($F_{\odot}=1.e6\ \rm{erg.cm^{-2}.s^{-1}}$,$H_{f}=7\ \rm{Mm}$) for a loop in
quasi-steady thermodynamic equilibrium (see section 7.1.1).

The resulting quasi-steady ISAM solution is shown in Figure 7.8 for the number
densities, while the temperature profiles are very close to the ($1.4e6,7$)
ISAM run already presented in Figure 7.2. In the chromosphere, the neutral
Hydrogen density mostly follows a hydrostatic law of which the slope is
determined by the local temperature. As we shall see below, low-FIP elements
that are already ionized in the chromosphere are significantly coupled with
protons through the Coulomb collisions, where the strength of the interaction
is highly controlled by the proton density. As already discussed in section
6.4, the photoionization of neutral Hydrogen that I implemented in ISAM only
constitutes a rough estimate of the photoionization rate in the upper
chromosphere. Although a proper treatment of radiative transfer would be
necessary for a better description of the ionization/recombination processes
in the chromosphere, Carlsson & Stein (2002) have shown that the problem can
be reduced to a neutral Hydrogen atom with only a few levels and transition
rates, that is at reach for future mid-term implementations in ISAM. For the
purpose of this thesis where we focus our attention on the upper chromosphere
and corona where most of the composition is expected to settle, the current
implementation of photoionization in ISAM is accurate enough by providing a
mid profile of the proton density to the Al-C7 model. When reaching the
transition region and corona, ionization of neutral Hydrogen mostly occurs
through collision with electrons that is commonly called the "coronal
approximation". In this approximation the ionization/recombination processes
are well known and can be described accurately using analytical fits and
tabulated data that I implemented in ISAM (see section 6.4). Similar formulas
are used in the ChiantiPy spectral code that leads to a proton density in the
corona that matches closely the ISAM solution.

Figure 7.8:  Number densities of electrons (dashed red line), protons (solid
green line) and neutral Hydrogen (solid black line) computed by ISAM. The
neutral Hydrogen and electron density profiles from the AL-C7 model are marked
by black and red star symbols respectively. The proton density computed by the
ChiantiPy spectral code given the ISAM neutral Hydrogen profile is plotted as
green crosses.

We present in Figure 7.9 the collisional couplings in the Hydrogen-proton-
electron atmosphere solved in ISAM. That includes the charge-exchange resonant
interaction between neutral Hydrogen and protons (see section 6.5.4), the two-
body elastic electron-neutral interaction (see section 6.5.4), and the Coulomb
interaction between protons and electrons (see section 6.5.3). The few protons
that are ionized in the mid/upper chromosphere are closely coupled with the
dominant neutral plasma. To first order the resonant collision frequency of
protons with neutrals scales as $n_{H}T^{1/2}$, and is hence mostly controlled
by the decrease of the neutral Hydrogen density throughout the chromosphere as
ionization proceeds. In the upper chromosphere, the electrons produced from
ionization of neutral Hydrogen become dominant and prevail on the collisional
coupling with protons through the Coulomb interaction. This proton-electron
coupling then rapidly weakens throughout the transition region where the hot
coronal temperatures strongly diminish the efficiency of Coulomb collisions
which scale as $n_{e}T^{-3/2}$.

Figure 7.9:  Collision frequencies of neutral Hydrogen with electrons and
protons (dashed and solid black lines), and of protons with electrons and
neutral Hydrogen (dashed and solid red lines).

After a relaxation time of $\simeq 90\ \rm{min}$, neutral Hydrogen and protons
have almost completely settle in the chromosphere and corona respectively,
which are in a quasi hydrostatic equilibrium as shown in Figure 7.10. This
equilibrium is reached through a perpetual adjustment of the internal pressure
of the system until the gravity pull is balanced. That is fulfilled in the
chromosphere where neutrals are dominant, and also in the corona where both
the protons, and electrons through the electrostatic field (eq 6.13),
contribute at counterbalancing the gravity. Furhermore, the quasi-neutrality
condition assumed in ISAM (eq 6.14a) imposes that $n_{e}=n_{p}$ in a pure
Hydrogen plasma. Therefore any discrepancy between the proton and electron
contributions to the thermal force is induced by different profiles of proton
and electron temperatures. Such discrepancy is visible just above the
transition region and is likely the result of the electrons that carry more
thermal energy into that region thanks to their enhanced heat flux (see Figure
7.7).

Figure 7.10:  Forces acting on neutral Hydrogen (left), protons (middle) and
on both (right). All forces are first multiplied by the ionization fraction
($n_{X}/n_{[X]}$), and then normalized by the absolute value of gravity.
Positive forces are directed outward.

A junction between these two main hydrostatic equilibria occurs at the
transition region. Contrary to neutral Hydrogen, protons face a strong
downward pressure force in the transition region that acts as a barrier to the
transfer of protons into the corona. Therefore a differential flow appears
that generates a net friction force between protons and neutrals, which reads
$m_{p}n_{p}\nu_{pH}(u_{H}-u_{p})$ for protons or inversely
$m_{H}n_{H}\nu_{Hp}(u_{p}-u_{H})$ for neutrals where polarization effects are
negligible (see eq 6.33). Note that there is no friction between protons and
electrons since we assume a current-free ambipolar flow in ISAM (see eq 6.14b)
and hence $u_{e}=u_{p}$ in an atmosphere that is only constituted of protons.

A thermal force can arise from thermal diffusion effects between charged
species that interact through Coulomb collisions because of the dependency of
the collision frequency on the temperature. Basically, charged particles will
be tempted to move towards regions of higher temperatures where Coulomb
collisions are less frequent, that is from the cool chromosphere to the hot
corona. In a loop that remains highly collisional with small pressure
anisotropies, the expression of the thermal force that results from the
interaction of an ion $i$ with another specie $t$ (i.e. an ion, electron or
neutral) can be reduced to (see eq 6.33):

$\displaystyle
n_{i}m_{i}\left[\nu_{it}\frac{z_{it}}{k_{b}T_{it}(m_{i}+m_{t})}\left(m_{t}\frac{q_{i}}{n_{i}}-m_{i}\frac{q_{t}}{n_{t}}\right)+Z_{i}\frac{m_{e}}{m_{i}}\nu_{et}\frac{z_{et}}{k_{b}T_{et}(m_{e}+m_{t})}\left(m_{t}\frac{q_{e}}{n_{e}}-m_{e}\frac{q_{t}}{n_{t}}\right)\right]\quad\text{for
}t\neq e$ (7.1a) $\displaystyle
n_{i}m_{i}\left[\nu_{ie}\frac{z_{ie}}{k_{b}T_{ie}(m_{i}+m_{e})}\left(m_{e}\frac{q_{i}}{n_{i}}-m_{i}\frac{q_{e}}{n_{e}}\right)+Z_{i}\frac{m_{e}}{m_{i}}\nu_{ei}\frac{z_{ei}}{k_{b}T_{ei}(m_{e}+m_{i})}\left(m_{i}\frac{q_{e}}{n_{e}}-m_{e}\frac{q_{i}}{n_{i}}\right)\right]\quad\text{for
}t=e$ (7.1b)

which accounts for the contributions from the electrostatic polarization field
(eq 6.13).

The thermal force cancels out exactly in the proton-electron Coulomb
interaction ($i=p$ in eq 7.1b), since we assume elastic collisions
$n_{p}m_{p}\nu_{pe}=n_{e}m_{e}\nu_{ep}$ and quasi-neutrality $n_{p}=n_{e}$.
Furthermore, collisions of protons with neutrals are too rare in the
transition region to produce a significant thermal force, that is also
explained by the fact that protons and neutrals have similar normalized heat
fluxes $q_{H}/n_{H}$ and $q_{p}/n_{p}$ in that region. That justifies our
treatment of proton-neutral collisions in ISAM as ideal Maxwell molecule
interactions which do not include the thermal force ($z_{it}=0$ in eq 6.33).
Therefore the thermal force likely does not play any role in a solar
atmosphere that is only constituted of neutral Hydrogen, protons and
electrons.

As noted above, there is an ambipolar flow that forms between neutral Hydrogen
and protons that is shown in Figure 7.11. When expressed in the frame
associated to the center of mass of neutrals and protons, this ambipolar flow
becomes evident with speed profiles that are perfectly opposite. That does not
mean that there is no mean flow (plotted on the right-hand side y-axis)
despite being close to a quasi-steady state solution. As noted by Killie et
al. (2005) and Killie & Lie-Svendsen (2007), this ambipolar flow remains
persistent in the ionization layer of Hydrogen as long as the temperature
rises at the top of the transition region during chromospheric evaporation
phases. The ionization equilibria that was previously established in this
region is then perturbed where ionization of neutrals is enhanced due to the
higher temperature that hence produces a net outflow to fulfil the increased
demand in neutrals. Such evaporation phases have been frequently observed and
modeled in dynamic loops that undergo TNE cycles as already discussed in
section 7.1.1. Chromospheric evaporation phases can also be initiated by
heating pulses near the transition region during magnetic reconnection events
for instance. Whenever this ambipolar flow is present, protons flow down the
transition region and hence form a "barrier" to any element that is already
ionized in this region. That effect is expected to weaken in open solar wind
solutions where most of the energy is carried away from the transition region,
and hence would explain why closed-field and open-field plasma show different
compositions overall. This will have a major impact on the transfer of heavy
ions through the transition region where a filtering of heavy elements will
naturally occur according to their FIP as we shall see in the next section.

Figure 7.11:  Bulk velocities of neutral Hydrogen and protons (dashed and
dotted black lines) expressed in the center of mass
$v_{G}=(m_{H}u_{H}+m_{p}u_{p})/(m_{H}+m_{p})$. The total mass flux of the
Hydrogen element is plotted on a logarithmic scale that is shown on the right-
hand side and where positive and negative values are denoted by black and red
colors respectively.  Figure 7.12:  Total ionization fractions of Oxygen and
Hydrogen.

### 7.2 Separation of heavy ions: application to Oxygen

Except for Helium that makes about $5-10\%$ of the composition of the solar
atmosphere, heavy elements can be considered as minor/trace species that have
no feedback on the Hydrogen-proton-electron plasma. In this case, minor
species can be solved at a second stage in ISAM for a fixed Hydrogen-proton-
electron atmosphere. That allows us to greatly accelerate the simulations by
removing the hard time-resolution constrain from the electrons. Although the
energy and heat flux transport equations for electrons are not solved in this
case, the electron density and bulk velocity are still re-computed at each
time step to ensure the condition of quasi-neutrality and ambipolar flows (eq
6.14a-6.14b). In the following, the background atmosphere is taken to be the
pure Hydrogen-proton-electron atmosphere presented in section 7.1.3. The
photospheric abundances relative to Hydrogen from Avrett & Loeser (2008, Table
2) are set at the base of the loop. Practically, that determines at the base
of the loop either the density of the neutral or of the first ionization state
according to whether the element has a high or low FIP respectively (see also
section 6.8.2).

Oxygen is 16 times heavier than Hydrogen but still has a FIP of $\simeq 13.62\
\rm{eV}$ that is very close to that of Hydrogen $13.60\ \rm{eV}$. Another
peculiarity of Oxygen is its strong coupling with Hydrogen through the
resonant charge-exchange reactions $O+H^{+}\leftrightarrow O^{+}+H$. Therefore
Oxygen has a total ionization fraction in the chromosphere that closely
follows that of Hydrogen as shown in Figure 7.12, where about $40\%$ of the
element is in an ionized state at the base of the transition region at around
$2000\ \rm{km}$. For a first application I limit myself to the first four
ionized states of Oxygen. In practice, the system could be readily extended to
include all ionisation states of Oxygen, but then that would require
additional data on the ionization/recombination rates and on the collision
frequencies for which I already dedicated a significant time to gather and
implement in ISAM (see section 6.5 and 6.4).

Figure 7.13:  FIP bias of Oxygen relative to Hydrogen
$(n_{[O]}/n_{[H]})_{alt}/(n_{[O]}/n_{[H]})_{PHO}$ function of altitude. Colors
are indicative of the proportion of each specie (neutral and ions) in the
total Oxygen abundance.

As already discussed in section 1.3.4, the level of chromospheric mixing is
likely determinant to the amount of heavy ions that can get into the corona
through the transition region. I consider first a case of a stratified
chromosphere. By starting from a loop that is totally empty in Oxygen, I let
the loop fill in Oxygen from the lower boundary until the composition in
Oxygen starts to settle. The result is shown in Figure 7.13 for a loop that
run about $11\ \rm{hr}$. The common FIP bias is used as an indicator of the
enhancement or depletion of heavy ions with respect to the photospheric
abundances, that is defined as the ratio of the relative abundances with
respect to Hydrogen between a certain altitude and the base of the loop.
According to this definition, the FIP bias equals one at the lower boundary
where photospheric abundances are assumed. Then a FIP bias that is either
larger or lower than one means either a enrichment or depletion with respect
to the photospheric abundances. A first look at Figure 7.13 indicates a strong
depletion of Oxygen in a large portion of the loop with a depletion down to
$\approx 10^{-4}$. The fraction of Oxygen that manages to get through the
transition region then slightly accumulates higher up in the corona at around
$7-8\ \rm{Mm}$. The global depletion of Oxygen is likely a consequence of its
mass being 16 times greater than that of Hydrogen and as a result Oxygen is
much hardly lifted up in the atmosphere. To better understand which state of
Oxygen contribute to the FIP bias locally, the proportion of each specie
within the Oxygen element is color plotted under the curve. The thicker the
area, the more abundant is the Oxygen specie at that altitude.

I found this format convenient to analyse the role of each specie in the total
fractionation of Oxygen relative to Hydrogen. Therefore I can concentrate my
analysis on neutral Oxygen and $O^{4+}$ which constitute most of the
chromosphere and corona respectively in the present ISAM run. I should also
mention that Oxygen is likely to be ionized at a higher level and that the
higher ionization levels of Oxygen will be accounted for in a subsequent
paper. The forces at play in the overall Oxygen system are plotted in Figure
7.14 using the same normalizing procedure as in Figure 7.10. Here the thermal
and friction forces have been summed up together for a better readability of
the graph. Their summed contribution then allows us to better see the
influence of external species on the Oxygen system through collisions. For
completeness, the separate contributions of friction and thermal effects to
the total external collisional force applying to neutral Oxygen and $O^{4+}$
are shown in the middle and right-hand side panels of Figure 7.14.

Figure 7.14:  Left panel: summed up contributions of all forces that apply to
the Oxygen element using the same normalizing procedure as in Figure 7.10. The
particle-particule interactions that contribute the most to the total friction
and thermal forces that apply to neutral Oxygen (middle panel) and $O^{4+}$
(right panel) are also shown.

In contrast to the analysis made in section 7.1.3 for the pure Hydrogen
atmosphere, Oxygen is only partially stratified in the chromosphere where the
collisional coupling with external species substitutes partially the pressure
force to sustain the gravity pull. If at opposite the pressure force had
completely fulfilled its role at balancing the gravity, Oxygen would have
fully stratified in the chromosphere. According to the scale height
$k_{b}T_{s}/(m_{s}g)$ of Oxygen that is $16$ times shorter than that of
Hydrogen, that would have led to a depletion of Oxygen relative to Hydrogen of
about $8\times 10^{-52}$ at the top of the chromosphere that is much more
significant than that simulated in ISAM ($\approx 10^{-4}$, see Figure 7.13).
The collisional friction of Oxygen with protons is the dominant force that
prevents neutral Oxygen from fully stratifying in the chromosphere as shown in
the middle panel of Figure 7.14.

In the corona, thermal diffusion effects becomes dominant through the Coulomb
interaction between $O^{4+}$ and protons that generate a net outward thermal
force that overcomes the gravity pull. The abundance of Oxygen then
progressively increases that reduces the depletion with respect to Hydrogen.
This lift up of Oxygen proceeds up to $\simeq 7400\ \rm{km}$ where the total
net outward force from the collisional coupling with protons becomes equal to
the gravity pull. Past this point the stratification of Oxygen takes over and
leads to an even greater depletion at the loop-top.

The analysis becomes much more complex in the transition region where the
ionization of Oxygen proceeds rapidly to the fourth-state. The attribution of
more grid points within this thin layer should help at a better description of
the forces at play, but at the cost of greater computational times. We can say
at first order that Oxygen ions seem to be slightly retained from entering the
corona due to the coupling with the down-streaming protons, the so-called
proton "barrier" mentioned above and in Killie et al. (2005); Killie & Lie-
Svendsen (2007)), although this barrier is likely overcome here by the thermal
diffusion effects that are associated to both protons and electrons.

To conclude on the Oxygen balance in this given Hydrogen-proton-electron
atmosphere, the strong collisional coupling of neutral Oxygen and $O^{4+}$
through charge-exchange and Coulomb interactions with protons prevents Oxygen
from stratifying completely in the chromosphere and corona. That leads up to a
significant depletion of Oxygen relative to Hydrogen, but that still remains
limited compared to the case of a full stratification where particle-particle
interactions would not be accounted for. In the presented ISAM simulation,
this stratification of Oxygen takes over $11\ \rm{hr}$ to establish in a non-
perturbed chromosphere. Our results agree with a previous study from Killie &
Lie-Svendsen (2007) who also show significant variations of the Oxygen
abundance throughout the upper chromosphere and corona. However, such
variations are not supported by the observations and therefore suggest the
existence of a mixing process in the chromosphere that counteracts the
settling of Oxygen over time. As discussed several times throughout this
thesis (see e.g. 1.3.4), the chromosphere is known to be highly dynamic and
many sources can provide this mixing including turbulence, wave-particle
interactions and flux emergence. The impact of these mixing processes on the
abundance of Oxygen will be investigated in a subsequent work.

### 7.3 Conclusion

We have presented in this chapter a first application of the Irap Solar
Atmospheric Model (ISAM) to the study of composition along coronal loops. That
required a significant amount of involvement and determination of myself to
get the ISAM model working in such specific cases. The ISAM model has been
recently proven to be successful at modeling various solar wind regimes
(Lavarra et al., 2022). I brought significant improvements to the initial
model to account for the peculiarity of coronal loops, and to allow for a more
comprehensive treatment of minor species in the solar atmosphere. All of these
changes have been described in detail in chapter 6.

The case of a coronal loop composed of Hydrogen only has been first
investigated in section 7.1, as a baseline to build up our expertise on the
interpretation of the forces at play in the low solar atmosphere. As
introduced in section 1.3, we basically found that both the chromosphere and
corona satisfy their own hydrostatic equilibria. However, we have seen in
section 7.1.1 that the coronal part of the loop can be perturbed and enter in
a thermal non-equilibrium state where the plasma successively undergoes
cooling/heating phases, and bounces back and forth between the transition
region and the loop-top. As noted in past studies on the FIP effect (Killie et
al., 2005; Killie & Lie-Svendsen, 2007), an increase of the temperature at the
base of the corona can convect more thermal energy through the transition
region and hence fuels an ambipolar flow between neutral Hydrogen and protons
in that region. I have also observed this ambipolar flow in the pure Hydrogen
atmosphere presented in section 7.1.3, where the down-streaming protons act as
a "barrier" for the heavy elements that are already ionized in the transition
region (such as low-FIP elements), and hence those heavy ions are more likely
to be dragged down by the protons through Coulomb collisions.

By including Oxygen in this pure Hydrogen background atmosphere as a trace
element, I have shown in section 7.2 that Oxygen is able to get through the
proton "barrier" where thermal effects are sufficient to overcome the proton
drag. That was also noted in Killie & Lie-Svendsen (2007) and is permitted by
the fact that Oxygen ionizes at almost the same rate as Hydrogen. This barrier
is supposed to be much more effective for low-FIP elements that are completely
ionized in the lower transition region however (Killie & Lie-Svendsen, 2007).
As a consequence, an inverse FIP effect across the transition region is
favored where low-FIP (and high-FIP) elements are depleted (and enriched) in
the corona, an inverse FIP effect that is not observed in the solar corona but
in active stellar coronae though (see the review from Laming, 2015). This
paradox seems inherent to models that consider diffusion effects alone.

Preliminary tests with the inclusion of low-FIP Magnesium and high-FIP Helium
in ISAM tend to supports an inverse FIP effect throughout the transition
region as well, where a significant portion of the Magnesium available in the
upper chromosphere is prevented from entering the corona. However, these
preliminary tests also show that this inverse FIP effect likely does not
persist very long, where higher up in the corona Magnesium turns to be well
enriched with respect to Hydrogen in the end, thanks to a strong upward
thermal force. Therefore the consistency of the modelled abundances is
strongly conditioned by the height that is considered, where composition
diagnostics of coronal loops done by spectroscopy are usually derived from
spectral lines that are relevant to distinct altitudes in the solar
atmosphere.

Furthermore, the impact of chromospheric mixing on the coronal loop
composition has not been investigated in this chapter. The code itself can
nonetheless be readily adapted to treat such cases, for which I already
performed first attempts. In addition, ISAM could be exploited to generate
acoustic shocks at the base of the loop that may then propagate photospheric
abundances throughout the chromosphere. As noted by Laming (2009), wave-
particle interactions through the ponderomotive force can also furnish the
upper chromosphere in photospheric abundances. The ponderomotive force may
also be essential at preventing the inverse FIP effect that tends to be
obtained when only diffusion effects are accounted for. The reader may have
noted how complex the diffusion effects alone are to interpret. Therefore the
results presented in this chapter will serve as a baseline for future
applications of ISAM that will consider the contribution from wave-particle
interactions. I must highlight nonetheless that I have already made
significant preliminary advances towards this goal, including the solving of
an additional transport equation in ISAM for the propagation of Alfvén waves
(see section 6.6.2.1) and the call to an external turbulence code called
Shell-Atm (see section 6.6.3).

## Chapter 8 Conclusion and future perspectives

Aside from the major breakthroughs that have been made recently from the
measurements taken in situ at the Parker Solar Probe (PSP) of the nascent
solar wind, we have shown in chapter 4 that the WISPR white-light imager on
board PSP significantly enriched our knowledge of the structure and formation
of the slow solar wind (SSW). By imaging the SSW from a vantage point closer
to the source, WISPR has depicted a SSW of an unexpected complexity, with
small-scale details that are often smeared out from near 1 AU remote
observations due to line-of-sight integration effects. While WISPR allowed to
scrutinize the slow solar wind at scales down to the thin heliospheric plasma
sheet (HPS) (i.e. $\lesssim 5^{\circ}$), WISPR has especially portrayed a slow
solar wind that emerges from multiple sources distributed further away from
the HPS. Therefore WISPR corroborates a more general picture where the source
regions of the slow solar wind are diverse and are likely dictated by the
topology of the magnetic field lower in the corona. In particular, that has
been formulated in the quasi-stationnary theory (section 1.4.1) which shows
that dense and slow winds should form along magnetic fields that have expanded
significantly all the way up from the surface to the corona. This effect has
been further quantified in chapter 4 by exploiting the high-resolution MULTI-
VP MHD simulations, where the expansion rate of the coronal magnetic field
controls the energy that fuels the solar wind. Given a magnetogram of the
surface magnetic field that is sufficiently resolved, this modeling framework
generated a myriad of streams which in turn produce a fine-structuring of the
slow solar wind similarly to the last observations from WISPR. However, that
may not be the only source of all the fine striation observed by WISPR lately,
and the contribution from the magnetic field has still to be evaluated
precisely.

By greatly reducing the gap between the source and the measurement point, the
PSP mission has already provided a wealth of precious data that incite to
rethink the current formation theories of the slow solar wind. Regions of high
expansion factors that are favorable to the formation of a quasi-stationary
slow solar wind are also propitious environment for magnetic reconnection to
occur, these regions being more generally termed the (quasi-)separatrix layers
or the S-web. The large affluence of magnetic reconnection events in the
corona has long been suspected as a major source of the observed variability
of the SSW. In chapter 5, we considered dynamic theories that may explain part
of the variability of the slow wind observed both remotely and in situ at PSP.
As a step forward to the first "blobs" observed by SoHO 25 years ago, we
suggested a more general picture of "blobs" that form hand in hand with flux
rope structures. The presence of such structures have been suspected from in
situ measurements taken at STEREO and more recently at PSP and SolO, but PSP-
WISPR allowed for the first time to observe remotely these flux ropes in
greater depth as shown in section 5.2. Advanced WindPredict-AW simulations
have been exploited in section 5.3 to investigate the generation of such flux
ropes from reconnection at the tip of streamers. Density structures with
multiple periodicities have then been produced with time scales that are
coherent with the observations. By simulating their aspect in WISPR images I
have found a close similarity with actual flux ropes that have been observed
by WISPR. However, WISPR observations unveiled the presence of likely smaller-
scale density fluctuations which have also been revealed in a recent deep-
field STEREO-COR2 campaign. The origin of these structures remains largely
unknown but we showed in section 5.4 that they can be in part generated by
heating pulses at the base of the corona. This constitutes an elegant picture
of the formation of the SSW where the dynamic theories provide the necessary
energy inputs, through the abundance of magnetic reconnection events that are
known to occur in the low corona, and with the associated generation and
propagation of density fluctuations into the solar wind.

While WISPR captured plenty of transient structures, slow solar winds with a
low variability have also been observed. That has been further noted in an in-
depth analysis of in situ measurements taken at PSP combined with a precised
assessment of the connectivity that I furnished for this study (section 5.1).
Slow winds with different bulk properties have been identified that suggest
not a _time-dependent_ but _spatial-dependent_ variability. Their difference
appeared to be inherent to the magnetic topology of their source region and
hence corroborates the picture drawn in chapter 4 of a SSW that is textured by
the complex structure of the open coronal magnetic field.

Although the novel observations made by PSP allowed us to pinpoint more
precisely the sources of the slow solar wind and of its associated forming
mechanisms, subsequent studies should benefit from the large amount of data
collected by both PSP and now Solar Orbiter (SolO) to extend the observations
that we made on a case by case basis. To benefit the most from these novel
measurements, we have highlighted the necessity of developing new methods that
employ forward modeling techniques coupled with advanced MHD models of the
solar atmosphere, but also novel tools that allow for a more systematic
evaluation of the model performances. However, the models employed in this
thesis still face significant difficulties at explaining the most recent
observations from PSP in all their complexity. We especially noted in chapter
4 that WISPR now provides more stringent tests to current solar wind models.

As introduced in section 1.2.2.2 and 1.2.2.3, a variability of the SSW is also
observed in term of its composition in heavy ions. As coronal loops have been
observed to have similar composition signatures than those measured in situ in
the solar wind, we have shown in chapter 7 that a high variability of the
distribution of heavy ions along coronal loops is also to be expected. I
contributed to the development of a multi-specie model of the solar atmosphere
called ISAM (presented in chapter 6), that solves the transport of Hydrogen
and minor species in a self-consistent manner from the chromosphere to the
corona. ISAM also includes a comprehensive treatment of interactions between
species that allowed us to investigate in detail the role of collisional
effects in the FIP effect, where I have shown that protons play a significant
role in the separation of Oxygen for instance. The strong collisional coupling
of Oxygen with Hydrogen through friction and thermal effects prevents the full
stratification of Oxygen, that would otherwise be much more marked with a
dramatic depletion of Oxygen being 16 times heavier than Hydrogen. Coulomb
interactions with protons also tend to favor an inverse FIP effect throughout
the transition region where only mid to high FIP elements would have access to
the corona by being blind to the "barrier" made by the down-streaming protons
in the transition region. This effect is likely to be regulated in dynamic
loops that undergo successive evaporation/cooling phases, and to be boosted up
during magnetic reconnection events that deposit great amount of energy at the
base of the corona. Furthermore, this "local" inverse FIP effect seems to be
greatly overcome higher up in the corona as suggested by some preliminary
tests that include the low-FIP Magnesium and high-FIP Helium. Although our
work only focused on the analysis of the diffusion effects alone onto Oxygen,
ISAM has already been found powerful to test some theories of the FIP effect.
Therefore I have laid in chapter 7 the baseline for subsequent applications of
ISAM to study the FIP effect in coronal loops at greater depth, by preparing
for the inclusion of wave-particle interactions for instance.

## Chapter 9 Conclusion et perspectives futures

En plus des percées majeures réalisées récemment grâce aux mesures in situ du
vent solaire naissant effectuées par la sonde Parker Solar Probe (PSP), nous
avons montré dans le chapitre 4 que l’imageur en lumière blanche WISPR à bord
de PSP a considérablement enrichi nos connaissances sur la structure et la
formation du vent solaire lent (SSW). En observant le SSW depuis un point
d’observation plus proche de la source, WISPR a dépeint un SSW d’une
complexité inattendue, avec des détails à petites échelles qui sont souvent
absents dans les observations realisées à une distance de 1 UA en raison des
effets d’intégration le long de la ligne de visée. Alors que WISPR a permis de
scruter le vent solaire lent à des échelles allant jusqu’à la fine couche de
plasma héliosphérique (HPS) (i.e. $\lesssim 5^{\circ}$), WISPR a surtout
dépeint un vent solaire lent qui émerge de plusieurs sources distribuées plus
loin de la HPS. Par conséquent, WISPR corrobore une image plus générale où les
régions sources du vent solaire lent sont diverses et sont probablement
contrôlées par la topologie du champ magnétique plus bas dans la couronne. En
particulier, cela a été formulé dans la théorie quasi-stationnaire (section
2.4.1) qui montre que les vents denses et lents devraient se former le long
des champs magnétiques qui subissent une forte expansion depuis la surface
jusqu’à la couronne. Cet effet a été davantage quantifié dans le chapitre 4 en
exploitant des simulations MHD MULTI-VP à haute résolution, où le taux
d’expansion du champ magnétique coronal contrôle l’énergie qui alimente le
vent solaire. Pour un magnétogramme donné du champ magnétique de surface qui
est suffisamment résolu, ce cadre de modélisation a permis de générer une
myriade d’écoulements qui, à leur tour, ont produit une structure fine du vent
solaire lent comme le révèlent en partie les dernières observations de WISPR.
Toutefois, cela n’est probablement pas la seule source de striation fine
observée par WISPR dernièrement, où la contribution du champ magnetique reste
encore à être évaluée précisement.

En réduisant considérablement l’écart entre la source et les points de
mesures, la mission PSP a déjà fourni beaucoup de données précieuses qui
incitent à repenser les théories actuelles de formation du vent solaire lent.
Les régions où les facteurs d’expansion sont élevés et favorables à la
formation d’un vent solaire lent quasi-stationnaire sont également des
environnements propices à la reconnexion magnétique, ces régions étant plus
généralement appelées les couches (quasi-)séparatrices ou la S-web. La grande
affluence d’événements de reconnexion magnétique dans la couronne a longtemps
été soupçonnée d’être une source majeure de la variabilité observée du vent
solaire lent. Dans le chapitre 5, nous avons considéré des théories dynamiques
qui peuvent expliquer une partie de la variabilité du vent lent observé à
distance et in situ à PSP. Pour aller plus loin que les premiers "blobs"
observés par SoHO il y a 25 ans, nous avons suggéré une image plus générale de
"blobs" qui se forment en même temps que des structures magnétiques torsadées.
La présence de telles structures a été suspectée par des mesures in situ
prises à STEREO et plus récemment à PSP et SolO, mais PSP-WISPR a permis pour
la première fois d’observer à distance ces structures magnétiques torsadées de
plus près comme le montre la section 5.2. Les simulations avancées de
WindPredict-AW ont été exploitées dans la section 5.3 pour étudier la
génération de ces champs torsadés à partir de la reconnexion magnétique à
l’extrémité des streamers. Des structures de densité à périodicités multiples
ont alors été produites avec des échelles de temps cohérentes avec les
observations. En simulant leur aspect dans les images de WISPR, j’ai trouvé
une grande similarité avec des structures torsadées réelles qui ont été
observées par WISPR. Cependant, les observations de WISPR ont révélé la
présence de fluctuations de densité à plus petites échelles, qui ont également
été mises en évidence lors d’une récente campagne STEREO-COR2 en champ
profond. L’origine de ces structures reste largement inconnue mais nous avons
montré dans la section 5.4 qu’elles peuvent être en partie générées par du
chauffage impulsif à la base de la couronne. Ceci constitue une image élégante
de la formation du vent solaire lent où les théories dynamiques fournissent
l’énergie nécessaire à ce chauffage impulsif, à travers la reconnexion
magnétique qui est connue pour se produire abondamment dans la basse couronne,
genérant par ailleurs des fluctuations de densité qui se propagent ensuite
dans le vent solaire.

Alors que WISPR a capturé de nombreuses structures transitoires, des vents
solaires lents avec une faible variabilité ont également été observés. Cela a
été noté dans une analyse approfondie des mesures in situ prises à PSP
combinée à une évaluation précise de la connectivité que j’ai fournie pour
cette étude (section 5.1). Des vents lents avec des propriétés globales
différentes ont été identifiés, ce qui suggère non pas une variabilité
_temporelle_ mais _spatiale_. Leur différence semble être propre à la
topologie magnétique de leur région source et corrobore donc l’image dressée
dans le chapitre 4 d’un vent solaire lent qui est texturé par la structure
complexe du champ magnétique coronal ouvert.

Bien que les nouvelles observations réalisées par PSP nous aient permis de
déterminer plus précisément les sources du vent solaire lent et les mécanismes
de formation associés, les études ultérieures devraient bénéficier de la
grande quantité de données collectées par PSP et maintenant Solar Orbiter
(SolO) pour étendre les observations que nous avons faites au cas par cas.
Pour tirer le meilleur parti de ces nouvelles mesures, nous avons mis en
évidence la nécessité de développer de nouvelles méthodes qui utilisent des
techniques d’imagerie synthétique couplées à des modèles MHD avancés de
l’atmosphère solaire, mais aussi de nouveaux outils qui permettent une
évaluation plus systématique des performances des modèles. Cependant, les
modèles employés dans cette thèse rencontrent encore des difficultés
importantes pour expliquer les observations les plus récentes de PSP dans
toute leur complexité. Nous avons particulièrement noté dans le chapitre 4 que
WISPR fournit maintenant des tests plus rigoureux aux modèles de vent solaire
actuels.

Comme présenté dans les sections 2.2.2.2 et 2.2.2.3, une variabilité du vent
lent est également observée en terme de sa composition en ions lourds. Comme
il a été observé que les boucles coronales ont des signatures de composition
similaires à celles du vent solaire lent mesurées in situ, nous avons montré
dans le chapitre 7 qu’une grande variabilité de la distribution des ions
lourds le long des boucles coronales est également à prévoir. J’ai contribué
au développement d’un modèle multi-espèces de l’atmosphère solaire appelé
ISAM, qui résout le transport de l’Hydrogène et des espèces mineures d’une
manière auto-consistante depuis la chromosphère jusqu’à la couronne. ISAM
comprend également un traitement complet des interactions entre espèces qui
nous a permis d’étudier en détail le rôle des effets collisionnels sur l’effet
FIP, où j’ai notamment montré que les protons jouent un rôle primordial dans
la séparation de l’Oxygène par exemple. Le fort couplage collisionnel de
l’Oxygène avec l’Hydrogène par effets de friction et de diffusion thermique
empêche la stratification complète de l’Oxygène, qui serait autrement beaucoup
plus marquée avec un fort appauvrissement de l’Oxygène étant 16 fois plus
lourd que l’Hydrogène. Les interactions Coulombiennes avec les protons ont
également tendance à favoriser un effet FIP inverse à travers la région de
transition où seuls les éléments à FIP moyen ou élevé auraient accès à la
couronne, en étant invisibles à la "barrière" formée par les protons qui
descendent dans la région de transition. Cet effet est susceptible d’être
régulé dans les boucles dynamiques qui subissent des phases successives
d’évaporation et de refroidissement, et d’être amplifié lors d’événements de
reconnexion magnétique qui déposent une grande quantité d’énergie à la base de
la couronne. Par ailleurs, cet effet FIP inverse "local" pourrait être
fortement compensé plus haut dans la couronne comme le suggère des tests
préliminaires qui incluent le Magnesium (faible FIP) et l’Helium (haut FIP).
Bien que notre travail ne se soit concentré que sur l’analyse des seuls effets
de diffusion sur l’Oxygène, ISAM s’est déjà révélé puissant pour tester
certaines théories de l’effet FIP. C’est pourquoi le chapitre 7 servira de
base pour des applications ultérieures d’ISAM qui étudieront plus en
profondeur l’effet FIP dans les boucles coronales, en préparant notamment
l’intégration des interactions ondes-particules.

##  Appendix A Technical specificities of ISAM

### A.1 Scaling of the equations

All physical moments of the VDF that are solved in ISAM are scaled as follows:

$\displaystyle\bar{n}_{s}$ $\displaystyle=\frac{n_{s}}{n_{0}}$ (A.1a)
$\displaystyle\bar{u}_{s}$ $\displaystyle=\frac{u_{s}}{c_{s,0}}$ (A.1b)
$\displaystyle\bar{T}_{s}^{\parallel}$
$\displaystyle=\frac{T_{s}^{\parallel}}{T_{0}}$ (A.1c)
$\displaystyle\bar{T}_{s}^{\perp}$ $\displaystyle=\frac{T_{s}^{\perp}}{T_{0}}$
(A.1d) $\displaystyle\bar{\gamma}_{s}^{\parallel}$
$\displaystyle=\frac{q_{s}^{\parallel}}{p_{s}^{\parallel}}\sqrt{\frac{m_{s}}{k_{b}T_{s}^{\parallel}}}$
(A.1e) $\displaystyle\bar{\gamma}_{s}^{\perp}$
$\displaystyle=\frac{q_{s}^{\perp}}{p_{s}^{\perp}}\sqrt{\frac{m_{s}}{k_{b}T_{s}^{\parallel}}}$
(A.1f)

where here $c_{s,0}=\sqrt{k_{b}T_{0}/m_{s}}$ and
$p_{s}^{\parallel,\perp}=n_{s}k_{b}T_{s}^{\parallel,\perp}$. In addition to
the temperature and density, we consider the solar gravitational acceleration
constant as a reference. These scaling parameters equal:

$\displaystyle n_{0}$ $\displaystyle=1\times 10^{20}\ m^{-3}$ (A.2a)
$\displaystyle T_{0}$ $\displaystyle=1\times 10^{5}\ K$ (A.2b) $\displaystyle
G_{0}$ $\displaystyle=GM_{\odot}/R_{\odot}^{2}\ m.s^{-2}\simeq 273.9638\
m.s^{-2}$ (A.2c)

From this basic set we define the spatial and temporal characteristic scales
as:

$\displaystyle t_{0}$ $\displaystyle=c_{0,0}/G_{0}$ (A.3a) $\displaystyle
r_{0}$ $\displaystyle=c_{0,0}t_{0}$ (A.3b)

where $c_{0,0}=\sqrt{k_{b}T_{0}/m_{0}}$ is the thermal speed of the specie of
reference. The specie of reference is chosen as the ion with the lowest mass
which is therefore the proton with mass $m_{0}=m_{p}\simeq 1,007uma\simeq
1.6726\times 10^{-27}\ kg$. More generally, all atomic masses are normalized
in the dalton or unified atomic mass unit $\mu_{s}=m_{s}/uma$ where $uma\simeq
1.6605\times^{-27}\ kg$.

Since the Alfvén-wave energy density $\mathcal{E}=\epsilon/\rho$ scales as the
square of the velocity, its associated normalized quantity is defined as:

$\bar{\mathcal{E}}=\mathcal{E}/c_{p,0}^{2}$ (A.4)

where for recall $c_{p,0}=\sqrt{k_{b}T_{0}/m_{p}}$ is the reference thermal
speed for protons.

### A.2 Mass ratios for collision terms

#### A.2.1 Neutral-neutral collisions

$\displaystyle z_{st}$ $\displaystyle=-\frac{1}{5}$ (A.5a) $\displaystyle
A^{*0p}_{st}$ $\displaystyle=3\left(1-z_{st}\frac{m_{t}}{m_{s}+m_{t}}\right)$
(A.5b) $\displaystyle A^{*0t}_{st}$
$\displaystyle=\left(1-z_{st}\frac{m_{t}}{m_{s}+m_{t}}\right)$ (A.5c)
$\displaystyle D^{*1pp}_{st}$
$\displaystyle=-\frac{3}{5}\left(5\left(\frac{m_{s}}{m_{s}+m_{t}}\right)^{2}+2\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}+4\frac{m_{s}}{m_{s}+m_{t}}\frac{m_{t}}{m_{s}+m_{t}}\right)$
(A.5d) $\displaystyle D^{*1pt}_{st}$
$\displaystyle=\frac{3}{5}\left(4\frac{m_{s}}{m_{s}+m_{t}}+\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.5e) $\displaystyle D^{*1tp}_{st}$
$\displaystyle=\frac{1}{10}\left(4\frac{m_{s}}{m_{s}+m_{t}}+\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.5f) $\displaystyle D^{*1tt}_{st}$
$\displaystyle=-\frac{1}{10}\left(30\left(\frac{m_{s}}{m_{s}+m_{t}}\right)^{2}+13\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}+28\frac{m_{s}}{m_{s}+m_{t}}\frac{m_{t}}{m_{s}+m_{t}}\right)$
(A.5g) $\displaystyle D^{*4pp}_{st}$
$\displaystyle=\frac{9}{5}\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}$ (A.5h)
$\displaystyle D^{*4pt}_{st}$
$\displaystyle=\frac{9}{5}\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}$ (A.5i)
$\displaystyle D^{*4tp}_{st}$
$\displaystyle=\frac{3}{10}\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}$ (A.5j)
$\displaystyle D^{*4tt}_{st}$
$\displaystyle=\frac{3}{2}\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}$ (A.5k)

#### A.2.2 Ion-neutral (Maxwell) collisions

$s$ and $t$ stand for ion and neutral respectively. The reverse interaction is
obtained by flipping the two indices.

$\displaystyle z_{st}$ $\displaystyle=0$ (A.6a) $\displaystyle A^{*0p}_{st}$
$\displaystyle=3\left(1-z_{st}\frac{m_{t}}{m_{s}+m_{t}}\right)$ (A.6b)
$\displaystyle A^{*0t}_{st}$
$\displaystyle=\left(1-z_{st}\frac{m_{t}}{m_{s}+m_{t}}\right)$ (A.6c)
$\displaystyle D^{*1pp}_{st}$
$\displaystyle=-\frac{3}{5}\left(5\left(\frac{m_{s}}{m_{s}+m_{t}}\right)^{2}+2\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}+4\frac{m_{s}}{m_{s}+m_{t}}\frac{m_{t}}{m_{s}+m_{t}}\right)$
(A.6d) $\displaystyle D^{*1pt}_{st}$
$\displaystyle=\frac{3}{5}\left(4\frac{m_{s}}{m_{s}+m_{t}}+\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.6e) $\displaystyle D^{*1tp}_{st}$
$\displaystyle=\frac{1}{10}\left(4\frac{m_{s}}{m_{s}+m_{t}}+\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.6f) $\displaystyle D^{*1tt}_{st}$
$\displaystyle=-\frac{1}{10}\left(30\left(\frac{m_{s}}{m_{s}+m_{t}}\right)^{2}+13\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}+28\frac{m_{s}}{m_{s}+m_{t}}\frac{m_{t}}{m_{s}+m_{t}}\right)$
(A.6g) $\displaystyle D^{*4pp}_{st}$
$\displaystyle=\frac{9}{5}\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}$ (A.6h)
$\displaystyle D^{*4pt}_{st}$
$\displaystyle=\frac{9}{5}\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}$ (A.6i)
$\displaystyle D^{*4tp}_{st}$
$\displaystyle=\frac{3}{10}\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}$ (A.6j)
$\displaystyle D^{*4tt}_{st}$
$\displaystyle=\frac{3}{2}\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}$ (A.6k)

#### A.2.3 Coulomb collisions

$\displaystyle z_{st}$ $\displaystyle=\frac{3}{5}$ (A.7a) $\displaystyle
A^{*0p}_{st}$ $\displaystyle=3\left(1-z_{st}\frac{m_{t}}{m_{s}+m_{t}}\right)$
(A.7b) $\displaystyle A^{*0t}_{st}$
$\displaystyle=\left(1-z_{st}\frac{m_{t}}{m_{s}+m_{t}}\right)$ (A.7c)
$\displaystyle D^{*1pp}_{st}$
$\displaystyle=-\frac{3}{70}\left(70\left(\frac{m_{s}}{m_{s}+m_{t}}\right)^{2}+2\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}+35\frac{m_{s}}{m_{s}+m_{t}}\frac{m_{t}}{m_{s}+m_{t}}\right)$
(A.7d) $\displaystyle D^{*1pt}_{st}$
$\displaystyle=\frac{3}{35}\left(49\frac{m_{s}}{m_{s}+m_{t}}+10\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.7e) $\displaystyle D^{*1tp}_{st}$
$\displaystyle=\frac{1}{70}\left(49\frac{m_{s}}{m_{s}+m_{t}}+10\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.7f) $\displaystyle D^{*1tt}_{st}$
$\displaystyle=-\frac{1}{35}\left(105\left(\frac{m_{s}}{m_{s}+m_{t}}\right)^{2}+8\left(\frac{m_{t}}{m_{s}+m_{t}}\right)^{2}+77\frac{m_{s}}{m_{s}+m_{t}}\frac{m_{t}}{m_{s}+m_{t}}\right)$
(A.7g) $\displaystyle D^{*4pp}_{st}$
$\displaystyle=\frac{3}{70}\left(-21\frac{m_{s}}{m_{s}+m_{t}}+16\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.7h) $\displaystyle D^{*4pt}_{st}$
$\displaystyle=\frac{9}{35}\left(-7\frac{m_{s}}{m_{s}+m_{t}}+6\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.7i) $\displaystyle D^{*4tp}_{st}$
$\displaystyle=\frac{3}{70}\left(-7\frac{m_{s}}{m_{s}+m_{t}}+6\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.7j) $\displaystyle D^{*4tt}_{st}$
$\displaystyle=\frac{3}{35}\left(-7\frac{m_{s}}{m_{s}+m_{t}}+5\frac{m_{t}}{m_{s}+m_{t}}\right)\frac{m_{t}}{m_{s}+m_{t}}$
(A.7k)

### A.3 Implementation of the improved radiative cooling recipe in ISAM

#### A.3.1 Upper chromosphere: the Carlsson et al. (2012)’s recipe

The optically thin radiative loss profiles $L_{X_{m}}(T)$ have been directly
extracted from figures 2, 4 and 5 in Carlsson & Leenaarts (2012). For CaII and
MgII and temperatures below $6000\ K$, I replaced the $L_{X_{m}}(T)$ values by
those plotted in figures 12 and 13 in Carlsson & Leenaarts (2012) that account
for radiative heating in the same transition lines. The empirical escape
probability $E_{X_{m}}(\tau)$ that represents the medium opacity to radiation
is extracted from figures 6,7 and 8 in Carlsson & Leenaarts (2012). I also
fetched their ionization fractions $n_{X_{m}}/n_{X}$ from figures 9, 10 and
11. For the $A_{X}$ quantities I use the coronal abundances of element $X$
relative to Hydrogen taken from Schmelz et al. (2012). These recipes have been
tabulated for HI, CaII and MgII and for chromospheric temperatures between
$~{}3000\ K$ and $30000\ K$. The remaining input parameter to the recipes is
the optical depth parameter $\tau$ which is calculated by this formula:

$\tau=\begin{dcases*}-4\times 10^{-14}\int_{z=+\infty}^{z=0}n_{HI}dz&for HI\\\
-\int_{z=+\infty}^{z=0}\rho dz\quad(g.cm^{-2})&for CaII and MgII\end{dcases*}$
(A.8)

where we take $\rho$ as the total mass density.

#### A.3.2 Transition region and corona: ChiantiPy

The RadLoss module of ChiantiPy is used to compute the radiative losses for a
given transition line of an atom $X$, electronic density and temperature
array. The user can also ask for integrated radiative losses for all
transitions of an atom or directly the summed up contributions from all lines
of all elements including H, He, C, O, Ne, N, Mg, Si, S, Fe, Na, Al, Ar, Ca
and Ni (ordered by decreasing relative abundance).

As a first approach I ran ChiantiPy to get the total radiative losses for all
the above mentioned elements which can take up to several tens of minutes to
compute all transition lines. As in equation 6.61 this implies to provide also
ChiantiPy with ionization fractions and abundances.

I selected the default ionization equilibria provided in the Chianti database
(Dere et al., 2009; Dere, 2007) which agrees very well with the one given by
Mazzotta et al. (1998a). This is not surprising as some of the ionization and
recombination rates collected in the Chianti database have been fetched from
Mazzotta et al. (1998a). We further notice that most of these rates are also
being used in ISAM (see section 6.4).

For the relative abundances I use the coronal abundances from Schmelz et al.
(2012) that are provided in the file "sun_coronal_2012_schmelz_ext.abund" in
the Chianti database.

### A.4 Numerical treatment of collisions and heating/cooling sources

The collision terms are purely local in the sense that they do not involve
coupling between adjacent altitude points. As a consequence the problem can be
formatted as a matrix system:

$\displaystyle\frac{Z^{n}-Z^{0}}{\Delta t}$
$\displaystyle=\frac{\tilde{Z}-Z^{0}}{\Delta t}+AZ^{n}+B$ (A.9)
$\displaystyle\text{where }Z$ $\displaystyle=\begin{vmatrix}n_{1}\\\ u_{1}\\\
T_{1}^{\parallel}\\\ T_{1}^{\perp}\\\ q_{1}^{\parallel}\\\ q_{1}^{\perp}\\\
\vdots\end{vmatrix}$

where $Z$ is the solution vector of size $6\times k$ for $k$ species resolved.
In the above formula the superscript 0, $\tilde{}$ and n stands for the
initial solution, the solution calculated by LCPFCT, and the final solution
respectively. The square matrix $A$ contains all collision terms where the VDF
moments from the solution vector are introduced. Because electrons are solved
separately than neutrals and ions, the moments for electrons are not included
in the solution vector $Z$. Their contribution to the collision terms is then
accounted for in vector $B$.

### A.5 Numerical resolution of the Alfvén-wave propagation and dissipation

The method is very similar with the one already described in Appendix A.4:

$\frac{\mathcal{E}^{n}-\mathcal{E}^{0}}{\Delta
t}=\underbrace{\frac{\tilde{\mathcal{E}}-\mathcal{E}^{0}}{\Delta
t}}_{C}-\frac{Q_{h}}{\rho}\Biggr{|}^{n}$ (A.10)

where $\mathcal{E}=\epsilon/\rho$ is the actual quantity that we solve in
LCPFCT.

The dissipation term given in equation 6.51 can be approximated as:

$\frac{Q_{h}^{\pm}}{\rho}\Biggr{|}^{n}\simeq\frac{\sqrt{\mathcal{E}^{\pm
0}}}{\mathcal{L}}\mathcal{E}^{\pm n}$ (A.11)

Similarly for the dissipation term given in equation 6.51 and that includes
some form of Alfvén-wave reflection:

$\frac{Q_{h}}{\rho}\Biggr{|}^{n}\simeq
c_{d}\frac{u_{p}^{0}+v_{A}^{0}}{v_{A}^{0}}\left|\frac{\partial}{\partial
s}v_{A}\right|^{0}\mathcal{E}^{n}$ (A.12)

In both cases the dissipation term can be formulated as follows:

$\frac{Q_{h}}{\rho}\Biggr{|}^{n}=B\mathcal{E}^{n}$ (A.13)

where we dropped the $\pm$ subscripts for simplicity.

Therefore the linear superposition stage is equivalent at solving a first
order partial differential equation:

$\frac{\partial\mathcal{E}}{\partial t}\Biggr{|}^{n}+B\mathcal{E}^{n}=C$
(A.14)

with solution:

$\mathcal{E}^{n}=\exp(B\Delta t)+\frac{C}{B}$ (A.15)

### A.6 Hyperbolocity criteria

For a given grid profile, the numerical time step is systematically adjusted
to meet the following conditions for hyperbolicity, below for the transport of
parallel energy:

$\left(|u_{s}|+\left|\frac{3}{2}\frac{q_{s}^{\parallel}}{p_{s}^{\parallel}}\right|\right)\frac{\Delta
t}{\Delta r}\lesssim 1$ (A.16)

or similarly for the transport of the perpendicular energy:

$\left(|u_{s}|+\left|\frac{q_{s}^{\perp}}{p_{s}^{\perp}}\right|\right)\frac{\Delta
t}{\Delta r}\lesssim 1$ (A.17)

##  Appendix B List of publications in high impact journals

### B.1 Main publications

Poirier N., Rouillard A.P., Kouloumvakos A., Przybylak A., Fargette N., Pobeda
R., Réville V., Pinto R.F., Indurain M., Alexandre M., 2021, Exploiting white-
light observations to improve estimates of magnetic connectivity, FrASS,
Volume 8, id.84, ADS LINK

Poirier N., Kouloumvakos A., Rouillard A.P., Pinto R.F., Vourlidas A.,
Stenborg G., Valette E., Howard R.A., Hess P., Thernisien A., Rich N., Griton
L., Indurain M., Raouafi N.-E., Lavarra M., Réville V., 2020, Detailed Imaging
of Coronal Rays with the Parker Solar Probe, ApJS, Volume 246, Issue 2, id.60,
ADS LINK

Rouillard A.P., Poirier N., Lavarra M., Bourdelle A., Dalmasse K.,
Kouloumvakos A., Vourlidas A., Kunkel V., Hess P., Howard R.A., Stenborg G.,
Raouafi N.-E., 2020, Modeling the Early Evolution of a Slow Coronal Mass
Ejection Imaged by the Parker Solar Probe, ApJS, Volume 246, Issue 2, id.72,
ADS LINK

Badman S.T., Brooks D.H., Poirier N., Warren H.P., Petrie G., Rouillard A.P.,
Nick Arge C., Bale S.D., de Pablos Agüero D., Harra L., Jones S.I.,
Kouloumvakos A., Riley P., Panasenco O., Velli M., Wallace S., 2022,
Constraining Global Coronal Models with Multiple Independent Observables, ApJ,
Volume 932, Issue 2, id.135, ADS LINK

Griton L., Pinto R.F., Poirier N., Kouloumvakos A., Lavarra M., Rouillard
A.P., 2020, Coronal Bright Points as Possible Sources of Density Variations in
the Solar Corona, ApJ, Volume 893, Issue 1, id.64, ADS LINK

Griton L., Rouillard A.P., Poirier N., Issautier K., Moncuquet M., Pinto R.F.,
2021, Source-dependent Properties of Two Slow Solar Wind States, ApJ, Volume
910, Issue 1, id.63, ADS LINK

Pinto R.F., Poirier N., Rouillard A.P., Kouloumvakos A., Griton L., Fargette
N., Kieokaew R., Lavraud B., Brun A.S., 2021, Solar wind rotation rate and
shear at coronal hole boundaries. Possible consequences for magnetic field
inversions, A&A, Volume 653, id.A92, ADS LINK

Lavarra M, Rouillard A.P., Poirier N., Blelly P.-L., Pinto R.F., Réville V.,
2022, Testing a multi-specie model of the fast and slow solar wind, in review

### B.2 Other publications

Howard R.A., Vourlidas A., Bothmer V., Colaninno R.C., DeForest C.E.,
Gallagher B., Hall J.R., Hess P., Higginson A.K., Korendyke C.M., Kouloumvakos
A., Lamy P.L., Liewer P.C., Linker J., Linton M., Penteado P., Plunkett S.P.,
Poirier N., Raouafi N.E., Rich N., Rochus P., Rouillard A.P., Socker D.G.,
Stenborg G., Thernisien A.F., Viall N.M., 2019, Near-Sun observations of an
F-corona decrease and K-corona fine structure, Nature, Volume 576, Issue 7786,
p.232-236, ADS LINK

Korreck K.E., Szabo A., Nieves-Chinchilla T., Lavraud B., Luhmann J., Niembro
T., Higginson A.K., Alzate N., Wallace S., Paulson K., Rouillard A.P.,
Kouloumvakos A., Poirier N., Kasper J.C., Case A.W., Stevens M.L., Bale S.D.,
Pulupa M., Whittlesey P., Livi R., Goetz K., Larson D., Malaspina D.M., Morgan
H., Narock A.A., Schwadron N.A., Bonnell J., Harvey P., Wygant J., 2020,
Source and Propagation of a Streamer Blowout Coronal Mass Ejection Observed by
the Parker Solar Probe, ApJS, Volume 246, Issue 2, id.69, ADS LINK

Réville V., Fargette N., Rouillard A.P., Lavraud B., Velli M., Strugarek A.,
Parenti S., Brun A.S., Shi C., Kouloumvakos A., Poirier N., Pinto R.F., Louarn
P., Fedorov A., Owen C.J., Génot V., Horbury T.S., Laker R., O’Brien H.,
Angelini V., Fauchon-Jones E., Kasper J.C., 2022, Flux rope and dynamics of
the heliospheric current sheet. Study of the Parker Solar Probe and Solar
Orbiter conjunction of June 2020, A&A, Volume 659, id.A110, ADS LINK

Kouloumvakos A., Vourlidas A., Rouillard A.P., Roelof E.C., Leske R., Pinto
R.F., Poirier N., 2020, The Solar Origin of Particle Events Measured by Parker
Solar Probe, ApJ, Volume 899, Issue 2, id.107, ADS LINK

Réville V., Rouillard A.P., Velli M., Verdini A., Buchlin Éric, Lavarra M.,
Poirier N., 2021, Investigating the origin of the FIP effect with a shell
turbulence model, FrASS, Volume 8, id.2, ADS LINK

Rouillard A.P., Kouloumvakos A., Vourlidas A., Kasper J., Bale S., Raouafi
N.-E., Lavraud B., Howard R.A., Stenborg G., Stevens M., Poirier N., Davies
J.A., Hess P., Higginson A.K., Lavarra M., Viall N.M., Korreck K., Pinto R.F.,
Griton L., Réville V., Louarn P., Wu Y., Dalmasse K., Génot V., Case A.W.,
Whittlesey P., Larson D., Halekas J.S., Livi R., Goetz K., Harvey P.R.,
MacDowall R.J., Malaspina D., Pulupa M., Bonnell J., de Witt T.D., Penou E.,
2020, Relating Streamer Flows to Density and Magnetic Structures at the Parker
Solar Probe, ApJS, Volume 246, Issue 2, id.37, ADS LINK

Rouillard A.P., Pinto R.F., Vourlidas A., De Groof A., Thompson W.T., Bemporad
A., Dolei S., Indurain M., Buchlin E., Sasso C., Spadaro D., Dalmasse K.,
Hirzberger J., Zouganelis I., Strugarek A., Brun A.S., Alexandre M., Berghmans
D., Raouafi N.-E., Wiegelmann T., Pagano P., Arge C.N., Nieves-Chinchilla T.,
Lavarra M., Poirier N., Amari T., Aran A., Andretta V., Antonucci E.,
Anastasiadis A., Auchère F., Bellot Rubio L., Nicula B., Bonnin X., Bouchemit
M., Budnik E., Caminade S., Cecconi B., Carlyle J., Cernuda I., Davila J.M.,
Etesi L., Espinosa Lara F., Fedorov A., Fineschi S., Fludra A., Génot V.,
Georgoulis M.K., Gilbert H.R., Giunta A., Gomez-Herrero R., Guest S.,
Haberreiter M., Hassler D., Henney C.J., Howard R.A., Horbury T.S., Janvier
M., Jones S.I., Kozarev K., Kraaikamp E., Kouloumvakos A., Krucker S., Lagg
A., Linker J., Lavraud B., Louarn P., Maksimovic M., Maloney S., Mann G.,
Masson A., Müller D., Önel H., Osuna P., Orozco Suarez D., Owen C.J.,
Papaioannou A., Pérez-Suárez D., Rodriguez-Pacheco J., Parenti S., Pariat E.,
Peter H., Plunkett S., Pomoell J., Raines J.M., Riethmüller T.L., Rich N.,
Rodriguez L., Romoli M., Sanchez L., Solanki S.K., St Cyr O.C., Straus T.,
Susino R., Teriaca L., del Toro Iniesta J.C., Ventura R., Verbeeck C., Vilmer
N., Warmuth A., Walsh A.P., Watson C., Williams D., Wu Y., Zhukov A.N., 2020,
Models and data analysis tools for the Solar Orbiter mission, A&A, Volume 642,
id.A2, ADS LINK

Lavraud B., Fargette N., Réville V., Szabo A., Huang J., Rouillard A.P., Viall
N., Phan T.D., Kasper J.C., Bale S.D., Berthomier M., Bonnell J.W., Case A.W.,
Dudok de Wit T., Eastwood J.P., Génot V., Goetz K., Griton L.S., Halekas J.S.,
Harvey P., Kieokaew R., Klein K.G., Korreck K.E., Kouloumvakos A., Larson
D.E., Lavarra M., Livi R., Louarn P., MacDowall R.J., Maksimovic M., Malaspina
D., Nieves-Chinchilla T., Pinto R.F., Poirier N., Pulupa M., Raouafi N.-E.,
Stevens M.L., Toledo-Redondo S., Whittlesey P.L., 2020, The Heliospheric
Current Sheet and Plasma Sheet during Parker Solar Probe’s First Orbit, ApJL,
Volume 894, Issue 2, id.L19, ADS LINK

##  Appendix C List of communications

### C.1 Orals

Simulating the FIP effect in coronal loops: modeling with a high-order multi-
specie approach, July 16-24 2022, COSPAR 44th Scientific Assembly.

Simulating the FIP effect in coronal loops: modeling with a high-order multi-
specie approach, June 28 - July 1 2022, The 10th Coronal Loops Workshop.

Interface entre la chromosphère et la couronne solaire: modélisation avec une
approche 16-moments multi-espèces, May 16-20 2022, Colloque du PNST.

Exploiting white-light observations to improve estimates of magnetic
connectivity, May 27 2021, SolO ISWG: SW sources and connection science.

Exploiting white-light observations to improve estimates of magnetic
connectivity, April 2 2021, LESIA science seminar.

Using modeling tools to interpret the coronal rays observed by WISPR from E1
to E6, April 14 2021, WISPR Consortium meeting - virtual.

The CONNECT-TOOL: update on the white-light based selection of the best
coronal models, October 26-27 2020, MADAWG meeting.

Using white-light observations to constrain coronal models: application to E6,
October 15 2020, PSP SWG meeting.

The CONNECT-TOOL: current state/limitations & next developments, July 22 2020,
MADAWG meeting.

Parker Solar Probe - First Results (co-presenting with Naïs Fargette),
February 2020, Journées scientifiques et techniques de l’IRAP.

Techniques to exploit remote-sensing observations from a rapidly moving
spacecraft: application to Parker Solar Probe and Solar Orbiter, January 2020,
ISSI meeting: Exploring the Solar Wind In Regions Closer Than Ever Observed
Before.

The Forming Slow Solar Wind Imaged along Streamer Rays by the Wide-Angle
Imager on Parker Solar Probe, December 9-13 2019, AGU Fall Meeting, ADS LINK.

Interpreting the first WISPR observations, June 19-20 2019, First WISPR
science meeting.

### C.2 Hybrid

Simulating the FIP effect in coronal loops using a multi-species kinetic-fluid
model, May 23-27 2022, EGU General Assembly Conference.

Simulating the FIP effect in coronal loops using a multi-species kinetic-fluid
model, September 9 2021, ESPM-16 virtual edition.

Simulating the FIP effect in coronal loops using a multi-species kinetic-fluid
model, April 19-30 2021, EGU General Assembly Conference - Online edition, ADS
LINK.

The forming slow solar wind imaged along streamer rays by the wide-angle
imager on Parker Solar Probe, May 4-8 2020, EGU General Assembly Conference -
Online edition, ADS LINK.

### C.3 Posters

The many faces of Nicolas’s PhD thesis, May 2021, IRAP PhD day.

Simulating the FIP Effect in Coronal Loops Using a Multi-Species Kinetic-Fluid
Model, December 1-17 2020, AGU Fall Meeting - Online edition, ADS LINK.

Evolution of the internal structure of CME flux ropes from the Sun to 1AU,
April 7-12 2019, EGU General Assembly Conference, ADS LINK.

###### Table of contents (full)

  1. Acknowledgements
  2. Remerciements
  3. Abstract
  4. Résumé
  5. 1 Introduction
    1. 1.1 General considerations on the enigmatic origin of the slow wind
    2. 1.2 Observations of the slow wind and its candidate sources
      1. 1.2.1 Structure of the solar atmosphere
        1. 1.2.1.1 The heliospheric plasma and current sheets
        2. 1.2.1.2 The quiet and active Sun
        3. 1.2.1.3 Streamers and pseudo-streamers
        4. 1.2.1.4 White-light synoptic maps of the streamer belt
        5. 1.2.1.5 Spectroscopy of the low solar atmosphere
      2. 1.2.2 Properties of the slow solar wind
        1. 1.2.2.1 The intermittent nature of the slow wind
        2. 1.2.2.2 Composition of the slow wind
        3. 1.2.2.3 Variability of the SSW
    3. 1.3 Basics of the low solar atmosphere
      1. 1.3.1 The chromosphere, transition region and corona
      2. 1.3.2 Heating of the solar atmosphere
      3. 1.3.3 The partially ionized chromosphere
      4. 1.3.4 Extraction of heavy ions from the chromosphere
    4. 1.4 Theories on the origin of the slow solar wind
      1. 1.4.1 The quasi-stationnary theory
      2. 1.4.2 Dynamic theories of the slow solar wind
        1. 1.4.2.1 Magnetic reconnection at the tip of streamers
        2. 1.4.2.2 Interchange reconnection at open-closed boundaries
        3. 1.4.2.3 Release of magnetic flux ropes from the tip of streamers
        4. 1.4.2.4 Open-closed boundaries as potential source regions of the slow wind
  6. 2 Introduction (français)
    1. 2.1 Considérations générales sur l’origine énigmatique du vent lent
    2. 2.2 Observation du vent lent et de ses régions sources potentielles
      1. 2.2.1 Structure de l’atmosphère solaire
        1. 2.2.1.1 La couche de courant et de plasma héliosphérique
        2. 2.2.1.2 Le Soleil calme et actif
        3. 2.2.1.3 Streamers et pseudo-streamers
        4. 2.2.1.4 Cartes synoptiques en lumière blanche des streamers
        5. 2.2.1.5 Spectroscopie de la basse atmosphère solaire
      2. 2.2.2 Propriétés du vent solaire lent
        1. 2.2.2.1 La nature intermittente du vent lent
        2. 2.2.2.2 Composition du vent lent
        3. 2.2.2.3 Variabilité du vent lent
    3. 2.3 Principes fondamentaux de la basse atmosphère solaire
      1. 2.3.1 La chromosphere, région de transition et couronne
      2. 2.3.2 Chauffage de l’atmosphère solaire
      3. 2.3.3 La chromosphère partiellement ionisée
      4. 2.3.4 Extraction des ions lourds depuis la chromosphère
    4. 2.4 Théories sur l’origine du vent solaire lent
      1. 2.4.1 La théorie quasi-stationnaire
      2. 2.4.2 Théories dynamiques du vent solaire lent
        1. 2.4.2.1 Reconnection magnétique à l’extrémité des streamers
        2. 2.4.2.2 Reconnection d’interchange aux interfaces ouvert-fermé
        3. 2.4.2.3 Libération de champs torsadés à l’extrémité des streamers
        4. 2.4.2.4 Les frontières ouvert-fermé comme régions sources potentielles du vent lent
  7. 3 Instrumentation and tools
    1. 3.1 Instrumentation
      1. 3.1.1 The SoHO-LASCO imaging suite
      2. 3.1.2 The STEREO-SECCHI imaging suite
      3. 3.1.3 The PSP-WISPR white-light telescopes
      4. 3.1.4 Synoptic photospheric magnetic maps
        1. 3.1.4.1 The Wilcox Solar Observatory
        2. 3.1.4.2 The Global Oscillation Network Group
        3. 3.1.4.3 The Air force Data Assimilative Flux Transport model
    2. 3.2 Supporting tools
      1. 3.2.1 The Magnetic Connectivity Tool
      2. 3.2.2 The propagation tool
      3. 3.2.3 AMDA and CLWeb
      4. 3.2.4 SolarSoft and Astropy
    3. 3.3 External models
      1. 3.3.1 The Potential Field Source Surface (PFSS) model
        1. 3.3.1.1 Solving the magnetostatic equation
        2. 3.3.1.2 Reconstruction of the coronal magnetic field
        3. 3.3.1.3 Limitations of PFSS models
      2. 3.3.2 The WindPredict-AW model
      3. 3.3.3 The Multiple Flux tube Solar Wind Model (MULTI-VP)
    4. 3.4 Constraining coronal and heliospheric models of the solar wind
      1. 3.4.1 Exploiting White-Light Observations to Improve Estimates of Magnetic Connectivity (Poirier et al., 2021)
      2. 3.4.2 Constraining Global Coronal Models with Multiple Independent Observables (Badman et al., 2022)
  8. 4 The fine structure of streamer rays imaged by PSP-WISPR
    1. 4.1 The Thomson scattering theory
    2. 4.2 Producing WISPR synthetic images
    3. 4.3 Analysis of the first observations of WISPR with the MULTI-VP model (Poirier et al. 2020)
      1. 4.3.1 Comparison with 1 AU observatories
      2. 4.3.2 WL synoptic map of the first WISPR-I observations
      3. 4.3.3 Interpreting the large-scale structures
      4. 4.3.4 Numerical setup
      5. 4.3.5 WISPR-I synthetic images
      6. 4.3.6 Interpreting the small-scale structures
      7. 4.3.7 Conclusion
    4. 4.4 A quick tour of subsequent WISPR-I observations
    5. 4.5 Conclusion
  9. 5 Variability of the slow solar wind
    1. 5.1 Variability of the SSW as measured in situ at PSP
    2. 5.2 Variability of the SSW as observed by PSP-WISPR
    3. 5.3 Quasi-periodic density structures generated by magnetic reconnection at the HCS
      1. 5.3.1 Idealized 2.5-D setup: the tearing instability
      2. 5.3.2 Realistic 3-D setup: comparison with PSP-WISPR observations
    4. 5.4 Propagating density structures triggered by transient heating at the coronal base
    5. 5.5 Conclusion
  10. 6 The Irap Solar Atmosphere Model (ISAM)
    1. 6.1 Legacy
    2. 6.2 Origin: the Boltzmann equation
    3. 6.3 The 1-D transport equations
    4. 6.4 Partial ionization
      1. 6.4.1 Photoionization
      2. 6.4.2 Radiative recombination and dielectronic recombination
      3. 6.4.3 Collisional and auto-ionization
      4. 6.4.4 Charge exchange reactions
      5. 6.4.5 Limitations and future improvements
    5. 6.5 Collisions between species
      1. 6.5.1 General collision terms
      2. 6.5.2 Neutral-neutral collisions
      3. 6.5.3 Collisions between charged particles: Coulomb collisions
      4. 6.5.4 Ion-neutral collisions
    6. 6.6 Heating models
      1. 6.6.1 Ad-hoc heating prescription
      2. 6.6.2 Heating based on the dissipation of Alfvén waves following the WKB approximation
        1. 6.6.2.1 Alfvén-wave propagation in the WKB approach
        2. 6.6.2.2 Alfvén-wave dissipation
        3. 6.6.2.3 Plasma heating
      3. 6.6.3 The Shell-Atm turbulence model
      4. 6.6.4 Limitations and future improvements
    7. 6.7 Radiative cooling
      1. 6.7.1 Ad-hoc prescription
      2. 6.7.2 Improved recipe for radiative cooling/heating in the upper chromosphere
      3. 6.7.3 Improved optically thin radiative cooling from Chianti
      4. 6.7.4 Limitations and future improvements
    8. 6.8 Numerical method
      1. 6.8.1 A Flux-Corrected Transport algorithm to solve for conservative equations
      2. 6.8.2 Boundary conditions
      3. 6.8.3 Grid definition
  11. 7 Separation of heavy ions in coronal loops
    1. 7.1 The background Hydrogen, protons and electrons atmosphere
      1. 7.1.1 Thermodynamics of coronal loops
      2. 7.1.2 Including the upper chromosphere
      3. 7.1.3 Collisional coupling between Hydrogen and protons
    2. 7.2 Separation of heavy ions: application to Oxygen
    3. 7.3 Conclusion
  12. 8 Conclusion and future perspectives
  13. 9 Conclusion et perspectives futures
  14. A Technical specificities of ISAM
    1. A.1 Scaling of the equations
    2. A.2 Mass ratios for collision terms
      1. A.2.1 Neutral-neutral collisions
      2. A.2.2 Ion-neutral (Maxwell) collisions
      3. A.2.3 Coulomb collisions
    3. A.3 Implementation of the improved radiative cooling recipe in ISAM
      1. A.3.1 Upper chromosphere: the Carlsson et al. (2012)’s recipe
      2. A.3.2 Transition region and corona: ChiantiPy
    4. A.4 Numerical treatment of collisions and heating/cooling sources
    5. A.5 Numerical resolution of the Alfvén-wave propagation and dissipation
    6. A.6 Hyperbolocity criteria
  15. B List of publications in high impact journals
    1. B.1 Main publications
    2. B.2 Other publications
  16. C List of communications
    1. C.1 Orals
    2. C.2 Hybrid
    3. C.3 Posters
  17. D List of initials and abbreviations

##  Appendix D List of initials and abbreviations

  * •

ACE: Advanced Composition Explorer

  * •

ADAPT: Air force Data Assimilative Flux Transport

  * •

AIA: Atmospheric Imaging Assembly

  * •

AU: astronomical unit

  * •

CBP: coronal bright point

  * •

CDPP: Centre de Données de la Physique des Plasmas

  * •

CE: charge-exchange

  * •

CIR: corotating interaction regions

  * •

CME: coronal mass ejection

  * •

DI: direct ionization

  * •

DR: dielectronic recombination

  * •

EA: excitation-autoionization

  * •

EIS: EUV imaging spectrometer

  * •

EIT: Extreme ultraviolet Imager Telescope

  * •

ESA: European Space Agency

  * •

EUV: extreme ultraviolet

  * •

FIP: first ionisation potential

  * •

FITS: Flexible Image Transport System

  * •

FOV: field-of-view

  * •

FSW: fast solar wind

  * •

GONG: Global Oscillation Network Group

  * •

HCS: heliospheric current sheet

  * •

HPS: heliospheric plasma sheet

  * •

IDL: Interactive Data Language

  * •

IMP: Interplanetary Monitoring Platform

  * •

IRAP: Institut de Recherche en Astrophysique et Planétologie

  * •

ISAM: Irap Solar Atmospheric Model

  * •

ISSI: International Space Science Institute

  * •

LASCO: Large-Angle and Spectrometric Coronagraph

  * •

LOS: line-of-sight

  * •

LTE: local thermal equilibrium

  * •

MADAWG: Modelling and Data Analysis Working Group

  * •

MAS: Magnetohydrodynamics Around a Sphere

  * •

MCT: Magnetic Connectivity Tool

  * •

MHD: magneto-hydro-dynamics

  * •

MULTI-VP: Multiple Flux tube Solar Wind Model

  * •

NASA: National Aeronautics and Space Administration

  * •

NSO: National Solar Observatory

  * •

PI: photoionization

  * •

PSP: Parker Solar Probe

  * •

RR: radiative recombination

  * •

SCS: Schatten Current Sheet

  * •

s/c: spacecraft

  * •

SDO: Solar Dynamics Observatory

  * •

SoHO: Solar and Heliospheric Observatory

  * •

SolO: Solar Orbiter

  * •

SSW: slow solar wind

  * •

STEREO: Solar-TErrestrial RElations Observatory

  * •

SWICS: Solar Wind Ion Composition Spectrometer

  * •

TNE: thermal non-equilibrium

  * •

TR: transition region

  * •

WCS: World Coordinate System

  * •

WISPR: Wide-Field Imager for Parker Solar Probe

  * •

WKB: Wentzel–Kramers–Brillouin

  * •

WL: white-light

  * •

WSA: Wang-Sheeley-Arge

  * •

WSO: Wilcox Solar Observatory

  * •

3-D: three-dimensional

  * •

2-D: two-dimensional

###### List of Figures

  1. 1.1  The solar wind speed as sampled by Ulysses during its first (left panel) and second (right panel) passage. The large-scale configuration of the solar corona is illustrated with composite images assembled from: solar disk images taken by the SoHO Extreme ultraviolet Imager Telescope at $195$$\mathrm{\SIUnitSymbolAngstrom}$, Mauna Loa K-coronameter images ($7000$-$9500$$\mathrm{\SIUnitSymbolAngstrom}$) of the inner corona, and WL images from the SoHO LASCO-C2 coronagraph. The bottom panel shows the evolution of the sunspot number that is a good indicator of the level of solar activity. Figure adapted from McComas et al. (2003, Figure 1).
  2. 1.2 Properties of the slow and fast solar wind. Figure taken from Cranmer et al. (2017, Table 1).
  3. 1.3 Processed photography of the total solar eclipse of 2019. Credits: Nicolas Lefaudeux - https://hdr-astrophotography.com.
  4. 1.4  Illustration of a highly structured corona during period of high solar activity. Top panel: combined $193$$\mathrm{\SIUnitSymbolAngstrom}$ and $211$$\mathrm{\SIUnitSymbolAngstrom}$ EUV image taken by SDO-AIA on 2013 May 29. Bottom panel: a potential field source surface (PFSS, see section 3.3.1) reconstruction of the coronal magnetic field for Carrington rotation 2137, where open field lines colored in blue and grey denote opposite polarities, and where the HCS is plotted as a red sheet. Figure extracted from Sanchez-Diaz et al. (2017b, Figure 1).
  5. 1.5 Comparison between the observed corona (left) captured during the 21 August 2017 total solar eclipse, and an associated prediction of the magnetic field structure (right). Adapted from Figure 1 in Mikić et al. (2018). Credit for image a: ©2017 Miloslav Druckmüller, Peter Aniol, Shadia Habbal.
  6. 1.6  Cartoon of the magnetic field structure of the solar corona. Figure taken from Pellegrin et al. (2021), with prior permission by the author.
  7. 1.7 Example of a narrow open-field corridor (green lines) forming along an extension of a polar coronal hole (shaded grey area). Contours of the magnetic field amplitude are color plotted at the photosphere. Figure taken from Antiochos et al. (2011, Figure 4).
  8. 1.8  Panel (a): a 3-D reconstruction of the coronal magnetic field on 2018 November 5 using a PFSS extrapolation method (see section 3.3.1). The magnetic map used as an input to the PFSS extrapolation is displayed in a grey scale at the solar surface. The open and closed magnetic field lines are depicted in yellow and orange, respectively. The polarity inversion line (i.e. the baseline of the HCS) is plotted as a red line at a height of $2.1\ R_{\odot}$. Panel (b): a zoomed-in view of a pseudo-streamer anchored in an isolated and small equatorial coronal hole (blue arrow). Extreme ultraviolet emissions in the $193$$\mathrm{\SIUnitSymbolAngstrom}$ wavelength are shown at the surface.
  9. 1.9 Latitude-longitude Carrington maps of the photospheric magnetic field (1st row), of the modeled HCS (2nd row) and white-light emissions (3rd row), and of the real white-light observations taken by SoHO LASCO-C2, over Carrington rotation CR1919 (left) and CR1935 (right) that are typical of a solar minimum (left) and maximum (right) coronal configuration. Figure taken from Wang et al. (2000, Figure 2).
  10. 1.10 2-D map of Silicium (Si) over Surfur (S) abundances ratio on the solar disk. Figure taken from Brooks et al. (2015, Figure 3).
  11. 1.11  Example of streamer blobs observed by SoHO-LASCO, using a running difference technique to enhance density fluctuations in the corona where bright (or dark) regions are indicative of an enhanced (or depleted) density respectively. Figure taken from Sheeley et al. (1997, Figure 1).
  12. 1.12  Brightness fluctuations in the upper solar corona detected during a high-cadence campaign of the STEREO-A COR2 coronagraph. Figure taken from DeForest et al. (2018, Figure 12).
  13. 1.13  Overview of dynamic structures in the slow solar wind from the large (top) to the small (bottom) spatial scales with their associated references mentioned in the text.
  14. 1.14 Variation of abundances for the main constituents measured in the solar wind as a function of the FIP. Abundances are relative to the abundance of Oxygen. Figure taken from Peter (1998, Figure 1).
  15. 1.15 Temporal variation of the low FIP Magnesium (Mg) abundance with respect to Oxygen (O), of the solar wind speed (here taken as the speed of doubly ionized Helium), and of the Oxygen temperature. Figure taken from Geiss et al. (1995, Figure 7).
  16. 1.16 In situ measurements from ACE of the Oxygen $O^{7+}/O^{6+}$ and Carbon $C^{6+}/C^{5+}$ charge state ratios function of the solar wind speed (blue dots). Data corresponding to fast, slow Alfvénic and slow non-Alfvénic winds as identified by D’Amicis & Bruno (2015) are indicated by green, red and black dots respectively. Simulations from the Irap Solar Atmosphere Model (ISAM) for these three solar wind regimes are shown as red crosses along with their associated uncertainties in black (see Lavarra et al., 2022, for further details). Figure taken from Lavarra et al. (2022, Figure 1).
  17. 1.17 Variation of the Oxygen $O^{7+}/O^{6+}$ charge state ratio and of the solar wind speed over time from ACE in situ measurements taken at near 1 AU. Figure taken from Liewer et al. (2004, Figure 6).
  18. 1.18 Global picture of the structure of the solar atmosphere. Figure taken from Wedemeyer-Böhm et al. (2009, Figure 16).
  19. 1.19 Left panel: Electron density and temperature in the different layers of the solar atmosphere. Right panel: Typical range of the plasma beta parameter. Figure taken from Aschwanden (2005).
  20. 1.20 Overall structure of the solar atmosphere with its associated physical properties and dominant processes. Figure adapted from the habilitation thesis of Dr. Alexis Rouillard (2021).
  21. 1.21 Model of the solar chromosphere and low corona for the averaged quiet Sun, from the AL-C7 model of Avrett & Loeser (2008). Both the total density of Hydrogen $n_{[H]}$, and the density of neutral Hydrogen alone $n_{HI}$ are plotted.
  22. 1.22 Variation of abundances relative to Oxygen as a function of the FIP, given by the velocity-drift model of Peter (1996). Figure taken from Peter (1996, Figure 1).
  23. 1.23 Correspondence between expansion factors (computed between the photosphere and a coronal height of $2.5\ R_{\odot}$) and solar wind speeds measured in situ at 1 AU. Figure taken from Wang & Sheeley (1990, Table 1).
  24. 1.24 Closed (black) and open (multi-color) magnetic field lines traced from a time-steady solution of the polytropic MHD conservation equations, computed by the Magnetohydrodynamics Around a Sphere (MAS) code (Linker et al., 1999; Riley et al., 2001). Photospheric boundary conditions were from Carrington Rotation 2058 (June-July 2007). Colors of open field lines correspond to the Wang & Sheeley (1990) expansion factor $f_{ss}=(R_{\odot}/R_{ss})^{2}(B_{\odot}/B_{ss})$: $f_{ss}\leq 4$ (purple), $f_{ss}\simeq 6$ (blue), $f_{ss}\simeq 10$ (green), $f_{ss}\simeq 15$ (gold), $f_{ss}\geq 40$ (red). Figure taken from Cranmer et al. (2017, Figure 1).
  25. 1.25 Sketch of different reconnection scenarios for the release of closed-field plasma in the slow solar wind near the tip of streamers by: (a) interchange reconnection (Crooker et al., 2003), (b) reconnection in the HCS (Wang et al., 1998), and (c) reconnection between a pair of coronal loops (Gosling et al., 1995). Figure taken from Sanchez-Diaz (2017, Figure 1.10).
  26. 1.26 Connectivity map of PSP during its 6th passage close to the Sun (26-29 of September 2020). The northern and southern polar coronal holes are represented as red and cyan shaded areas respectively. The magnetic pressure $B^{2}/(2\mu_{0})$ is plotted as dark contours at $0.02\ R_{\odot}\simeq 14\ \rm{Mm}$ above the photosphere, and represent the open-closed boundaries associated with the supergranular network. The PSP trajectory (blue diamonds) is projected ballistically to a height of $1.2\ R_{\odot}$ (blue crosses), and then tracked down to $0.02\ R_{\odot}$ (blue circles) above the photosphere using a 3-D reconstruction of the magnetic field. The position of the HCS predicted by the 3-D reconstruction model is plotted as a solid black line at $1.2\ R_{\odot}$. Figure taken from Bale et al. (2021, Figure 1).
  27. 1.27  Schematic of the generation of magnetic flux ropes and streamer blobs in a HCS that is seen face-on (panel b) and edge-on (panel c). The grey shaded areas depict either the flux ropes in panel b, or the streamer blobs in panel c. Figure extracted from Sanchez-Diaz et al. (2017a, Figure 12).
  28. 1.28 Latitude-longitude Carrington map of the squashing factor (color plotted in a logarithmic scale) at $10\ R_{\odot}$ derived from 3-D magnetohydrodynamic simulation run with the MAS code (see Antiochos et al., 2011, and references therein). Figure taken from Antiochos et al. (2011, Figure 7).
  29. 1.29 Schematic of the distribution of fast and slow solar winds in the meridional plane. Figure taken from Wang et al. (2000, Figure 12).
  30. 2.1  La vitesse du vent solaire telle que mesurée par le satellite Ulysses lors de son premier (panneau de gauche) et deuxième (panneau de droite) passage. Les grandes structures de la couronne solaire sont illustrées par des images composites assemblées à partir: d’images du disque solaire prises par le télescope SoHO-EIT dans l’ultraviolet extrême à $195$$\mathrm{\SIUnitSymbolAngstrom}$, d’images de la couronne interne par le K-coronamètre de Mauna Loa ($7000$-$9500$$\mathrm{\SIUnitSymbolAngstrom}$), et d’images dans le domaine visible provenant du coronagraphe SoHO LASCO-C2. Le panneau inférieur montre l’évolution du nombre de taches solaires qui est un bon indicateur du niveau d’activité solaire. Figure adaptée de McComas et al. (2003, figure 1).
  31. 2.2 Propriétés du vent solaire lent et rapide. Figure prise de Cranmer et al. (2017, Table 1).
  32. 2.3 Photographie (retravaillée) de la couronne solaire lors de l’éclipse solaire totale de 2019. Crédits: Nicolas Lefaudeux - https://hdr-astrophotography.com.
  33. 2.4  Illustration d’une couronne solaire hautement structurée durant une période de forte activité. Panneau supérieur: images en EUV prises à $193$$\mathrm{\SIUnitSymbolAngstrom}$ et $211$$\mathrm{\SIUnitSymbolAngstrom}$ par SDO-AIA le 29 Mai 2013. Panneau inférieur: une reconstruction à champ potentiel (PFSS, see section 3.3.1) du champ magnétique coronal pour la rotation Carrington 2137, où les lignes de champ ouvertes sont colorées en bleu et gris suivant leur polarité, et où la HCS est affichée comme une surface rouge. Figure prise de Sanchez-Diaz et al. (2017b, Figure 1).
  34. 2.5 Comparaison entre une image réelle de la couronne (gauche) prise lors de l’éclipse solaire totale d’Août 2017, et une reconstruction associée (droite). Figure adapatée de la figure 1 de Mikić et al. (2018). Crédits pour l’image a: ©2017 Miloslav Druckmüller, Peter Aniol, Shadia Habbal.
  35. 2.6  Schéma de la structure du champ magnétique coronal. Figure prise de Pellegrin et al. (2021), avec permission de l’auteur.
  36. 2.7 Exemple d’un étroit corridor de champ ouvert (lignes vertes) se formant le long d’une extension d’un trou coronal polaire (zone grise). Les contours de l’amplitude du champ magnétique sont tracés en couleur au niveau de la photosphère. Figure prise de Antiochos et al. (2011, Figure 4).
  37. 2.8  Panneau (a) : une reconstruction 3-D du champ magnétique coronal du 5 Novembre 2018 à l’aide d’une méthode d’extrapolation PFSS (voir section 3.3.1). La carte magnétique utilisée comme entrée pour l’extrapolation PFSS est affichée en échelle de gris à la surface du Soleil. Les lignes de champ magnétique ouvertes et fermées sont représentées en jaune et en orange respectivement. La ligne d’inversion de polarité (c’est-à-dire la ligne de base de la HCS) est représentée par une ligne rouge à une hauteur de $2.1\ R_{\odot}$. Panneau (b) : une vue agrandie d’un pseudo-streamer ancré dans un petit trou coronal équatorial isolé (flèche bleue). Est montré également une vue projetée sur la surface d’observations réalisées en ultraviolet extrême dans la longueur d’onde de $193$$\mathrm{\SIUnitSymbolAngstrom}$.
  38. 2.9  Cartes de Carrington en latitude et en longitude du champ magnétique photosphérique (1ère ligne), de la HCS prédite (2ème ligne) et des émissions de synthèse en WL (3ème ligne), ainsi que des observations réelles en WL réalisées par SoHO LASCO-C2, sur la rotation de Carrington CR1919 (gauche) et CR1935 (droite) qui sont typiques d’une configuration coronale minimale (gauche) et maximale (droite) du Soleil. Figure prise de Wang et al. (2000, Figure 2).
  39. 2.10 Carte 2-D du rapport d’abondance Silicium (Si) sur Soufre (S) du disque solaire. Figure prise de Brooks et al. (2015, Figure 3).
  40. 2.11  Exemple de blobs de streamers observés par SoHO-LASCO, en utilisant une technique de "running difference" (voir texte) pour mettre en valeur les fluctuations de densité dans la couronne où les régions brillantes (ou sombres) indiquent respectivement une densité accrue (ou plus faible). Figure tirée de Sheeley et al. (1997, Figure 1).
  41. 2.12  Fluctuations de luminosité dans la haute couronne solaire détectées lors d’une campagne à haute cadence du coronographe STEREO-A COR2. Figure tirée de DeForest et al. (2018, Figure 12).
  42. 2.13 Aperçu des structures dynamiques dans le vent solaire lent, depuis les grandes (en haut) jusqu’aux petites (en bas) échelles spatiales, avec leurs références associées qui ont été mentionnées dans le texte.
  43. 2.14 Variation des abondances des principaux constituants mesurés dans le vent solaire en fonction du FIP. Les abondances sont relatives à l’abondance de l’Oxygène. Figure tirée de Peter (1998, Figure 1).
  44. 2.15 Variation temporelle de l’abondance en Magnésium (Mg, bas FIP) par rapport à l’Oxygène (O), de la vitesse du vent solaire (prise ici comme la vitesse de l’Hélium doublement ionisé), et de la température de l’Oxygène. Figure tirée de Geiss et al. (1995, Figure 7).
  45. 2.16 Mesures in situ prises par ACE du rapport d’état de charge de l’Oxygène $O^{7+}/O^{6+}$ et du Carbone $C^{6+}/C^{5+}$ en fonction de la vitesse du vent solaire (points bleus). Les mesures correspondant à des vents rapides, lents Alfvéniques et lents non Alfvéniques identifiés par D’Amicis & Bruno (2015) sont indiquées par les points verts, rouges et noirs respectivement. Des simulations de l’état de charge pour ces trois régimes de vents, obtenues à partir du modèle d’atmosphère solaire de l’IRAP (ISAM), sont montrées par les croix rouges avec leurs incertitues associées en noir (voir Lavarra et al., 2022, pour plus de détails). Figure prise de Lavarra et al. (2022, Figure 1).
  46. 2.17 Variation temporelle du rapport d’état de charge $O^{7+}/O^{6+}$ et de la vitesse du vent solaire dans des mesures in situ prises par ACE depuis 1 UA. Figure prise de Liewer et al. (2004, Figure 6).
  47. 2.18 Structure globale de l’atmosphère solaire. Figure prise de Wedemeyer-Böhm et al. (2009, Figure 16).
  48. 2.19 Panneau de gauche : densité et température des électrons dans les différentes couches de l’atmosphère solaire. Panneau de droite : plage typique de variation du paramètre bêta associé au plasma. Figure tirée de Aschwanden (2005).
  49. 2.20 Structure globale de l’atmosphère solaire avec ses propriétés physiques associées et ses processus dominants. Figure prise de l’habilitation à diriger des thèses de Dr. Alexis Rouillard (2021).
  50. 2.21 Modèle de la chromosphère solaire et de la basse couronne pour le Soleil calme moyenné, à partir du modèle AL-C7 de Avrett & Loeser (2008). La densité totale d’Hydrogène $n_{[H]}$, et celle de l’Hydrogène neutre seul $n_{HI}$ sont représentées.
  51. 2.22 Variation de l’abondance relative d’Oxygen en fonction du FIP, générée depuis le modèle à vitesse différentielle de Peter (1996). Figure prise de Peter (1996, Figure 1).
  52. 2.23 Correspondances entre les facteurs d’expansion (calculés entre la photosphère et une hauteur coronale de $2.5\ R_{\odot}$) et la vitesse du vent solaire mesurée in situ à 1 UA. Figure prise de Wang & Sheeley (1990, Tableau 1).
  53. 2.24 Lignes de champ magnétique fermées (noir) et ouvertes (multicolore) tracées à partir d’une solution stationnaire aux équations de conservation MHD polytropiques, calculée par le code "Magnetohydrodynamics Around a Sphere" (MAS) (Linker et al., 1999; Riley et al., 2001). La condition limite au niveau de la photosphère corresponds à la rotation Carrington 2058 (Juin-Juillet 2007). Les couleurs des lignes de champ ouvertes correspondent au facteur d’expansion de Wang & Sheeley (1990) pour des valeurs: $f_{ss}=(R_{\odot}/R_{ss})^{2}(B_{\odot}/B_{ss})$: $f_{ss}\leq 4$ (violet), $f_{ss}\simeq 6$ (bleu), $f_{ss}\simeq 10$ (vert), $f_{ss}\simeq 15$ (or), $f_{ss}\geq 40$ (rouge). Figure prise de Cranmer et al. (2017, Figure 1).
  54. 2.25 Esquisse de différents scénarios de reconnexion pour la libération de plasma en champ fermé dans le vent solaire lent à l’extrémité des streamers par: (a) la reconnexion d’interchange entre lignes ouvertes et fermées (Crooker et al., 2003), (b) la reconnexion dans la HCS (Wang et al., 1998), et (c) la reconnexion entre pairs de boucles coronales (Gosling et al., 1995). Figure tirée de Sanchez-Diaz (2017, Figure 1.10).
  55. 2.26 Carte de connectivité de PSP lors de son 6ème passage près du Soleil (26-29 septembre 2020). Les trous coronaux polaires nord et sud sont représentés respectivement par des zones de couleur rouge et cyan. La pression magnétique $B^{2}/(2\mu_{0})$ est représentée par des contours noirs à $0.02\ R_{\odot}\simeq 14\ \rm{Mm}$ au-dessus de la photosphère, et représente les frontières ouvert-fermé associées au réseau supergranulaire. La trajectoire de PSP (losanges bleus) est projetée de manière balistique à une hauteur de $1.2\ R_{\odot}$ (croix bleues), puis tracée jusqu’à $0,02\ R_{\odot}$ (cercles bleus) au-dessus de la photosphère en utilisant une reconstruction 3-D du champ magnétique. La position de la HCS prédite par le modèle de reconstruction 3-D est représentée par une ligne noire pleine à $1,2\ R_{\odot}$. Figure tirée de Bale et al. (2021, Figure 1).
  56. 2.27 Schéma de la génération de champs torsadés et de blobs de streamers dans une HCS vue de face (panneau b) et de côté (panneau c). Les zones grisées représentent soit les champs torsadés dans le panneau b, soit les blobs de streamer dans le panneau c. Figure extraite de Sanchez-Diaz et al. (2017a, Figure 12).
  57. 2.28 Carte Carrington en latitude-longitude du squashing factor (couleur tracée sur une échelle logarithmique) à $10\ R_{\odot}$ dérivée à partir de la simulation magnéto-hydrodynamique 3-D du code MAS (Antiochos et al., 2011). Figure prise de Antiochos et al. (2011, Figure 7).
  58. 2.29 Schéma de la distribution des vents solaires rapides et lents dans le plan méridional. Figure tirée de Wang et al. (2000, Figure 12).
  59. 3.1 Schematics of the LASCO-C2 and LASCO-C3 coronagraphs. The top panel shows the path followed by the incoming light from the target until it reaches the detector. The bottom panel illustrates how the residual light that is diffracted at the edges of the external occulter D1 and apertures is deviated to not reach the detector. Figure taken from Brueckner et al. (1995, Figure 3))
  60. 3.2 Combined field-of-view of the SECCHI telescopes on board STEREO-A. (Credit: STEREO/GSFC)
  61. 3.3 Position of the WISPR white-light imager on the ram side of PSP (NASA/JHUAPL).
  62. 3.4 Merged image from the inner WISPR-I and WISPR-O outer telescopes around the 8th perihelion on 2021-04-29 at 00:34 UT. PSP was located at a heliocentric distance of $16.5\ R_{\odot}$. The helioprojective grid (centered at PSP) is plotted with elongation angles given as the abscissa. Image produced by the WISPR team.
  63. 3.5 Panel a: 3-D rendering of WISPR (taken from Vourlidas et al. (2016, Figure 9)). Panel b: schematic of WISPR (taken from Vourlidas et al. (2016, Figure 15)).
  64. 3.6 Snapshots from the MCT website with input date 2021-11-21 at 12:00 UT. The polarity inversion line which is obtained from the coronal model is plotted as a dashed red line. The dark line denotes the visible portion of the solar disk from the Earth perspective. Colored dots represent the likelihood of spacecraft connectivity to the surface. Colored crosses are vertical projections of the spacecraft location onto the solar disk. Top panel: GONG-ADAPT (2nd realization) synoptic map of the surface magnetic field which has been used for the PFSS reconstruction of the coronal magnetic field. Middle panel: EUV synoptic map at $171$ $\mathrm{\SIUnitSymbolAngstrom}$ from the SDO-AIA. Bottom panel: white-light synoptic map at $2.5\ R_{\odot}$ from the SoHO LASCO-C2 coronagraph.
  65. 3.7 Schematic illustrating how magnetic connectivity is established in the MCT, showing the location of a spacecraft (S/C), the intersection of the Parker Spiral (B) and the photospheric/low corona magnetic footpoint of field line (A). Schematic taken from Owens & Forsyth (2013, Figure 1).
  66. 3.8 PFSS reconstruction of the coronal magnetic field using a photospheric magnetic field map from WSO (Carrington rotation CR2149) and a source surface height of $2.5\ R_{\odot}$. Open and closed magnetic field lines are depicted in orange and blue respectively. The solar equator is represented by a dashed green line. The line of polarity inversion or neutral line is traced in purple and is the baseline of the heliospheric current sheet. Figure 7 that I produced for Griton et al. (2020).
  67. 3.9 Comparison of the HCS predicted by a model (solid red line) with the HCS derived from white-light observations of the streamer belt (dashed black line) taken by SoHO LASCO-C2 between 2018, October 30 and 2018, November 13. The angular deviations between the two lines that are used to compute the WL metric are represented by the blue arrows. Results from the WL metric are given in the legend, of which the WL global score (in %). Figure 2 that I produced for Badman et al. (2022).
  68. 3.10  Top panel: Prediction of magnetic polarity measurements at PSP from a PFSS model. The simulated polarity switch sign when the spacecraft moves across the modeled HCS (dark solid line). Bottom panel: Actual magnetic polarity measured by PSP during its first passage to the Sun (October/November 2018). In both real and virtual modes, the PSP’s trajectory is projected closer to the Sun (at around $2.0-2.5\ R_{\odot}$) by following the Parker spiral (also known as a ballistic model). That way the measured magnetic polarities can be compared with the magnetic sectors predicted by PFSS models. Figure adapted from Badman et al. (2022, Figure 3).
  69. 3.11 Top panel: Coronal holes distribution (dark shaded areas) as predicted by a PFSS model. Bottom panel: EUV synoptic map at $193$ $\mathrm{\SIUnitSymbolAngstrom}$ built from AIA-EUVI observations during Carrington rotation CR2210. Darker regions in the EUV map are identified as coronal holes with a white contour. Figure adapted from Badman et al. (2022, Figure 1).
  70. 4.1  Region of sensitivity of WISPR-I compared to LASCO-C2, defined as the LOS portions that contain $99\%$ of the total light received by the detectors. Figure 14 that I produced for the Poirier et al. (2020) paper.
  71. 4.2  A comparison between WISPR-I images with LASCO-C3 images at around 00:06 UT on 2018 November 1 (top panels) and STA COR-2A and HI1 images at around 00:11 UT on 2018 November 6 (bottom panels). The red boxes overplotted at the left panels represent the WISPR-I field of view for comparison with those observations. Figure taken from Poirier et al. (2020, Figure 4).
  72. 4.3  Carrington map from LASCO-C3 WL observations during the time interval 2018 October 20 to November 14. The likely scanned region by WISPR during its first observing window is represented by the green dotted line. Figure taken from Poirier et al. (2020, Figure 5).
  73. 4.4  Latitude vs. time map from WISPR-I images taken during the first encounter: from 2018 November 1, 00:45 UT, to 2018 November 10, 17:29 UT. The y-axis is the heliographic latitude in degrees. Figure 6 that I produced for the Poirier et al. (2020) paper.
  74. 4.5  A comparison between synthetic WL images (panels a and b) and a WISPR-I image (panel c) on 2018 November 3, at 06:55 UT. The synthetic images produced by MHD data from the MULTI-VP model and two different source magnetograms; WSO magnetogram of Carrington rotation CR2210 (panel a) and GONG-ADAPT magnetogram of 2018 November 5, 12:00 UT (panel b). The arrows superimposed at the images are the same as in Figure 4.4, and are indicative of the position of the features discussed in the text. Figure 8 that I produced for the Poirier et al. (2020) paper.
  75. 4.6  Three WISPR WL synthetic images on 2018 November 3 at 06:55 UT from the MVP-ADAPT run. They correspond to integration along the LOS over (a) the full span, (b) the foreground only, and (c) the background only. The arrows’ color coding is the same as in the previous figures. Figure 10 that I produced for the Poirier et al. (2020) paper.
  76. 4.7  Panels (a) and (b) show, respectively, the Carrington maps of the simulated density and magnetic field polarity at $15\ R_{\odot}$ from the MVP-ADAPT simulation. The magenta line traces the intersection of the Thomson sphere with the Carrington map. The cone of integration, defined by the intersection between the field of view of WISPR-I and the map, is shown in black (yellow) in panel a (b). Figure 11 from the Poirier et al. (2020) paper.
  77. 4.8 Synthetic latitude vs. time maps from the MVP-ADAPT run. They have been generated from (a) the full synthetic images, (b) the foreground subimages only, and (c) the background subimages only. The arrows’ color coding is the same as in the previous figures. Figure 12 that I produced for the Poirier et al. (2020) paper.
  78. 4.9  WISPR-I WL synoptic maps for encounters 1, 2, 4, 5, 6, 8 and 9.
  79. 5.1 Top panel: Carrington map from SDO-AIA 193$\mathrm{\SIUnitSymbolAngstrom}$ EUV observations, on March 31, 2019, at 00:00 UT. Bottom panel: Carrington map from SOHO LASCO-C3 WL observations, on April 3, 2019, at 12:00 UT and at a height of 5 $R_{\odot}$. The PSP trajectory (dark solid-line) is projected at the height of the Carrington maps (and at 2.1 $R\odot$ for the AIA EUV map), using the Parker Spiral (Parker, 1958) and the solar wind speed measured in situ at PSP. Along the PSP projected trajectory, some PSP projected positions for several dates and times are depicted with colored stars and arrows. On the zoom-in views in the middle panels, the plasma electron density $n_{e}(35.6R\odot)$ (in $cm^{-3}$) taken from quasi-thermal-noise (QTN) data of the FIELDS-RFS instrument (see Moncuquet et al., 2020, for a description of the method) is plotted along the PSP projected trajectory with a logarithmic color scale. The red and dark dashed-lines represent the neutral line from an optimized PFSS reconstruction with a source surface height of $2.1\ R_{\odot}$ and a GONG-ADAPT photospheric magnetic map (2nd realization) of March 24, 2019, at 00:00 UT. Open field regions of positive (negative) polarity from the PFSS model are represented at the photosphere by a semi-transparent blue (red) layer. Figure 8 that I produced for Griton et al. (2021).
  80. 5.2  Left column: Samples of magnetic field lines from the optimized PFSS magnetic field model with the source surface at $2.1\ R_{\odot}$, in the plane cut through $18^{\circ}$ longitude. PSP certainly crossed magnetic field lines connected to field lines A (state 1) and B (state 2). Middle column: plasma density (top), speed (middle) and temperature profiles (bottom) computed by MULTI-VP along 5 flux tubes around field line A (in red) and around field line B (in blue). Right column: same as the middle column but for the expansion factor (top), heating profile (middle) and magnetic field intensity profiles (bottom). Figure taken from (Griton et al., 2021, Figure 9).
  81. 5.3  A sample of several density fluctuations observed by WISPR-I.
  82. 5.4  A WISPR-I synoptic map extracted at $10\ R_{\odot}$ and focused on the inbound of the 8th passage of PSP on April 25-27, 2021. The density perturbations that have been identified in Figure 5.3 are denoted here using the same colors.
  83. 5.5  Snapshots of two WindPredict-AW 2.5-D simulations of Réville et al. (2020a) illustrating the generation of quasi-periodic structures from the tearing instability, with two distinct Lundquist numbers $S$. The latitudinal component of the Alfvén speed is color plotted with a sample of magnetic field lines in grey.
  84. 5.6  3-D structure of a streamer flux rope generated in the WindPredict-AW 3-D simulation of June 14, 2020 described in Réville et al. (2022). This figure has been extracted from Réville et al. (2022, Figure 7).
  85. 5.7  Synthetic WISPR-I running difference images of the formation and propagation of a "streamer blob" associated to a flux rope topology from the WindPredict-AW 3-D simulation of June 14, 2020 described in Réville et al. (2022).
  86. 5.8  Carrington map of the simulated density at $12\ R_{\odot}$ from the June 14, 2020 simulation. The position of PSP on June 7 23:42 is marked by the cyan star symbol with a rough estimate of the WISPR-I FOV indicated by the magenta arrows. At that time, PSP was located at $\simeq 28.4\ R_{\odot}$ in heliocentric distance. The dashed red lines indicate the portions of the HPS that are likely captured by WISPR-I.
  87. 5.9  Synthetic WISPR-I map for the 5th passage of PSP to the Sun from the WindPredict-AW 3-D simulation of June 14, 2020 described in Réville et al. (2022). The map is extracted at $12R_{\odot}$.
  88. 5.10  Same as figure 5.7 but for a flux rope generated in the simulation of August 14 2021.
  89. 5.11  Carrington map of the simulated density at $6.5\ R_{\odot}$ from the August 14 2021 simulation. The position of PSP on August 9 05:07 is marked by the cyan star symbol with a rough estimate of the WISPR-I FOV indicated by the magenta arrows. At that time, PSP was located at $\simeq 17.5\ R_{\odot}$ in heliocentric distance. The dashed red lines indicate the portions of the HPS that are likely captured by WISPR-I. The grey dashed line denotes the HCS where magnetic polarity switches sign in the simulation.
  90. 5.12  Synthetic WISPR-I map for the 9th passage of PSP close to the Sun, from the WindPredict-AW 3-D simulation of August 14 2021. The map is extracted at $5\ R_{\odot}$. To ease the interpretation each streamer ray is identified by a dashed red line and a number where the associated portion of the HPS producing the brightness is identified in Figure 5.11 using the same numbers.
  91. 5.13  Panel a: mean (averaged) synthetic image over times $t=15\mathrm{h}$ to $t=30\mathrm{h}$ from simulation 9 (see Griton et al., 2020). Panel b: synthetic image at time $t=24\mathrm{h}$. Panel c: synthetic base difference image at time $t=24\mathrm{h}$ with respect to the mean background image shown in panel a.
  92. 6.1  Radiative losses as defined in the MULTI-VP model (dashed black line) (Pinto & Rouillard, 2017), and as derived from the improved method (solid black line) that includes both the semi-empirical recipe of Carlsson & Leenaarts (2012) (for $2000<T<30000\ \rm{K}$) and ChiantiPy (for $T>30000\ \rm{K}$) that are introduced later in section 6.7.2 and 6.7.3. The improved method has also a dependence on the column mass, which here is retrieved from the ISAM run presented in section 7.1.2.
  93. 6.2 Distribution of grid points in ISAM for a small loop of half-length $16\ \rm{Mm}$ and apex height $10\ \rm{Mm}$.
  94. 7.1  Stability criteria of coronal loops function of their half-length $L$ (x-axis) and heating scale height $H_{f}$ (y-axis) (both units are in Mm). The 1MK, 3MK and 10MK labelled curves correspond to the criteria from Winebarger et al. (2003a) where the critical heating scale height is $H^{*}_{f}(L)\approx A(T)L^{\delta}(T)$ with $A(T)$ and $\delta(T)$ that are tabulated data. The criteria from Serio et al. (1981) $H^{*}_{f}(L)=0.5L$ and Aschwanden et al. (2001) $H^{*}_{f}(L)=\sqrt{L}$ are also plotted. Figure adapted from Winebarger et al. (2003a, Figure 6).
  95. 7.2  Left panel: comparison of the electronic temperature function of altitude, between ISAM runs that have different coronal heating parameters ($F_{\odot},H_{f}$) and after $26$ min of simulation time. Right panel: same comparison but for the time evolution of the loop-top electronic temperature.
  96. 7.3 Example of a full TNE cycle in a dynamic loop of half-length $153\ \rm{Mm}$, base flux heating $F_{\odot}=1.e6\ \rm{erg.cm^{-2}.s^{-1}}$ and heating scale height $H_{f}=2.1\ \rm{Mm}$ following the same format as in Figure 7.2.
  97. 7.4  Total temperatures of electron (dashed red line), protons (solid green line) and neutral Hydrogen (solid blue line) computed by ISAM. The neutral Hydrogen and protons temperatures overlap everywhere in the loop which remains highly collisional. The temperature fit from the AL-C7 model is given as black star symbols.
  98. 7.5  Contributions from the most prevalent chromospheric lines to the average quiet-Sun temperature of the chromosphere. Figure taken from Vernazza et al. (1981, Figure 1).
  99. 7.6  Total (solid black line), chromospheric (dashed black line) and coronal (dotted black line) ah-hoc heating laws adopted in ISAM. The chromospheric heating prescription of Janse et al. (2006) is plotted as a dashed red line.
  100. 7.7  Total heat fluxes of electrons (solid black line), protons (solid green line) and neutral Hydrogen (solid red line). The classical Spitzer-Härm heat flux $F_{c}=-\kappa_{0}T^{5/2}\nabla_{s}T$ with a conductivity $\kappa_{0}=9.2\times 10^{-7}\ erg.cm^{-1}.s^{-1}.K^{-7/2}$ is plotted as a long dashed black line. The approximation of the total electron heat flux solved in ISAM following Lavarra et al. (2022, eq 30-31) is traced as a short dashed black line. Positive heat fluxes are directed outward.
  101. 7.8  Number densities of electrons (dashed red line), protons (solid green line) and neutral Hydrogen (solid black line) computed by ISAM. The neutral Hydrogen and electron density profiles from the AL-C7 model are marked by black and red star symbols respectively. The proton density computed by the ChiantiPy spectral code given the ISAM neutral Hydrogen profile is plotted as green crosses.
  102. 7.9  Collision frequencies of neutral Hydrogen with electrons and protons (dashed and solid black lines), and of protons with electrons and neutral Hydrogen (dashed and solid red lines).
  103. 7.10  Forces acting on neutral Hydrogen (left), protons (middle) and on both (right). All forces are first multiplied by the ionization fraction ($n_{X}/n_{[X]}$), and then normalized by the absolute value of gravity. Positive forces are directed outward.
  104. 7.11  Bulk velocities of neutral Hydrogen and protons (dashed and dotted black lines) expressed in the center of mass $v_{G}=(m_{H}u_{H}+m_{p}u_{p})/(m_{H}+m_{p})$. The total mass flux of the Hydrogen element is plotted on a logarithmic scale that is shown on the right-hand side and where positive and negative values are denoted by black and red colors respectively.
  105. 7.12  Total ionization fractions of Oxygen and Hydrogen.
  106. 7.13  FIP bias of Oxygen relative to Hydrogen $(n_{[O]}/n_{[H]})_{alt}/(n_{[O]}/n_{[H]})_{PHO}$ function of altitude. Colors are indicative of the proportion of each specie (neutral and ions) in the total Oxygen abundance.
  107. 7.14  Left panel: summed up contributions of all forces that apply to the Oxygen element using the same normalizing procedure as in Figure 7.10. The particle-particule interactions that contribute the most to the total friction and thermal forces that apply to neutral Oxygen (middle panel) and $O^{4+}$ (right panel) are also shown.

style=

References

## References

  * int (1999) 1999, Atomic and Plasma–Material Interaction Data for Fusion, Atomic and Plasma–Material Interaction Data for Fusion No. 8 (Vienna: INTERNATIONAL ATOMIC ENERGY AGENCY).  https://www.iaea.org/publications/4710/atomic-and-plasma-material-interaction-data-for-fusion
  * Alazraki & Couturier (1971) Alazraki, G., & Couturier, P. 1971, Astronomy and Astrophysics, 13, 380
  * Alipour & Safari (2015) Alipour, N., & Safari, H. 2015, The Astrophysical Journal, 807, 175, doi: 10.1088/0004-637X/807/2/175
  * Antiochos (1994) Antiochos, S. K. 1994, Advances in Space Research, 14, 139, doi: 10.1016/0273-1177(94)90173-2
  * Antiochos et al. (2011) Antiochos, S. K., Mikić, Z., Titov, V. S., Lionello, R., & Linker, J. A. 2011, The Astrophysical Journal, 731, 112, doi: 10.1088/0004-637X/731/2/112
  * Antolin et al. (2015) Antolin, P., Vissers, G., Pereira, T. M. D., Rouppe van der Voort, L., & Scullion, E. 2015, The Astrophysical Journal, 806, 81, doi: 10.1088/0004-637X/806/1/81
  * Arge et al. (2013) Arge, C. N., Henney, C. J., Hernandez, I. G., et al. 2013, in American Institute of Physics Conference Series, Vol. 1539, Solar Wind 13, ed. G. P. Zank, J. Borovsky, R. Bruno, J. Cirtain, S. Cranmer, H. Elliott, J. Giacalone, W. Gonzalez, G. Li, E. Marsch, E. Moebius, N. Pogorelov, J. Spann, & O. Verkhoglyadova, 11–14, doi: 10.1063/1.4810977
  * Arge et al. (2010) Arge, C. N., Henney, C. J., Koller, J., et al. 2010, AIP Conference Proceedings, 1216, 343, doi: 10.1063/1.3395870
  * Arge et al. (2011) Arge, C. N., Henney, C. J., Koller, J., et al. 2011, in Astronomical Society of the Pacific Conference Series, Vol. 444, 5th International Conference of Numerical Modeling of Space Plasma Flows (ASTRONUM 2010), ed. N. V. Pogorelov, E. Audit, & G. P. Zank, 99
  * Arnaud & Raymond (1992) Arnaud, M., & Raymond, J. 1992, The Astrophysical Journal, 398, 394, doi: 10.1086/171864
  * Arnaud & Rothenflug (1985) Arnaud, M., & Rothenflug, R. 1985, Astronomy and Astrophysics Supplement, 60, 425
  * Aschwanden (2005) Aschwanden, M. J. 2005, Physics of the Solar Corona. An Introduction with Problems and Solutions (2nd edition)
  * Aschwanden (2014) Aschwanden, M. J. 2014, in Encyclopedia of the Solar System (Third Edition), third edition edn., ed. T. Spohn, D. Breuer, & T. V. Johnson (Boston: Elsevier), 235–259, doi: https://doi.org/10.1016/B978-0-12-415845-0.00011-6
  * Aschwanden et al. (2000a) Aschwanden, M. J., Alexander, D., Hurlburt, N., et al. 2000a, The Astrophysical Journal, 531, 1129, doi: 10.1086/308483
  * Aschwanden et al. (2000b) Aschwanden, M. J., Nightingale, R. W., & Alexander, D. 2000b, The Astrophysical Journal, 541, 1059, doi: 10.1086/309486
  * Aschwanden et al. (2001) Aschwanden, M. J., Schrijver, C. J., & Alexander, D. 2001, The Astrophysical Journal, 550, 1036, doi: 10.1086/319796
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, Annual Review of Astronomy and Astrophysics, 47, 481, doi: 10.1146/annurev.astro.46.060407.145222
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, Astronomy and Astrophysics, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, The Astronomical Journal, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Athay (1986) Athay, R. G. 1986, The Astrophysical Journal, 308, 975, doi: 10.1086/164565
  * Auchère et al. (2016) Auchère, F., Froment, C., Bocchialini, K., Buchlin, E., & Solomon, J. 2016, The Astrophysical Journal, 827, 152, doi: 10.3847/0004-637X/827/2/152
  * Aulanier et al. (2005) Aulanier, G., Pariat, E., & Démoulin, P. 2005, Astronomy and Astrophysics, 444, 961, doi: 10.1051/0004-6361:20053600
  * Aulanier et al. (2006) Aulanier, G., Pariat, E., Démoulin, P., & DeVore, C. R. 2006, Solar Physics, 238, 347, doi: 10.1007/s11207-006-0230-2
  * Avrett & Loeser (2008) Avrett, E. H., & Loeser, R. 2008, The Astrophysical Journal Supplement Series, 175, 229, doi: 10.1086/523671
  * Badman et al. (2022) Badman, S. T., Brooks, D. H., Poirier, N., et al. 2022, The Astrophysical Journal, 932, 135, doi: 10.3847/1538-4357/ac6610
  * Baker et al. (2015) Baker, D., Brooks, D. H., Démoulin, P., et al. 2015, The Astrophysical Journal, 802, 104, doi: 10.1088/0004-637X/802/2/104
  * Baker et al. (2009) Baker, D., van Driel-Gesztelyi, L., Mandrini, C. H., Démoulin, P., & Murray, M. J. 2009, The Astrophysical Journal, 705, 926, doi: 10.1088/0004-637X/705/1/926
  * Bale et al. (2021) Bale, S. D., Horbury, T. S., Velli, M., et al. 2021, The Astrophysical Journal, 923, 174, doi: 10.3847/1538-4357/ac2d8c
  * Banks & Kockarts (1973) Banks, P. M., & Kockarts, G. 1973, Aeronomy.
  * Berghmans et al. (2021) Berghmans, D., Auchère, F., Long, D. M., et al. 2021, Astronomy and Astrophysics, 656, L4, doi: 10.1051/0004-6361/202140380
  * Biermann (1948) Biermann, L. 1948, Zeitschrift für Astrophysik, 25, 161
  * Biermann (1951) —. 1951, Zeitschrift für Astrophysik, 29, 274
  * Bird & Edenhofer (1990) Bird, M. K., & Edenhofer, P. 1990, in Physics of the Inner Heliosphere I, ed. R. Schwenn & E. Marsch, 13
  * Blelly & Schunk (1993) Blelly, P. L., & Schunk, R. W. 1993, Annales Geophysicae, 11, 443
  * Bø et al. (2013) Bø, I. M. T., Esser, R., & Lie-Svendsen, Ø. 2013, The Astrophysical Journal, 769, 60, doi: 10.1088/0004-637X/769/1/60
  * Boe et al. (2020) Boe, B., Habbal, S., & Druckmüller, M. 2020, The Astrophysical Journal, 895, 123, doi: 10.3847/1538-4357/ab8ae6
  * Bohlin (1970) Bohlin, J. D. 1970, Solar Physics, 12, 240, doi: 10.1007/BF00227121
  * Boqueho (2005) Boqueho, V. 2005, PhD thesis.  http://www.theses.fr/2005TOU30269
  * Boris et al. (1993) Boris, J. P., Landsberg, A. M., Oran, E. S., & Gardner, J. H. 1993, LCPFCT - Flux-Corrected Transport Algorithm for Solving Generalized Continuity Equations, Technical report.  https://apps.dtic.mil/sti/citations/ADA265011
  * Brooks et al. (2015) Brooks, D. H., Ugarte-Urra, I., & Warren, H. P. 2015, Nature Communications, 6, 5947, doi: 10.1038/ncomms6947
  * Brooks & Warren (2011) Brooks, D. H., & Warren, H. P. 2011, The Astrophysical Journal Letters, 727, L13, doi: 10.1088/2041-8205/727/1/L13
  * Brueckner et al. (1995) Brueckner, G. E., Howard, R. A., Koomen, M. J., et al. 1995, Solar Physics, 162, 357, doi: 10.1007/BF00733434
  * Buchlin & Velli (2007) Buchlin, E., & Velli, M. 2007, The Astrophysical Journal, 662, 701, doi: 10.1086/512765
  * Bumba & Howard (1969) Bumba, V., & Howard, R. 1969, Solar Physics, 7, 28, doi: 10.1007/BF00148402
  * Burgers (1969) Burgers, J. M. 1969, Flow Equations for Composite Gases
  * Carlsson (2007) Carlsson, M. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 368, The Physics of Chromospheric Plasmas, ed. P. Heinzel, I. Dorotovič, & R. J. Rutten, 49.  https://arxiv.org/abs/0704.1509
  * Carlsson & Leenaarts (2012) Carlsson, M., & Leenaarts, J. 2012, Astronomy and Astrophysics, 539, A39, doi: 10.1051/0004-6361/201118366
  * Carlsson & Stein (2002) Carlsson, M., & Stein, R. F. 2002, The Astrophysical Journal, 572, 626, doi: 10.1086/340293
  * Chandran et al. (2011) Chandran, B. D. G., Dennis, T. J., Quataert, E., & Bale, S. D. 2011, The Astrophysical Journal, 743, 197, doi: 10.1088/0004-637x/743/2/197
  * Chapman & Cowling (1970) Chapman, S., & Cowling, T. G. 1970, The mathematical theory of non-uniform gases. an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases
  * Chapman & Zirin (1957) Chapman, S., & Zirin, H. 1957, Smithsonian Contributions to Astrophysics, 2, 1
  * Cranmer et al. (2017) Cranmer, S. R., Gibson, S. E., & Riley, P. 2017, Space Science Reviews, 212, 1345, doi: 10.1007/s11214-017-0416-y
  * Cranmer & van Ballegooijen (2005) Cranmer, S. R., & van Ballegooijen, A. A. 2005, The Astrophysical Journal Supplement Series, 156, 265, doi: 10.1086/426507
  * Crooker et al. (2012) Crooker, N. U., Antiochos, S. K., Zhao, X., & Neugebauer, M. 2012, Journal of Geophysical Research (Space Physics), 117, A04104, doi: 10.1029/2011JA017236
  * Crooker et al. (2003) Crooker, N. U., Larson, D. E., Kahler, S. W., Lamassa, S. M., & Spence, H. E. 2003, Geophysical Research Letters, 30, 1619, doi: 10.1029/2003GL017036
  * D’Amicis & Bruno (2015) D’Amicis, R., & Bruno, R. 2015, The Astrophysical Journal, 805, 84, doi: 10.1088/0004-637X/805/1/84
  * D’Amicis et al. (2019) D’Amicis, R., Matteini, L., & Bruno, R. 2019, Monthly Notices of the Royal Astronomical Society, 483, 4665, doi: 10.1093/mnras/sty3329
  * De Groof et al. (2004) De Groof, A., Berghmans, D., van Driel-Gesztelyi, L., & Poedts, S. 2004, Astronomy and Astrophysics, 415, 1141, doi: 10.1051/0004-6361:20034252
  * de Patoul et al. (2015) de Patoul, J., Foullon, C., & Riley, P. 2015, The Astrophysical Journal, 814, 68, doi: 10.1088/0004-637X/814/1/68
  * DeForest et al. (2018) DeForest, C. E., Howard, R. A., Velli, M., Viall, N., & Vourlidas, A. 2018, The Astrophysical Journal, 862, 18, doi: 10.3847/1538-4357/aac8e3
  * Del Zanna et al. (2021) Del Zanna, G., Dere, K. P., Young, P. R., & Landi, E. 2021, The Astrophysical Journal, 909, 38, doi: 10.3847/1538-4357/abd8ce
  * Demars & Schunk (1979) Demars, H. G., & Schunk, R. W. 1979, Journal of Physics D Applied Physics, 12, 1051, doi: 10.1088/0022-3727/12/7/011
  * Demoulin et al. (1997) Demoulin, P., Bagala, L. G., Mandrini, C. H., Henoux, J. C., & Rovira, M. G. 1997, Astronomy and Astrophysics, 325, 305
  * Démoulin et al. (1996) Démoulin, P., Priest, E. R., & Lonie, D. P. 1996, Journal of Geophysical Research, 101, 7631, doi: 10.1029/95JA03558
  * Dere (2007) Dere, K. P. 2007, Astronomy and Astrophysics, 466, 771, doi: 10.1051/0004-6361:20066728
  * Dere et al. (1997) Dere, K. P., Landi, E., Mason, H. E., Monsignori Fossi, B. C., & Young, P. R. 1997, Astronomy and Astrophysics Supplement, 125, 149, doi: 10.1051/aas:1997368
  * Dere et al. (2009) Dere, K. P., Landi, E., Young, P. R., et al. 2009, Astronomy and Astrophysics, 498, 915, doi: 10.1051/0004-6361/200911712
  * Dmitruk et al. (2002) Dmitruk, P., Matthaeus, W. H., Milano, L. J., et al. 2002, The Astrophysical Journal, 575, 571, doi: 10.1086/341188
  * Domingo et al. (1995) Domingo, V., Fleck, B., & Poland, A. I. 1995, Solar Physics, 162, 1, doi: 10.1007/BF00733425
  * Doschek & Warren (2019) Doschek, G. A., & Warren, H. P. 2019, The Astrophysical Journal, 884, 158, doi: 10.3847/1538-4357/ab426e
  * Druckmüller et al. (2014) Druckmüller, M., Habbal, S. R., & Morgan, H. 2014, The Astrophysical Journal, 785, 14, doi: 10.1088/0004-637X/785/1/14
  * Duvall et al. (1977) Duvall, T. L., J., Wilcox, J. M., Svalgaard, L., Scherrer, P. H., & McIntosh, P. S. 1977, Solar Physics, 55, 63, doi: 10.1007/BF00150874
  * Elsasser (1950) Elsasser, W. M. 1950, Physical Review, 79, 183, doi: 10.1103/PhysRev.79.183
  * Evans (1948) Evans, J. W. 1948, Journal of the Optical Society of America (1917-1983), 38, 1083
  * Eyles et al. (2009) Eyles, C. J., Harrison, R. A., Davis, C. J., et al. 2009, Solar Physics, 254, 387, doi: 10.1007/s11207-008-9299-0
  * Fargette et al. (2022) Fargette, N., Lavraud, B., Rouillard, A. P., et al. 2022, arXiv e-prints, arXiv:2203.14591.  https://arxiv.org/abs/2203.14591
  * Fargette et al. (2021) —. 2021, The Astrophysical Journal, 919, 96, doi: 10.3847/1538-4357/ac1112
  * Feldman et al. (1998) Feldman, U., Schühle, U., Widing, K. G., & Laming, J. M. 1998, The Astrophysical Journal, 505, 999, doi: 10.1086/306195
  * Ferland et al. (1998) Ferland, G. J., Korista, K. T., Verner, D. A., et al. 1998, The Publications of the Astronomical Society of the Pacific, 110, 761, doi: 10.1086/316190
  * Ferland et al. (2017) Ferland, G. J., Chatzikos, M., Guzmán, F., et al. 2017, Revista Mexicana de Astronomía y Astrofísica, 53, 385.  https://arxiv.org/abs/1705.10877
  * Fisher & Sime (1984) Fisher, R., & Sime, D. G. 1984, The Astrophysical Journal, 287, 959, doi: 10.1086/162753
  * Fisk (1996) Fisk, L. A. 1996, Journal of Geophysical Research, 101, 15547, doi: 10.1029/96JA01005
  * Fontenla et al. (2002) Fontenla, J. M., Avrett, E. H., & Loeser, R. 2002, The Astrophysical Journal, 572, 636, doi: 10.1086/340227
  * Fox et al. (2016) Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, Space Science Reviews, 204, 7, doi: 10.1007/s11214-015-0211-6
  * Freeland & Handy (1998) Freeland, S. L., & Handy, B. N. 1998, Solar Physics, 182, 497, doi: 10.1023/A:1005038224881
  * Froment et al. (2015) Froment, C., Auchère, F., Bocchialini, K., et al. 2015, The Astrophysical Journal, 807, 158, doi: 10.1088/0004-637X/807/2/158
  * Galsgaard & Nordlund (1996) Galsgaard, K., & Nordlund, Å. 1996, Journal of Geophysical Research, 101, 13445, doi: 10.1029/96JA00428
  * Geiss (1982) Geiss, J. 1982, Space Science Reviews, 33, 201, doi: 10.1007/BF00213254
  * Geiss & Buergi (1986) Geiss, J., & Buergi, A. 1986, Astronomy and Astrophysics, 159, 1
  * Geiss et al. (1995) Geiss, J., Gloeckler, G., & von Steiger, R. 1995, Space Science Reviews, 72, 49, doi: 10.1007/BF00768753
  * Génot et al. (2021) Génot, V., Budnik, E., Jacquey, C., et al. 2021, Planetary and Space Science, 201, 105214, doi: 10.1016/j.pss.2021.105214
  * Gibson et al. (2003) Gibson, S. E., Foster, D. J., Guhathakurta, M., Holzer, T., & St. Cyr, O. C. 2003, Journal of Geophysical Research (Space Physics), 108, 1444, doi: 10.1029/2003JA009994
  * Gloeckler et al. (1992) Gloeckler, G., Geiss, J., Balsiger, H., et al. 1992, Astronomy and Astrophysics Supplement, 92, 267
  * Golub & Pasachoff (2009) Golub, L., & Pasachoff, J. M. 2009, The Solar Corona
  * Gosling et al. (1995) Gosling, J. T., Birn, J., & Hesse, M. 1995, Geophysical Research Letters, 22, 869, doi: 10.1029/95GL00270
  * Greisen & Calabretta (2002) Greisen, E. W., & Calabretta, M. R. 2002, Astronomy and Astrophysics, 395, 1061, doi: 10.1051/0004-6361:20021326
  * Gringauz et al. (1964) Gringauz, K. I., Bezrukikh, V. V., Balandina, S. M., Ozerov, V. D., & Rybchinskiy, R. E. 1964, Planetary and Space Science, 12, 87, doi: 10.1016/0032-0633(64)90070-4
  * Gringauz et al. (1961) Gringauz, K. I., Kurt, V. G., Moroz, V. I., & Shklovskii, I. S. 1961, Soviet Astronomy, 4, 680
  * Griton et al. (2020) Griton, L., Pinto, R. F., Poirier, N., et al. 2020, The Astrophysical Journal, 893, 64, doi: 10.3847/1538-4357/ab7b76
  * Griton et al. (2021) Griton, L., Rouillard, A. P., Poirier, N., et al. 2021, The Astrophysical Journal, 910, 63, doi: 10.3847/1538-4357/abe309
  * Gudiksen et al. (2011) Gudiksen, B. V., Carlsson, M., Hansteen, V. H., et al. 2011, Astronomy and Astrophysics, 531, A154, doi: 10.1051/0004-6361/201116520
  * Gudiksen & Nordlund (2005) Gudiksen, B. V., & Nordlund, Å. 2005, The Astrophysical Journal, 618, 1020, doi: 10.1086/426063
  * Guhathakurta et al. (1996) Guhathakurta, M., Holzer, T. E., & MacQueen, R. M. 1996, The Astrophysical Journal, 458, 817, doi: 10.1086/176860
  * Habbal et al. (1985) Habbal, S. R., Ronan, R., & Withbroe, G. L. 1985, Solar Physics, 98, 323, doi: 10.1007/BF00152464
  * Hansen et al. (1976) Hansen, R. T., Hansen, S. F., & Sawyer, C. 1976, Planetary and Space Science, 24, 381, doi: 10.1016/0032-0633(76)90051-9
  * Hansteen & Leer (1995) Hansteen, V. H., & Leer, E. 1995, Journal of Geophysical Research, 100, 21577, doi: 10.1029/95JA02300
  * Hansteen et al. (1997) Hansteen, V. H., Leer, E., & Holzer, T. E. 1997, The Astrophysical Journal, 482, 498, doi: 10.1086/304111
  * Hansteen & Velli (2012) Hansteen, V. H., & Velli, M. 2012, Space Science Reviews, 172, 89, doi: 10.1007/s11214-012-9887-z
  * Harten (1983) Harten, A. 1983, Journal of Computational Physics, 49, 357, doi: 10.1016/0021-9991(83)90136-5
  * 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
  * Hénoux (1998) Hénoux, J.-C. 1998, Space Science Reviews, 85, 215, doi: 10.1023/A:1005194215611
  * Hess et al. (2020) Hess, P., Rouillard, A. P., Kouloumvakos, A., et al. 2020, The Astrophysical Journal Supplement Series, 246, 25, doi: 10.3847/1538-4365/ab4ff0
  * Hess et al. (2021) Hess, P., Howard, R. A., Stenborg, G., et al. 2021, Solar Physics, 296, 94, doi: 10.1007/s11207-021-01847-9
  * Higginson et al. (2017) Higginson, A. K., Antiochos, S. K., DeVore, C. R., Wyper, P. F., & Zurbuchen, T. H. 2017, The Astrophysical Journal Letters, 840, L10, doi: 10.3847/2041-8213/aa6d72
  * Higginson & Lynch (2018) Higginson, A. K., & Lynch, B. J. 2018, The Astrophysical Journal, 859, 6, doi: 10.3847/1538-4357/aabc08
  * Hill (2018) Hill, F. 2018, Space Weather, 16, 1488, doi: 10.1029/2018SW002001
  * Hoeksema (1984) Hoeksema, J. T. 1984, PhD thesis, Stanford Univ., CA.
  * Hoeksema & Scherrer (1987) Hoeksema, J. T., & Scherrer, P. H. 1987, The Astrophysical Journal, 318, 428, doi: 10.1086/165379
  * Hollweg (1974) Hollweg, J. V. 1974, Journal of Geophysical Research, 79, 1539, doi: 10.1029/JA079i010p01539
  * Hollweg (1986) —. 1986, Journal of Geophysical Research, 91, 4111, doi: 10.1029/JA091iA04p04111
  * Howard & Koomen (1974) Howard, R. A., & Koomen, M. J. 1974, Solar Physics, 37, 469, doi: 10.1007/BF00152504
  * Howard et al. (2008) Howard, R. A., Moses, J. D., Vourlidas, A., et al. 2008, Space Science Reviews, 136, 67, doi: 10.1007/s11214-008-9341-4
  * Howard & DeForest (2012) Howard, T. A., & DeForest, C. E. 2012, The Astrophysical Journal, 752, 130, doi: 10.1088/0004-637X/752/2/130