# Impact of the solar activity on the propagation of ICMEs:
Simulations of hydro, magnetic and median ICMEs at minimum and maximum of
activity

Barbara Perri Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM,
91191, Gif-sur-Yvette, France  Centre for mathematical Plasma Astrophysics, KU
Leuven, 3001 Leuven, Belgium  Brigitte Schmieder Centre for mathematical
Plasma Astrophysics, KU Leuven, 3001 Leuven, Belgium  LESIA, Observatoire de
Paris, France  Pascal Démoulin LESIA, Observatoire de Paris, France
Laboratoire Cogitamus, rue Descartes, 75005 Paris, France  Stefaan Poedts
Centre for mathematical Plasma Astrophysics, KU Leuven, 3001 Leuven, Belgium
Institute of Physics, University of Maria Curie-Skłodowska, Pl. M. Curie-
Skłodowskiej 5, 20-031 Lublin, Poland  Florian Regnault Space Science Center,
University of New Hampshire, Durham, New Hampshire, United States

###### Abstract

The propagation of Interplanetary Coronal Mass Ejections (ICMEs) in the
heliosphere is influenced by many physical phenomena, related to the internal
structure of the ICME and its interaction with the ambient solar wind and
magnetic field. As the solar magnetic field is modulated by the 11-year dynamo
cycle, our goal is to perform a theoretical exploratory study to assess the
difference of propagation of an ICME in typical minimum and maximum activity
backgrounds. We define a median representative CME at 0.1 au, using both
observations and numerical simulations, and describe it using a spheromak
model. We use the heliospheric propagator European Heliospheric FORecasting
Information Asset (EUHFORIA) to inject the same ICME in two different
background wind environments. We then study how the environment and the
internal CME structure impact the propagation of the ICME towards Earth, by
comparison with an unmagnetized CME. At minimum of activity, the structure of
the heliosphere around the ecliptic causes the ICME to slow down, creating a
delay with the polar parts of the ejecta. This delay is more important if the
ICME is faster. At maximum of activity, a southern coronal hole causes a
northward deflection. For these cases, we always find that the ICME at maximum
of activity arrives first, while the ICME at minimum of activity is actually
more geo-effective. The helicity sign of the ICME is also a crucial parameter
but at minimum of activity only, since it affects the magnetic profile and the
arrival time of up to 8 hours.

solar wind — solar cycle — coronal mass ejections — space weather

##  1 Introduction

Space weather is the ability to anticipate sudden events from the Sun and
their impact on our planet (Schrijver et al., 2015). Coronal Mass Ejections
(or CMEs) are considered one of the main drivers of strong space weather
events (Gosling, 1993): they consist of large-scale ejections of plasma and
magnetic field by the Sun, which then travels through the inner heliosphere
until they eventually reach our planet (Webb & Howard, 2012). CMEs that have
an impact on Earth are called geo-effective (Koskinen & Huttunen, 2006). In
particular, when they interact with the Earth magnetosphere, they can generate
magnetic storms that have consequences for our atmosphere (polar lights) and
ground (induced currents) (Pulkkinen, 2007). CMEs are an important concern for
space weather due to their frequency (more than 10 times a day launched in all
directions during intense phases of solar activity, see Robbrecht et al.
(2009)), and their impact on our technology (severe electrical damages due to
induced currents, see Pirjola (2005)). The most powerful recorded event was
the Carrington event on 1st of September 1859, where a CME traveled at 2300
km/s and reached Earth within 17 hours (Tsurutani et al., 2003); the cost of
the impact of such an event happening nowadays has been evaluated at trillions
of dollars (Schrijver, 2015).

CMEs originate mostly from active regions, which are regions where the
magnetic field is particularly intense, and stored in sheared and twisted
structures (usually flux-ropes, see Démoulin (2008); Schmieder et al. (2015)).
Eventually, these structures become unstable, and the plasma trapped inside is
released into the heliosphere as magnetic ejectas (MEs, Winslow et al., 2015).
When observed close to the Sun, most CMEs are characterized with a bright
frontal loop, a dark cavity, and an embedded bright core (Illing & Hundhausen,
1985) possibly corresponding to the erupting filament (House et al., 1981).
Images are available using white-light images via Thomson scattering to follow
their initial propagation (Davies et al., 2013). The in situ counterpart of
CMEs have been historically called Interplanetary Coronal Mass Ejections
(ICMEs). Their most important components are both hydrodynamic and magnetic
(Dumbović et al., 2015). The speed and density of the ICME, contribute to the
dynamic pressure and also contain signatures of the propagation of the
interplanetary shock at the front of the ICME. Interplanetary shocks alone can
drive geomagnetic activity (Oliveira & Samsonov, 2018), which is why the most
simple models do not include an internal magnetic field structure for the
ICME. However, it has been shown that it is its magnetic field amplitude and
orientation which are driving the strongest geomagnetic storms (especially its
$B_{z}$ component which favors dayside reconnection with the magnetopause,
Lugaz et al., 2016). The interplanetary shock is followed by a compressed and
heated region called the sheath (Kilpua et al., 2017). This region is caused
by the accumulation of solar wind at the front of the ICME, as well as the
expansion of the following ME (Kaymaz & Siscoe, 2006). An ME is characterized
by a strong and smooth magnetic field, as well as low temperature and low
plasma beta (ratio of the thermal pressure over the magnetic pressure, Wang et
al., 2005). Other diagnostics can also be used when available (Zurbuchen &
Richardson, 2006). When the magnetic ejecta shows rotation of its magnetic
field, and has a proton temperature lower by a factor two than the typical
solar wind with the same speed, it is categorized as a magnetic cloud (MC), as
it is associated with the existence of a flux-rope (Burlaga et al., 1981;
Burlaga, 1995). Such configuration only happen for about one third of ICMEs at
1 au (Wu & Lepping, 2011). However, this may be due to the limitation of
having only one measurement point (Jian et al., 2006; Kilpua et al., 2011).
The recent era allows for multi-spacecraft coordination thought the
heliosphere to try to better quantify the 3D geometry of ICMEs (Möstl et al.,
2022).

As ICMEs propagate through the heliosphere, they interact with the various
structures they encounter, and as a result evolve. For example, ICMEs will
naturally expand as they travel, and usually it can be approximated by a self-
similar expansion (Démoulin et al., 2008; Gulisano et al., 2010, 2012; Chané
et al., 2021; Verbeke et al., 2022). Concerning their trajectory, CMEs can
suffer deflections both in latitude and longitude, because of their
interaction with the magnetic field of coronal holes, helmet streamers and the
heliospheric current sheet (HCS) (Gopalswamy & Mäkelä, 2014). This tends to
focus CMEs towards the Earth latitudes (especially at minimum of activity, see
Zuccarello et al. (2012)). Recent studies have shown that the strength and
sign of the ambient magnetic field can influence their drift (Asvestari et
al., 2022) due to tilting instabilities (Bellan, 2000). It is also linked to
the sign of helicity of the source region (Green et al., 2007). Concerning
their speed, ICMEs can be accelerated or decelerated through their interaction
with the ambient solar wind (Gopalswamy et al., 2000) which causes drag-like
effects (Cargill et al., 1995). Concerning their magnetic flux, it can be
reduced due to magnetic reconnection, which will lead to magnetic erosion at
the front of the ICME (Dasso et al., 2007; Ruffenach et al., 2012). As they
propagate, ICMEs are more likely to become more and more complex as a result
of their interaction with solar wind structures (Winslow et al., 2015, 2022;
Scolini et al., 2022, 2023), and to display aging processes that will
contribute to their deformation (Démoulin et al., 2020). They will also change
as a result of interactions with specific structures, such as high-speed
streams (HSSs) (Fenrich & Luhmann, 1998; Heinemann et al., 2019; Scolini et
al., 2021b), stream-interaction regions (SIRs) or other ICMEs (Lugaz et al.,
2005; Scolini et al., 2020). For more details, see reviews by Lavraud &
Rouillard (2014) and Shen et al. (2022), and references within.

The medium in which the ICMEs propagate is far from simple to describe, as it
shows great complexity and variability. The interplanetary medium is
influenced both by the magnetic field close to the Sun, and then the solar
wind further away from it. The solar magnetic field is generated inside the
Sun by dynamo effect (Moffatt, 1978; Parker, 1993; Brun & Browning, 2017), and
then bathes the entire heliosphere following the Parker spiral due to the
rotation of the star (Owens & Forsyth, 2013). The solar wind is made of plasma
particles continuously emitted by the Sun (Parker, 1958). It has two main
components, one slow and one fast, whose source mechanisms differ (see Cranmer
et al. (2007), and Viall & Borovsky (2020) and references within). The short-
term variability of the interplanetary medium is caused by reconnection
effects in the lower corona (causing the switchbacks observed by Parker Solar
Probe, see Kasper et al. (2019)), and transient events (perturbations caused
by SIRs, HSSs, previous ICMEs…). The long-term variability on the other hand
is due to the solar activity cycle (Hathaway, 2015). Indeed, the solar dynamo
is cyclic with a period of 22 years (Weiss, 1990), which generates periods of
low magnetic activity, called minima, where the field is mostly dipolar, and
periods of high magnetic activity, called maxima, where the field becomes more
multipolar (Hoeksema, 1984; DeRosa et al., 2012). This modulation has a direct
effect on the structure of the corona and inner heliosphere. At minimum of
activity, the corona is very structured with stable equatorial helmet
streamers, thus confining the slow solar wind to the equator and the fast wind
at the poles; at maximum of activity, the corona is more complex with pseudo-
streamers and streamers emerging at various latitudes, and thus the solar wind
has slow and fast streams at all latitudes (McComas et al., 2003, 2008).

Because of all these complex interactions, the impact of solar activity on
ICME propagation is still unclear. We know that CMEs/ICMEs are more frequent
(Gopalswamy et al., 2003), faster (Hundhausen et al., 1990; Dasso et al.,
2012) and more magnetized (Wu & Lepping, 2011) at maximum of activity, and yet
the most powerful events recorded (like the Carrington event for example) did
not necessarily happen during these periods (because they are probably due to
ICME-ICME interactions) (Chapman, 2023). There are other interesting
relationships between ICMEs and the solar cycle. The sign of the helicity can
be estimated using hemispheric rules that depend on the solar cycle polarity
(Bothmer & Schwenn, 1998). Gopalswamy et al. (2003) noted that high-latitude
CMEs do not occur during polarity reversals phases of the solar cycle, due to
the lack of closed field lines close to the poles. Studies have analyzed the
dependence of specific properties with the phase of the solar cycle. Dasso et
al. (2012) showed that the expansion of the ICME is not related to the solar
cycle, but the amount of helicity of the magnetic cloud is. Jian et al. (2011)
showed that during solar minima, shocks are more important due to a slower
solar wind. Finally, Regnault et al. (2020) analyzed 20 years of ACE data and
compared the internal properties of ICMEs based on the solar activity phase:
they confirmed that ICMEs during maximum were faster, but otherwise parameters
were similar; what changes is that at maximum, they observe a larger
distribution of ICME parameters, just like in Wu & Lepping (2016) where they
used Wind data.

Our goal with this paper is to explore more quantitatively the impact of solar
activity on the propagation and geo-effectiveness of ICMEs. In this first
exploratory study, we will inject the same ICME in two different activity
background using numerical simulations, and quantify the differences and their
origins, as a first step towards understanding what effects to expect.

The article is organized as follows. In Section 2, we describe the numerical
set-up behind the European Heliospheric FORecasting Information Asset
(EUHFORIA) code, for both the coronal and heliospheric part, as well as the
CME/ICME modeling. In Section 3, we explain our choice of boundary conditions,
first for the solar wind, then for the CME insertions. In Section 4, we
present two limit cases: a first one with only solar wind to quantify the
background for the next study of ICME propagation, and a second one where the
injected CME is purely hydrodynamic, to analyze the effect of the solar wind
speed. In Section 5, we present the results for a magnetized CME inspired by a
real event. In Section 6, we present the results for a ICME representing a
median case based on statistical analysis of ICME parameters. Finally, in
Section 7, we present our discussion and conclusion.

##  2 Description of EUHFORIA simulations

We use the numerical 3D MHD code EUHFORIA (Pomoell & Poedts, 2018). It is
divided into two numerical domains with different treatments: a semi-empirical
model for the coronal part (from 1 $R_{\odot}$ to 21.5 $R_{\odot}$), and an
MHD model for the heliospheric part (from 0.1 to 2 au). The limit between the
two domains is set at 0.1 au because at this distance the solar wind is
superfast, making the coupling one-way (the heliospheric part cannot back-
react on the coronal part).

###  2.1 Coronal part

The coronal part uses synoptic magnetic maps as inputs, which are observations
of the photospheric radial magnetic field used to drive the simulation. They
are called synoptic because they cover the 360∘ of the solar surface, but they
are not necessarily synchronic (which means that the solar observations
displayed on the map were not necessarily taken at the same date, for example
the observations can last a full solar rotation and thus have a 27-day gap)
(Riley et al., 2014). The default set-up of EUHFORIA can use two types of
magnetic maps: the ones from the Global Oscillation Network Group (GONG)
(Harvey et al., 1996) and the ones from the GONG Air Force Data Assimilative
Photospheric Flux Transport Model (GONG-ADAPT) (Arge et al., 2010; Hickmann et
al., 2015). From this input, the magnetic field global configuration is
derived using two models. First, we use a potential-field source surface
(PFSS) model up to the source surface $R_{ss}$ of 2.6 $R_{\odot}$ (Altschuler
& Newkirk, 1969). This allows us to compute a current-free configuration by
assuming the magnetic field is potential below the source surface, and purely
radial afterwards. Then it is coupled to a Schatten current sheet (SCS) model
from 2.3 $R_{\odot}$ up to 21.5 $R_{\odot}$ (Schatten et al., 1969). The SCS
model starts slightly below the source surface to reduce possible kinks due to
incompatibility between the two models (McGregor et al., 2008). The SCS field
is required to vanish at infinity, in order to extend the magnetic field
radially while maintaining a thin structure for the heliospheric current sheet
(HCS). This method provides results in better agreement with observations
(with the Ulysses mission, for example, see Pinto & Rouillard (2017)).

Once the magnetic configuration is computed, a semi-empirical Wang-Sheeley-
Arge method is used to compute the radial velocity (Wang & Sheeley, 1990; Arge
et al., 2003), using the following formula (in km/s):

$v_{r}(f,d)=240+\frac{675}{(1+f)^{0.222}}\left[1.0-0.8\,\rm{exp}\left(-\left(\frac{d}{0.02}\right)^{1.25}\right)\right]^{3},$
(1)

where $d$ is the angular distance of the footpoint of the magnetic field line
to the closest coronal hole boundary, and $f$ is the flux tube expansion
factor between the photosphere and the source surface. This law is adapted
from van der Holst et al. (2010); McGregor et al. (2011). Since the solar wind
continues to accelerate beyond 0.1 au, we deduct a constant value of 50 km/s
from equation (1) to prevent the wind speed from being systematically
overestimated. Furthermore, we enforce the final speed to be in the range
$v_{r}\in[275,625]$ km/s by capping it (McGregor et al., 2011). Finally, we
apply a rotation of 10∘ to the obtained solar wind speed map at 0.1 au to take
into account the approximated solar rotation that is not included in the
magnetic field model.

Based on this computed velocity, the density $n$, temperature $T$ and radial
magnetic field $B_{r}$ are computed at 0.1 au using the following relations
(Pomoell & Poedts, 2018):

$n=n_{fsw}\left(v_{fsw}/v_{r}\right)^{2},$ (2)
$T=T_{fsw}\left(\rho_{fsw}/\rho\right),$ (3)
$B_{r}=\rm{sgn}\left(B_{corona}\right)B_{fsw}\left(v_{r}/v_{fsw}\right),$ (4)

with $v_{fsw}=675$ km/s, $n_{fsw}=300\,\rm{cm}^{-3}$, $T_{fsw}=0.8$ MK,
$\rho_{fsw}=0.5n_{fsw}m_{p}$ ($m_{p}$ being the proton mass), $B_{sfw}=300$ nT
and $\rm{sgn}\left(B_{corona}\right)$ being the sign of the magnetic field as
given originally by the coronal model at 0.1 au. All these values are
consistent with a fast solar wind (hence the abbreviation $fsw$) (Odstrčil &
Pizzo, 1999). The number density prescription ensures a constant kinetic
energy density on the spherical surface at 0.1 au. The plasma thermal pressure
is chosen to be constant at 0.1 au (equal to 3.3 nPa), which sets the fast
wind temperature. The reconstruction of the magnetic field based on the radial
speed instead of the PFSS+SCS avoids the open-flux problem denoted by Linker
et al. (2017). These conditions are similar to the ones described in Odstrčil
& Pizzo (1999).

###  2.2 Heliospheric part

Using the boundary conditions provided by the coronal part, the heliospheric
part of EUHFORIA then computes the solar wind all the way to 2 au by solving
the 3D time-dependent ideal MHD equations augmented with gravity. The
equations are solved in the heliocentric Earth equatorial (HEEQ) frame, which
is described as the frame with its Z-axis aligned with the rotation axis of
the Sun, and its X-axis defined by the intersection of the solar equatorial
plane and the solar central meridian of date as seen from the Earth. Although
the chosen frame is not inertial, we choose to omit the Coriolis and
centrifugal terms which should be the result of the orbital motion of Earth,
as their contribution is actually negligible. A value of 1.5 is selected for
the polytropic index as in Odstrcil et al. (2004). This value is close to the
ones found for protons in all solar winds with Helios and Parker Solar Probe
(Dakeyo et al., 2022). The fact that we use a reduced index closer to 1 than
5/3 is a simple way of modelling coronal heating and as a result acceleration
of the solar wind (Pomoell & Vainio, 2012).

A relaxation phase is first performed (set to 14 days in our case to avoid any
spurious effect caused by the initial transient at the outer boundary
condition). Then the forecast phase begins at the date set by the magnetic
map. The forecast phase last 7 days in our cases, which is more than enough
for the ICME to cross the entire domain.

The default set-up has a 2∘ angular resolution. The computational domain
extends from 0.1 to 2 au in the radial direction and spans 120∘ in latitude
and 360∘ in longitude. This means that the solar poles are truncated by 30∘ in
each hemisphere. To solve the MHD equations, we use a finite volume method
combined with a constrained transport approach. To obtain a scheme which is
both robust and second-order accurate, we use an approximate Riemann solver
with standard piece-wise linear reconstruction (Kissmann & Pomoell, 2012;
Pomoell & Vainio, 2012). At the outer radial boundary, we use open boundary
conditions implemented via a simple extrapolation, whereas at the latitudinal
boundaries we use symmetric reflection boundary conditions.

One last point of detail we would like to discuss here is the difference
between EUHFORIA outputs and proton data. As explained in Scolini et al.
(2021a), EUHFORIA uses a single-fluid approach, and thus makes no difference
between the different particle populations (such as protons, electrons,
$\alpha$ particles, etc.). In order to compare EUHFORIA outputs to
observational data, we must thus make some assumptions. We will consider the
proton and electron populations as the two primary contributors to solar wind
plasma, and we assume that the two species have the same temperature:
$T=T_{p}=T_{e}$. To further ensure the quasi-neutrality of the plasma at all
locations and times in the heliosphere, we also assume the two species have
the same number density: $n_{p}=n_{e}$. This means that the plasma density in
EUHFORIA is $n=n_{p}+n_{e}=2n_{p}$. As a result, EUHFORIA thermal pressure and
$\beta$ plasma parameter is also twice that of the proton population:
$P_{th}=P_{p,th}+P_{e,th}=n_{p}k_{B}T_{p}+n_{e}k_{B}T_{e}=2P_{p,th}$ and
$\beta=P_{th}/P_{mag}=2P_{p,th}/P_{mag}=2\beta_{p}$. The plasma total pressure
is then computed as: $P_{tot}=P_{mag}+P_{th}=P_{mag}+2P_{p,th}$. The plasma
temperature is retrieved using: $T=T_{p}=P/nk_{B}=P_{p}/n_{p}k_{B}$. Since
$m_{p}\gg m_{e}$, we can assume that $v\approx v_{p}$. To avoid confusion,
EUHFORIA quantities will hereafter be denoted without indexes, while proton
quantities will be denoted with index $p$. This approximation is of course
different from known observations, but it would require to move from ideal MHD
to at least two-fluid resistive MHD to overcome it, which is beyond the scope
of this paper (Priest, 2014).

###  2.3 CME modeling

EUHFORIA offers the possibility to inject a coronal mass ejection (CME) at 0.1
au inside the heliospheric part in a time-dependent way, in order to model its
propagation and interaction with the modeled ambient solar wind. There are
various models available to represent the CME. In this study, we will be using
two of them.

First, we use the cone model, described in Pomoell & Poedts (2018) and similar
to Odstrčil & Pizzo (1999). The CME is treated as a hydrodynamic cloud and is
characterized by a constant angular width (which then defines an initial
radius at the injection point), propagation direction and radial speed. The
user can also prescribe the injection point, defined via its coordinates in
HEEQ frame. The cross-section is assumed to be circular, and the CME density,
pressure and radial speed are constant.

The other model we use is the linear-force-free spheromak (LFFS) model,
described in Verbeke et al. (2019) and similar to Kataoka et al. (2009);
Shiota & Kataoka (2016). It is a magnetized model, but unlike the model from
Gibson & Low (1998), the CME completely goes through the boundary without
footpoints left attached to the solar surface. The CME, considered to be a
sphere of radius $r_{0}$ upon the time of its injection, is launched outward.
The velocity of the CME is chosen to be constant everywhere within the CME,
and always oriented along the given propagation direction. As a result, the
total velocity vector $v$ is not purely radial at the inner simulation
boundary, but contains also a latitudinal and longitudinal component as well.
This means that the total speed of the CME $v_{3D}$ can be decomposed into two
components: the radial speed $v_{rad}$ and the expansion speed $v_{exp}$. In
our case, we prescribe only the radial speed at the injection point. It
requires the same physical input parameters as the cone model, plus three
additional magnetic parameters: the handedness $H$ of the spheromak (which
determines the polarity of the magnetic field inside the CME); the tilt angle
$\tau$ (measured from the $z$-axis in the $yz$-plane); the total toroidal flux
$F$ (related to the magnetic field strength $B_{0}$). The spheromak
orientation is thus given by the injection point, combined with the tilt that
gives the axis of symmetry of the spherical structure (the magnetic
configuration is shown in appendix in Figure 16). For example, in HEEQ
coordinates of EUHFORIA, if the tilt is equal to 0, then the axis of symmetry
is the vertical $z$-axis; if the tilt is equal to 90∘, the axis of symmetry is
the horizontal $y$-axis. Compared to Euler angles, the tilt angle corresponds
to the elevation angle. In the current implementation, we do not need
additional angles (like the heading or bank angles) that would allow us to
rotate around the axis of symmetry, since the spheromak exhibits symmetry in
the azimuthal direction $\phi$. The direction of propagation of the spheromak
is finally set to be perpendicular to the 0.1 au boundary surface at the
injection point.

Figure 1:  Comparison between the two selected magnetic maps of the radial
magnetic field $B_{r}$ for minimum (left panel) and maximum (middle panel) of
activity. The left panel shows a GONG synoptic map corresponding to the 15th
of December 2008, representative of a minimum of activity (dipolar field with
very few active regions). The middle panel shows a GONG-ADAPT map
corresponding to the 20th of March 2015, representative of a maximum of
activity (multipolar field with intense active regions). We also show for
information the GONG map corresponding to the date chosen for maximum of
activity (right panel). We can thus see that there is more difference between
minimum and maximum of activity than between GONG and GONG-ADAPT. Positive
polarity of the magnetic field is shown in red, negative polarity in blue. The
color bar has been set symmetric and saturated to better visualize the
structures (between -20 and 20 G). Note that the y-axis for the left panel is
in sine latitude.

##  3 Boundary conditions

In this section, we discuss how we selected the boundary conditions at 0.1 au
for our heliospheric study. We first explain in subsection 3.1 the parameters
related to the choice of the solar wind background, and then in subsection 3.2
the parameters related to the ICME initialization.

###  3.1 Solar wind for minimum and maximum of activity

The idea behind this study is to use a solar wind background that would be
more realistic than in previous studies using EUHFORIA: for example, Scolini
et al. (2021c) used an ideal analytical background, and Asvestari et al.
(2022) even approximated the magnetic field as a monopole (which is an
impossible configuration due to the Maxwell equations). We especially aim to
reflect the complexity brought by the 11-year modulation of the solar cycle.
To do so, we first selected two dates that correspond to typical minimum and
maximum of solar activity.

For the minimum, we selected the Carrington Rotation number 2077, set between
November 20 and December 17, 2008. This corresponds to the end of solar cycle
23. This date is indeed a well-known case used for calibration of coronal
models at minimum of activity due to its very quiet magnetic field during this
period (Rušin et al., 2010; van der Holst et al., 2014; Wiegelmann et al.,
2017; Perri et al., 2022). It has even been chosen as the International Space
Weather Action Team (ISWAT) validation benchmark for solar-wind models (Reiss
et al., 2022). At this date, only the GONG magnetograms are available; the
corresponding magnetogram can be found in the left panel of Figure 1. It
exhibits typical features from minimum of activity, such as strong flux
concentration at the poles with different polarities in the two hemispheres,
which is a marker of the dipolar-dominant structure of the solar magnetic
field at this period. There are also very few active regions, so that we
mostly see the super granulation pattern typical of the quiet Sun.

Figure 2: Comparison of the EUHFORIA input boundary condition at 0.1 au for
minimum (left panel) and maximum (right panel) of activity. The first row
shows the radial velocity $v_{r}$ in km/s, the second row the number density
$N$ in $\rm{cm}^{-3}$, the third row the temperature $T$ in $10^{6}\ K$, the
fourth row the radial magnetic field $B_{r}$ in nT. All these profiles were
derived from the magnetic maps displayed in Figure 1 using Wang-Sheeley-Arge
and PFSS+SCS empirical relations. X-axis is in Stonyhurst longitude, y-axis in
latitude.

For maximum of activity, we selected the date of March 20, 2015. This date
corresponds to the maximum of solar cycle 24, and has been extensively used as
a benchmark date for many coronal models (Yeates et al., 2018). After 2010,
the GONG-ADAPT maps are available, so we use it for the maximum as it reduces
the probability of having spurious features at the solar poles thanks to the
additional post-processing of ADAPT. This causes a difference in the provider
and thus the processing of the input map. However, it guarantees that the
final solution does not suffer from numerical artifacts. We also show in
Figure 1 the difference between GONG and GONG-ADAPT at maximum of activity. We
can see that there is a difference in amplitude for the most intense active
region, sometimes a difference in polarity for the quiet Sun and a different
method for filling the solar poles. However, the general structure is still
very similar, and these differences are still less than the difference between
minimum and maximum of activity. This means that the differences we may see in
the final CME solutions are indeed mainly due to the difference in solar
activity. We will dedicate a specific study on the impact of the input
magnetic map on the final CME solution, but for now it is out of the scope of
this paper. The map can be seen in the right panel of Figure 1. At maximum of
activity, many active regions are present at the surface of the Sun between
-0.5 and 0.5 in sine latitude (which corresponds to around -40 and 40∘ in
latitude), creating intense magnetic field configurations that dominate over
the poles and the quiet Sun.

The implication of these two different magnetic maps for the coronal part of
EUHFORIA can be seen in Figure 2. We represent the boundary condition computed
by the semi-empirical coronal model which will serve as boundary condition
(BC) for the heliospheric MHD model. The left panel shows the BC for the
selected minimum of activity, the right panel shows the BC for the selected
maximum of activity. In each panel, the first row corresponds to the radial
velocity in km/s, the second to the number density in $\rm{cm}^{-3}$, the
third to the temperature in $10^{6}\ K$ and the last one to the radial
magnetic field component in nT. At minimum of activity (left panel), the
corona is very structured, with fast hot wind at the poles and slow dense cold
wind at the equator. The magnetic field is very much dipolar, with a negative
polarity at the northern pole and positive polarity at the southern pole. The
HCS between the two polarity is almost in the ecliptic with few disturbances.
At maximum of activity, slow and fast wind are much more mixed at all
latitudes. The HCS exhibits a more complex pattern, with the negative polarity
from the southern hemisphere almost filling completely the map between
longitudes 0 and 130. There is also an incursion of positive polarity in the
negative polarity around longitude -50. These different BCs will result in
different backgrounds for propagation of the CMEs, as will be detailed in
Section 4.1. Note that no optimization was performed here to constrain the
wind with observations: first, because we do not aim at reproducing a specific
event, this is foremost a theoretical study; second, because we want to keep
an operational set-up to evaluate the implication for the space-weather
forecasts done using EUHFORIA that use these automatic parameters.

Case | Hydro CME | Reference CME | Median CME
---|---|---|---
Type of CME model | Cone | LFFS | LFFS
Latitude of center [deg. HEEQ] | 0.0 | 0.0 | 0.0
Longitude of center [deg. HEEQ] | 0.0 | 0.0 | 0.0
Radius [$R_{\odot}$] | 15.0 | 15.0 | 15.0
Speed [km/s] | 1400 | 763 | 541
Mass density [kg/m3] | 1.0 $10^{-18}$ | $1.0\,10^{-18}$ | $2.0\,10^{-18}$
Temperature [K] | $8.0\,10^{5}$ | $2.4\,10^{5}$ | $6.2\,10^{5}$
Handedness | N/A |  $\pm$1 |  $\pm$1
Tilt [deg.] | N/A | 0.0 | 0.0
Flux [Wb] | N/A | $1.0\,10^{14}$ | $2.3\,10^{13}$
Table 1: Summary of the CME parameters used at 0.1 au as a boundary condition
for the EUHFORIA runs. For each case, we specify the model used, the center of
the injected CME with its latitude and longitude in HEEQ coordinate, the
radius of the inserted CME, the speed, mass density, and temperature. For the
magnetized CMEs, we also specify the handedness, tilt and magnetic flux.

###  3.2 Definition of the CME parameters

At 0.1 au, we also have to define the parameters of the CME to inject in the
simulation. In this study, we have chosen three cases to explore. All the
corresponding parameters can be found summarized in Table 1.

The first case is based on a real event that took place on July 12, 2012. This
case has been extensively studied in Scolini et al. (2019) and Scolini et al.
(2021a) using a magnetized LFFS model. The parameters for this CME were
derived from observations, using white-light coronagraph images to constrain
the geometrical and kinetic parameters, and photospheric and low-corona
observations of the source active region to constrain the magnetic parameters
(for more details, see Scolini et al., 2019). The resulting parameters are
summarized in column labeled Reference CME of Table 1. They have been adapted
in the scope of this study: we have adjusted the center of the CME to be along
the Sun-Earth axis (which yields 0∘ in HEEQ latitude and longitude), and we
have selected the same average initial radius at injection for all cases of
$15\,R_{\odot}$ (to avoid geometric factors contributing to the variation in
mass). This value corresponds to an average of the values found by GCS fitting
for the events studied in Scolini et al. (2019), which were 10.5, 14.5, 16.8
and 18.0 $R_{\odot}$. The injection point being on the Sun-Earth line in HEEQ
coordinates, the CME direction of propagation is towards Earth, to simulate
optimal initial conditions for a full hit. The tilt has also been set to 0 to
study a CME where the full magnetic ejecta propagates along the ecliptic
plane. The speed, mass density, temperature, handedness and flux are however
the same as in Scolini et al. (2019).

We also wanted to try a limit case with the same CME, but using the cone model
instead of the LFFS model. This means that the CME is purely hydrodynamic
(without an inner magnetic field). To obtain the corresponding parameters, we
used this time the DONKI database111https://kauai.ccmc.gsfc.nasa.gov/DONKI/
which recommended a speed around 1400 km/s. The lower initial speed is due to
the fact that we prescribe only the input radial speed, which is equal to the
full 3D speed inferred from observations for a cone model ($v_{rad}=v_{3D}$),
but equal to the difference between the 3D speed and the expansion speed for a
spheromak ($v_{rad}=v_{3D}-v_{exp}$). For the reference case, the CME
geometric parameters were derived using a GCS model (Graduated Cylindrical
Shell model, see Thernisien et al. (2009); Thernisien (2011)), which gave a
full 3D speed $v_{3D}$ of 1266 or 1352 km/s depending on the fitting. The
expansion speed of the CME is derived using empirical relations from Dal Lago
et al. (2003) and Schwenn et al. (2005), which is an alternative when 3D
reconstruction of the event is not possible due to a single spacecraft
configuration for the observation (more details in Scolini et al. (2019)).
These relations give: $v_{rad}=0.43v_{3D}$ in the case of a spheromak, which
explains the difference in input speed of a factor close to 2. The DONKI
database values are thus consistent with the values obtained from Scolini et
al. (2019). This is explained in more details in Scolini et al. (2019). We
then completed the input values with the default EUHFORIA mass density and
temperature (respectively $1.0\,10^{-18}$ kg/m3 and $8.0\,10^{5}$ K, as
specified in Pomoell & Poedts (2018)). All other geometric parameters were
kept identical to limit the number of free parameters of the study. This is
summed up in the column labeled Hydro CME of Table 1.

Figure 3: Comparison of the ICME parameters obtained at Earth for the median
case at minimum (left panel) and maximum (right panel) of activity. The first
row is the total magnetic amplitude in nT, the second row the total velocity
amplitude in km/s, the third row the temperature in $10^{4}$ K, the fourth row
the number density for the protons in $\rm{cm}^{-3}$ and the last row $\beta$
parameter of the plasma. A yellow rectangle highlights the region
corresponding to the sheath, and a blue rectangle the region corresponding to
the magnetic cloud. The method used to derive these regions is explained in
more details in Appendix A.

Finally, we also wanted to have a magnetized case which would be more
representative of an average CME at Earth. To obtain such values, we combined
two studies. First, we used the statistical study from Regnault et al. (2020),
where 20 years of ACE data have been analyzed using the superposed epoch
analysis (SEA) method to derive the distribution of ICME parameters at Earth.
From this study, we can extract the median value of the distribution, which
yields the following parameters:

$v_{r}\approx 450\,\rm{km/s},N\approx 5\,\rm{cm}^{-3},T\approx
4\,10^{4}K,B_{r}\approx 10\,\rm{nT}.$ (5)

By using the median, we select a set of parameters so that 50% of ICMEs have
values under this set, and 50% above. We could have also used the mode (which
is the most probable set of parameters), but due to the fact that it is a log
normal distribution, it would have been biased towards slower events that
would have been too close to our background wind speed to allow proper
analysis. Once we have these values at 1 au, we need to extrapolate them back
to 0.1 au. To do so, we use the scaling laws derived by Scolini et al. (2021a)
by studying the radial evolution of ICMEs in EUHFORIA:

$v_{r}\propto r^{-0.08},N_{p}\propto r^{-2.38},T_{p}\propto
r^{-1.19},B_{r}\propto r^{-1.9}.$ (6)

Combining these two set of parameters, we can obtain the corresponding values
at 0.1 au for a median CME:

$v_{r}=541\,\rm{km/s},N_{p}=1.2\,10^{3}\,\rm{cm}^{-3},T_{p}=6.2\,10^{5}\,K,B_{r}=7.9\,10^{2}\,\rm{nT}.$
(7)

We then convert these values to obtain the parameters needed for the input
configuration file of EUHFORIA, using the following relations:

$\rho=m_{p}N_{p}\times
10^{6},\phi_{t}=\frac{2B_{0}}{\alpha}\left[-\rm{sin}\left(\alpha
r_{0}\right)+\int_{0}^{\alpha r_{0}}\frac{\rm{sin}\,t}{t}dt\right],$ (8)

with $m_{p}$ being the proton mass, $\alpha r_{0}=4.4934$ (which corresponds
to prescribing the radial component of the magnetic field to be 0 on the
boundary of the spheromak) and $r_{0}=15R_{\odot}$ (for more details, see
Verbeke et al. (2019)). The resulting parameters are summed up in the column
labeled Median CME of Table 1. Compared to the real event, we can see that the
median CME is slower, denser and hotter, and less magnetized than the
reference case.

We finally check with EUHFORIA simulations that this approach yields correct
results by analyzing the values of the ICME obtained at Earth for the median
CME, both at minimum and maximum of activity. Results can be visualized in
Figure 3. The minimum of activity case in on the left panel, the maximum of
activity case on the right panel. For each case, we plot the total magnetic
amplitude in nT, the total velocity amplitude in km/s, the temperature in
$10^{4}$ K, the number density for the protons in $\rm{cm}^{-3}$ and the
$\beta$ parameter of the plasma, defined as
$\beta=P_{th}/P_{mag}=2P_{p,th}/P_{mag}=2\beta_{p}$ (thermal pressure over
magnetic pressure). The panel is made to be easily comparable with the
analysis from Regnault et al. (2020), which has the same layout for the SEA.
On top of the curves, we plot a yellow rectangle to highlight the region
corresponding to the sheath and a blue rectangle for the region corresponding
to the magnetic ejecta. The method used to derive these regions is explained
in more details in Appendix A. In the magnetic ejecta, we obtain on average
the following values for the minimum of activity case: $v_{r}\approx 420$
km/s, $N\approx 5\,\rm{cm}^{-3}$, $T\approx 5\,10^{4}$ K, $B\approx 8$ nT.
Similarly, we obtain, for the maximum of activity case: $v_{r}\approx 450$
km/s, $N\approx 10\,\rm{cm}^{-3}$, $T\approx 4\,10^{4}$ K, $B\approx 10$ nT.
These values are consistent with the targeted valued at 1 au (cf. equation
(5)), which means these cases can indeed be classified as representative of a
median ICME as seen at Earth.

Figure 4: Comparison of the heliospheric wind background at minimum (left
panel) and maximum (right panel) of activity. The first row is the radial wind
speed in km/s, the second row the number density in $\rm{cm}^{-3}$ normalized
to 1 au and the last row the radial magnetic field in nT. For each row, we
show the ecliptic (view from above) and meridional (view from the side
crossing Earth) views in the HEEQ frame. Various planets and satellites are
also shown in the bottom legend.  Figure 5: Comparison of the 3D heliospheric
current sheet (HCS) structure for the selected minimum (left panel) and
maximum of activity (right panel). The HCS contour is shown in dark blue. For
context, the ecliptic plane with the radial velocity is shown in transparency.
The position of the Earth is indicated by a blue sphere. Both figures are in
HEEQ frame. Figure 6: Comparison of the time evolution of the background solar
wind parameters at 0.11 au (left panel) and 1 au (right panel) along the Sun-
Earth axis. The first row shows the total velocity in km/s, the second row the
number density in $\rm{cm}^{-3}$ and the last row the radial magnetic field
component in nT. The curves for the minimum of activity are in blue, while the
curves for the maximum of activity are in orange. Grey rectangles have been
added to show the time intervals when the background solar wind is likely to
interact with the ICME.

##  4 Limit cases

Before analyzing further the differences between the ICMEs, we want to analyze
two limit cases to better understand the context of our simulations. In
Section 4.1, we will first present a case with only the background wind (no
CME inserted) to better quantify the differences and similarities between
without and with CME. In Section 4.2, we will then inject a hydrodynamic CME
(described in column labeled Hydro CME of Table 1) to quantify the impact of
the hydrodynamic parameters alone.

###  4.1 Wind-only simulation

In this first limit case, we do not inject any CME, we just let the wind
background evolve for 7 days to see how it behaves without perturbation. In
Figure 4, we can see the heliospheric part of EUHFORIA with the modeling of
the ambient solar wind background. We show the radial wind speed, the number
density and the radial magnetic field, both in the ecliptic and meridional
(including Earth) planes. This allows us to get a first qualitative look at
the similarities and differences between the selected minimum and maximum of
activity. Since these are realistic solar wind backgrounds, we can see a lot
of substructures in the wind. One constrain on the dates is that any large-
scale structures (such as high-speed streams where the wind speed reaches 500
km/s or more) is not Earth directed and thus not geo-effective, to limit their
interference with the propagation of the CME towards Earth. The dates have
been adjusted so that the sector in the ecliptic plane is dominantly negative
(in blue in the bottom row) in both cases.

There are however expected differences that are relevant to this study,
because they are representative of what we expect from minimum and maximum of
activity configurations. At minimum of activity, the meridional view shows a
very organized wind structure, with slow dense wind in the ecliptic plane, and
fast less dense wind near the poles. This is a logical consequence of the
structures observed in the coronal boundary condition in Section 3.1. At
maximum of activity, the solar wind organization is much more complex. For the
magnetic field, we also see that at minimum of activity, the current sheet
location near the ecliptic results in the polarity sectors being very clearly
defined: at Earth location, the northern hemisphere is negative while the
southern hemisphere is positive. At maximum of activity, the polarity remains
negative in latitude, but a positive sector will cross the Earth later on in
the simulation (see the bottom right panel of Figure 4). This kind of fast
polarity switch is expected at maximum of activity and is one of the features
we are interested in.

To focus on the magnetic field structure, in Figure 5, we represent the HCS in
3D. We can clearly see its quasi-ecliptic shape at minimum of activity, with
the Earth being positioned slightly above it. At maximum of activity and in
the inertial frame, the current sheet is very distorted, so that the Earth
falls into one sector. The contour on the left-side of the picture is the
positive polarity incursion from the ecliptic view, and it may interfere with
the CME propagation during its latest stages.

Finally, in Figure 6, we show a more quantitative view of the background solar
wind with 1D evolution plots. We plot the same quantities as in Figure 4, but
along the Sun-Earth axis using virtual satellites. The left panel at 0.11 au
gives a view of what the background looks like at the injection point of the
CME, while the right panel shows the background at 1 au to show the ambient
medium at Earth. Minimum and maximum of activity data are over-imposed,
respectively in blue and orange. Grey rectangles have been added to highlight
the regions of interest in each panel for the interaction with the ICME. We
notice in the right panel that the wind structures are indeed very similar at
Earth, except for $|B_{r}|$ which is more important at maximum of activity (as
expected since the solar activity is increasing). Close to injection point
(which means on the left panel), the wind speed is low in both cases (around
300 km/s). The main difference lies in the polarity inversion: at minimum of
activity, the magnetic field polarity remains mostly negative, getting weak
around 1 au (Figure 6 right panel), while at maximum of activity the disturbed
HCS allows for a polarity switch from negative to positive as the ICME is
progressing outward.

Figure 7:  Comparison of the cone CME propagation for the minimum (top panel)
and maximum (bottom panel) of activity cases. We show the density in
$\rm{cm}^{-3}$, normalized to 1 au to better visualize the CME ejecta as an
under-dense structure. We show the moment when the ejecta is reaching Earth
(symbolized by a blue circle circled by a white line to the right of the Sun).
For each case, we show the ecliptic (view from above, left panel), meridional
(view from the side including Earth, middle panel) and spherical (view at 1
au, right panel) views in the HEEQ frame. The magnetic field line connecting
the Earth to the Sun is shown in dotted line. The polarity of the magnetic
sectors is shown at the edges of the frames for reference (red for positive,
blue for negative), separated by the HCS shown as a white line inside the
domain. Animated versions of this figure are available in the online version,
showing the full propagation of the cone CME from 0.1 au to the Earth for both
the minimum and maximum of activity backgrounds, with the evolution of the
radial magnetic field, density and radial velocity. Figure 8: Comparison of
the time evolution of the main physical quantities at 1 au at Earth position
for the propagation of the cone CME. The time is the number of physical days
that lasted the simulation. The blue line shows the minimum of activity case,
while the orange line shows the maximum of activity case. From top to bottom,
we show the radial velocity (in km/s), the proton number density (in
$\rm{cm}^{-3}$), the temperature (in units of $10^{4}$ K), the total magnetic
field (in nT) and the plasma beta parameter. The arrival of the shock is shown
by a vertical line (light blue for minimum of activity case, red for maximum
of activity case). The crossing of the ejecta is shown by a colored window
(light purple for minimum of activity, light red for maximum of activity).

###  4.2 Hydrodynamic cone CME

In this second limit case, we will now inject a CME, but a purely hydrodynamic
one. To do so, we use the cone model, detailed in Section 2.3, and with the
parameters described in the column labeled Hydro CME of Table 1. As explained
in Section 3.2, this corresponds to the typical parameters prescribed to study
the event of the 2nd of July 2012, which is our reference case due to the
great number of studies using EUHFORIA on it. This was a single-CME event that
triggered a strong magnetic storm on Earth, which also explains why it is
relevant for our study. The parameters have been adjusted to make the CME
fully directed towards Earth in order to maximize the geo-effectiveness of the
event. We inject the same cone CME in the two wind backgrounds corresponding
to minimum and maximum of activity. This allows us to quantify the impact of
the background solar wind, in a context where there is no magnetic interaction
possible between the CME and the background (because the CME does not possess
an intrinsic magnetic structure).

The results are displayed in Figure 7. We show the density in $\rm{cm}^{-3}$
to better visualize the CME ejecta as an under-dense structure. We show the
moment when the ejecta is reaching Earth (symbolized by a blue circle to the
right of the Sun). For each case, we show the ecliptic (view from above),
meridional (view from the side) and spherical (view at 1 au) views in the HEEQ
frame. From the ecliptic view, we can see that the CME structures seem rather
similar, although the shock for the maximum of activity case is better
defined. The difference is more visible in the meridional view.

For the minimum of activity case, the CME is actually cut in half, with a
dragging middle structure and two accelerated lobes over and under the current
sheet. This can easily be understood if we put this result in perspective with
the wind structure at minimum of activity described in the previous section:
since at minimum of activity the heliosphere is very organized, the parts of
the CME caught in the current sheet are slowed down by the equatorial slow
wind, while the parts outside the current sheet are accelerated by the fast
solar wind. We can wonder how realistic it is to have a CME injected right in
the middle of the current sheet at 0.1 au. Although CMEs have very little
chances of being generated at the equator because they may face magnetic
trapping (Sahade et al., 2022), they can be channeled towards the current
sheet if they form close to the border of a streamer or pseudo-streamer
(Zuccarello et al., 2012). Also, this effect is dominant at minimum of
activity thanks to the dipolar structure of the solar magnetic field, which
justifies even more this result for the minimum case (Lavraud & Rouillard,
2014). Similar configurations have been found for simulations of stellar CMEs
associated with extremely dipolar stars (and hence very organized
astrospheres, Alvarado-Gómez et al., 2022). At maximum of activity, on the
other hand, the CME is much more compact, but undergoes a deflection towards
the northern hemisphere.

In the spherical view, we can see more clearly the impact of the ICME at
Earth. At minimum of activity, we see the full hit of the central part of the
ejecta with an under-dense structure at the same longitude and latitude than
Earth. At maximum of activity, detecting the ICME signature is more difficult
because of the complex structure of the heliosphere. The large under-dense
structure north to the Earth is the ICME, while the largest one in the
southern hemisphere is the trace of the open southern coronal hole.

To be more quantitative, we show in Figure 8 the time evolution of the main
physical quantities at the position of the Earth at 1 au. We do this by using
a virtual satellite within the simulation. From top to bottom, we show the
radial velocity (in km/s), the proton number density (in $\rm{cm}^{-3}$), the
temperature (in units of $10^{4}$ K), the total magnetic field (in nT), and
the plasma beta parameter. The blue line shows the minimum of activity case,
while the orange line shows the maximum of activity case. The horizontal axis
shows the number of physical days after the insertion of the CME for both
cases. The arrival of the shock is shown by a vertical line (light blue for
minimum of activity case, red for maximum of activity case). The crossing of
the ejecta is shown by a colored window (light purple for minimum of activity,
light red for maximum of activity). Although the two CMEs have very different
3D structures, their 1D profile at Earth are actually not so different in
density, temperature, and especially radial velocity. Because it is a cone
CME, the detection of the ICME and its internal borders is a bit more
challenging due to the lack of internal magnetic structure, but the global
overview of all these physical quantities allow us to make estimations. The
initial shock of the ICME is visible in velocity, but more clear in density
and in the total magnetic field. From the radial velocity, we see that the
shock is rather gradual, starting around day 4. The CME at minimum of activity
is a bit faster, going over 400 km/s in the sheath. This is probably due to
the fact that it was a full hit, compared to the maximum of activity case
which was a flank hit. We do show the magnetic field in this case, but we
remind the reader that it is not representative of the polarity of the ICME
itself, but rather of the accumulation of magnetic field from the solar wind
background in front of it due to the propagation of the shock. The temperature
is mostly shown for context of the variations of the $\beta_{p}$ parameter.
The plasma beta parameter is shown as to visually support our settings of the
boundaries of the ejecta for both cases. Because of the clear shock in density
and total magnetic field, we can clearly say that the ICME traveling at
maximum of activity arrives faster (before 4 days, vs after 4 days for the
minimum case). However, if we look at the $\beta$ parameter, we can see that
the sheath region is shorter at minimum of activity (it passes Earth in only
3.83 hours), so that in the end the magnetic ejecta arrives faster at minimum
than at maximum (4.28 days vs. 4.76 days). The ejecta duration is rather
similar (a bit more than 2 days), so logically the ejecta ends sooner for the
minimum of activity. All these results are summarized in Table 2.

Please note that these results are specific to the background chosen, so it is
not clear how generalized they can be. It can also be slightly affected by our
method selected to detect the ICME borders (see Appendix A for more details).

In conclusion, these two analysis show that for our purely hydrodynamic CME,
the change in activity we used caused mostly a geometric deviation : our
minimum case CME is decelerated in the ecliptic plane and accelerated out of
it due to the heliosphere structure; our maximum case presents a deviation
towards the northern hemisphere. In the 1D profiles, the ICME shock at maximum
of activity seems to arrive faster, but the ME arrives later than in minimum.
It also reinforces the limitation of one-vantage in situ measurement, since
our two different CMEs actually produced similar speed 1D profiles at Earth
which do not reflect the major geometrical difference in 3D.

Figure 9:  Comparison of the reference spheromak CME propagation for the
minimum (top panel) and maximum (bottom panel) of activity cases. In this
case, both CMEs have positive handedness. We show the density in
$\rm{cm}^{-3}$, normalized to 1 au to better visualize the CME ejecta as an
under-dense structure. We show the moment when the ejecta is reaching Earth
(symbolized by a blue circle to the right of the Sun). For each case, we show
the ecliptic (view from above, left panel), meridional (view from the side,
middle panel) and spherical (view at 1 au, right panel) views in the HEEQ
frame. The magnetic field line connecting the Earth to the Sun is shown in
dotted line. The polarity of the magnetic sectors is shown at the edges of the
frames for reference (red for positive, blue for negative), separated by the
HCS shown as a white line inside the domain. Animated version of this figure
are available in the online version, showing the full propagation of the
reference spheromak CME from 0.1 au to the Earth for both the minimum and
maximum of activity backgrounds, with the evolution of the radial magnetic
field, density and radial velocity.  Figure 10: Comparison of the time
evolution of the main physical quantities at 1 au at Earth position for the
propagation of the reference spheromak CME with positive handedness. The time
is the number of physical days that lasted the simulation. The blue line shows
the minimum of activity case, while the orange line shows the maximum of
activity case. From top to bottom, we show the radial velocity (in km/s), the
proton number density (in $\rm{cm}^{-3}$), the temperature (in units of
$10^{4}$ K), the total magnetic field (in nT), the $z$ components of the
magnetic field in HEEQ coordinates (in nT) and the plasma beta parameter. The
arrival of the shock is shown by a vertical line (light blue for minimum of
activity case, red for maximum of activity case). The crossing of the ejecta
is shown by a colored window (light purple for minimum of activity, light red
for maximum of activity). We also show the vertical deviation at 5 and 10
degrees north and south of the ecliptic plane as colored shaded areas around
the curves.

##  5 Reference spheromak case from real event

In this next section, we now use a different modeling for the CME with
intrinsic magnetic field, which is the spheromak CME (described in Section
2.3). With the previous section isolating the hydrodynamic effects, this
allows us to better quantify the impact of the interaction of the CME magnetic
field with the surrounding background. It becomes especially important that
the magnetic field is more intense and more complex at maximum of activity. As
explained in Section 3.2, we will first use parameters inspired by a true
event from July 12 2012 which has already been extensively studied with
EUHFORIA, and that will act as a reference case. The input parameters are
summed up in Table 1. With the internal magnetic field comes three new
parameters: the tilt, flux intensity and handedness.

In this first configuration, we inject a spheromak with positive handedness,
which means the inner flux-rope is a right-handed helix at injection. In order
to get a more compact study, negative handedness is only reported in the next
section (Section 6.3). For more information about the handedness, please refer
to appendix B. The resulting CMEs can be seen in Figure 9. We recover mainly
similar effects observed in Section 4.2: at minimum of activity the CME
appears to be cut in half due to the highly-organized structure of the
heliosphere, while at maximum of activity the CME is deflected to the northern
hemisphere and thus producing a flank hit. When we compare it with Figure 7,
we can also see some differences. In both cases, the CME appears more
structured and coherent, which is due to the inner magnetic field allowing for
a more cohesive structure. This also produces a more defined shock at the
front of the CME in both cases, especially visible in the density structure
with a dark ring showing the corresponding over-density (the same color scale
is used in both figures). At minimum of activity, the dislocation of the CME
is less pronounced, due to the inner magnetic field counteracting the solar
wind influence. This is especially visible in the spherical view where this
time the under-dense structure is visible not only at Earth latitude, but also
north and south of it. At maximum of activity, the CME is more elongated in
latitude and presents a slight asymmetry, probably due to the more complex
magnetic interaction and reconnection with the irregular HCS magnetic
configuration.

To be more quantitative, we also show the 1D evolution at Earth in Figure 10,
similar to Figure 8. To complete this figure, we also show this time the
vertical deviation at 5 and 10 degrees north and south of the ecliptic plane
as colored shaded areas around the curves, similar to what was done in Figure
17 of Scolini et al. (2019). This allows us to display some estimation of the
uncertainty for the ICME profile around the Earth position. We can see in this
case that the shock is indeed more defined in all the physical quantities,
arriving at Earth around 2.75 days. The ICME is thus arriving faster than in
the pure hydro case, even though the initial speed of the CME is much less
(but we recall from section 3.2 that the full 3D speed is actually the same).
Here the CME at maximum of activity still arrives first, but with only a
2-hour lead compared to the CME at minimum of activity, as seen more clearly
in the radial velocity. The sheath is then followed by a magnetic ejecta
clearly visible in the $B_{z}$ component, until the parameters return to the
initial solar wind before the shock. Once again, the speed of the magnetic
ejecta is slightly higher for the minimum of activity case, especially in the
deviation at 5 and 10 degrees because of the acceleration of the polar solar
wind. Density remains similar in both cases. The magnetic field component,
this time representative of the structure of the inner flux-rope, is however
different. At minimum of activity, $B_{z}$ rises first to 20 nT, before going
to a -8 nT phase at the end of the ejecta. On the other hand, at maximum of
activity $B_{z}$ always remain positive, but under 20 nT. This difference in
$B_{z}$ is very likely due to the difference in geometry discussed above,
which results in a different impact parameter for both cases. In this
scenario, none of the CMEs are geo-effective, as their $B_{z}$ are both mostly
positive.

This case shows that with an internal magnetic field, we retrieve similar
results for the geometry of the propagation of the CMEs, except that the
ejections are more cohesive thanks to their internal flux-rope. In this case,
the CMEs arrive at a very similar time at Earth, with only a 2-hour lead in
the maximum of activity case.

##  6 Median spheromak case

The previous results were interesting, but we can easily argue that they may
hold only for the specific case we are studying. Although we do realize and
acknowledge that our results are intrinsically dependent on the choice of our
solar wind background and CME model, we include one last case that would aim
at being slightly broader. To do so, we have defined in Section 3.2 a median
CME, based on the statistical analysis of ACE data from Regnault et al.
(2020), combined with extrapolations from 1 au to 0.1 au by Scolini et al.
(2021a). The parameters selected are detailed in Table 1. We recall that
compared to the previous case, the median CME is slower (541 vs. 763 km/s at
injection), more dense (2 vs. 1 $10^{-18}$ $\rm{kg.m^{-3}}$ at injection),
hotter (6.2 vs. 2.4 $10^{5}$ K at injection) and less magnetized (2.3 vs 10
$10^{13}$ Wb flux at injection). We will now perform the same study as in the
previous section, but for this median CME, in order to see how general our
results can be extrapolated.

Figure 11: Comparison of the time evolution of the main physical quantities at
1 au at Earth position for the propagation of the median spheromak CME with
positive handedness. The time is the number of physical days that lasted the
simulation. The blue line shows the minimum of activity case, while the orange
line shows the maximum of activity case. From top to bottom, we show the
radial velocity (in km/s), the proton number density (in $\rm{cm}^{-3}$), the
temperature (in units of $10^{4}$ K), the total magnetic field (in nT), the
$z$ component of the magnetic field in HEEQ coordinates (in nT) and the plasma
beta parameter. The arrival of the shock is shown by a vertical line (light
blue for minimum of activity case, red for maximum of activity case). The
crossing of the ejecta is shown by a colored window (light purple for minimum
of activity, light red for maximum of activity). We also show the vertical
deviation at 5 and 10 degrees north and south of the ecliptic plane as colored
shaded areas around the curves. Animated movies corresponding to this figure
are available in the online version, showing the full propagation of the
median spheromak CME with positive handedness in 2D cuts (equatorial,
meridional and spherical) from 0.1 au to the Earth for both the minimum and
maximum of activity backgrounds, with the evolution of the radial magnetic
field, density and radial velocity.

###  6.1 Positive handedness

Once again, we start by injecting a spheromak CME with positive handedness
(H=+1) in both solar wind backgrounds. For clarity, we will first focus on the
1D profiles, visible in Figure 11. Similar to Figure 10, the CME at maximum of
activity arrives slightly in advance, but it could be due to the fact that the
shock is more steep (especially visible in the velocity). What is surprising
is that, contrary to the reference case where both CMEs had mostly positive
$B_{z}$, in this case the CME at maximum of activity has a positive $B_{z}$
while the CME at minimum of activity has a clear negative component (end of
day 3). Moreover, the ICME at minimum of activity has a longer ejecta, which
means it takes more time to get back to a normal solar wind state after the
crossing of the structure (the ejecta lasts 2.1 days, vs. 1.6 days at maximum
of activity). This means that the median case is more geo-effective than the
reference one. This is surprising, because the reference case was based on a
CME event which has proven to be geo-effective by interaction with the wind
background (Hu et al., 2016; Marubashi et al., 2017; Gopalswamy et al., 2018;
Scolini et al., 2019), although it had been initially underestimated by the
space-weather community (Webb & Nitta, 2017). This negative $B_{z}$ is also
significant in magnitude, up to 15 nT. This is also surprising since the
median CME is slower and less magnetized than the reference case. This
highlights the fact that the geo-effectiveness of a CME does not depend only
on the input CME parameters, but also on its interaction with its background
and especially the HCS.

Figure 12: Comparison of the meridional view of the ICME at minimum of
activity depending on the input radial speed. Each panel shows the meridional
cut passing across Earth in the simulation (in the $x-z$ plane in HEEQ
coordinates) when the ICME reaches Earth (blue point at 1 au). We show the
density in order to visualize the magnetic ejecta as an under-density. The
left panel shows the case for the median spheromak CME (Section 6.1), the
middle panel the case for the reference spheromak CME (Section 5) and the
right panel the case for the cone CME (Section 4.2). Each panel has the input
CME speed at 0.1 au as a label. See Table 1 for a reminder of all the
differences between the models. We see then that the faster the CME is, the
more delay we observe between the equatorial and the polar parts of the
ejecta, which means it is caused by the equatorial solar wind deceleration.

###  6.2 Slicing effect at minimum of activity

Another interesting result we can derive from this study is a comparison of
the ICME profiles at minimum of activity. We focus on this specific phase of
the cycle because surprisingly, this is where we see the most differences
between our cases. We would have expected the ICME at maximum of activity to
show more disparity because of the complex wind background, but actually this
complexity constrains the magnetic ejecta. As a result, the input parameters
or even the modeling of the ICME (cone vs. spheromak) has little effect on the
final profiles (in 2D or 1D, as can be seen in the previous figures),
affecting only the size of the ejecta (by 0.1 au at most) but not its shape.

At minimum of activity however, the high structuring of the corona allows for
more distinct effects that we can quantify. In Figure 12, we compare the
meridional profile for our three cases described in Table 1 at minimum of
activity. From left to right, we show the median spheromak, the reference
spheromak and the cone model. We show here the cuts in density to visualize
the sheath as an over-density and the magnetic ejecta as an under-density.

We find again this slicing effect that we noticed before, where the high-
latitude fast wind carries faster the northern and southern part of the ICME,
while the equatorial part is slowed down by the slow equatorial wind. These
three cases allow us to understand that this effect is not due to the modeling
of the ICME, because we find it for both the cone and spheromak models
(although the effect is reduced for the spheromak, probably because its
internal magnetic structure makes it less sensitive to the wind background).
The amount of slicing is highly sensitive to the input speed of the CME. The
median and reference spheromaks have the same modeling, and yet the median
case barely shows this effect. The major difference is the input speed: the
spheromak case has an input speed closer to the fast wind, while the median
case has an input speed closer to the slow wind. This indicates that this
effect is not due so much to the acceleration caused by the fast wind, but
rather by the slow-down caused by the slow equatorial wind. Then, as the ICME
is faster, its equatorial part is much more slowed down, causing this
geometric separation. The effect is even stronger for the cone model: the
input speed appears to be even more important, but we recall that the
definition of the input speed is slightly different for the cone model (as it
does not have an expansion speed as in the spheromak model we also recall that
the absence of an internal magnetic field makes the ICME more sensitive to the
background wind.

This difference of behaviors in speed was also seen in ACE data in Regnault et
al. (2020) for example. We know that at maximum of activity and during the
declining phase, active regions have higher magnetic fluxes, which triggers
more extreme events. But we could assume beyond this hypothesis that, even if
an extreme event were to happen at minimum of activity, it would be slowed
down drastically by the solar wind configuration and thus results in lower
speed detected at Earth, just like what is seen in the ACE data (see also Chi
et al. (2016) and Wu & Lepping (2016)). This result shows that the structuring
of the wind itself has a strong influence on the ICME propagation, and not
just the wind speed or specific structures at stream interfaces, and calls for
more careful and realistic modeling of the solar corona.

Figure 13:  Comparison of the time evolution of the main physical quantities
at 1 au at Earth position for the propagation of the median spheromak CME with
negative handedness. The time is the number of physical days that lasted the
simulation. The blue line shows the minimum of activity case, while the orange
line shows the maximum of activity case. From top to bottom, we show the
radial velocity (in km/s), the proton number density (in $\rm{cm}^{-3}$), the
temperature (in units of $10^{4}$ K), the total magnetic field (in nT), the
$z$ components of the magnetic field in HEEQ coordinates (in nT) and the
plasma beta parameter. The arrival of the shock is shown by a vertical line
(light blue for minimum of activity case, red for maximum of activity case).
The crossing of the ejecta is shown by a colored window (light purple for
minimum of activity, light red for maximum of activity). We also show the
vertical deviation at 5 and 10 degrees north and south of the ecliptic plane
as colored shaded areas around the curves. Animated movies corresponding to
this figure are available in the online version, showing the full propagation
of the median spheromak CME with negative handedness in 2D cuts (equatorial,
meridional and spherical) from 0.1 au to the Earth for both the minimum and
maximum of activity backgrounds, with the evolution of the radial magnetic
field, density and radial velocity. Figure 14: Comparison of the meridional
profile of the ICME for positive (left panel) and negative (middle panel)
handedness. We show the $x-z$ plane in HEEQ coordinates when the ICME reaches
Earth (blue point at 1 au). We show the proton number density in
$\rm{cm}^{-3}$. We can see that for the negative handedness case, the CME is
elongated with a double sheath, probably due to the acceleration it gets from
initial reconfiguration at 0.1 au. In the right panel, we show the radial
evolution of the mean speed inside the ejecta for positive (green) and
negative handedness (red).

###  6.3 Negative handedness

One final parameter study we undergo is to switch the handedness from positive
to negative ($H=-1$) for the median case. In order to better compare all the
cases and discuss our conclusions, we have summarized all the arrival times
derived in the previous figures in table 2.

CME Case |  Activity level |  Shock arrival time (days) |  Ejecta arrival time (days) |  Ejecta end time (days) |  Sheath duration (hours) |  Ejecta duration (days)
---|---|---|---|---|---|---
Cone |  Minimum |  $4.12$ |  $4.28$ |  $6.44$ |  $3.83$ |  $2.16$
Cone |  Maximum |  $3.78$ |  $4.76$ |  $7.25$ |  $23.67$ |  $2.49$
Reference |  Minimum |  $2.73$ |  $3.07$ |  $5.73$ |  $8.17$ |  $2.66$
Reference |  Maximum |  $2.67$ |  $3.17$ |  $6.36$ |  $12.01$ |  $3.19$
Median (positive H) |  Minimum |  $3.58$ |  $3.82$ |  $6.31$ |  $5.67$ |  $2.49$
Median (positive H) |  Maximum |  $3.51$ |  $3.73$ |  $5.39$ |  $5.17$ |  $1.66$
Median (negative H) |  Minimum |  $3.31$ |  $3.49$ |  $6.51$ |  $4.17$ |  $3.02$
Median (negative H) |  Maximum |  $3.65$ |  $3.75$ |  $5.85$ |  $2.33$ |  $2.10$
Table 2: Table summarizing the key times found for each CME case and each
activity level (minimum or maximum of activity). We indicate the shock arrival
time (in days), the ejecta arrival time (in days), the ejecta end time (in
days), and deduce from them the sheath duration (in hours) and the ejecta
duration (in days).

We plot the 1D profiles at Earth in Figure 13, with the same labels and
visualization as before. This figure should be compared with Figure 11 to see
the effect of the handedness. Here we can see a slight numerical artifact
produced by EUHFORIA because of strong shocks where the velocity drops
suddenly before rising again. It can disrupt locally certain physical
quantities (like the $\beta_{p}$ plasma parameter), but this effect is purely
local and does not impact the overall results and conclusions. At maximum of
activity, the change of handedness causes the ICME to be slightly delayed at
Earth, reaching 1 au after 3.65 days of propagation instead of 3.51 days for
positive handedness. This can be explained by the fact that the ICME speed has
been reduced (the speed at the shock is around 400 km/s vs. 500 km/s
previously). The ejecta also lasts longer, with a return to solar wind
parameters after 2.1 days instead of 1.66. More surprisingly, the ICME becomes
slightly geo-effective, with a visible negative $B_{z}$ after day 4. However,
these effects are minor compared to the minimum of activity case.

Once again, the case with the most disparity is surprisingly the case at
minimum of activity. The most surprising feature is that the ICME at minimum
of activity now arrives first, before the ICME at maximum of activity (this is
the only case we have shown where this happens): the shock is move to day 3.31
instead of 3.58. The shock is also better defined, with a steepened slope. As
for maximum of activity, the ICME also becomes more geo-effective: for
negative handedness, the $B_{z}$ reaches -20 nT and stays negative, which
would be conditions for a mild geomagnetic storm; whereas for positive
handedness, the $B_{z}$ component would reach only -15 nT, but immediately go
back to positive afterwards.

The above result is interesting, because it shows several properties of this
case: the handedness can affect the ICME arrival time as well as its geo-
effectiveness; these effects are more visible at minimum of activity than at
maximum. One explanation we have for this phenomenon is shown in Figure 14. We
show the meridional profile of the ICME hitting Earth for positive (left
panel) and negative handedness (middle panel). For positive helicity, the ICME
structure is typical, with an expanding spherical sheath followed by the
magnetic ejecta. But for negative helicity, the sheath is more concentrated
towards the equator, while the magnetic ejecta structure is not so visible. It
almost seems that there is a second sheath forming behind the main one, with
an overextended perturbation of the ejecta in latitude.

Although the two injected CMEs have the same input speed, the one with
negative helicity is accelerated immediately after injection (at only 0.11 au,
for an injection at 0.1 au), as seen in the right panel of Figure 14. This
suggests that the rapid reconfiguration of the spheromak after injection to
adjust to the wind background leads in the negative handedness case to the
separation of the ICME, between an accelerated first part and then a
decelerated one. Since we only changed the magnetic structure, we assume that
this is the result of reconnection effects with the heliospheric magnetic
field. That would also explain why this effect is more visible at minimum of
activity, which would be because the current sheet is closer to the equator,
which means closer to the injection point as well as the propagation path of
the ICME.

Such effects of the impact of the handedness have been previously investigated
by Chané et al. (2005, 2006). They have indeed showed that the handedness
itself can change the geo-effectiveness of the ICME as well as its arrival
time, but only for a purely dipolar background configuration. Our results seem
to extend this result even for more realistic backgrounds: at minimum of
activity, the background is still dipolar enough so that the result can apply,
but at maximum of activity the background becomes too complex and the
influence of the handedness becomes less dominant. It is however difficult to
generalize completely this result, as the CME in our case is only injected at
0.1 au, thus missing the propagation in the lower corona; for a more accurate
result and better comparison, we would need to incorporate this phase as well,
such as in Talpeanu et al. (2022).

For verification, we have performed the same inversion of handedness for the
reference spheromak case. We have found similar effects, although reduced
compared to the median case (as expected due to the more intense internal
magnetic field that makes the ICME less sensitive to the background). We
retrieved the acceleration of the ICME at minimum of activity, although the
internal separation was more difficult to see. This means that our results do
apply to more extreme cases that can be seen at Earth. Finally, since the
spheromak model is not connected to the Sun, it is more likely than other
models to self-reconfigure, especially magnetically. It would be interesting
to see if we could reproduce the same result with a CME model that has legs
connected back to the Sun such as Fri3D (Isavnin, 2016; Maharana et al.,
2022), Gibson & Low model (Gibson & Low, 1998), or the modified Titov-Démoulin
model (Titov et al., 2014; Regnault et al., 2023; Linan et al., 2023).

##  7 Discussion and Conclusion

In this study, we investigate the role of the solar cycle on the propagation
of ICMEs using numerical simulations. To do so, we start with a theoretical
study that has an exploratory purpose. We select two dates that were
representative of solar minimum (15th of December 2008) and solar maximum
(20th of March 2015), based on previous studies. Then we use synoptic maps
(GONG and GONG-ADAPT respectively) to drive the EUHFORIA model (Pomoell &
Poedts, 2018) to compute the corresponding state of the heliosphere. Finally,
we inject the same ICME within these two backgrounds, and quantify the
differences and their origins, in order to better understand what to expect
from a propagation at minimum versus a propagation at maximum of activity. We
use several modelings for the ICME in order to test the robustness of our
results: first a cone model to check for the hydrodynamical effects, then a
linear force-free spheromak model to see the effect of an internal magnetic
field. We also use parameters that were based on a true event (the CME that
caused the geomagnetic storm on the 12ht of July 2012), as well as parameters
derived to obtain a median ICME (based on ACE observations and EUHFORIA
scaling, see Regnault et al. (2020) and Scolini et al. (2021a)).

We showed that the solar wind backgrounds selected were yielding similar
speeds at Earth, but with very different structures. At minimum of activity,
because the magnetic field of the Sun is mostly dipolar, the inner heliosphere
is very organized, with slow wind at the equator and fast wind at the poles,
while the HCS is close to the ecliptic plane. At maximum of activity on the
contrary, coronal holes are more frequent at lower latitudes, leading to slow
and fast wind everywhere, as well as a magnetic field with more complex
structures and more polarity reversals.

With a purely hydrodynamical ICME, we observe that the 1D speed profile at
Earth is very similar, though the 3D structure of the ICME is very different.
At minimum, the ICME is a direct hit at Earth, although its core is being
slowed down by the slow equatorial wind, while the rest is accelerated by the
fast polar wind. At maximum, the ICME is a flank hit, due to a northern-
oriented deflection caused by a fast wind stream originating from a coronal
hole. The ICME at maximum of activity arrives first at Earth with a 10-hour
lead. These geometrical results remain true for a magnetized ICME, although
the inner magnetic field allows for the ICME to suffer less deformation and
much less drag along the propagation since the ICME reaches 1 au faster with a
lower initial velocity. In this case, we can compare the geo-effectiveness of
the two ICMEs by checking how negative their $B_{z}$ component is at Earth.
For a positive handedness (H=+1), none are geo-effective (positive $B_{z}$),
but for a negative handedness (H=-1), it is the ICME at minimum of activity
which is actually the most geo-effective (-25 vs. -20 nT). The ICME at maximum
of activity still arrives first at Earth, but only with a 2-hour lead. These
results remain true even for a median ICME. In this case, the difference in
geo-effectiveness is even larger (-25 vs. -5 nT). This could explain why fast
halo CMEs observed in solar maximum activity (2002) were poorly geoeffective
(Schmieder et al., 2020). There is also a more distinct difference in arrival
time: for a positive handedness, the ICME at maximum of activity arrives first
with a 3-hour lead; with a negative handedness, it is the ICME at minimum of
activity that arrives first with a 8-hour lead. We recover results obtained in
Chané et al. (2006) for the minimum of activity, but show that the influence
of the CME handedness is less dominant at maximum of activity. This could
affect forecasts as it suggests that providing the handedness of the CME is
only crucial at minimum of activity. This seems to be due to the self-
reconfiguration of the ICME at injection, influenced by the dipolar magnetic
background which accelerates the most favorable initial condition. Another
interesting result we obtained is that the deformation of the ICME at minimum
of activity, caused by the structuring of the solar wind, depends on the speed
of the ICME: the faster it is, the more important this effect will be, because
the core of the ICME will be decelerated down even more noticeably.

In conclusion, we have shown that the same ICME will propagate very
differently during solar minimum and maximum. The main factors are the
organization of the solar wind, that can cause slow-downs or accelerations,
but also the organization of the heliospheric magnetic field, that can cause
magnetic reconnection so allowing the overtaken plasma to be less stack in
front of the ICME. In the cases studied, the ICME at minimum of activity was
often the most geo-effective, which shows that the most powerful events will
not necessarily happen at maximum of activity. This reinforces the need to
quantify in the most precise way the coronal holes locations to anticipate
deflections, as well as the HCS position and the ICME handedness to anticipate
reconnection effects.

This study is a first step towards better understanding and quantifying the
impact of the solar cycle on ICME propagation. This will become more and more
important, as solar cycle 25 is on the rise, and space-weather forecasting
facilities aim at delivering more and more reliable forecasts. To reach this
goal, there are of course many ways to widen the scope of this study. A first
natural step would be to include more intermediate states along a solar cycle
(for example, following the previous cycle number 24) and see how the same
ICME propagates. This would allow to better identify the key features that
alter the propagation and when they change. Also, we could use other CME
models (Fri3D, Gibson & Low, etc.(Maharana et al., 2022; Linan et al., 2023)),
as well as other heliospheric models (ENLIL…) (Odstrcil et al., 2004)to test
even further the robustness of our results. We could also focus on reproducing
specific events to validate our understanding of the key features of the
heliosphere and their interaction with the ICME. Recent solar missions (such
as PSP and Solar Orbiter) as well as future ones (such as PUNCH and Vigil)
will even provide more data to add better constraints on the features of the
inner heliosphere and the observed ICMEs. Finally, an important step would be
to include the CME initialization inside the solar corona, in order to take
into account the impact of the structures close to the Sun on the early
propagation of the CME (similar to Lynch et al. (2022) for example).

The authors would like to thank Anwesha Maharana and Camilla Scolini for
interesting discussions and important feedback. This project has also received
funding from the European Union’s Horizon 2020 research and innovation program
under grant agreement No. 870405 (EUHFORIA 2.0) and the ESA project
”Heliospheric modelling techniques“ (Contract No. 4000133080/20/NL/CRS). SP
acknowledges support via the projects C14/19/089 (C1 project Internal Funds KU
Leuven), G.0B58.23N (FWO-Vlaanderen), SIDC Data Exploitation (ESA Prodex-12),
and Belspo project B2/191/P1/SWiM. The resources and services used in this
work were provided by the VSC (Flemish Supercomputer Centre), funded by the
Research Foundation - Flanders (FWO) and the Flemish Government. Data were
acquired by GONG instruments operated by NISP/NSO/AURA/NSF with contribution
from NOAA. This work utilizes data produced collaboratively between AFRL/ADAPT
and NSO/NISP. We recognise the collaborative and open nature of knowledge
creation and dissemination, under the control of the academic community as
expressed by Camille Noûs at http://www.cogitamus.fr/indexen.html.

##  Appendix A Determination of the CME shock, sheath and magnetic ejecta

An ICME exhibits different substructures that are essential to distinguish,
due to the different underlying physics in each one of them. In this study, we
need to be able to distinguish between the CME-driven shock, the CME sheath
and the magnetic ejecta (also called magnetic cloud in some studies, e.g. when
the presence of a flux-rope can be confirmed). To do so, we need to determine
the shock time $t_{shock}$ (which is the beginning of the sheath), the end of
the sheath and thus beginning of the magnetic ejecta time $t_{in}$, and the
end of the magnetic ejecta time $t_{out}$.

Figure 15:  Examples of detection of the shock, sheath and magnetic cloud for
various cases. For each case, the first row is the total magnetic amplitude in
nT, the second row the total velocity amplitude in km/s, the third row the
temperature in $10^{4}$ K, the fourth row the number density for the protons
in $\rm{cm}^{-3}$ and the last row $\beta$ parameter of the plasma. The left
panel shows the median case at minimum of activity at 1 au. The right panel
shows the same case, but at 0.4 au instead of 1 au. A yellow rectangle
highlights the region corresponding to the sheath, and a blue rectangle the
region corresponding to the magnetic cloud.

To determine $t_{shock}$ at various distances for various case, we use a
modified version of the criterion used in Scolini et al. (2021a):

$\left(v(t_{i})-v(t_{i}-\Delta t)\geqslant v_{thresh}\right)\quad
OR\quad\left(\frac{n(t_{i})}{n(t_{i}-\Delta t)}\geqslant
n_{thresh}\right)\quad OR\quad\left(\frac{B(t_{i})}{B(t_{i}-\Delta
t)}\geqslant B_{thresh}\right),$ (A1)

where $t_{i}$ is a generic time in the time series, $\Delta t$ is a time delay
to compare to the steady wind state before the event , and $v_{thresh}$,
$n_{thresh}$ and $B_{thresh}$ are threshold parameters to identify the shock.
Scolini et al. (2021a) set their own thresholds to the following values:
$v_{thresh}=20$ km/s, $n_{thresh}=1.2$, $B_{thresh}=1.2$. However, these were
for a comparison between the CME run and the corresponding wind-only
simulation, which means they used a wind model for reference. In our case, we
use only one CME run and compare present time with previous time data. Their
values were also optimized for a specific wind background set at maximum of
activity. This means that our threshold values need to be different. We have
thus adjusted these values by trying different combinations, and selected the
most robust and efficient ones. These final parameter values are: $\Delta
t=10$ (with 10 minutes between each output, this means an interval of around
1.6 hours), $v_{thresh}=30$ km/s, $n_{thresh}=1.5$, $B_{thresh}=1.5$
($n_{thresh}$ and $B_{thresh}$ do not have units because they are ratios).
Since we have a minimum of activity configuration, the HCS is close to the
equatorial place and as a result, the magnetic field can become very close to
0 locally, producing false detection of the shock because of these low values.
To avoid this, we have set an additional threshold of 1 G for the local
magnetic field. On top of the automatic selection given by this criterion,
systematic visual verification has been made to ensure the validity of the
results.

For $t_{in}$, we use a criterion from the $\beta$ plasma. This criterion is
adapted from in situ solar wind measurements (Lepping et al., 2005) and has
already been used in Scolini et al. (2021a). In the observations, the sheath
ends (and the magnetic ejecta begins) when $\beta_{p,obs}\leqslant 0.3$. For
EUHFORIA simulations, the threshold can range from 0.1 to 1. Scolini et al.
(2021a) found that the value of 0.5 yielded good results in their cases. For
our cases, it is the value of 0.1 which usually gives the best results.
However, we sometimes had to adjust it manually in order to get a ME selection
consistent with the other physical quantities. For clarity, we have indicated
the value of $\beta$ used for the border selection in table 3.

CME Case | Activity level |  $\beta$ threshold | Corresponding Figure
---|---|---|---
Cone | Minimum | $4.0$ | 8
Cone | Maximum | $0.15$ | 8
Reference | Minimum | $0.1$ | 10
Reference | Maximum | $0.1$ | 10
Median (positive H) | Minimum | $0.25$ | 11
Median (positive H) | Maximum | $0.1$ | 11
Median (negative H) | Minimum | $0.5$ | 13
Median (negative H) | Maximum | $0.1$ | 13
Table 3: Table of the values of the $\beta$ parameter used as threshold for
detection of the borders of the CME sheath.

$t_{out}$ is set when the $\beta$ parameter goes above the same threshold as
for $t_{in}$. We include an exception for reconnection effects that can occur
within the magnetic ejecta, and that usually generate perturbations of the
$\beta$ for only a few points. The goal is to not take into account spikes in
the $\beta$ parameter.

Figure 15 shows how these various criteria manage to adapt to different cases
to detect automatically all the substructures. The left panel shows the same
case as the left panel in Figure 3, which is the median CME at minimum of
activity at 1 au. There is some reconnection happening within the magnetic
ejecta, because the $\beta$ spikes above 0.2 at December 18 around noon.
However the magnetic ejecta is not over yet, and our criterion can adapt to
these cases. The right panel shows the same case, but at 0.4 au instead of 1
au. Adjustment of $\Delta t$ can be needed for artificial satellites closer to
the injection point, because the shock will steepen with the distance, we
usually then divide it by 2.

Figure 16:  3D representation of the magnetic configuration of the initial
spheromak. We represent three sets of magnetic field lines: the blue ones
correspond to the outer edge of the spheromak, the red ones to the axis of the
internal flux rope, and the white ones to the orientation of the flux rope
field lines. In the left panel, the handedness is positive since the flux-rope
corresponds to a right-handed helix. In the right panel, the handedness is
negative since the flux-rope corresponds to a left-handed helix. Credits:
Camilla Scolini.

##  Appendix B Handedness of a spheromak

The magnetic field in the spheromak model is expressed as follows (in the
local spherical coordinate $(r,\theta,\phi)$ frame in which the origin is the
center of the spheromak):

$\displaystyle B_{r}$ $\displaystyle=2B_{0}\frac{j_{1}\left(\alpha
r\right)}{\alpha r}\rm{cos}\,\theta,$ (B1) $\displaystyle B_{\theta}$
$\displaystyle=-B_{0}\left[\frac{j_{1}\left(\alpha r\right)}{\alpha
r}+\partial_{r}j_{1}\left(\alpha r\right)\right]\rm{sin}\,\theta,$ (B2)
$\displaystyle B_{\phi}$ $\displaystyle=HB_{0}j_{1}\left(\alpha
r\right)\rm{sin}\,\theta,$ (B3)

where $B_{0}$ is a parameter determining the magnetic field strength,
$j_{1}(x)$ is the spherical Bessel function of order one, $\alpha$ is chosen
so that $\alpha r_{0}$ is the first zero of $j_{1}(x)$, which yields $\alpha
r_{0}\approx 4.4934$ ($r_{0}$ is the radius of the spheromak), and $H$ is the
handedness.

The handedness is a dimensionless parameter that can be equal to either +1 or
-1. Only the azimuthal component of the magnetic field is affected by the
handedness. In Figure 16, we show a 3D representation of the resulting
spheromak magnetic field (in the meridional plane) for positive and negative
handedness. It can be linked to the sign of the magnetic helicity of the
originating active region, which is conserved over time (Berger, 2005). The
magnetic helicity quantifies how much the magnetic field is sheared and
twisted. The handedness only retains its sign. Spheromaks with a positive
handedness ($H=+1$) exhibit a right-handed helix flux-rope, while spheromaks
with a negative handedness ($H=-1$) exhibit a left-handed helix flux-rope
(shown by the torus located in the central part of the spheromak). Handedness
can be estimated using empirical relationships such as hemispheric rules
(majority of positive handedness in the southern hemisphere and negative
handedness in the northern hemisphere, Bothmer & Schwenn, 1998; Pevtsov et
al., 2008), or by analyzing morphological features or the photospheric
magnetic field of the active region or/and erupting filament (Démoulin &
Pariat, 2009; Palmerio et al., 2017).

There is sometimes a confusion with other names for quantities that have the
same or a similar function. The name ”chirality” is used interchangeably with
handedness (as for example in Shiota & Kataoka (2016) or Scolini et al.
(2019)). Confusion may arise when the handedness is referred to as a ”helicity
parameter”, as in Jin et al. (2017). However the handedness is not the
helicity, it is only its sign.

## References

  * Altschuler & Newkirk (1969) Altschuler, M. D., & Newkirk, G. 1969, Sol. Phys., 9, 131, doi: 10.1007/BF00145734
  * Alvarado-Gómez et al. (2022) Alvarado-Gómez, J. D., Cohen, O., Drake, J. J., et al. 2022, ApJ, 928, 147, doi: 10.3847/1538-4357/ac54b8
  * Arge et al. (2010) Arge, C. N., Henney, C. J., Koller, J., et al. 2010, in American Institute of Physics Conference Series, Vol. 1216, Twelfth International Solar Wind Conference, ed. M. Maksimovic, K. Issautier, N. Meyer-Vernet, M. Moncuquet, & F. Pantellini, 343–346, doi: 10.1063/1.3395870
  * Arge et al. (2003) Arge, C. N., Odstrcil, D., Pizzo, V. J., & Mayer, L. R. 2003, in American Institute of Physics Conference Series, Vol. 679, Solar Wind Ten, ed. M. Velli, R. Bruno, F. Malara, & B. Bucci, 190–193, doi: 10.1063/1.1618574
  * Asvestari et al. (2022) Asvestari, E., Rindlisbacher, T., Pomoell, J., & Kilpua, E. K. J. 2022, ApJ, 926, 87, doi: 10.3847/1538-4357/ac3a73
  * Bellan (2000) Bellan, P. M. 2000, Spheromaks: a practical application of magnetohydrodynamic dynamos and plasma self-organization, doi: 10.1142/p121
  * Berger (2005) Berger, M. A. 2005, Highlights of Astronomy, 13, 85
  * Bothmer & Schwenn (1998) Bothmer, V., & Schwenn, R. 1998, Annales Geophysicae, 16, 1, doi: 10.1007/s00585-997-0001-x
  * Brun & Browning (2017) Brun, A. S., & Browning, M. K. 2017, Living Reviews in Solar Physics, 14, 4, doi: 10.1007/s41116-017-0007-8
  * Burlaga et al. (1981) Burlaga, L., Sittler, E., Mariani, F., & Schwenn, R. 1981, J. Geophys. Res., 86, 6673, doi: 10.1029/JA086iA08p06673
  * Burlaga (1995) Burlaga, L. F. 1995, Interplanetary magnetohydrodynamics, 3
  * Cargill et al. (1995) Cargill, P. J., Chen, J., Spicer, D. S., & Zalesak, S. T. 1995, Geophys. Res. Lett., 22, 647, doi: 10.1029/95GL00013
  * Chané et al. (2005) Chané, E., Jacobs, C., van der Holst, B., Poedts, S., & Kimpe, D. 2005, A&A, 432, 331, doi: 10.1051/0004-6361:20042005
  * Chané et al. (2021) Chané, E., Schmieder, B., Dasso, S., et al. 2021, A&A, 647, A149, doi: 10.1051/0004-6361/202039867
  * Chané et al. (2006) Chané, E., van der Holst, B., Jacobs, C., Poedts, S., & Kimpe, D. 2006, A&A, 447, 727, doi: 10.1051/0004-6361:20053802
  * Chapman (2023) Chapman, S. C. 2023, Frontiers in Astronomy and Space Sciences, 9, 426, doi: 10.3389/fspas.2022.1037096
  * Chi et al. (2016) Chi, Y., Shen, C., Wang, Y., et al. 2016, Sol. Phys., 291, 2419, doi: 10.1007/s11207-016-0971-5
  * Cranmer et al. (2007) Cranmer, S. R., van Ballegooijen, A. A., & Edgar, R. J. 2007, ApJS, 171, 520, doi: 10.1086/518001
  * Dakeyo et al. (2022) Dakeyo, J.-B., Maksimovic, M., Démoulin, P., Halekas, J., & Stevens, M. L. 2022, ApJ, 940, 130, doi: 10.3847/1538-4357/ac9b14
  * Dal Lago et al. (2003) Dal Lago, A., Schwenn, R., & Gonzalez, W. D. 2003, Advances in Space Research, 32, 2637, doi: 10.1016/j.asr.2003.03.012
  * Dasso et al. (2012) Dasso, S., Démoulin, P., & Gulisano, A. M. 2012, in Comparative Magnetic Minima: Characterizing Quiet Times in the Sun and Stars, ed. C. H. Mandrini & D. F. Webb, Vol. 286, 139–148, doi: 10.1017/S1743921312004759
  * Dasso et al. (2007) Dasso, S., Nakwacki, M. S., Démoulin, P., & Mandrini, C. H. 2007, Sol. Phys., 244, 115, doi: 10.1007/s11207-007-9034-2
  * Davies et al. (2013) Davies, J. A., Perry, C. H., Trines, R. M. G. M., et al. 2013, ApJ, 777, 167, doi: 10.1088/0004-637X/777/2/167
  * Démoulin (2008) Démoulin, P. 2008, Annales Geophysicae, 26, 3113, doi: 10.5194/angeo-26-3113-2008
  * Démoulin et al. (2020) Démoulin, P., Dasso, S., Lanabere, V., & Janvier, M. 2020, A&A, 639, A6, doi: 10.1051/0004-6361/202038077
  * Démoulin et al. (2008) Démoulin, P., Nakwacki, M. S., Dasso, S., & Mandrini, C. H. 2008, Sol. Phys., 250, 347, doi: 10.1007/s11207-008-9221-9
  * Démoulin & Pariat (2009) Démoulin, P., & Pariat, E. 2009, Advances in Space Research, 43, 1013, doi: 10.1016/j.asr.2008.12.004
  * DeRosa et al. (2012) DeRosa, M. L., Brun, A. S., & Hoeksema, J. T. 2012, ApJ, 757, 96, doi: 10.1088/0004-637X/757/1/96
  * Dumbović et al. (2015) Dumbović, M., Devos, A., Vršnak, B., et al. 2015, Sol. Phys., 290, 579, doi: 10.1007/s11207-014-0613-8
  * Fenrich & Luhmann (1998) Fenrich, F. R., & Luhmann, J. G. 1998, Geophys. Res. Lett., 25, 2999, doi: 10.1029/98GL51180
  * Gibson & Low (1998) Gibson, S. E., & Low, B. C. 1998, ApJ, 493, 460, doi: 10.1086/305107
  * Gopalswamy et al. (2018) Gopalswamy, N., Akiyama, S., Yashiro, S., & Xie, H. 2018, Journal of Atmospheric and Solar-Terrestrial Physics, 180, 35, doi: 10.1016/j.jastp.2017.06.004
  * Gopalswamy et al. (2000) Gopalswamy, N., Lara, A., Lepping, R. P., et al. 2000, Geophys. Res. Lett., 27, 145, doi: 10.1029/1999GL003639
  * Gopalswamy et al. (2003) Gopalswamy, N., Lara, A., Yashiro, S., & Howard, R. A. 2003, ApJ, 598, L63, doi: 10.1086/380430
  * Gopalswamy & Mäkelä (2014) Gopalswamy, N., & Mäkelä, P. 2014, in Astronomical Society of the Pacific Conference Series, Vol. 484, Outstanding Problems in Heliophysics: From Coronal Heating to the Edge of the Heliosphere, ed. Q. Hu & G. P. Zank, 63.  https://arxiv.org/abs/1310.8506
  * Gosling (1993) Gosling, J. T. 1993, J. Geophys. Res., 98, 18937, doi: 10.1029/93JA01896
  * Green et al. (2007) Green, L. M., Kliem, B., Török, T., van Driel-Gesztelyi, L., & Attrill, G. D. R. 2007, Sol. Phys., 246, 365, doi: 10.1007/s11207-007-9061-z
  * Gulisano et al. (2012) Gulisano, A. M., Démoulin, P., Dasso, S., & Rodriguez, L. 2012, A&A, 543, A107, doi: 10.1051/0004-6361/201118748
  * Gulisano et al. (2010) Gulisano, A. M., Démoulin, P., Dasso, S., Ruiz, M. E., & Marsch, E. 2010, A&A, 509, A39, doi: 10.1051/0004-6361/200912375
  * Harvey et al. (1996) Harvey, J. W., Hill, F., Hubbard, R. P., et al. 1996, Science, 272, 1284, doi: 10.1126/science.272.5266.1284
  * Hathaway (2015) Hathaway, D. H. 2015, Living Reviews in Solar Physics, 12, 4, doi: 10.1007/lrsp-2015-4
  * Heinemann et al. (2019) Heinemann, S. G., Temmer, M., Farrugia, C. J., et al. 2019, Sol. Phys., 294, 121, doi: 10.1007/s11207-019-1515-6
  * Hickmann et al. (2015) Hickmann, K. S., Godinez, H. C., Henney, C. J., & Arge, C. N. 2015, Sol. Phys., 290, 1105, doi: 10.1007/s11207-015-0666-3
  * Hoeksema (1984) Hoeksema, J. T. 1984, PhD thesis, Stanford University, California
  * House et al. (1981) House, L. L., Wagner, W. J., Hildner, E., Sawyer, C., & Schmidt, H. U. 1981, ApJ, 244, L117, doi: 10.1086/183494
  * Hu et al. (2016) Hu, H., Liu, Y. D., Wang, R., Möstl, C., & Yang, Z. 2016, ApJ, 829, 97, doi: 10.3847/0004-637X/829/2/97
  * Hundhausen et al. (1990) Hundhausen, A. J., Sime, D. G., & Low, B. C. 1990, in Galactic and Intergalactic Magnetic Fields, ed. R. Beck, P. P. Kronberg, & R. Wielebinski, Vol. 140, 16
  * Illing & Hundhausen (1985) Illing, R. M. E., & Hundhausen, A. J. 1985, J. Geophys. Res., 90, 275, doi: 10.1029/JA090iA01p00275
  * Isavnin (2016) Isavnin, A. 2016, ApJ, 833, 267, doi: 10.3847/1538-4357/833/2/267
  * Jian et al. (2006) Jian, L., Russell, C. T., Luhmann, J. G., & Skoug, R. M. 2006, Sol. Phys., 239, 393, doi: 10.1007/s11207-006-0133-2
  * Jian et al. (2011) Jian, L. K., Russell, C. T., & Luhmann, J. G. 2011, Sol. Phys., 274, 321, doi: 10.1007/s11207-011-9737-2
  * Jin et al. (2017) Jin, M., Manchester, W. B., van der Holst, B., et al. 2017, ApJ, 834, 173, doi: 10.3847/1538-4357/834/2/173
  * Kasper et al. (2019) Kasper, J. C., Bale, S. D., Belcher, J. W., et al. 2019, Nature, 576, 228, doi: 10.1038/s41586-019-1813-z
  * Kataoka et al. (2009) Kataoka, R., Ebisuzaki, T., Kusano, K., et al. 2009, Journal of Geophysical Research (Space Physics), 114, A10102, doi: 10.1029/2009JA014167
  * Kaymaz & Siscoe (2006) Kaymaz, Z., & Siscoe, G. 2006, Sol. Phys., 239, 437, doi: 10.1007/s11207-006-0308-x
  * Kilpua et al. (2017) Kilpua, E., Koskinen, H. E. J., & Pulkkinen, T. I. 2017, Living Reviews in Solar Physics, 14, 5, doi: 10.1007/s41116-017-0009-6
  * Kilpua et al. (2011) Kilpua, E. K. J., Jian, L. K., Li, Y., Luhmann, J. G., & Russell, C. T. 2011, Journal of Atmospheric and Solar-Terrestrial Physics, 73, 1228, doi: 10.1016/j.jastp.2010.10.012
  * Kissmann & Pomoell (2012) Kissmann, R., & Pomoell, J. 2012, SIAM Journal on Scientific Computing, 34, A763, doi: 10.1137/110834329
  * Koskinen & Huttunen (2006) Koskinen, H. E. J., & Huttunen, K. E. J. 2006, Space Sci. Rev., 124, 169, doi: 10.1007/s11214-006-9103-0
  * Lavraud & Rouillard (2014) Lavraud, B., & Rouillard, A. 2014, in Nature of Prominences and their Role in Space Weather, ed. B. Schmieder, J.-M. Malherbe, & S. T. Wu, Vol. 300, 273–284, doi: 10.1017/S1743921313011095
  * Lepping et al. (2005) Lepping, R. P., Wu, C. C., & Berdichevsky, D. B. 2005, Annales Geophysicae, 23, 2687, doi: 10.5194/angeo-23-2687-2005
  * Linan et al. (2023) Linan, L., Regnault, F., Perri, B., et al. 2023, arXiv e-prints, arXiv:2305.02089, doi: 10.48550/arXiv.2305.02089
  * Linker et al. (2017) Linker, J. A., Caplan, R. M., Downs, C., et al. 2017, ApJ, 848, 70, doi: 10.3847/1538-4357/aa8a70
  * Lugaz et al. (2016) Lugaz, N., Farrugia, C. J., Winslow, R. M., et al. 2016, Journal of Geophysical Research (Space Physics), 121, 10,861, doi: 10.1002/2016JA023100
  * Lugaz et al. (2005) Lugaz, N., Manchester, W. B., I., & Gombosi, T. I. 2005, ApJ, 634, 651, doi: 10.1086/491782
  * Lynch et al. (2022) Lynch, B. J., Al-Haddad, N., Yu, W., Palmerio, E., & Lugaz, N. 2022, Advances in Space Research, 70, 1614, doi: 10.1016/j.asr.2022.05.004
  * Maharana et al. (2022) Maharana, A., Isavnin, A., Scolini, C., et al. 2022, Advances in Space Research, 70, 1641, doi: 10.1016/j.asr.2022.05.056
  * Marubashi et al. (2017) Marubashi, K., Cho, K. S., & Ishibashi, H. 2017, Sol. Phys., 292, 189, doi: 10.1007/s11207-017-1204-2
  * McComas et al. (2008) McComas, D. J., Ebert, R. W., Elliott, H. A., et al. 2008, Geophys. Res. Lett., 35, L18103, doi: 10.1029/2008GL034896
  * McComas et al. (2003) McComas, D. J., Elliott, H. A., Schwadron, N. A., et al. 2003, Geophys. Res. Lett., 30, 1517, doi: 10.1029/2003GL017136
  * McGregor et al. (2008) McGregor, S., Hughes, W., Arge, C., & Owens, M. 2008, Journal of Geophysical Research: Space Physics, 113
  * McGregor et al. (2011) McGregor, S. L., Hughes, W. J., Arge, C. N., Owens, M. J., & Odstrcil, D. 2011, Journal of Geophysical Research (Space Physics), 116, A03101, doi: 10.1029/2010JA015881
  * Moffatt (1978) Moffatt, H. K. 1978, Magnetic field generation in electrically conducting fluids
  * Möstl et al. (2022) Möstl, C., Weiss, A. J., Reiss, M. A., et al. 2022, ApJ, 924, L6, doi: 10.3847/2041-8213/ac42d0
  * Odstrcil et al. (2004) Odstrcil, D., Riley, P., & Zhao, X. P. 2004, Journal of Geophysical Research (Space Physics), 109, A02116, doi: 10.1029/2003JA010135
  * Odstrčil & Pizzo (1999) Odstrčil, D., & Pizzo, V. J. 1999, J. Geophys. Res., 104, 483, doi: 10.1029/1998JA900019
  * Oliveira & Samsonov (2018) Oliveira, D. M., & Samsonov, A. A. 2018, Advances in Space Research, 61, 1, doi: 10.1016/j.asr.2017.10.006
  * Owens & Forsyth (2013) Owens, M. J., & Forsyth, R. J. 2013, Living Reviews in Solar Physics, 10, 5, doi: 10.12942/lrsp-2013-5
  * Palmerio et al. (2017) Palmerio, E., Kilpua, E. K. J., James, A. W., et al. 2017, Sol. Phys., 292, 39, doi: 10.1007/s11207-017-1063-x
  * Parker (1958) Parker, E. N. 1958, ApJ, 128, 664, doi: 10.1086/146579
  * Parker (1993) —. 1993, ApJ, 408, 707, doi: 10.1086/172631
  * Perri et al. (2022) Perri, B., Leitner, P., Brchnelova, M., et al. 2022, arXiv e-prints, arXiv:2205.03341.  https://arxiv.org/abs/2205.03341
  * Pevtsov et al. (2008) Pevtsov, A. A., Canfield, R. C., Sakurai, T., & Hagino, M. 2008, ApJ, 677, 719, doi: 10.1086/533435
  * Pinto & Rouillard (2017) Pinto, R. F., & Rouillard, A. P. 2017, ApJ, 838, 89, doi: 10.3847/1538-4357/aa6398
  * Pirjola (2005) Pirjola, R. 2005, Advances in Space Research, 36, 2231, doi: 10.1016/j.asr.2003.04.074
  * Pomoell & Poedts (2018) Pomoell, J., & Poedts, S. 2018, Journal of Space Weather and Space Climate, 8, A35, doi: 10.1051/swsc/2018020
  * Pomoell & Vainio (2012) Pomoell, J., & Vainio, R. 2012, ApJ, 745, 151, doi: 10.1088/0004-637X/745/2/151
  * Priest (2014) Priest, E. 2014, Magnetohydrodynamics of the Sun, doi: 10.1017/CBO9781139020732
  * Pulkkinen (2007) Pulkkinen, T. 2007, Living Reviews in Solar Physics, 4, 1, doi: 10.12942/lrsp-2007-1
  * Regnault et al. (2020) Regnault, F., Janvier, M., Démoulin, P., et al. 2020, Journal of Geophysical Research (Space Physics), 125, e28150, doi: 10.1029/2020JA028150
  * Regnault et al. (2023) Regnault, F., Strugarek, A., Janvier, M., et al. 2023, A&A, 670, A14, doi: 10.1051/0004-6361/202244483
  * Reiss et al. (2022) Reiss, M. A., Muglach, K., Mullinix, R., et al. 2022, arXiv e-prints, arXiv:2201.13447.  https://arxiv.org/abs/2201.13447
  * Riley et al. (2014) Riley, P., Ben-Nun, M., Linker, J. A., et al. 2014, Sol. Phys., 289, 769, doi: 10.1007/s11207-013-0353-1
  * Robbrecht et al. (2009) Robbrecht, E., Berghmans, D., & Van der Linden, R. A. M. 2009, ApJ, 691, 1222, doi: 10.1088/0004-637X/691/2/1222
  * Ruffenach et al. (2012) Ruffenach, A., Lavraud, B., Owens, M. J., et al. 2012, Journal of Geophysical Research (Space Physics), 117, A09101, doi: 10.1029/2012JA017624
  * Rušin et al. (2010) Rušin, V., Druckmüller, M., Aniol, P., et al. 2010, A&A, 513, A45, doi: 10.1051/0004-6361/200912778
  * Sahade et al. (2022) Sahade, A., Cécere, M., Sieyra, M. V., et al. 2022, A&A, 662, A113, doi: 10.1051/0004-6361/202243618
  * Schatten et al. (1969) Schatten, K. H., Wilcox, J. M., & Ness, N. F. 1969, Sol. Phys., 6, 442, doi: 10.1007/BF00146478
  * Schmieder et al. (2015) Schmieder, B., Aulanier, G., & Vršnak, B. 2015, Sol. Phys., 290, 3457, doi: 10.1007/s11207-015-0712-1
  * Schmieder et al. (2020) Schmieder, B., Kim, R. S., Grison, B., et al. 2020, Journal of Geophysical Research (Space Physics), 125, e27529, doi: 10.1029/2019JA027529
  * Schrijver (2015) Schrijver, C. J. 2015, Space Weather, 13, 524, doi: 10.1002/2015SW001252
  * Schrijver et al. (2015) Schrijver, C. J., Kauristie, K., Aylward, A. D., et al. 2015, Advances in Space Research, 55, 2745, doi: 10.1016/j.asr.2015.03.023
  * Schwenn et al. (2005) Schwenn, R., dal Lago, A., Huttunen, E., & Gonzalez, W. D. 2005, Annales Geophysicae, 23, 1033, doi: 10.5194/angeo-23-1033-2005
  * Scolini et al. (2021a) Scolini, C., Dasso, S., Rodriguez, L., Zhukov, A. N., & Poedts, S. 2021a, A&A, 649, A69, doi: 10.1051/0004-6361/202040226
  * Scolini et al. (2019) Scolini, C., Rodriguez, L., Mierla, M., Pomoell, J., & Poedts, S. 2019, A&A, 626, A122, doi: 10.1051/0004-6361/201935053
  * Scolini et al. (2021b) Scolini, C., Winslow, R. M., Lugaz, N., & Poedts, S. 2021b, ApJ, 916, L15, doi: 10.3847/2041-8213/ac0d58
  * Scolini et al. (2021c) —. 2021c, ApJ, 916, L15, doi: 10.3847/2041-8213/ac0d58
  * Scolini et al. (2023) —. 2023, ApJ, 944, 46, doi: 10.3847/1538-4357/aca893
  * Scolini et al. (2022) Scolini, C., Winslow, R. M., Lugaz, N., et al. 2022, ApJ, 927, 102, doi: 10.3847/1538-4357/ac3e60
  * Scolini et al. (2020) Scolini, C., Chané, E., Temmer, M., et al. 2020, ApJS, 247, 21, doi: 10.3847/1538-4365/ab6216
  * Shen et al. (2022) Shen, F., Shen, C., Xu, M., et al. 2022, Reviews of Modern Plasma Physics, 6, 8, doi: 10.1007/s41614-022-00069-1
  * Shiota & Kataoka (2016) Shiota, D., & Kataoka, R. 2016, Space Weather, 14, 56, doi: 10.1002/2015SW001308
  * Talpeanu et al. (2022) Talpeanu, D. C., Poedts, S., D’Huys, E., & Mierla, M. 2022, A&A, 658, A56, doi: 10.1051/0004-6361/20214197710.48550/arXiv.2111.14909
  * Thernisien (2011) Thernisien, A. 2011, ApJS, 194, 33, doi: 10.1088/0067-0049/194/2/33
  * Thernisien et al. (2009) Thernisien, A., Vourlidas, A., & Howard, R. A. 2009, Sol. Phys., 256, 111, doi: 10.1007/s11207-009-9346-5
  * Titov et al. (2014) Titov, V. S., Török, T., Mikic, Z., & Linker, J. A. 2014, ApJ, 790, 163, doi: 10.1088/0004-637X/790/2/163
  * Tsurutani et al. (2003) Tsurutani, B. T., Gonzalez, W. D., Lakhina, G. S., & Alex, S. 2003, Journal of Geophysical Research (Space Physics), 108, 1268, doi: 10.1029/2002JA009504
  * van der Holst et al. (2010) van der Holst, B., Manchester, W. B., I., Frazin, R. A., et al. 2010, ApJ, 725, 1373, doi: 10.1088/0004-637X/725/1/1373
  * van der Holst et al. (2014) van der Holst, B., Sokolov, I. V., Meng, X., et al. 2014, ApJ, 782, 81, doi: 10.1088/0004-637X/782/2/81
  * Verbeke et al. (2019) Verbeke, C., Pomoell, J., & Poedts, S. 2019, A&A, 627, A111, doi: 10.1051/0004-6361/201834702
  * Verbeke et al. (2022) Verbeke, C., Schmieder, B., Démoulin, P., et al. 2022, Advances in Space Research, 70, 1663, doi: 10.1016/j.asr.2022.06.013
  * Viall & Borovsky (2020) Viall, N. M., & Borovsky, J. E. 2020, Journal of Geophysical Research (Space Physics), 125, e26005, doi: 10.1029/2018JA026005
  * Wang et al. (2005) Wang, C., Du, D., & Richardson, J. D. 2005, Journal of Geophysical Research (Space Physics), 110, A10107, doi: 10.1029/2005JA011198
  * Wang & Sheeley (1990) Wang, Y. M., & Sheeley, N. R., J. 1990, ApJ, 355, 726, doi: 10.1086/168805
  * Webb & Nitta (2017) Webb, D., & Nitta, N. 2017, Sol. Phys., 292, 142, doi: 10.1007/s11207-017-1166-4
  * Webb & Howard (2012) Webb, D. F., & Howard, T. A. 2012, Living Reviews in Solar Physics, 9, 3, doi: 10.12942/lrsp-2012-3
  * Weiss (1990) Weiss, N. O. 1990, Philosophical Transactions of the Royal Society of London Series A, 330, 617, doi: 10.1098/rsta.1990.0042
  * Wiegelmann et al. (2017) Wiegelmann, T., Petrie, G. J. D., & Riley, P. 2017, Space Sci. Rev., 210, 249, doi: 10.1007/s11214-015-0178-3
  * Winslow et al. (2015) Winslow, R. M., Lugaz, N., Philpott, L. C., et al. 2015, Journal of Geophysical Research (Space Physics), 120, 6101, doi: 10.1002/2015JA021200
  * Winslow et al. (2022) Winslow, R. M., Scolini, C., Jian, L. K., et al. 2022, Frontiers in Astronomy and Space Sciences, 9, 1064175, doi: 10.3389/fspas.2022.1064175
  * Wu & Lepping (2011) Wu, C.-C., & Lepping, R. P. 2011, Sol. Phys., 269, 141, doi: 10.1007/s11207-010-9684-3
  * Wu & Lepping (2016) —. 2016, Sol. Phys., 291, 265, doi: 10.1007/s11207-015-0806-9
  * Yeates et al. (2018) Yeates, A. R., Amari, T., Contopoulos, I., et al. 2018, Space Sci. Rev., 214, 99, doi: 10.1007/s11214-018-0534-1
  * Zuccarello et al. (2012) Zuccarello, F. P., Bemporad, A., Jacobs, C., et al. 2012, ApJ, 744, 66, doi: 10.1088/0004-637X/744/1/66
  * Zurbuchen & Richardson (2006) Zurbuchen, T. H., & Richardson, I. G. 2006, Space Sci. Rev., 123, 31, doi: 10.1007/s11214-006-9010-4