side of the Sun, thus fully complementing the observations
from the point of view of the Earth.

The impact of including far-side ARs in operational space weather modelling is
thus so far untested. So in order to anticipate the value of upcoming data
products from Solar Orbiter, which incorporate far-side observations, we
performed numerical experiments. As a first step, we selected a case when
solar activity is at a minimum, which allowed us to study a singular AR and to
already quantify its impact in this simple configuration. We therefore
examined a period of time during Solar Orbiter’s cruise phase in which an AR
emerged on the far-side of the Sun but in view of Solar Orbiter. The aim is to
quantify how the inclusion of this region in the input synoptic map influences
the resulting coronal and heliospheric wind simulation, which in turn are
likely to affect CME propagation and space weather forecasts at Earth.

The article is organized as follows. In Section 2, we describe our methodology
for taking into account the far-side AR in the driving of the coronal models.
In Section 3, we briefly describe the models used in this study, that is, the
potential field source surface (PFSS) extrapolation and the two variants of
our 3D MHD solar corona model used in this study, namely, Wind Predict (WP)
and Wind Predict with Alfvén Waves (WP-AW). Detailed descriptions of each
coronal model can be found in Appendix B. We present a comparison of our
results with the remote-sensing observations to constrain the impact of the AR
on the inner corona, focusing on closed and open magnetic structures in
Section 4 and Section 5, respectively. In Section 6, we present a comparison
of our results with the in situ observations to constrain the impact of the AR
on the inner heliosphere. To conclude, our findings are discussed in Section
7, where we offer explanations for the differences observed between the models
before discussing future perspectives to this work in Section 8.

##  2 Methodology

###  2.1 Selection of the event with Solar Orbiter data

Figure 1: Observations of AR12803 during February 2021. The left panel shows
the orbital configuration of the various satellites from February 13 to
February 22, 2021 in the Carrington frame synchronized with the surface of the
Sun. Solar Orbiter was covering the far-side of the Sun during this period.
The upper-right panel shows observations by Solar Orbiter/EUI 174 Å between
February 13 and 15, 2021, where one can see the emergence of the AR. The
lower-right panel shows observations by SDO/AIA 171 Å between February 19 and
21, 2021 when the AR finally entered the Earth field of view. In both cases,
the AR of interest is located inside the red grid.

The goal of this study is to quantify the impact of an emerging far-side AR
observed by Solar Orbiter on data-driven 3D simulations of the heliosphere.
Such an event happened in early 2021 during Solar Orbiter cruise phase. On
February 13, 2021, an AR emerged on the far-side of the Sun, and was observed
by Solar Orbiter EUI instrument (Extreme Ultraviolet Imager, see Rochus et al.
(2020)) while the satellite was in opposition with the Earth. The AR reached
its maximum amplitude on February 15 and finally rotated in the field of view
of the Earth on February 20, where it was labeled AR12803. This sequence is
illustrated in Figure 1. The left panel shows the orbital configuration of the
various satellites from February 13 to February 22, 2021, as viewed from above
in the Sun’s rotating frame of reference. Solar Orbiter (in red) was covering
the far-side of the Sun (Earth in green) during this period. The upper-right
panel shows observations by Solar Orbiter/EUI 174 Å between February 13 and
15, 2021, where the emergence of the AR inside the red grid can be seen. The
lower-right panel shows observations by the Atmospheric Imaging Assembly (AIA)
(Lemen et al., 2012) onboard the Solar Dynamics Observatory (SDO) (Pesnell et
al., 2012) with the 171 Å channel, between February 19 and 21, 2021, when the
AR finally entered the Earth field of view.

We focused specifically on this period because it was close to solar cycle 25
minimum of activity (December 2020). This means that there were no major
erupting structure that could change the global configuration of the
heliosphere over the studied period between February 15 and 22, 2021. Thus,
the underlying assumption of our quasi-static models, that the photospheric
magnetic field does not evolve much and can be reproduced using only a steady-
state solution, is reasonable.

###  2.2 Input magnetic maps selection

In order to quantify the impact of this far-side emerging AR, our approach in
this paper is to compare coronal models initiated with different input data.
We first use a synoptic map describing the magnetic field at the surface of
the Sun before the emergence (so without the AR), and compare it with
simulations driven by a second synoptic map that does include the far-side
region after its emergence.

Solar Orbiter has a magnetograph dedicated to measuring the surface solar
magnetic field called PHI (Polarimetric and Helioseismic Imager; see Solanki
et al. (2020)). The ideal way to assess the impact of the far-side AR would be
to use a magnetic map relying on old asynchronous data and a second map with
updated PHI measurements. Unfortunately, since the event selected was during
Solar Orbiter’s cruise phase and since there are no official HMI/PHI or
GONG/PHI merged products yet, we cannot directly use the Solar Orbiter data.
Instead, we selected two GONG-ADAPT magnetic
maps111https://gong.nso.edu/adapt/maps/ at specific dates to emulate the
emergence of the far-side AR. This is shown in Figure 2. The first map is the
ADAPT map from February 15, 2021, at 12:00 UTC, which was made using data
assimilation from previous observations, but only from the Earth field of
view. This means that, although the AR was emerged at that time, it will not
be visible on the magnetic map. The other input data we chose was again an
ADAPT map, but from February 22 at 22:00 UTC. At this date, the AR was visible
from Earth, and thus is visible also in the map (around 280 degrees of
longitude and 25 degrees of latitude, region marked with an orange box on
Figure 2). It was the main change in the magnetic field configuration over
this period of time, so that the rest of the map outside the AR is very
similar between both maps. For each ADAPT map, we have used only the first
realization (number 0 in the FITS file). Both maps are in Carrington frames,
which means that the longitudes only depend on the current Carrington
rotation. For clarity, we also show on the bottom panel the filtered maps that
we use to initialize the various models (spherical harmonics reconstruction of
the field with $\ell_{max}=30$). This reconstruction filters out the smallest
and most intense structures, but we can still see clearly the emergence of the
far-side AR of interest. As can be seen in Figure 2, we could thus use the map
from February 22 as if it were an updated map of February 15 with the far-side
AR included. The AR would be located at the same Carrington longitude
corresponding to the far-side, and we could then run our simulations and see
how the inner heliosphere is affected.

Figure 2: Analysis of the magnetic input maps selected for the study. The left
panels show the maps for February 15, 2021, and the middle panels the maps for
February 22, 2021. The top row shows the original maps (both GONG-ADAPT maps
in Carrington longitudes), while the bottom row shows the filtered maps used
to initialize the simulations (spherical harmonics reconstruction up to
$\ell_{max}=30$). The far-side AR location is marked with an orange box. In
the right panel is a histogram analysis of the radial magnetic field
distribution for original (top) and filtered maps (bottom), without (blue) and
with (orange) the far-side AR.

###  2.3 Input magnetic maps analysis

Before running the simulations, we can already quantify the impact of the far-
side AR on the input maps. This is the purpose of the right panels of Figure
2, where we plot the histogram distribution (with 100 bins) of the radial
magnetic field in the previous maps. The top panel is for the original maps,
while the bottom panel is for the filtered maps. In blue, we show the
distribution without the far-side AR, and in orange the distribution with the
AR included. We observed that the general distribution is as expected: a
Gaussian centered around zero. For the original maps, the wings extend up to
20 G, while for the filtered maps they extend only up to 10 G (as a result of
the filtering of the most intense structures). We observed that the
distribution is also slightly altered, with two bumps around $\pm 7.5$ G that
may bring it closer to a Bessel function than a Gaussian. Our focus is however
on the effect of the inclusion of the AR. In both histograms, one can see that
it depopulates the levels of the smallest values around zero and increases the
values in the wings in an almost uniform way. Therefore, we observed a global
increase of the intensity of the magnetic field but no specific peak
associated with one value. We have also computed the unsigned flux of the
radial magnetic field in the region of interest, shown by an orange rectangle
on Figure 2. We find that the inclusion of the AR in the original map leads to
an increase in flux in the region of interest of a factor of 1.95
($2.79\,10^{22}$ Mx vs. $1.4\,10^{22}$ Mx). For the filtered maps, the factor
is very similar with 2.11 ($1.34\,10^{22}$ Mx vs. $6.4\,10^{21}$ Mx). Though
the filtering of the map impacts the general flux of the map and the AR with a
factor two of reduction for both, its effect is thus negligible for the
relative increase of the flux. Compared to the total unsigned flux of the
original map, the flux in the AR represents only 8.3% (total unsigned flux of
$3.4\,10^{23}$ Mx). For the filtered map, this value goes down to 4% of the
total unsigned flux ($2.4\,10^{23}$ Mx). This gives us an estimate of the
order of magnitude of the changes we can expect if we have a linear response
from the corona.

##  3 Numerical models

###  3.1 Parameters of models

In this study, we use three different models, the analytical method called
PFSS extrapolation and two different versions of the WP MHD model: the
polytropic version of WP and the Alfvén wave-driven version, WP-AW. These 3D
MHD coronal models are based on the multi-physics PLUTO code (Mignone et al.,
2007).

The PFSS is the fastest method used for coronal forecasting (converges in tens
of minutes), but is only semi-empirical (Schrijver & DeRosa, 2003).

The WP model was first introduced and described for star-planet interactions
in Strugarek et al. (2014), and for stellar applications in Réville et al.
(2015a, b). It has since been adapted specifically for the solar case (Réville
& Brun, 2017) and for spherical geometry (Perri et al., 2018). It has also
been used as a reference case to calibrate other wind models (Perri et al.,
2022). WP is faster to run than WP-AW (it converges in less than 6 hours on
200 cores) and is thus an operational space weather forecasting
model,222https://swe.ssa.esa.int/kul-cmpa-federated but is less physically
realistic than WP-AW.

The WP-AW model has been validated against both in situ (Réville et al., 2020,
2021) and remote-sensing observations (Parenti et al., 2022). It has already
been improved with new features, such as the inclusion of a realistic
transition region (Réville et al., 2021), (Réville et al., 2022).

We would like to stress that none of these models have been fine-tuned for
this specific case study in this work. Our approach is to use them in a set-up
as close as operational as possible in order to both understand how the
magnetic connectivity is modified by adding a new far side AR as well as
estimate the impact of these potential modifications for space weather
forecasting. The only adjustment made was for the WP model, where the
polytropic index $\gamma$ was set to 1.03 instead of 1.05 in order to guaranty
a better opening of the equatorial coronal holes.

Because these models are well established, we focus here only on their main
parameters relevant to this study, which can be found in Table 1. For each
model, we specify the input parameters related to the input magnetic map (cut-
off harmonics $\ell_{max}$ and multiplying factor $\mu$). We also specify
parameters regarding the simulation domain and the grid resolution. Finally,
we specify the parameters that are characteristic of each model. We end with
the main numerical methods used, when it is relevant.

Input parameters | PFSS | WP | WP-AW
---|---|---|---
$\ell_{max}$ (cut-off harmonics of input map) | 30 | 30 | 25
$\mu$ (multiplying factor of input map) | 1 | 0.3 | 1
Domain ($r[R_{\odot}]\times\theta[\rm{rad}]\times\varphi[\rm{rad}]$) | $[0,2.5]\times[0,\pi]\times[0,2\pi]$ | $[0,30]\times[0,\pi]\times[0,2\pi]$ | $[0,30]\times[0.05,3.09]\times[0,2\pi]$
Grid ($r\times\theta\times\varphi$) | $12\times 92\times 184$ | $172\times 86\times 172$ | $182\times 160\times 320$
Specific parameters | $R_{ss}=2.5\,R_{\odot}$ | $\gamma=1.03,\frac{c_{s}}{v_{esc}}=0.26$ | $F_{h}=2\,10^{4}\,\rm{cgs},\kappa_{0}=9\,10^{-7}\,\rm{cgs},$
|  |  | $\delta v=12\rm{km/s},T_{c}=2\,10^{4}\,K$
Numerical methods | N/A | HLLD, divergence cleaning | HLL, constrained transport
Table 1: Table summarizing the main modeling parameters for the three
different models tested in this work (PFSS, WP, and WP-AW). For each model, we
specify the input parameters related to the input magnetic map (cut-off
harmonics $\ell_{max}$ and multiplying factor $\mu$). We also specify
parameters regarding the simulation domain and the grid resolution. Finally,
we specify the parameters that are characteristic of each model. We end with
the main numerical methods used, when it is relevant. For a more precise
definition of each parameter, please see Appendix B.

From this table, we can make a few remarks. The filtered magnetograms where
reconstructed with an $\ell_{max}$ of 25 for WP-AM because it is the best
compromise between accuracy and speed of computation, with no major structure
being erased. The WP model needs a multiplying factor of 0.3 applied to the
input magnetic map to stay within its robustness parameter space, but once
again this has very little effect on the realism of the global magnetic
structure. The PFSS model only computes the configuration up until the
$R_{ss}$ because after that the magnetic field lines are simply forced to be
radial. A detailed description of the models’ underlying physics and numerics
can be found in Appendix B for further details and explanations.

###  3.2 Overview of the different models

Figure 3: Three-dimensional overview of the magnetic field structure derived
from the various models without (top row) and with the far-side AR (bottom
row). The left panels show the PFSS results, the middle panels the WP results
and the right panels the WP-AW results. Magnetic field lines are gray when
closed, red when open with a positive polarity and blue when open with a
negative polarity. The HCS is also shown as a continuous black line around the
Sun. We indicate the location of the eastern and western limbs with the letter
E and W respectively. We also indicate whether a streamer is a helmet streamer
contributing to the HCS (S) or a pseudo-streamer (P) (Wang et al., 2007), and
we show their number. Arrows point out HCS structures that break apart from
the main line.

We can have a first qualitative look at the various solutions in Figure 3. We
show here the 3D solutions, focusing on the magnetic field structure (the
physical quantity common to all three models) in the inner corona up to
$2.5\,R_{\odot}$. We show results from the simulations without (top row) and
with the far-side AR included (bottom row). The left panels show the PFSS
results, the middle panels the WP results and the right panels the WP-AW
results. Magnetic field lines are gray when closed, red when open with a
positive polarity and blue when open with a negative polarity. The
heliospheric current sheet (HCS) is also shown as a continuous black line
around the Sun. We also indicate whether a streamer is a helmet streamer
contributing to the HCS (S) or a pseudo-streamer (P), and we attribute them
numbers. We recall that helmet streamers separate coronal holes of opposite
magnetic polarity, while pseudo-streamers overlie twin loop arcades and thus
separate coronal holes of the same polarity (Wang et al., 2007).

For the PFSS model (panels a and d of Figure 3), we observed that the HCS is
rather flat and fully continuous, which means that open field lines are mostly
located at the poles and closed field lines mostly near the equator. The
strongest latitudinal modulation of the HCS is on the eastern limb, at least
without the far-side AR (panel a). We observed one large helmet streamer on
the western limb (S1) surrounded by two pseudo-streamers (P1 and P2) and one
helmet streamer (S2) with a pseudo-streamer in the southern hemisphere (P3) on
the eastern limb. With the inclusion of the far-side AR (panel d), the HCS
becomes even flatter, even on the western limb. This is surprising since the
AR is located closer to the eastern limb, so opposite to the western limb.
With the far-side AR, the eastern streamers appear more complex, which is
expected.

For the WP model (panels b and e), we observed that the HCS has larger
northward excursions from the equator near the eastern and western limbs. The
increased complexity compared to the PFSS model is a result of the additional
physics in the MHD model. For the case without the far-side AR (panel b), one
can see a streamer structure similar to the PFSS case (panel a) but with
closed field lines that are more extended and more concentrated along the
equator. We can also see that in this case, we have only one pseudo-streamer
(P1) and two streamers (S1 and S2) on the eastern limb, while the pseudo-
streamer on the western limb is absent. When we include the far-side AR (panel
e), the HCS is flattened but fragmented, with a large flux tube opening in the
south on the eastern limb (indicated by an arrow). The northern eastern
streamers shift so that they are now entangled in the line of sight (which
creates the impression that S2 and S1 have merged).

Finally, for the WP-AW model (panels c and f), we observed that the results
are similar to the WP model as they are both MHD models, but even more complex
due to the additional physics. Without the far-side AR (panel c), the HCS is
very modulated northward near both limbs, but remains continuous. On the
western limb, we have like in the PFSS case one helmet streamer (S4) and one
pseudo-streamer (P1). On the eastern limb, we have now three helmet streamers
(S1, S2 and S3) with no pseudo-streamer. With the inclusion of the far-side AR
(bottom panel), the HCS also becomes more flattened but fragmented (with an
even bigger southern flux tube opening than the WP model on the eastern limb,
indicated by an arrow). The western streamers remain unaffected, while the
eastern streamers S2 and S3 seem to merge, while S1 becomes a pseudo-streamer
(P2).

From this quick overview, we observed that all three models result in slightly
different coronal structures, but that each of them is sensitive to the
inclusion of the far-side AR. Surprisingly, including the AR does not only
change the eastern limb (which is the closest one to the AR location), but
also affects the western limb for all models. Changes on the western limb are
crucial for space weather forecasting, as this is the side that is more likely
to be magnetically connected to Earth due to the shape of the Parker spiral.
This shows that ARs can have strong non-local effects on the global magnetic
structure, and can thus modify the magnetic structure of the Sun at longitudes
that are going to be impactful for Earth forecasting. In order to provide a
more quantitative analysis, we next move on to other systematic comparisons
between the three models.

##  4 Impact of the far-side active region on closed magnetic structures

###  4.1 White-light emissions as a magnetic proxy

In order to go beyond this quick overview, we would need to use observations
of the magnetic field structure of the corona for validation. However, there
are no direct observations of the coronal magnetic field available. Instead,
we can infer it, for example by using white-light (WL) polarized brightness
(pB) emissions images for comparison, as they are a good proxy of the magnetic
structure of the corona. Such a proxy has already been used in previous
studies, such as Wagner et al. (2022), Perri et al. (2023) or Kuźma et al.
(2023) in order to validate the number and location of streamers and pseudo-
streamers.

The WL pB solar emission is caused by Thomson scattering, that is, the
scattering of photospheric radiation by free coronal electrons. This process
is tied to the density of the corona, but the resulting images usually show
the imprint of the magnetic field lines in the corona structure (with closed
field lines generating more emissions than the open ones because they are
denser).

We used data from two observatories. The first one is the COronal Solar
Magnetism Observatory (COSMO) K-coronograph (K-Cor) from the High-Altitude
Observatory (HAO) (de Wijn et al., 2012). K-Cor captures scattered white-light
brightness images between 1.05 and 3 solar radii. We selected L1.5 data that
are fully calibrated to $B/B_{0}$ in the 720-750 nm spectral range ($B_{0}$
being the mean solar brightness). These data were processed for instrumental
effect and calibration as well as to remove sky polarization, corrected for
sky transmission. The data are 10-minute averaged time series from February
15, 2021, at 18:01 UTC.333https://doi.org/10.5065/d69g5jv8 The images are
$1024\times 1024$ pixels in size.

We also used data from the Large Angle and Spectrometric Coronagraph
Experiment (LASCO) C2 coronagraph (Brueckner et al., 1995) aboard the Solar
Heliospheric Observatory (SoHO) (Domingo et al., 1995). LASCO/C2 captures
scattered white-light brightness images between 2.2 and 6.5 solar radii. These
pB data are in the spectral range 540-640 nm and were retrieved using the
LASCO/C2 legacy archive (Lamy et al., 2020) hosted by the MEDOC data center.
These data were processed to remove instrumental effect and sky polarization,
corrected for sky transmission, and calibrated to $10^{-10}B/B_{0}$ units. The
data were taken at 15:01 UTC on February 15, 2021. The images are $512\times
512$ pixels in size.

These images allowed us to capture the structure of the corona at a given
time, but they can also be put together to give a more extended view of the
temporal variations. A proxy map developed by Poirier et al. (2021) (used
already in Badman et al. (2022) and Perri et al. (2023)) uses LASCO/C2 data at
around $3\;R_{\odot}$ to assemble them as a synoptic map over a Carrington
rotation. This map gives an estimate of the streamer belt (SB), which can be
assumed to host the HCS and act then as a proxy for it. It is however not a
one to one comparison: the white-light brightness images capture the pseudo-
streamers as well as the HCS streamer belt. Other effects can affect the
detection, such as variations of brightness due to the orbital configuration
between the Sun and Earth, or the passage of CMEs at the time of observation.
The thickness is also not the same, as the HCS is usually thinner than the SB
(for more details, see Poirier et al. 2021).

Figure 4: Comparison between white-light pB synoptic maps and streamers
extracted from simulations for models without (top row) and with (bottom row)
the far-side AR. The background is a synoptic map made using LASCO/C2 white-
light brightness images over the corresponding Carrington rotation (CR2240).
With the chosen color scale, the SB appears as dark regions. The HCS extracted
from simulations is shown as a blue line. The pseudo-streamers extracted from
simulations are shown as reddish regions. The longitude of the far-side AR on
February 15, 2021, is shown with a vertical red line. We keep the same
nomenclature as in Figure 3 for the identification of streamers and pseudo-
streamers.

###  4.2 Comparison with models

In Figure 4, we show the comparison between white-light pB synoptic maps and
streamers extracted from simulations for models without (top row) and with
(bottom row) the far-side AR. The background is a synoptic map made using
LASCO/C2 WL pB images over the corresponding Carrington rotation (CR2240). The
HCS extracted from simulations is shown as a blue line (delimiting the
separation between positive and negative polarity in the $B_{r}$ field). The
longitude of the far-side AR on February 15, 2021, is shown with a vertical
red line. We keep the same nomenclature as in Figure 3 for the identification
of streamers and pseudo-streamers.

WL analysis | HCS maximum deviation | HCS mean deviation | HCS standard deviation | Pseudo-streamers agreement
---|---|---|---|---
PFSS (no AR) | 47 degrees | 17 degrees | 14 degrees | –
PFSS (AR) | 40 degrees | 15 degrees | 11 degrees | \- 28%
WP (no AR) | 20 degrees | 6 degrees | 5 degrees | –
WP (AR) | 32 degrees | 7 degrees | 8 degrees | \- 0.9%
WP-AW (No AR) | 34 degrees | 9 degrees | 8 degrees | –
WP-AW (AR) | 43 degrees | 9 degrees | 10 degrees | \+ 9%
Table 2: Summary of the changes observed in the closed magnetic field
structures obtained by comparing with the WL images. We quantify for the three
models (PFSS, WP and WP-AW) the changes in the HCS and the pseudo-streamers.
For the HCS, we show the maximum, mean and standard deviation, while for the
pseudo-streamers we show the variation of the agreement with the observations.

With the chosen color map (which is an inverse gray map), the HCS is hosted in
the darkest features exhibited. In order to validate our simulations, we would
then need the blue line to go through all the darkest regions of the white-
light emissions. However, if the blue line does not match the dark line, it
can also mean that it is a pseudo-streamer instead of the HCS. This is why, in
addition to the HCS, we also extract from all models a proxy for pseudo-
streamers. The pseudo-streamer proxy was designed with the same idea as behind
the squashing factor (Titov, 2007): here we compute the latitudinal
displacement between neighboring footpoints of the magnetic field lines at 3
$R_{\odot}$, and if they find themselves separated by a sufficient amount
above a certain threshold, then it probably means that they surround a pseudo-
streamer structure. This allowed us to then estimate the location of pseudo-
streamers, that we can then compare with the LASCO images. Using this proxy,
pseudo-streamers are highlighted as pale red regions by selecting latitudinal
displacements above $\sim 10$ degrees (darker shades of red then indicate
larger displacements). These additional red patches should then match the
complementary dark regions that are not reached by the HCS. This is visible
for example in panel a: the HCS matches the darkest features from longitudes 0
to 180, and then it oscillates between two dark branches north and south of
the equator. With the pseudo-streamer proxy, we observed that the PFSS model
considers that the northern branch is the HCS between longitudes 200 and 280,
while the southern branch is actually matched by a pseudo-streamer to explain
the dark region. It then reverses twice between longitudes 280 and 360,
showing how difficult it is to distinguish between the HCS and pseudo-
streamers. We also want to add a word of caution regarding the data. The map
is synoptic but not synchronic, and hence the further away we go in both
directions from the vertical red line, the more outdated the data are going to
be, explaining why pseudo-streamers around 60 degrees are going to differ
significantly from the models. We can then quantify the quality of the HCS and
pseudo-streamers described in the models by comparison with the WL data. For
the HCS, we can compute the path that goes through the darkest pixels of the
map (which is the most likely to be the HCS path) and compute the deviation
with the HCS extracted from the simulations. For the pseudo-streamers, we can
compute the percentage of covering between the ones extracted from the
simulations and the ones from the simulations (using a simple threshold
extraction to obtain the gray structures). Since this pseudo-streamer proxy is
new and has not been fully quantified yet, here we focus more on the
variations induced by the inclusion of the far-side AR than the absolute
values. Results are summarized in Table 2.

We next describe the changes induced by the far-side AR for all models on the
HCS and pseudo-streamers, and compare them with white-light observations for
validation. For the PFSS model (panels a and d), we observed that without the
AR (panel a), the fit of the darkest regions by the HCS and pseudo-streamers
is already good between longitudes 180 and 360. We see that the inclusion of
the far-side AR (panel d) brings little changes to the structure of the HCS.
It is already quite flat without the AR (maximum amplitude of 50 degrees), and
becomes even flatter after emergence (maximum amplitude of 30 degrees). This
is opposite to what one would expect since the inclusion of the AR should
increase the complexity of the magnetic configuration. The HCS corresponds to
the darkest regions of the white-light map, although the biggest region on the
bottom right is barely reached. The region affected the most is between
longitudes 120 and 240, which is actually eastward of the far-side AR
location. In Table 2, we observed that the inclusion of the far-side AR tends
to reduce all deviations to the HCS, which means that the general description
is more accurate. Pseudo-streamers appear to match quite well the SB, although
they have a more curved shape than the dark WL structures. When we include the
far-side AR, this curvature is even accentuated in the vicinity of the AR
(close to the vertical red line). We observed that the inclusion of the far-
side AR tends to decrease the quality of the description of the pseudo-
streamers by 28% due to this increase in their curvature.

For the WP model (panels b and e), the structure of the HCS is more complex
due to the MHD effects, which means that we have the plasma dynamics and the
field structure is more physically consistent. Without the AR (panel b), we
see more sharp modulations of the SB (maximum amplitude of 50 degrees). With
the AR (panel e), the SB becomes more flattened (maximum amplitude of 25
degrees), with the southern dark region between longitudes 150 and 240 no
longer being reached. Instead, an open tube opens in this region, disrupting
the HCS. In Table 2, we observed that the inclusion of the far-side AR tends
to slightly increase the mean and standard deviations, due to a significant
increase in the maximum deviation. Pseudo-streamers are a bit more northward
than for the PFSS, thus not matching perfectly with the SB. Including the far-
side AR seems to reduce this discrepancy, but causes the same shift as seen
before. We observed that the inclusion of the far-side AR tends to barely
affect the quality of the description of the pseudo-streamers (with a slight
decrease of 0.9%).

For the WP-AW model (panels c and f), we see similar effects. It is the model
that best reproduces the path across the darkest regions. However, the
inclusion of the far-side AR (panel f) tends to flatten it and remove the same
region as for WP, which also turns into an open flux tube. In Table 2, we
observed that the inclusion of the far-side AR tends to increase the maximum
and standard deviation, but does not affect the mean deviation. We recall that
the model is not fine-tuned, so these results could be improved with a
dedicated parameter study (which is, however, not the object of this paper).
Pseudo-streamers are even more northward and curved, but the inclusion of the
far-side AR seems to reduce this discrepancy. We observed that the inclusion
of the far-side AR tends to increase the quality of the description of the
pseudo-streamers (with an increase of 9% for the quality of the agreement).
WP-AW is the only model that shows this significant improvement in pseudo-
streamers, due to the more complex underlying physics.

In conclusion, in all models the inclusion of the far-side AR tends to flatten
the HCS. The PFSS is the model that reproduces best the pseudo-streamers, but
without the far-side AR, as its inclusion introduces deviations. On the
contrary, the WP-AW model performs best with the inclusion of the far-side AR.
This shows the importance of the modelling behind the magnetic field
structure.

###  4.3 Synthetic emissions for Wind Predict with Alfvén waves

For the PFSS and WP model, it is not possible to go beyond this proxy
comparison with WL data because these models do not have a realistic enough
thermodynamic structure to simulate WL emissions. It was, however, possible
for the WP-AW model to produce white-light pB synthetic images for
quantitative comparison with the observations. To do so, we used the TomograPy
package from Barbey et al. (2013) as described in Parenti et al. (2022). This
package uses as input the 3D density distribution from the global model to
calculate the white-light coronal emission due to Thomson scattering. Then it
integrates the result along the line of sight to simulate the detection by a
specific instrument.

Figure 5: Comparison of white-light synthetic images with COSMO K-Cor and
LASCO/C2 data of February 15 for the WP-AW model. The first line shows the
results for COSMO K-Cor while the second line shows the results for LASCO/C2.
The first column shows the results of the simulation for the input map of
February 15 without the far-side AR, the second column shows the results of
the simulation for the input map of February 22 with the inclusion of the far-
side AR and the third column shows the real data for comparison. Each line has
its own colorbar indicated to the right.

First, we are going to comment on the magnetic structure inferred from the
white-light images from the observations on February 15 (visible in Figure 5
in the right column). On the western limb (on the right side of the image),
the emissivity is dominated by a massive helmet streamer located near the
equator. At least one pseudo-streamer could also be present slightly
southward, but may not be on the plane perpendicular to the line of sight and
is less intense than the main streamer. On the eastern limb (on the left side
of the picture), we can clearly see in the LASCO data a three-fold structure.
The K-Cor data are not so clear, but it seems reasonable to assume that the
middle structure near the equator on the western limb is also a helmet
streamer, while the northern and southern structures on the eastern limb are
likely to be pseudo-streamers.

Figure 6: Comparison of 1D cuts of the synthetic white-light emission from the
WP-AW simulations with data from COSMO K-Cor (top line) and SOHO/LASCO C2
(bottom line) from February 15. Each panel shows the comparison at
respectively $1.2$, $1.4$, $1.6$, $2.25$, $2.7$ and $3.2$ solar radii. The
simulation with WP-AW of February 15 without the far-side AR is shown in blue,
while the simulation of February 22 with the far-side AR is shown in orange.
Data from K-Cor and LASCO is shown with a black line. The gray areas show the
error bars associated with the data. The x-axis shows the position angle in
degrees, with 0 being the north pole of the Sun and the angle progressing in
clock-wise direction (so that the western limb is shown from 0 to 180 degrees,
and the eastern limb is shown from 180 to 360 degrees). Streamer structures
visible in the data are highlihgted using vertical dotted lines.

In order to quantitatively see the streamers in both observations and
simulations, we perform latitudinal 1D cuts through the 2D images (both data
and simulations), which we present in Figure 6. Each panel shows the
comparison at respectively $1.2$, $1.4$, $1.6$, $2.25$, $2.7$ and $3.2$ solar
radii. The simulation with WP-AW of February 15 without the far-side AR is
shown in blue, while the simulation of February 22 with the far-side AR is
shown in orange. Data from K-Cor and LASCO is shown with a black line.
Streamer structures visible in the data are highlighted using vertical dotted
lines. The gray areas show the error bars associated with the data. As
explained in Parenti et al. (2022), for K-Cor data the background noise in the
pB is estimated at $3\,10^{-9}B/B_{0}$. For LASCO data, these error bars are
estimated based on the work from Frazin et al. (2012), where they are assumed
to be 15% in the whole field of view. More details can be found in Lamy et al.
(2020). The x-axis shows the position angle in degrees, with 0 being the north
pole of the Sun and the angle progressing in clock-wise direction (so that the
western limb is shown from 0 to 180 degrees, and the eastern limb is shown
from 180 to 360 degrees). As we said before, the K-Cor data corresponds to
structures close to the solar surface (between 1 and 3 solar radii, so with
the line-of-sight entanglement the three-fold structure of the western limb is
less visible (panels a, b and c, only two vertical dotted lines past 180
degrees). The large peak centered around 100 degrees is the large western
streamer visible on the other limb. The three eastern streamers are more
visible in the LASCO data (three vertical dotted lines past 180 degrees in
panels d, e and f), with three clear peaks at 230, 260 and 290 degrees at 3.2
$R_{\odot}$ (panel f). For both cases (with and without the far-side AR in
blue and orange) and both data (top and bottom panels), we observed that the
western streamer is always present with a peak at the right angle around 100
degrees, which means there is a good agreement in alignment between the data
and simulations. However, the intensity of the emission and the width of the
streamer is only correct for K-Cor synthetic data in the top panel (this is
not affected by the inclusion of the far-side AR). LASCO synthetic data in the
bottom panel tend to overestimate the WL intensity, which is consistent with
what was already found in Parenti et al. (2022). Regarding the eastern
streamers, we observed clearly that for the synthetic K-Cor observations (top
panel), the addition of the far-side AR does indeed generate an additional
streamer peak around 300 degrees, that is clearly visible at 1.2 $R_{\odot}$
but also further away from the Sun. This could be due to an increase of the
local density because of the inclusion of the far-side AR, which then leads to
an increase of intensity in the white-light pB images. As for the LASCO
synthetic data (bottom panel), it is less clear: The three streamers could be
present at 2.25 $R_{\odot}$ for both simulations but then merge into two
streamers further away from the Sun with also a tendency to overestimate the
intensity (even stronger effect with the inclusion of the far-side AR). In
conclusion, we demonstrate further here that the inclusion of the far-side AR
has an impact on the streamer structures, and that in particular it can
improve their number and position when compared to observations. Next, we try
to explain these changes by focusing on the open magnetic field distribution.

##  5 Impact of the far-side active region on open magnetic structures

###  5.1 Extreme ultra-violet maps as coronal holes proxy

In order to know more precisely where the changes in streamer shapes come
from, we examined the variations of open magnetic field lines in the lower
atmosphere. We compared this structure with extreme ultra-violet (EUV)
observations. This helped us identify the open field regions correlated with
the darker patches seen in the EUV in the atmosphere, which are known as
coronal holes (Cranmer, 2009).

The EUV data comes from the 193 Å channel from the Atmospheric Imaging
Assembly (AIA) instrument onboard the Solar Dynamics Observatory (SDO). They
are L1 data, meaning that they have been corrected for instrumental effects
and radiometrically calibrated (units of data number, or $Dn$). The images are
$4096\times 4096$ pixels in size.

In particular, we used synoptic maps of EUV data to have the most consistent
comparison with our models. These maps are constructed from full-disk images
of the Sun from SDO/AIA. Each full-disk image is interpolated onto a
Carrington longitude-latitude grid, then multiplied by a smoothly varying
weight function, centered on the central meridian, and summed to produce a
synoptic Carrington map. We use the official synoptic maps from SDO produced
for each Carrington rotation.444https://sdo.gsfc.nasa.gov/data/synoptic/ The
corresponding Carrington rotation for the studied period is CR2240, which
started on January 22 and ended on February 18, 2021.

The 193 Å band corresponds to the recommended wavelength to automatically
extract coronal hole boundaries (Wagner et al., 2022; Badman et al., 2022). It
is because it has a response function which peaks at the Fe XII temperature
around 1.5MK, which makes it excellent to capture the contrast between
magnetically open regions fuelling the fast solar wind, and hot, closed
magnetic structures. The CHs from EUV data can then be automatically extracted
using dedicated algorithms, such as the EZSEG algorithm developed by
Predictive Science Inc. (Caplan et al., 2016).555The software is available as
part of the EUV2CHM Matlab package from the Predictive Science Inc. website:
https://www.predsci.com/chd/. The details of the algorithm and its conversion
to Python can be found in Perri et al. (2023). We experimented with the
optimal input parameters, and found that the best result was obtained with a
connectivity of two neighbors, a first threshold at 20 DN and a second
threshold at 30 DN (values for SDO/AIA 193 Å picture).

###  5.2 Comparison with models

The WP-AW model produces a realistic solar atmosphere for which the EUV
emission can be simulated. This allowed us to recover the coronal dimming.
However the PFSS and WP models do not have a realistic enough solar atmosphere
to model the EUV emission, and thus recover the coronal hole dimming. However,
coronal holes also correspond to regions of open magnetic field lines, and
this information can be retrieved from all models. We then proceed for the
various models to find the boundaries between closed and open field lines at
the lower boundary of the domain. We use a sphere of $400\times 200$ seed
points at $1.01\;R_{\odot}$. We followed the field lines to see if they reach
the upper boundary of the computational domain (see Table 1 for the value for
each model). If they do, they are defined as open field lines. If not, they
are defined as closed field lines. This allowed us to retrieve contours of the
open field line regions that we could directly compare with the coronal hole
synoptic map. Similar comparisons have been performed in previous studies with
positive results (Badman et al., 2022).

We can thus plot in Figure 7 the comparison between CHs extracted from EUV
observations, and CHs extracted from the open magnetic field lines regions in
the models. Results extracted from observations are shown with white contours,
while results obtained from numerical simulations are shown in color. The
coronal holes corresponding to the simulation performed using the magnetic map
from February 15 without the AR are shown in blue, while the ones using the
magnetic map from February 22 with the AR are shown in yellow. We also
indicate the longitude corresponding to the AR on the far-side with a red
line, and the longitude corresponding to the central meridian as seen from the
Earth with a green line. We focus especially on the region between Carrington
longitudes 210° and 330°, as it is the region where the far-side AR is
located. We show for most panels the comparison with CR2240. However, since
the emergence was on the far-side on longitude 280, the data was not yet
updated on the asynchronous map, which means that the description of the CH on
the right side of the map is 27 days late. We then also include the comparison
of the WP-AW solution with CR2241 on the bottom right panel to show the
difference.

Figure 7: Comparison of the location of the coronal holes for the PFSS (top
left panel), WP (top right panel), and WP-AW model (bottom left panel) with
EUV observations by SDO/AIA 193 (background). We show the comparison with EUV
data from Carrington Rotation 2240. Just for the WP-AW model, we also show the
comparison with Carrington rotation 2241 (bottom right panel). Results
extracted from observations are shown with white contours, while results
obtained from numerical simulations are shown in color. The coronal holes
corresponding to the simulation performed using the magnetic map from February
15 without the AR are shown in blue, while the ones using the magnetic map
from February 22 with the AR are shown in yellow. We also indicate the
longitude corresponding to the AR on the far-side with a red line, and the
longitude corresponding to the central meridian as seen from the Earth with a
green line.

We begin with the PFSS model (panel a). We observed that without the far-side
AR (in blue), the PFSS model already provides a good description of the CH
borders. It tends to overestimate the northern and southern coronal hole
coverage by extending them to all longitudes, but this is the case for all
models. We can also argue that this is the region where the CH from the EUV
data are less reliable as well because of a lack of polar data, which can lead
to projection effects and associated errors (for more details, see Appendix
A). It also tends to overestimate the equatorial coronal holes. These are
usually the most difficult to reproduce due to their smaller size, given the
fact that we have considered the magnetic maps only up to $\ell_{max}=30$ in
this study. We can define a percentage of agreement for the definition of the
CHs, which includes the number of pixels that are found both in observation
and models to be inside or outside a CH divided by the total number of pixels.
We find that without the far-side AR, the percentage of agreement of the PFSS
with the EUV data extraction is 73%. Now with the inclusion of the far-side AR
(in yellow), we observed slight changes in the CH shapes. The biggest change
is probably concerning the equatorial CHs that almost disappear (except for
the lower-right one). We can also notice that the fragmentation of the CHs is
more realistic (especially visible between longitudes 50 and 100). With the
inclusion of the far-side AR, the percentage of agreement of the PFSS with the
observations remains similar at 72%. There are however global changes, as the
percentage of difference for the PFSS with and without the far-side AR is of
8% of the pixels. This is probably the most surprising result here: all the CH
borders are affected, not only the ones located close to the AR.

We now move on to the WP model (panel b). We recover similar properties
compared to the PFSS model: northern and southern CHs are more extended than
the data suggest, and equatorial CHs are more difficult to reproduce. Compared
to the PFSS model, the CH structures are thinner. We find that without the
far-side AR, the percentage of agreement of WP with the EUV data extraction is
76%. With the inclusion of the far-side AR, it remains similar at 74%. There
are however global changes, as the percentage of difference between WP results
with and without the far-side AR is of 6%. We observed a stronger shift of the
CH borders up to 10 degrees in longitude, globally towards the center of the
picture (longitude 180).

Finally, we analyze the results for the WP-AW model (panels c and d).
Properties are similar to what is described before, but we observed that the
matching for the CH borders is much better. For example, fragmentation for the
southern CHs is already present without the inclusion of the far-side AR. We
find that without the far-side AR, the percentage of agreement of WP-AW with
the EUV data extraction is 81% for panel c. With the inclusion of the far-side
AR, it remains similar at 80%. There are however global changes, as the
percentage of difference between WP with and without the far-side AR is of 7%.
The shift to the left close to the AR is even stronger (up to 20 degrees) and
matches way better with CR2241.

In conclusion, when we compare these results all together, we observe clearly
that improving the physics of the model improves the quality of the CH
description, with or without the far-side AR. Including the latter always
changes CH borders by around 7%. This change tends to lower by 1% the quality
of the CH description. However, what is interesting is that the change is
global, even though the AR emergence is localized. This shows that coronal
hole boundaries can evolve quickly even at minimum of activity, and that they
are globally sensitive to local perturbations. It also explains what we saw
earlier with the streamers: the inclusion of the far-side AR tends to reduce
the amount of open magnetic field close to the equator, which leads to a
global reconfiguration of the streamers.

###  5.3 Synthetic EUV emissions for Wind Predict with Alfvén waves

Figure 8: Comparison of synthetic EUV emission from WP-AW (top row) with EUV
synoptic maps from SDO/AIA-193A (bottom row). On the top row, the left panel
shows the results using the magnetic map from February 15 without the AR (blue
contours), while the right panel shows the results using the magnetic map from
February 22 with the AR (yellow contours). On the bottom row, we use the same
design as described in the last Figure, where we compare contours from data
(white) with WP-AW synthetic EUV contours without (blue) and with (yellow) the
far-side AR. We also indicate the longitude corresponding to the AR on the
far-side with a red line, and the longitude corresponding to the central
meridian as seen from the Earth with a green line. The comparison is made with
CR2240.

The polytropic approximation we used for the coronal heating in WP did not
allow us to use more refined techniques for comparison with EUV data since
only the magnetic field is described realistically. Synthetic EUV emissions
can, however, be generated from simulations to provide an accurate comparison
(Lionello et al., 2009). Using the calibrated response of a given EUV
instrument (e.g., SDO/AIA, see Landi et al., 2013) as a function of the number
density and the temperature of the plasma, the image is produced integrating
the volume response along the line of sight. Wind Predict-AW has been shown to
agree well with AIA 193 A images during solar minimum phases (Parenti et al.,
2022). To create synoptic maps, we use here the central meridian response and
perform a rotation of the simulation to account for all Carrington longitudes.
To extract the contours, we apply the same EZSEG algorithm as with the EUV
data, where we just adjusted the parameters to fit the synthetic emission
levels (first threshold at 14, second threshold at 16).

The results of this comparison can be seen in Figure 8. The top row shows
synthetic EUV emission from WP-AW, while the bottom row shows the comparison
directly with the EUV synoptic maps from SDO/AIA-193 for CR2240. For the top
row, the left panel shows the results using the magnetic map from February 15
without the AR (blue contours), while the right panel shows the results using
the magnetic map from February 22 with the AR (yellow contours). We observed
that the resolution is not the same as the previous results with the open
magnetic field. It was slightly downgraded for reasons related to computing
resources for the generation of the EUV synthetic maps, so we interpolated the
EUV data map to match it so that the comparison is relevant. At first glance,
we observed that the emergence of the far-side AR is very visible in the
synthetic EUV (panel b), with a clear enhancement of luminosity at the
longitude 280. The structure however appears to reflect multiple emergence and
not only one, which could be due to the spherical harmonics filtering applied,
or additional fainter surrounding magnetic structures. What is interesting in
this other approach is that we capture the north and south limits of the CHs
better. Instead of being extended to all longitudes, northern and southern CHs
are fragmented with limits that match the observations. This shows that EUV
emissions are not a one-to-one proxy for the magnetic field open regions,
which is something we can prove only with simulations (for more details
regarding this discussion, see Appendix A). Regarding the far-side inclusion,
we reached the same conclusions as with the open magnetic field, with less
equatorial CHs and shifting of northern CHs close to the AR.

##  6 Impact of the far-side active region for heliospheric connectivity

###  6.1 In situ magnetic measurements

In this section, we focus on large-scale magnetic structures that shape the
interplanetary medium and are crucial to space weather forecasting, as they
have an impact on the propagation of transients. Typically, such large
structures are more difficult to get with the PFSS model, and thus it is
extremely important to extract them from MHD models. These structures usually
require an additional current sheet model (such as the Schatten one; see for
example Pomoell & Poedts 2018), and even with it, it is usually challenging to
compute in situ magnetic field data far away from the Sun due to numerical
reconnection. For these kinds of structures, global observations are also more
difficult to get for comparison. Here, we focus on in situ observations in
order to validate and discuss the large-scale magnetic structures.

Figure 9: Comparison between Solar Orbiter/MAG measured radial field and
predictions for the WP model (top panel) and the WP-AW model (bottom panel).
The magnetic field corresponding to the simulation performed using the
magnetic map from February 15 without the far-side AR is shown in blue, while
the one using the magnetic map from February 22 with the far-side AR is shown
in orange. For context, we show the data over the full month of February 2021
(in gray), as well as their smoothing (in black dashed line). We also indicate
the date of the emergence of the far-side AR with a vertical red line.

To do so, we assess the in situ properties of the magnetic field along Solar
Orbiter trajectory which we plot in Figure 9 as measured by the MAG instrument
(Horbury et al., 2020), as it was passing on the far-side over the month of
February 2021 (in gray). We use the 1-minute cadence L2 data downloaded from
the AMDA database (Génot et al., 2021), with a quality flag oscillating
between two and three over this period. It is clear however that numerical
simulations will have difficulties reproducing this temporal resolution. This
is why we also show the smoothed data (in black dashed line), obtained by
applying a Savitzky-Golay filter with a window size of 1501 points and a 3rd-
order polynomial (Savitzky & Golay, 1964). For this configuration, positive
radial magnetic field means the spacecraft is connected to northern coronal
holes, while negative radial magnetic field means it is connected to southern
coronal holes.

###  6.2 Comparison with models

To extract similar information from the numerical simulations, we use a
virtual spacecraft that has the same trajectory in the corresponding frame of
the simulation, and we plot the radial magnetic field at its location. For
both models, we use a ballistic projection following the Parker spiral,
extrapolating from $20\,R_{\odot}$ the magnetic field assuming a 400 km/s
solar wind speed (Neugebauer et al., 1998). This value has been selected
because it is the most probable in the equatorial plane during a period close
to minimum of activity. We could have used the actual measurements of the
solar wind at Solar Orbiter for a more realistic result. However, SWA data are
not available during this time period (which corresponds to the cruise phase
of Solar Orbiter). Our simulations do not include the orbit of Solar Orbiter
into their numerical domains (see Table 1), so instead we have estimated the
solar wind speed at Solar Orbiter using the fast HUXt model (Riley & Lionello,
2011; Barnard & Owens, 2022). During this period, we find indeed a solar wind
speed around 400 km/s on average. The WP model tends to have more numerical
diffusion than WP-AW, and hence is more prone to numerical reconnections that
will decrease the magnetic field amplitude for non-physical reasons. This
means that we need to apply a multiplying factor of 20 to get similar
amplitudes to make better comparison with the data. We recall that here we are
more focused on the polarity of the field than its actual values, and that the
multiplying factor does not affect this property.

We compare results for the WP model (top panel) and the WP-AW model (bottom
panel) in Figure 9. The magnetic field corresponding to the simulation
performed using the magnetic map from February 15 without the far-side AR is
shown in blue, while the one using the magnetic map from February 22 with the
far-side AR is shown in orange. We also indicate the date of the emergence of
the far-side AR with a vertical red line. We see that, due to the lack of
resolution, both simulations have less variations of the polarity than the
real data, as expected, but do match the smoothed data. For the WP model, the
main differences are close to the date of the AR emergence: the positive
polarity lasts until February 17, and thus matches better the observations. It
is also the case around February 9, where the positive polarity is caught
earlier on. However, the positive polarity from February 21 is not captured
anymore (but this period is less critical, as it is far from the AR date and
thus the period of interest). For the WP-AW model, we see similar effects, but
stronger: the positive polarity patch from February 17 to 21 which is
completely missed without the AR is suddenly well captured with the inclusion
of the far-side AR. This means that the inclusion of far-side data could
improve both forecasting and hindcasting of the connectivity to the
spacecraft. However the effect of a single AR with the current resolution of
MHD codes is not clear, so confirmation with better resolution or with a more
active far-side would be needed in future work. We expect this effect to be
amplified when considering higher-resolution simulations, that will be able to
capture smaller and smaller scales and that are left for future work.

We summarize all the results found so far from all previous sections in Table
3. We move to the discussion section in order to attempt to explain all these
results.

Analysis |  Main results from the far-side AR inclusion
---|---
| Map analysis
---
Unsigned flux increase
Original map 8% of total map flux Filtered map 4% of total map flux
| Model analysis
---
3D overview
HCS
WL emissions
Coronal holes
EUV emissions
Connectivity
to spacecraft
PFSS Concentration of eastern limb streamers towards the equator Flattening of
HCS N/A From 73% to 72% agreement N/A N/A Wind Predict Merging of eastern limb
streamers Flattening and breaking of HCS N/A From 76% to 74% agreement N/A
Improvement of positive polarity connection Wind Predict-AW Radial extension
of eastern limb streamers Flattening and breaking of HCS Improvement of
southern pseudo-streamer emission From 81% to 80% agreement Improvement of
northern CH Improvement of positive polarity connection
Table 3: Table summarizing the main results for each section and models from
the study.

##  7 Discussion

We want to discuss here some of the results found above summarized in Table 3.
In particular, we want to bring forward some elements that may explain why the
local impact of this particular far-side AR is not necessarily as strong as
anticipated, and also why the streamer structure is so sensitive for most
models.

###  7.1 Magnetic precursor to the active region

Figure 10: SDO/AIA 171 observations of the AR precursor from January 26 to
February 3, 2021. We show for every two days the full-disk view of the Sun,
with the region of interest marked with a red grid. The grayscale makes
intense magnetic structures such as ARs appear black, so that one can see a
magnetic precursor at the same latitude as the AR examined in this study.

One obvious reason for the not so strong local impact of the far-side AR could
of course be its small flux, which we quantified to be only twice the
surrounding quiet Sun flux in section 2, and only 4 to 8% of the total
unsigned flux of the Sun. Some AR can increase the local unsigned flux by a
factor of at least 10, which of course would be expected to generate more
impact (DeRosa, 2018). Another explanation as to why the flux increase is so
low could be that the surrounding magnetic field is actually not so quiet due
to previous emergence. We can check this theory by monitoring the AR location
before its emergence. In Figure 10, we observed the region where the far-side
AR emerged when it was facing the Earth-side during the previous rotation,
more specifically from January 26 to February 3, 2021 (which is two weeks
prior to the AR emergence). We used SDO/AIA 171 Å data to track it, with a red
grid marking the region of interest. We show for every two days the full-disk
view of the Sun to put the region in context. This shows clearly that when the
region of interest was facing Earth, there was a decaying AR which was
present. This previous AR was labeled AR12800, while the new one that emerged
on the far-side that we studied was labeled AR12803. When monitoring Solar
Orbiter far-side data, we observed that AR12800 faded between February 3 and 7
and that AR12803 emerged at a similar location on February 15. This certainly
explains why the impact of the AR was reduced. Not only is the AR of medium
intensity for the solar standards, but it also emerged at the same location as
a magnetic precursor visible one week before. This shows the necessity of
putting ARs in perspective in time; the intensity of the AR is not the only
component that determines its impact, as the global solar context before its
emergence also has an influence.

###  7.2 Interpretation using the source surface radius

Figure 11: Study of the influence of the source surface radius ($R_{ss}$) for
the PFSS model. In the top row, we show EUV synoptic maps made from SDO/AIA
193 observations, where we superimpose the boundaries of open magnetic field
regions extracted from the PFSS model with yellow contours. In the bottom row,
we show the comparison between the LASCO/C2 white-light synoptic maps at 3
$R_{\odot}$ with the PFSS model using as input the ADAPT map of February 22,
2021, at 22:00UT (with the AR). From the PFSS, we extract the HCS (blue line)
as well as a proxy for pseudo-streamers (red patches). The different panels
show this comparison for various $R_{ss}$ values (from left to right:
1.8$R_{\odot}$, 2.2$R_{\odot}$, 2.6$R_{\odot}$ and 3.0$R_{\odot}$). We
indicate the longitude of the AR at the date with a vertical red line.

How can we explain the global impact of the AR? With the PFSS model, we can
perform simple tests where we vary the height of the source surface, which in
return changes the limit between open and closed magnetic field lines, as well
as the flux tube expansion rate. The result of this study is shown in Figure
11. In the top row, we show EUV synoptic maps made from SDO/AIA 193
observations, where we superimpose the boundaries of open magnetic field
regions extracted from the PFSS model with yellow contours. In the bottom row,
we show the comparison between the LASCO/C2 white-light synoptic maps at 3
$R_{\odot}$ with the PFSS model using as input the ADAPT map of February 22,
2021, at 22:00UT (with the AR). From the PFSS, we extract the HCS (blue line)
as well as a proxy for pseudo-streamers (red patches). The different panels
show this comparison for various $R_{ss}$ values (from left to right:
1.8$R_{\odot}$, 2.2$R_{\odot}$, 2.6$R_{\odot}$ and 3.0$R_{\odot}$). We
indicate the longitude of the AR at the date with a vertical red line.

We can clearly see in Figure 11 that by changing the height of the source
surface, we change the shape of the HCS which looks more similar to what is
obtained with the MHD models without the AR. There is also a critical value
where we transition from two streamers to three streamers at the longitude of
the AR. For 2.6 $R_{\odot}$ (which is close to the value used of 2.5 used in
the previous study), the line crosses the two pseudo-streamers, while for 1.8,
the shift of the pseudo-streamer makes the crossing unclear. A similar effect
can be observed with the PFSS model using the map without the AR as input.
Similarly, when we look at the CH structure evolution with the PFSS, we find
that for smaller $R_{ss}$, we have bigger CHs, while for larger $R_{ss}$,
equatorial CHs close and polar CH boundaries shift, just like with the
inclusion of the far-side AR. This analysis shows that the PFSS case without
the AR seem to be equivalent to a PFSS model with a lower source surface
(which means that magnetic field lines open closer to the solar surface), and
that the inclusion of the far-side AR has a similar effect as to increasing
the source surface height globally. This could explain most of the effects
described in the previous sections. With a lower source surface, it is easier
to open coronal holes (hence the equatorial CHs better reproduced) and to
reproduce pseudo-streamer structures. This allowed for a more complex HCS, but
this does not necessarily equate to a better connectivity estimation. All the
previous results can therefore be explained by the fact that the inclusion of
the far-side AR not only increases locally the magnetic flux, but also
globally increases the height of closed field loops, which explains the global
effects observed. Including missing far-side flux without modifying running
parameters to mitigate this effect might then yield results that do not match
better observations, especially for models that are not fully self-consistent
contrary to the WP-AW model. This discussion opens a reflection on the
necessity to adapt our typical run parameters to the input of magnetic flux,
which could be determined by exploratory parametric studies (which is of
course outside the scope of this paper).

##  8 Conclusion

In this work, we have estimated the impact of far-side magnetic field
structures observed by Solar Orbiter on 3D coronal and heliospheric wind
simulations. We analyzed the case of a far-side emerging AR on February 15,
2021, causing a local increase of magnetic flux by a factor two, which
corresponds to 8% of the total flux of the input map (4% of the processed
map). Since synchronic magnetic maps using Solar Orbiter/PHI data are not yet
available, we made a proxy experiment where we used two GONG-ADAPT magnetic
maps. The first one was made on February 15 and does not include the far-side
AR since it was not visible from Earth, while the second one was made on
February 22 and includes the far-side AR once it became visible on the Earth-
facing disk. We then used the magnetic maps to initialize global coronal
numerical simulations, with a filtering by spherical harmonic modes limited to
$\ell_{max}=30$ (25 for WP-AW). We used three different types of models: a
PFSS semi-empirical extrapolation, a polytropic 3D MHD WP model based on the
PLUTO code, and finally its upgrade with realistic description of the
transition region and the coronal heating using Alfvén waves (i.e., WP-AW).
For each case, we assessed the effect of the inclusion of the far-side AR
using comparisons with both remote-sensing observations, such as SDO/AIA EUV
maps or COSMO/K-Cor and SOHO/LASCO C2 white-light pB images, and in situ
diagnosis, such as from Solar Orbiter/MAG measurements. We compared all three
models between each other to assess the impact of the underlying physical
hypothesis on the accurate representation of the global corona and offered
some physical insights as to why these effects are observed. We summarize our
main conclusions with the following points.

Adopting a source surface of 2.5 $R_{\odot}$ (canonical value), the PFSS model
manages to reproduce quite well the distribution of coronal holes as well as
the HCS positioning, which is expected for this kind of low-activity
configuration (Kruse et al., 2021). To be more specific, the CH distribution
is correct at 73% without the AR and at 72% with the AR. The HCS has a
reduction of latitudinal amplitude of around 20 degrees, which makes it
globally more flat. The streamer distribution, however, appears to be better
reproduced without the inclusion of the far-side AR because it causes changes
in the streamers’ shape and position.

Both MHD models perform well for the distribution of coronal holes, with
similar results. To be more precise, the WP CH distribution is correct at 76%
without the AR and at 74% with the AR, while the WP-AW CH distribution is
correct at 81% without the AR and at 80% with the AR, which means both models
provide a better CH description than the PFSS. However, when it comes to the
streamers, the WP model does not seem to be improved by the inclusion of the
far-side AR, while the WP-AW model clearly is. This can be demonstrated
without ambiguity since the WP-AW model allowed us to produce synthetic EUV
and white-light emission from which we could make more quantitative
comparisons with the observations. The WP-AW model tends to overestimate the
white-light intensity but manages to recover the streamer structure with the
inclusion of the far-side AR. The EUV synthetic emissions also better recover
the borders of the EUV CHs without the longitudinal extent. Both models also
allow for in situ diagnosis, which is not possible with the PFSS model. Both
models exhibit a more flattened HCS with the inclusion of the AR, just like
the PFSS. They also show switches in connectivity, with the positive polarity
between February 15 and 20 being better captured with the inclusion of the
far-side AR.

Concerning the overall impact of the inclusion of the far-side AR, it does not
have a strong local impact, as coronal hole boundaries and streamer shapes are
only slightly affected. This may be due to the fact that this specific region
of interest was not very intense (flux only 8% of the total surface flux) and
also hosted the seat of a precursor AR two weeks before, whose influence may
still be felt. The strongest effect is actually non-local. The inclusion of a
single far-side AR of medium magnetic intensity drives big changes to the
global magnetic structure of the Sun. Coronal hole boundaries on the Earth-
facing side are affected as well, as the HCS shape may differ, and the
comprehension of the in situ connectivity changes. Our understanding of this
global change is that the inclusion of the far-side AR is equivalent to an
enhanced source surface with closed loops reaching higher layers of the
corona, which in turn closes the equatorial CHs and therefore affects the
streamers’ locations, the HCS, and the resulting connectivity. This means that
without information from the far-side, simulations of even the Earth-facing
corona may be biased, which shows the need to take them into account. However,
including the far-side AR without modifying the parameters of the simulation
works only for the most realistic models, such as WP-AW, and may require more
exploratory studies to improve forecasts for faster models.

This work attempts to prepare the community for the transition from
asynchronous synoptic maps as input to global coronal models to fully
synchronic maps enabled by the Solar Orbiter mission. We performed a
preliminary study in anticipation of this transition. A full test will only be
possible when such maps are completed, which hopefully should be in the near
future, for Solar Orbiter data at least (Sinjan et al., 2023). This study
experimented with a single far-side AR. Therefore, a more systematic analysis
would be required in order to validate our results. We have also made the
choice to study only one AR during quiet solar conditions. During periods of
higher solar activity, the photospheric magnetic field will evolve more
rapidly due to increased rates of flux emergence such that far-side
observations become more important. Finally, this study has been made with
three coronal models, and although they do cover a wide range of categories of
models (PFSS, MHD, MHD with realistic heating functions), it would be
interesting to see if such results hold for other models as well. Our future
work will focus on more Solar Orbiter events as well as the development of
even more diagnosis tools in order to further increase our capacity to compare
observations and models. We hope that this kind of systematic comparison can
also help improve the current models in both finding the best input parameters
and pointing out the most interesting physics to include. A very natural
follow-up work will be to use these results to find the best parameter space
for WP-AW with and without the inclusion of far-side data in order to provide
the best forecasting as well as the best understanding of solar and
heliospheric physics.

###### Acknowledgements.

This work was supported by the CNRS and INSU/PNST program, CNES Solar Orbiter
and “Météorologie de l’espace” funds, the ERC Synergy grant WholeSun
No.810218, and the ANR STORMGENESIS #ANR-22-CE31-0013-01. Computations were
carried out using CNRS IDRIS facility within the GENCI A0130410293 and
A010810133 allocations and a local mesocomputer founded by DIMACAV+. The
authors are grateful to A. Mignone and the PLUTO development team. Courtesy of
the Mauna Loa Solar Observatory, operated by the High Altitude Observatory, as
part of the National Center for Atmospheric Research (NCAR). NCAR is supported
by the National Science Foundation. This work makes use of the LASCO-C2 legacy
archive data produced by the LASCO-C2 team at the Laboratoire d’Astrophysique
de Marseille and the Laboratoire Atmosphères, Milieux, Observations Spatiales,
both funded by the Centre National d’Études Spatiales (CNES). LASCO was built
by a consortium of the Naval Research Laboratory, USA, the Laboratoire
d’Astrophysique de Marseille (formerly Laboratoire d’Astronomie Spatiale),
France, the Max-Planck-Institut für Sonnensystemforschung (formerly Max Planck
Institute für Aeronomie), Germany, and the School of Physics and Astronomy,
University of Birmingham, UK. SOHO is a project of international cooperation
between ESA and NASA. This work used data provided by the MEDOC data and
operations centre (CNES / CNRS / Univ. Paris-Saclay),
http://medoc.ias.u-psud.fr/. Courtesy of NASA/SDO and the AIA, EVE, and HMI
science teams. This work utilizes data produced collaboratively between
AFRL/ADAPT and NSO/NISP. Solar Orbiter is a mission of international
cooperation between ESA and NASA, operated by ESA. The EUI instrument was
built by CSL, IAS, MPS, MSSL/UCL, PMOD/WRC, ROB, LCF/IO with funding from the
Belgian Federal Science Policy Office (BELSPO/PRODEX PEA 4000112292); the
Centre National d’Etudes Spatiales (CNES); the UK Space Agency (UKSA); the
Bundesministerium für Wirtschaft und Energie (BMWi) through the Deutsches
Zentrum für Luft- und Raumfahrt (DLR); and the Swiss Space Office (SSO). Solar
Orbiter magnetometer data was provided by Imperial College London and
supported by the UK Space Agency. Data analysis was performed with the AMDA
science analysis system provided by the Centre de Données de la Physique des
Plasmas (CDPP) supported by CNRS, CNES, Observatoire de Paris and Université
Paul Sabatier, Toulouse.

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##  Appendix A Comparison between coronal hole boundaries derived from open
magnetic field lines and synthetic EUV emissions

Figure 12: Comparison between coronal hole boundaries derived from open
magnetic field lines (cyan) and synthetic EUV emissions (magenta). The left
panel corresponds to the WP-AW simulation without the far-side AR (February
15), while the right panel corresponds to the WP-AW simulation with the far-
side AR (February 22). Contours of the coronal holes extracted from the
background SDO/AIA 193 Å EUV synoptic map are shown in white. We also indicate
the longitude corresponding to the AR on the far-side with a red line, and the
longitude corresponding to the central meridian as seen from the Earth with a
green line. The comparison is made with CR2240.

As shown in section 5, there are different ways to estimate the boundaries of
a coronal hole: one can consider that it is a region of open magnetic field
lines, but one can also consider that it is a region of reduced EUV emission.
These two definitions usually refer to the same regions, but it is not always
the case, especially in numerical simulations.

Because of this, we have added in this appendix in Figure 12 the comparison
between coronal hole boundaries derived from open magnetic field lines (cyan)
and synthetic EUV emissions (magenta). The left panel corresponds to the WP-AW
simulation without the far-side AR (February 15), while the right panel
corresponds to the WP-AW simulation with the far-side AR (February 22).
Contours of the coronal holes extracted from the background SDO/AIA 193 Å EUV
synoptic map are shown in white. We also indicate the longitude corresponding
to the AR on the far-side with a red line, and the longitude corresponding to
the central meridian as seen from the Earth with a green line. The comparison
is made with CR2240. The WP-AW simulations are the only ones for which we can
perform such a comparison, as other models do not have a realistic enough
corona to generate realistic EUV synthetic emissions.

From Figure 12, one can see indeed that the two definitions overlap most of
the time but not always. We observed that for both simulations (with and
without the far-side AR), the equatorial CHs boundaries match the EUV data
boundaries better with the synthetic EUV emission rather than the open
magnetic field lines. Similarly, the polar CHs boundaries are less extended in
longitude when using the synthetic EUV emission. This comparison may also hold
for the real data, as the polar regions are more prone to line-of-sight
effects that tend to artificially enhance the EUV emission.

##  Appendix B Detailed description of the numerical models

###  B.1 Potential-field source surface extrapolation

The PFSS model (Altschuler & Newkirk 1969; Schatten et al. 1969; Schrijver &
DeRosa 2003), is an efficient means of extrapolating the coronal magnetic
field from each GONG-ADAPT map in the absence of non-potential structures.
Despite being an empirical model, it compares fairly well with MHD models
(Riley et al. 2006). The PFSS model requires that the coronal magnetic field
becomes radial at a given distance, called the source surface, and that there
are no electric currents present in the coronal magnetic field (i.e.,
$\nabla\times B=0$). For this reason, PFSS models fail to capture highly
twisted or stressed magnetic field structures in the low-corona (typically
associated with eruptive activity), although they adequately recover the
global structure of the corona and solar wind (e.g., Stansby et al. 2021). The
ADAPT-GONG maps, provided in latitude-longitude ($\theta$-$\phi$) coordinates,
are first broken down into spherical harmonics. We use a spherical harmonic
basis formed from the Legendre polynomial functions $P_{lm}(\cos\theta)$, with
degree $l$ and order $m$, given by

$\displaystyle Y_{lm}$ $\displaystyle=$ $\displaystyle
c_{lm}P_{lm}(\cos\theta)e^{im\phi},$ (1) $\displaystyle Z_{lm}$
$\displaystyle=$
$\displaystyle\frac{c_{lm}}{l+1}\frac{dP_{lm}(\cos\theta)}{d\theta}e^{im\phi},$
(2) $\displaystyle X_{lm}$ $\displaystyle=$
$\displaystyle\frac{c_{lm}}{l+1}P_{lm}(\cos\theta)\frac{im}{\sin\theta}e^{im\phi},$
(3)

with the normalization of

$\displaystyle c_{lm}=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}.$ (4)

The photospheric field, $B_{r}(\theta,\phi)$, is decomposed such that the
resulting coefficients $\epsilon_{lm}$ satisfy

$B_{r}(\theta,\phi)=\sum_{l=1}^{l_{max}}\sum_{m=-l}^{l}\epsilon_{lm}Y_{lm}(\theta,\phi),$
(5)

where $\epsilon_{lm}$ is the strength of each spherical harmonic mode,
extracted from the GONG-ADAPT maps (see methodology in Finley & Brun 2023).
The coronal field is extrapolated using these coefficients following

$\displaystyle
B_{r}(r,\theta,\phi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\alpha_{lm}(r)Y_{lm}(\theta,\phi),$
(6) $\displaystyle
B_{\theta}(r,\theta,\phi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\beta_{lm}(r)Z_{lm}(\theta,\phi),$
(7) $\displaystyle
B_{\phi}(r,\theta,\phi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\beta_{lm}(r)X_{lm}(\theta,\phi),$
(8)

where $r$ is radial distance, and the coefficients $\alpha_{lm}(r)$ and
$\beta_{lm}(r)$ represent the radial decay of each individual spherical
harmonic mode:

$\displaystyle\alpha_{lm}(r)=\epsilon_{lm}\frac{l(R_{*}/R_{ss})^{2l+1}(r/R_{*})^{l-1}+(l+1)(r/R_{*})^{-(l+2)}}{l(R_{*}/R_{ss})^{2l+1}+(l+1)},$
(9)
$\displaystyle\beta_{lm}(r)=(l+1)\epsilon_{lm}\frac{(R_{*}/R_{ss})^{2l+1}(r/R_{*})^{l-1}+(r/R_{*})^{-(l+2)}}{l(R_{*}/R_{ss})^{2l+1}+(l+1)}.$
(10)

Each PFSS model is computed with $l_{max}=30$, on a spherical grid with source
surface radii fixed at 2.5 solar radii. This value is the canonical value that
matches the structures in coronograph observations for most phases of the
solar cycle (Hoeksema et al. 1983). Some of the hypotheses can also be
refined, such as using a non-spherical source surface (Levine et al. 1982;
Panasenco et al. 2020) or a complementary current sheet model (Wang & Sheeley
1995). However, such refinements are beyond the scope of this paper, where we
simply want to assess the quality of a standard PFSS extrapolation.

###  B.2 Wind Predict

We now present our first 3D MHD model WP. The model solves the set of the
conservative ideal MHD equations composed of the continuity equation for the
density $\rho$, the momentum equation for the velocity field $\boldsymbol{v}$
with its momentum written $\boldsymbol{m}=\rho\boldsymbol{v}$, the energy
equation with $E$ the total energy and the induction equation for the magnetic
field $\boldsymbol{B}$:

$\frac{\partial}{\partial t}\rho+\nabla\cdot\rho\boldsymbol{v}=0,$ (11)
$\frac{\partial}{\partial
t}\boldsymbol{m}+\nabla\cdot(\boldsymbol{mv}-\boldsymbol{BB}+Ip)=\rho\boldsymbol{a},$
(12) $\frac{\partial}{\partial
t}E+\nabla\cdot((E+p)\boldsymbol{v}-\boldsymbol{B}(\boldsymbol{v}\cdot\boldsymbol{B}))=\boldsymbol{m}\cdot\boldsymbol{a},$
(13) $\frac{\partial}{\partial
t}\boldsymbol{B}+\nabla\cdot(\boldsymbol{vB}-\boldsymbol{Bv})=0,$ (14)

where $p$ is the total pressure (thermal and magnetic, such that
$p=p_{\rm{mag}}+p_{\rm{th}}$), $I$ is the identity matrix and $\boldsymbol{a}$
is a source term (gravitational acceleration in our case, which means
$\boldsymbol{a}=-\nabla\Phi$ and the gravity potential $\Phi$ is equal to
$-GM_{\odot}/r$). We also assumed $\nabla\cdot\boldsymbol{B}=0$. We used the
ideal equation of state:

$\rho e=p_{th}/(\gamma-1),$ (15)

where $p_{th}$ is the thermal pressure, $e$ is the internal energy per mass
and $\gamma$ is the adiabatic exponent. This gives for the energy : $E=\rho
e+\boldsymbol{m}^{2}/(2\rho)+\boldsymbol{B}^{2}/2$.

PLUTO solves normalized equations, using three variables to set all the
others: length, density and speed. If we note with $*$ the parameters related
to the star (the Sun in this case) and with $0$ the parameters related to the
normalization, we have $R_{*}/R_{0}=1$, $\rho_{*}/\rho_{0}=1$ and
$u_{kep}/U_{0}=\sqrt{GM_{*}/R_{*}}/U_{0}=1$, where $u_{kep}$ is the Keplerian
speed at the stellar surface and $G$ the gravitational constant. By choosing
the physical values of $R_{0}$, $\rho_{0}$ and $U_{0}$, one can deduce all the
other values given by the code in physical units. In our set-up, we choose the
following reference solar values: $R_{0}=R_{\odot}=6.96\ 10^{10}$ cm,
$\rho_{0}=\rho_{\odot}=1.67\ 10^{-16}\ \mathrm{g/cm}^{3}$ (which corresponds
to the density in the solar corona above 2.5 Mm, cf. Vernazza et al. 1981) and
$U_{0}=u_{kep,\odot}=4.37\ 10^{2}$ km/s.

One of the most critical aspect of coronal modeling is handling the coronal
heating. In WP, we use the polytropic approximation. This means that we use a
polytropic index equal to the adiabatic exponent $\gamma$ defined such as:
$p\sim\rho^{\gamma}$. With such an approximation, we end up with values of
$\gamma$ close to 1, which mimics a quasi-isothermal heating. Choosing
$\gamma\neq 5/3$ is a simplified way of taking into account heating and
thermal conduction, which are not modeled here. It is not possible to
reproduce the bimodal distribution of the solar wind with this approximation
(Hazra et al. 2021). However, it can successfully fit solar wind speed when we
isolate the various populations (Dakeyo et al. 2022; Shi et al. 2022).

Our wind simulations are then controlled by four parameters: the normalized
density $\rho_{s}$, the adiabatic exponent $\gamma$ for the polytropic wind,
the rotation of the star normalized by the escape velocity $u_{rot}/u_{esc}$
and the speed of sound normalized also by the escape velocity $c_{s}/u_{esc}$.
We note that the escape velocity is defined as
$u_{esc}=\sqrt{2}u_{kep}=\sqrt{2GM_{*}/R_{*}}$. For the rotation speed, we
take the solar value, which gives $u_{rot}/u_{esc}=2.93\ 10^{-3}$. For the
density, we choose a ratio of $\rho_{s}=1$. We choose to fix
$c_{s}/u_{esc}=0.26$, which corresponds to a $1.6\ 10^{6}$ K hot corona for
solar parameters and $\gamma=1.03$.

We use the spherical coordinates $(r,\theta,\phi)$. The numerical domain is a
sphere with the colatitude $\theta\in[0,\pi]$, the longitude
$\varphi\in[0,2\pi]$ and the radius $r\in[1,30]R_{\odot}$. We use a uniform
grid in latitude with 86 points and in longitude with 172 points, and a
stretched grid in radius with 256 points; the grid spacing is geometrically
increasing from $\Delta r/R_{*}=0.001$ at the surface of the star to $\Delta
r/R_{*}=0.74$ at the outer boundary.

Since PLUTO is a multi-physics and multi-solver code, we choose a finite-
volume method using an approximate Riemann Solver (here the HLLD solver which
stands for Harten-Lax-van Leer Discontinuities, cf. Einfeldt 1988). PLUTO uses
a reconstruct-solve-average approach using a set of primitive variables
$(\rho,\boldsymbol{u},p,\boldsymbol{B})$ to solve the Riemann problem
corresponding to the previous set of equations. We set the spatial order of
integration to second order accurate, using a piecewise TVD linear
reconstruction applied to primitive variables. We use the monotonized central
difference limiter (which is the least diffusive offered in PLUTO). We use the
splitting-field option in PLUTO, which causes a splitting between the magnetic
background field (which is curl-free) and the deviation field (which is a
perturbation of the background field and carries the magnetic energy). To
enforce the divergence-free property of the field, we use a hyperbolic
divergence cleaning, which means that the induction equation is coupled to a
generalized Lagrange multiplier in order to compensate the deviations from a
divergence-free field (Dedner et al. 2002). The default value of $0.1$ is used
for the parameter $\alpha$ which controls the rate at which monopoles are
damped.

At the latitudinal boundaries ($\theta=0$ and $\theta=\pi$), we set
axisymmetric boundary conditions. At the top radial boundary ($r=30R_{*}$), we
set an outflow boundary condition which corresponds to $\partial/\partial r=0$
for all variables, except for the radial magnetic field where we enforce
$\partial(r^{2}B_{r})/\partial r=0$. Because the wind has opened the field
lines, this ensures the divergence-free property of the field. At the bottom
radial boundary ($r=R_{*}$), we set a condition to be as close as possible to
a perfect rotating conductor. This means that we set the electric field to 0
inside the star by aligning the poloidal velocity on the poloidal magnetic
field. For the toroidal component, we set its radial derivative constant
(similar to Matt et al. 2012): $\partial^{2}B_{\varphi}/\partial r^{2}=0$. All
other quantities are kept constant, fixed at their initial values inside the
star. The Lagrangian multiplier from the divergence cleaning method is forced
to be constant across the boundary. We use the input magnetic map as the
bottom boundary condition for the radial magnetic field. We pass it on to the
code by decomposing it on spherical harmonics, and then reconstructing it
internally. We use a maximum degree of $\ell_{max}=30$ for the reconstruction.
We also need to avoid the instability regime of the code where regions with
$v_{A}/v_{esc}>3$ will cause negative pressures ($v_{A}$ being the Alfvén
speed). This means that we may need to apply a global scaling factor to the
input map in order to get convergence. In this case, the factor is 0.3. We
could also change the input density of the simulation, but tests have shown
that it changes more significantly the resulting structures and makes
comparisons with other codes more challenging.

We initialize the velocity field with a polytropic wind solution and the
magnetic field with a potential extrapolation of the field based on the input
map. We then let the simulation relax for 200 normalized times to reach a
converged steady state.

###  B.3 Wind Predict with Alfvén waves

This extension of the WP model has been described extensively in Réville et
al. (2020, 2021, 2022) and Parenti et al. (2022). We focus here on the
differences between the basic version of WP described above. This means that
if we do not mention a specific feature, then it is the same as in the WP
version.

The addition of Alfvén waves (AW) aims at modeling more accurately the coronal
heating by providing the additional source of energy and momentum necessary to
create the observed wind bimodality. This is achieved through both the
turbulence dissipation and the wave pressure, which provides additional
acceleration to the solar wind. The gamma index is now set to $5/3$, which is
the ratio of specific heat for a fully ionized hydrogen gas.

In this set-up, on top of the previous physical quantities, we propagate two
populations of parallel and antiparallel Alfvén waves from the boundary
conditions. We can thus define the Elsässer variables as

$\boldsymbol{z}^{\pm}=\delta\boldsymbol{v}\mp\rm{sign}(B_{r})\frac{\delta\boldsymbol{b}}{\sqrt{\mu_{0}\rho}},$
(16)

so that the sign $+$ ($-$) corresponds to the forward wave in a $+$ ($-$)
field polarity. We can also define the corresponding wave energy density:

$\varepsilon^{\pm}=\rho\frac{|z^{\pm}|^{2}}{4},$ (17)

as well as the corresponding dissipation terms:

$Q_{w}^{\pm}=\frac{\rho}{8}\frac{|z^{\pm|^{2}}}{\lambda}\left(\mathcal{R}|z^{\pm}|+|z^{\mp}|\right),$
(18)

where $\lambda$ is the dissipation length scale set to
$\lambda=\lambda_{\odot}\sqrt{B_{0}/|B|}$ ($\lambda_{\odot}$ is the
correlation length at the base of the domain set to $15,000\rm{km}$, which is
close to the size of supergranules, and $B_{0}=1$ G), and $\mathcal{R}=0.1$ is
a reflection coefficient used to generate turbulent dissipation in open
magnetic field regions. The wave energy propagation follows the Wentzel-
Kramers-Brillouin theory (WKB), which gives the following equation, solved
alongside the MHD ones:

$\frac{\partial\varepsilon^{\pm}}{\partial
t}+\nabla\cdot\left([\boldsymbol{v}\pm\boldsymbol{v_{A}}]\varepsilon^{\pm}\right)=-\frac{\varepsilon^{\pm}}{2}\nabla\cdot\boldsymbol{v}-Q_{w}^{\pm},$
(19)

where $\boldsymbol{v}\pm\boldsymbol{v}_{A}\equiv\boldsymbol{v}_{g}^{\pm}$ is
the group velocity of Alfvén wave packets.

As a result of these additional terms, the energy equation 13 is modified as
follows:

$\frac{\partial}{\partial
t}(E+\varepsilon+\rho\Phi)+\nabla\cdot\left[(E+p+\rho\Phi)\boldsymbol{v}-\boldsymbol{B}(\boldsymbol{v}\cdot\boldsymbol{B})+\boldsymbol{v}_{g}^{+}\varepsilon^{+}+\boldsymbol{v}_{g}^{-}\varepsilon^{-}\right]=Q,$
(20)

where $Q$ is a source term that we describe in more detail in the next
paragraph.

The source term $Q$ added to the energy equation can be decomposed into three
components: $Q=Q_{h}-Q_{c}-Q_{r}$, where $Q_{h}$ is an additional source of
heating, $Q_{c}$ is the thermal conduction and $Q_{r}$ models thin radiative
cooling. The $Q_{w}$ term, corresponding to the sum of the two dissipation
terms for both wave populations ($Q_{w}=Q_{w}^{+}+Q_{w}^{-}$), does not appear
explicitly here as a heating term because we solve for the total (wave +
fluid) energy equation (see the erratum of Réville et al. 2020). The wave
heating remains however the main source of heating in the domain (about 90%).

$Q_{h}$ is an ad hoc function modeling chromospheric or coronal processes that
would be different from waves, which can be written as

$Q_{h}=\frac{F_{h}}{H}\left(\frac{R_{\odot}}{r}\right)^{2}\rm{exp}\left(-\frac{r-R_{\odot}}{H}\right),$
(21)

where $H\sim 1R_{\odot}$ is the heating scale height, and $F_{h}$ is the
energy flux from the photosphere (equal to $2\times 10^{4}\
\rm{erg}.\rm{cm}^{-2}.\rm{s}^{-1}$) (Réville et al. 2020).

The term $Q_{r}$ corresponds to an optically thin radiation cooling
prescription, such as

$Q_{r}=n^{2}\Lambda(T),$ (22)

where $n$ is the electron density, $T$ is the electron temperature, and
$\Lambda(T)$ is a function that follows the prescription of Athay (1986). The
term $Q_{c}$ corresponds to the thermal conduction, written as

$Q_{c}=\nabla\cdot(\alpha\boldsymbol{q}_{s}+(1-\alpha)\boldsymbol{q}_{p}),$
(23)

where $\alpha=1/(1+(r-R_{\odot})^{4}/(r_{coll}-R_{\odot})^{4})$ creates a
smooth transition between the collision and collisionless regimes at a
specific height ($r_{coll}=5R_{\odot}$),
$\boldsymbol{q}_{s}=-\kappa_{0}T^{5/2}\nabla T$ is the usual Spitzer-Härm
collisional thermal conduction term ($\kappa_{0}=9\times 10^{-7}$ cgs), and
$\boldsymbol{q}_{p}=3/2p_{th}\boldsymbol{v}_{e}$ is the electron collisionless
heat flux prescribed in Hollweg (1986).

Another important modification is that, following Réville et al. (2021), the
model now includes the transition region. This means that most of the values
for the inner boundary conditions have to be changed. As a result, the
chromospheric temperature is set to $2\times 10^{4}$ K, and the density to
$2\times 10^{10}\ \rm{cm}^{-3}$. This also affects the new control parameters
of the simulation. The transverse velocity perturbation parameter, which
controls the amplitude of the Alfvén waves launched in the domain, is set to
$\delta v=12\ \rm{km}.\rm{s}^{-1}$.

The equations are solved using a HLL solver with parabolic reconstruction
method and minmod slope limiter (Einfeldt 1988). The non-solenoidal condition
on the magnetic field is ensured through the constrained transport method
instead of the divergence cleaning method. The equations are solved in the
rotating frame, and thus Coriolis and centrifugal forces are accounted for in
the momentum and energy equations. The azimuthal speed is thus set to zero in
the inner boundary condition.

The top boundary condition is now located at $30\ R_{\odot}$ because it is the
best compromise between the code accuracy and run time capabilities. For the
same reason, the maximum degree of spherical harmonics reconstruction has been
set to 25 instead of 30. The mesh is optimized in three parts: a uniform grid
from 1 to 1.004 $R_{\odot}$ with ten points to model the transition region,
then a first stretched segment with high resolution from 1.004 to 15
$R_{\odot}$ with 182 points to model accurately the solar corona, and then a
coarser stretched segment from 15 to 30 $R_{\odot}$ with 64 points for the
inner heliosphere. The grid has 160 points in latitude (and removes the 5
degrees closest to each poles to avoid numerical singularities) and 320 points
in longitude, and is regular in both these directions.