where $\mathcal{L}=\mathcal{L}_{\odot}/\sqrt{B}$ is a characteristic
transverse length of the turbulence cascade, that is usually set to match the
typical size of supergranules in the low corona $\mathcal{L}_{\odot}=0.022\
R_{\odot}\sqrt{B_{\odot}}\approx 15,000\ km\sqrt{B_{\odot}}$ (with $B_{\odot}$
in Gauss unit) (Verdini & Velli, 2007). In the above formulation, each Alfvén
wave population ($+,-$) gives a fraction of their energy to the turbulence
cascade that provide a total plasma heating rate $Q_{w}=Q_{w}^{+}+Q_{w}^{-}$.
Although counter-propagating waves naturally coexist in closed-field
geometries, it is not the case in open solar wind solutions for which a
specific strategy is adopted and is presented in section 6.6.2.2.

#### 3.3.3 The Multiple Flux tube Solar Wind Model (MULTI-VP)

MULTI-VP is a 3-D multi-tube magneto-hydrodynamic code that solves for the MHD
properties of the solar wind such as speed, density, and temperature by
solving a set of 1-D MHD conservation equations along individual flux tubes
(see Pinto & Rouillard, 2017). The model is typically run on thousands of
magnetic flux tubes to simulate the entire solar wind escaping the solar
atmosphere. The geometry of these flux tubes is an input of the model and can
be either idealized or extracted from global reconstructions of the coronal
magnetic field, that can be given by for instance full 3-D MHD (see e.g.
section 3.3.2) or PFSS extrapolations (see 3.3.1). The plasma is supposed to
be fully ionized, one-fluid, quasi neutral and isothermal. The energy equation
includes the effect of heating, thermal conduction, and radiative cooling,
which are essential in order to simulate a realistic solar wind mass flux
(Hansteen & Leer, 1995; Pinto & Rouillard, 2017).

The 1-D conservative equations solved in MULTI-VP can be expressed as follows
along the curvilinear abscissa $s$ of the magnetic flux tube (Pinto &
Rouillard, 2017):

$\displaystyle\frac{\partial}{\partial
t}\rho+\nabla_{\parallel}\cdot\left(\rho u\right)=0$ (3.4a)
$\displaystyle\frac{\partial}{\partial
t}u+u\nabla_{\parallel}u+\frac{\nabla_{\parallel}p_{th}}{\rho}+\frac{GM_{\odot}}{r^{2}}cos(\alpha)=\nu\nabla_{\parallel}^{2}u$
(3.4b) $\displaystyle\frac{\partial}{\partial
t}T+u\nabla_{\parallel}T+(\gamma-1)T\nabla_{\parallel}\cdot
u+\frac{\gamma-1}{\rho}\nabla_{\parallel}\cdot
F_{c}=-\frac{\gamma-1}{\rho}\left[\nabla_{\parallel}\cdot
F_{h}+\rho^{2}\Lambda(T)\right]$ (3.4c)

where $\rho=m_{p}n_{p}$ is the plasma mass density, $u$ the plasma bulk
velocity along the flux tube, $T$ the total plasma temperature,
$p_{th}=nk_{b}T$ the plasma thermal pressure, $k_{b}$ the Boltzmann constant,
$G$ the gravitational constant, $M_{\odot}$ the Sun mass, $\alpha=$ the angle
between the magnetic field $\hat{\mathbf{b}}$ and the vertical direction
$\hat{\mathbf{r}}$, $\nu$ the kinematic viscosity, $\gamma=5/3$ the adiabatic
index or ratio of specific heats, $F_{h}$ and $F_{c}$ the mechanical heating
and conductive heat fluxes, and $\Lambda(T)$ the radiative losses. Compared to
the full 3-D MHD equations (eq 3.2) described in 3.3.2, the 1-D set here does
not include explicit terms nor a specific equation for the magnetic field
since it is specified as an input to the model. But the magnetic field still
plays a role on the plasma via the divergence operator defined as:

$\nabla_{\parallel}\cdot(*)=\frac{1}{A}\nabla_{\parallel}(A*)=\frac{1}{A}\frac{\partial}{\partial
s}(A*)=B\frac{\partial}{\partial s}(*/B)$ (3.5)

where $\mathbf{B}=B(s)\hat{\mathbf{b}}$. This relation accounts for the effect
of the expansion of the flux tube’s cross-sectional area ($A(s)\propto
1/B(s)$) on the background plasma, that is sometimes called the geometric
effect.

The thermal energy of the plasma that is primarily heated in the corona is
conducted downward by the conductive heat flux $F_{c}$ until the heat gets
eventually evacuated in the lower and denser layer of the solar atmosphere by
strong radiative emissions. As in WindPredict-AW, the conductive heat flux is
not solved explicitly and hence must be approximated. The usual Spitzer–Härm
approximation is retained in the collisional part of the domain:

$F_{c}=-\kappa_{0}T^{5/2}\nabla_{s}T$ (3.6)

with $\kappa_{0}=9\times 10^{-7}\ erg.cm^{-1}.s^{-1}.K^{-7/2}$.

The dynamics of the winds simulated in MULTI-VP are highly dependent on the
balance between the energy inputs and losses along the flux tubes. As
introduced in section 1.3.2, MULTI-VP assumes that all the energy is
transported from the base of the flux tube to the corona in the form of a
mechanical flux. In the vein of the quasi-stationnary theory introduced in
section 1.4.1, MULTI-VP adopts an ad-hoc formulation of the heating flux
$F_{h}$ that is inspired from the work of Withbroe (1988) and where the
topology of the magnetic flux controls the dynamics of the simulated winds as
described in section 4.3.4.

Similarly to the WindPredict-AW model introduced in section 3.3.2, the usual
formulation of Athay (1986) is used to estimate radiative losses in an
optically thin medium, which is given in section 6.7.1. But the MULTI-VP
implementation includes an additional correction term to account for re-
absorption of the radiated emissions (radiative heating) by the optically
thick plasma in the chromosphere (Pinto & Rouillard, 2017).

### 3.4 Constraining coronal and heliospheric models of the solar wind

We have introduced in section 3.2 and 3.3 tools and models that were exploited
in this thesis by providing context as well as critical information necessary
to better understand observations of the solar wind as it propagates from the
inner corona to the heliosphere. Conversely, we show in this section that some
well-known features of the solar corona and the solar wind can be valuable
assets to improve and constrain these tools and models. As we gain insight and
a better understanding of the features that are observed remotely and measured
in situ we can place ever more stringent tests to the models.

In this context, a collaborative project was initiated between IRAP and
external collaborators, throughout the MADAWG and an international working
group entitled "Exploring The Solar Wind In Regions Closer Than Ever Observed
Before" led by Dr. Louise Harra and financed by the International Space
Science Institute (ISSI). My contribution to these projects were to develop
comparative procedures to evaluate in a systematic manner, coronal and
heliospheric models using white-light observations. These procedures are
discussed in section 3.4.1 and in greater detail in Poirier et al. (2021).

In Badman et al. (2022) we show that other types of observations, including
magnetic sectors measured in situ as well as the size and location of coronal
holes observed in ultraviolet imaging, are essential ingredients to benchmark
all aspects of the models. I have significantly contributed to this work by
providing the evaluation pipeline for the white-light observations. In section
3.4.2 I only give a summary of the main outcomes from this study while the
full published version can be found online (see Badman et al., 2022).

#### 3.4.1 Exploiting White-Light Observations to Improve Estimates of
Magnetic Connectivity (Poirier et al., 2021)

Figure 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).

As introduced in section 1.2.1, WL observations contain a wealth of valuable
information about the solar wind and the overall structure of the solar corona
and heliosphere, which can be compared with models in many different ways. For
the purpose of supporting the Solar Orbiter and Parker Solar Probe missions,
we based our benchmarking on the position of the HCS which draws an overall
picture of the structure of the heliosphere by marking the separation between
the positive and negative magnetic sectors. We exploit white-light synoptic
maps of the streamer belt (see section 1.2.1.4) and we extract an estimate
position of the HCS where the emissions are the brightest in the core of the
streamer belt, that is likely associated to the very dense plasma that
constitutes the HPS (see section 1.2.1.1).

On the modeling side, the HCS can be fetched from any models that provide a
3-D reconstruction of the magnetic field, as the surface where the magnetic
polarity switches sign. In practice we extract this boundary or neutral line
at a specific distance from the Sun, at the outer boundary of MHD models or at
the source surface height of PFSS models where the magnetic field becomes
purely radial.

In a study that I carried out with IRAP colleagues and which is described in
detail in Poirier et al. (2021), I define a comparative procedure to compare
the HCS extracted from the models and the WL observations of the streamer
belt. Basically I detect the core of the streamer rays and trace the line
where the brightness is maximum. I then compare this line with the polarity
inversion line (or neutral line, solid red line) that is extracted from the
models. At last I define a function that attributes a score according the
related positions of the two lines (see Poirier et al., 2021, eq 1-3). An
example is shown in Figure 3.9 for a PFSS model with a source surface height
set at $2.5\ R_{\odot}$.

One of the most relevant applications of this work is the ability to better
estimate how a spacecraft is magnetically connected to the Sun. Accurate
estimates of magnetic connectivities are essential to provide context to in
situ measurements and to link those data with their possible counterpart in
the lower solar atmosphere. The white-light comparative procedure has
therefore been implemented into the MCT introduced in section 3.2.1.

We have also highlighted the importance of using white-light observations to
better constrain the models. The comparative procedure allows us to select an
optimal photospheric map which is a critical parameter for most coronal
models. Almost all studies presented in this thesis have benefited from this
functionality.

#### 3.4.2 Constraining Global Coronal Models with Multiple Independent
Observables (Badman et al., 2022)

Figure 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).  Figure 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).

We have seen in section 3.4.1 that WL observations can be exploited to
evaluate coronal and heliospheric models based on the identification of the
HCS. However observations of the solar corona and heliosphere are not limited
to WL coronagraph images.

In situ measurements of the solar wind bulk properties and of the
interplanetary magnetic field are available at various places in the
heliosphere and distances to the Sun. Spacecrafts that cross the HCS show
clear inversion of the magnetic polarity, that is of the radial component of
the interplanetary magnetic field. When located at different points in the
heliosphere, space-based missions provide direct measurements, although
localized, of the spatial distribution of magnetic sectors over time. Such
temporal measurements can be simulated by inserting virtual probes into 3-D
reconstructions of the interplanetary magnetic field. An example is given in
Figure 3.10 for a PFSS model that is compared against real PSP measurements.

Much closer in, EUV observations of the solar disk provide a distribution of
the magnetically closed or open regions around the Sun. Open regions directly
connect to the interplanetary magnetic field and supply the heliosphere with
solar material. Those regions are cooler due to the energy that is evacuated
by the escaping plasma and usually appear much darker than magnetic closed
regions in EUV. Such dark regions are commonly called coronal holes and can be
directly compared to the open regions given by 3-D reconstructions of the
coronal magnetic field as illustrated in Figure 3.11.

To summarize, one can combine these different sets of observations to evaluate
various aspects of coronal and heliospheric models. Magnetic sectors measured
in situ allow to evaluate the models capability to capture HCS crossing with a
good timing, which can be a critical aspect for space weather applications.
Coronal hole observations are helpful to check whether a model can reproduce
or not the magnetic topology close to the surface, which can be essential for
connectivity science and as a support to the Solar Orbiter and Parker Solar
Probe operations. WL synoptic maps (see section 1.2.1.4) provide an additional
constraint at an intermediate height, between the inner corona and
heliosphere, when the coronal magnetic field stabilizes into a purely
spherical expansion.

The challenge addressed by Badman et al. (2022) then consists in defining a
general framework to compare the models with these three different sets of
observations. I contributed to this work by providing the evaluation pipeline
based on the white-light observations that is almost identical to the one
already presented in Poirier et al. (2021) and introduced in section 3.4.1.
The comparative procedures based on coronal hole observations and magnetic
sectors measured in situ are also explained in detail in Badman et al. (2022).
The reader is invited to refer to the full published version that is available
online.

In Badman et al. (2022) we applied our comparative framework to various models
of which: two PFSS models based on the open source package pfsspy151515pfsspy:
https://pfsspy.readthedocs.io/en/stable/ and on the standard implementation
from GONG (Harvey et al., 1996), the Wang-Sheeley-Arge (WSA) model that
alleviates the current-free limitation assumed in the classical PFSS
reconstructions with a Schatten Current Sheet (SCS) model (Schatten, 1972),
and the Magnetohydrodynamics Around a Sphere (MAS) model that is a full-
fledged 3-D MHD model that solves the magnetic field and the thermodynamics of
the plasma together (Linker et al., 1999; Riley et al., 2001) (see Badman et
al., 2022, for further details on the modeling framework).

In the following I summarize some of the main outcomes from this study, the
reader is also invited to refer to Badman et al. (2022) for further details.

  * •

Most of the time a compromise is needed to satisfy simultaneously the WL,
coronal hole and magnetic sector observations.

  * •

Although the typical $2.5\ R_{\odot}$ source surface height for PFSS models
does not always give the best scores, it is an optimal choice that performs
reasonably well in general.

  * •

Coronal hole observations favor PFSS models with low source surface heights
whereas high source surface heights match better with WL observations of the
streamer belt.

  * •

As expected, the WSA model circumvents this limitation thanks to a correction
that accounts for the effect of the transverse plasma pressure on the coronal
magnetic field, and hence encourages a more pronounced flattening of the HCS
in accordance with WL observations.

  * •

The MAS MHD model performs in general the best and for all three types of
observations. That highlights the advantage of inserting more physics into the
models at the cost of much heavier computations.

## Chapter 4 The fine structure of streamer rays imaged by PSP-WISPR

(This chapter contains material from the Poirier et al. (2020) paper.)

The appearance of coronal rays in white-light imagery taken from Earth’s
heliocentric distances evolves gradually over daily timescales even without
transient releases. This daily evolution is primarily due to the effect of
solar rotation that brings rays located at different heliocentric longitudes
(and latitudes) into the plane-of-the-sky of the observing telescope. On
longer timescales, the positions and brightness of coronal rays respond to
changes in the topology of the coronal magnetic field during the solar cycle
(Golub & Pasachoff, 2009). Despite decades of observations, the physical
mechanisms that produce these rays are still debated.

A source of difficulty resides in the nature of the observations themselves;
any WL image of the solar corona results necessarily from the integration of
sunlight that has been scattered by electrons situated along each line-of-
sight (LOS) of each pixel in the image. This observational constraint
complicates any interpretation of the 3-D structure of streamer rays and the
determination of their source closer to the surface of the Sun.

Because of the LOS effects, it is generally difficult to analyze the detailed
streamer topology from 1 AU. For this reason, the PSP is equipped with a
heliospheric imager WISPR that records the brightness of the corona from a
vantage point that is situated for the first time in the corona itself.
According to the Thomson scattering theory, as an imager gets closer to the
Sun, it becomes sensitive to plasma located over a more narrow region of the
solar atmosphere, acting as a microscope scrutinizing the fine-structure
coronal rays compared with instruments based at near 1 AU, that is discussed
further in section 4.1.

Since a component at least of the slow solar wind (SSW) appears to originate
in streamer rays, the next sections aim at evaluating whether the quasi-
stationary theory of the SSW (see section 1.4.1) can explain the appearance of
coronal rays in WISPR. We first introduce in section 4.1 the Thomson
scattering theory necessary to interpret WISPR observations. The novelty of
WISPR observations compared to typical 1 AU observatories is then discussed in
section 4.3.1. We focus on the 1st PSP encounter that I studied for the
Poirier et al. (2020) paper, and of which the main outcomes are summarized in
section 4.3.7. Finally we take a tour of the subsequent WISPR observations
taken over PSP’s first nine solar encounters, in section 4.4.

### 4.1 The Thomson scattering theory

Because of the LOS effects, WL remote-sensing observations only provide
limited information about the 3-D structure of the solar corona. Additional
inputs from the theory are often necessary to interpret WL observations in
greater detail.

The scattering of sunlight by coronal electrons has been formulated in the
Thomson scattering theory that has been reviewed recently in (Howard & Tappin,
2009; Howard & DeForest, 2012). Following this theory, the total intensity
received along the LOS of a detector from scattered electrons is as follows:

$I_{tot}=\int_{0}^{\infty}n_{e}z^{2}Gdz$ (4.1)

where $I_{tot}$ is the total intensity expressed in solar brightness unit
$B_{\odot}$, $n_{e}$ the electron density, $z$ the path length along the LOS.
The scattering factor $G$ can be expressed as a function of the van de Hulst
coefficients (see Howard & Tappin, 2009, eq 25-28) in which geometric effects
and the fall off of incident sunlight are encoded. Those geometric effects
include for instance the collimation of sunlight with distance from the Sun,
where the Sun is not considered as a point-source.

In the Poirier et al. (2020) paper, we argued that the brightness in WISPR-I
images originates from light mostly scattered by electrons situated close to a
surface called the "Thomson sphere" and that is located ahead of PSP
(Vourlidas & Howard, 2006). In fact, this effect is further described in
Howard & Tappin (2009) and Howard & DeForest (2012) who show that while the
scattering efficiency is minimal at the Thomson sphere, it is largely offset
by the combined fall off of the incident sunlight and of the electron density
with distance from the Sun (both as $1/r^{2}$). The Thomson sphere is
therefore the locus of points along all LOS where density and incident
brightness from the Sun are maximised and therefore produce a total scattered
intensity ($I_{tot}$) that is _maximum_ in this region. However, Howard &
DeForest (2012) warn of the fact that this _maximum_ at the Thomson sphere is
greatly smeared out. Therefore, a detector such as WISPR would not be only
sensitive to electrons that are concentrated near the Thomson sphere but
rather, to a broader region on either side of the Thomson sphere that is
called the "Thomson plateau" within which local intensities contribute almost
equally to the total light received by the detector. That is illustrated in
Figure 4.1 where the shaded areas account for $99\%$ of the total brightness
received by PSP WISPR-I and the SOHO LASCO-C2 coronagraph.

Figure 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.

Because brightness contributions can be significant along an extended portion
of the LOS within the Thomson plateau and even beyond for particularly dense
structures, it is often difficult to segregate which density structure
contributes to the total brightness. To alleviate this difficulty I adopt for
the following analyses a forward modeling strategy, where I exploit global 3-D
MHD simulations of the solar corona and wind to produce synthetic WL
observations. Then I carry out an in-depth analysis of each local contribution
to the total brightness, where one method consists in separating the
foreground and background contributions from either side of the Thomson sphere
and in generating a synthetic image for each one of these sub-domains as done
in Poirier et al. (2020).

### 4.2 Producing WISPR synthetic images

Two methods are generally employed to interpret the shape and origin of the
structures imaged by WL instruments, that are often based on the Thomson
scattering theory introduced in section 4.1.

First, inversion techniques derive the 3-D structure of the corona directly
from the imagery, they can infer the electron density by rotational tomography
(see e.g. Kramar et al., 2016a; Morgan & Cook, 2020) and even the magnetic
field direction via spectropolarimetry (see e.g Kramar et al., 2016b).

The second method, which we follow throughout this thesis, is the forward
modeling technique that consists in producing synthetic WL observations from a
given 3-D coronal structure. A comparison of the synthetic versus real
products then provide a baseline to interpret the imaged structures. A great
benefit of using this method is that we start from a modelled (known) coronal
structure and we can exploit it to understand how local density structures may
contribute to the total light received by a detector. In the Poirier et al.
(2020) paper we show that even simplistic PFSS reconstructions of the coronal
magnetic field alone (see section 3.3.1 for an introduction on PFSS models)
already give valuable insight to understand WL observations. We also resort to
extensive 3-D simulations such as MULTI-VP (see section 3.3.3) and
WindPredict-AW (see section 3.3.2) to reproduce a solar corona from density
cubes that are influenced by the forming solar wind. The Thomson scattering
theory (see section 4.1) is then followed to compute a synthetic image where
the simulated electron density is integrated along each LOS of an instrument.

For the construction of the synthetic WL images we use the following
procedure. For each WISPR-I image, we calculate the heliographic coordinates
of all pixels situated on a reference surface that we choose to be the Thomson
sphere. We define LOSs that start from PSP, pass through and beyond the
Thomson surface. Along each LOS, we interpolate the electron density of the
simulated 3-D datacube produced by any global MHD model such as MULTI-VP or
WindPredict-AW, and we use the Thomson scattering theory to calculate a total
intensity using equation 4.1. There are three major time-dependent effects
that are involved in this process.

First, because PSP-WISPR is a detector placed on a rapidly moving observatory
sweeping extended regions of the solar corona in only a few days, together
with a rapid variation of its distance to the Sun, the FOV of WISPR must be
updated very regularly. In order to carry out an accurate comparison between
observed and simulated WISPR images it is therefore critical to keep an
accurate tracking of WISPR’s pointing through usage of the World Coordinate
System (WCS).

Second, there is the time-dependency associated with the photospheric magnetic
map that is used as the inner boundary to all 3-D simulations of the solar
corona. For the sake of computational tractability we can generally afford to
run only one single full 3-D MHD simulation within an observing window of
WISPR. Such simulations are commonly termed as "stationary" or ”static”
because in that case only one photospheric magnetic map is set at the inner
boundary, that is maintained unchanged over the whole simulated time interval.

However, time-dependent simulations such as Multi-VP and WindPredict involve a
"dynamic" phase during which they evolve from an initial state until a
permanent (stationary) regime is reached. The latter constitutes the third
time-dependency that needs to be considered throughout the building process of
WISPR synthetic images.

In the case treated in Poirier et al. (2020) and in section 4.3 where we focus
on the quasi-stationary behavior of the solar wind, we run our forward
modelling technique on a single simulation snapshot but with LOSs that are
continuously updated throughout the observing window of WISPR-I.

To address the variability of the solar wind as done in Griton et al. (2020)
and in section 5.4 and 5.3, there remains an essential task that consists in
calibrating the timeline of the simulations that are used as inputs to build
the synthetic images. Because the cadence of the simulation outputs does not
necessarily match the one of WISPR, I select for each synthetic requested
WISPR image, a simulation snapshot that is the closest to the real observation
time. I prefer this method rather than performing time interpolation of the
3-D datacubes to avoid introducing non-physical structures in the simulations,
especially when simulating the propagation of density perturbations in the
solar wind (see section 5.4 and 5.3). Therefore when the cadence of the
simulation outputs is lower than WISPR ($\simeq 13-16\ \rm{min}$ for the cases
considered in this thesis) the same simulation snapshot can be re-used several
times in the building of subsequent synthetic images.

### 4.3 Analysis of the first observations of WISPR with the MULTI-VP model
(Poirier et al. 2020)

In this section, I focus on the first WISPR-I observations that I analysed in
detail in Poirier et al. (2020). Although PSP approached the Sun at much
closer distances during the next encounters, we show in Poirier et al. (2020)
that the WISPR observations taken during the first passage already enrich our
knowledge of the nature and structure of the slow solar wind. In the following
I only give a summary of the Poirier et al. (2020) study, the reader is
invited to refer to the full version of the paper published online for more
detail.

#### 4.3.1 Comparison with 1 AU observatories

We exploit level-3 WISPR images that capture primarily the K-corona which is
the photospheric light scattered by coronal electrons (see section 3.1.3). We
focus on the WISPR-I inner telescope where the WL signature from streamers is
the most distinguishable (see Figure 3.4 for a comparison of the WISPR-I and
WISPR-O FOV).

Comparing images from different instruments can be a challenging task,
especially from separate observatories. As seen in section 4.1, remote-sensing
instruments are in practice sensitive to local emissions from a broad region
that is called the Thomson plateau (Howard & DeForest, 2012). This is
particularly true for observations taken from near 1 AU. Figure 4.1 shows a
comparison of the regions likely observed by WISPR-I and LASCO-C3 on 2018
November 1 as predicted by the Thomson scattering theory. It shows that
WISPR-I and LASCO C3 imaged a similar coronal region. As we can see in Figure
4.1, the brightness recorded by WISPR-I originates from a broad region along
the LOS that extends from PSP to well behind the Thomson sphere. As PSP gets
closer to the Sun, this region shrinks. Nevertheless, Figure 4.1 shows that
WISPR-I already records plasma brightness from a smaller region than LASCO C3.

Figure 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).

Considering the above complications, we give in Figure 4.2 an overview of the
zoomed-in view offered by WISPR-I on the typical streamer structures observed
from near 1 AU observatories. In Figure 4.2, we compare WISPR-I with LASCO C3
and STEREO COR-2A observations for two selected days when the comparison was
optimal. On 2018 November 1 (top panels), the two brightest rays situated just
a few degrees north and south of the equatorial plane are imaged by both
spacecraft. In addition to these two bright features, a number of much fainter
rays are also visible at PSP, unveiling an apparent complex structuring of the
corona, which is not resolvable in LASCO-C3 images. Besides, we note that, on
2018 November 1, WISPR-I observations had not reached their highest resolution
for this encounter yet, as PSP was located at $\simeq 51\ R_{\odot}$, e.g.,
$\simeq 16\ R_{\odot}$ away from perihelion.

On 2018 November 6, the best alignment was with STA, and Figure 4.2 (bottom
panels) shows a comparison of COR-2A and WISPR-I images. The pair of bright
rays located just above the equatorial plane is not clearly distinguishable in
COR-2A despite the adequate resolution of the instrument to resolve such
structures. This is likely a LOS integration effect, suggesting that 1 AU
observatories may be not capturing the fine coronal structure. The zoomed-in
view that WISPR-I offers and the shrinking of the Thomson plateau, make it
possible to observe coronal rays in finer detail compared with near 1 AU
observations.

#### 4.3.2 WL synoptic map of the first WISPR-I observations

Following a similar approach as the one employed for years to exploit 1 AU
coronagraphic images (see section 1.2.1.4), I built synoptic WL maps of
WISPR-I images. However, because PSP is a fast-moving observatory that is
sometimes in corotation with the rotating solar corona, a latitude versus time
format was found in general to be more convenient than the usual latitude
versus Carrington longitude format. To illustrate this point, a visual
comparison is given in Figure 4.3 between a WISPR-I synoptic map that combines
images taken during the entire 1st encounter (i.e. about 10 days of coverage)
and the associated LASCO-C2 map assembled over half a synodic period (about
$27/2=13.5$ days). Therefore WISPR mostly scanned the same region of the solar
corona during the whole observing window. That is however less pronounced in
the recent closest approaches of PSP, where the quasi-corotating phase is
greatly reduced thanks to reduced periapsis distances.

Figure 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).

The building process of the WISPR-I maps is detailed in Poirier et al. (2020)
and is illustrated in Figure 16 of this paper. In summary I extract in each
WISPR-I image all pixels that are located at a specific distance to the Sun,
and then I map those pixels in a latitude versus time format. In this process,
one has firsthand to project the image on a specific plane of reference.
Motivated by the Thomson scattering theory introduced in section 4.1, a
natural choice is the Thomson sphere at the core of the broad Thomson plateau
where WL emissions are roughly expected to occur. For 1 AU observatories and
targets that are close to the Sun, one could simply use the plane-of-the-sky
to a good approximation.

Figure 4.4 presents a WISPR-I latitude versus time synoptic map for the first
encounter, combining about 10 days of observations. At first sight one can
distinguish many interesting features. We give a close-up look at those
features in the next sections and we investigate their possible origin.

Figure 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.

#### 4.3.3 Interpreting the large-scale structures

As mentioned earlier, PFSS reconstructions of the coronal magnetic field are
rather simplistic but valuable as they already provide some clues to interpret
the large-scale structures observed by WISPR.

In Figure 1.8(a), we present the results of a PFSS extrapolation based on the
GONG-ADAPT magnetogram produced for the 1st PSP encounter on 2018 November 5
at 12:00 UT. In this view, WISPR-I was observing plasma that originated near
the west limb of the Sun (i.e. on the right-hand side of the image). The open
and closed magnetic field lines are shown in yellow and orange, respectively,
and the position of the coronal neutral line, i.e. the origin of the HCS, at
the source surface is given by the red line.

The streamer stalk denoted by the black arrow is associated to the bright band
(orange arrows) located at $\approx 5\ \rm{deg}$ north in the WISPR-I map
shown in Figure 4.4.

Figure 1.8(b) presents a zoomed-in view of panel (a) centered to the south of
the streamer. The EUV map reveals the presence of an isolated coronal hole
(indicated by a blue arrow) at the southern edge of the regular streamer, and
that is associated with a low-lying cusp-like structure. This is a typical
pseudo-streamer, and it likely produces the bright band (green arrows) located
approximately at $\approx 10\ \rm{deg}$ south of the equator in the WISPR-I
map shown in Figure 4.4.

From November 3rd, the northern streamer starts to split in two individual
thin strands (red arrow in Figure 4.4) that persist until November 9th. The
splitting of streamer rays has already been observed from 1 AU observatories
but to a larger extent, and was generally associated to an inclination of the
HPS with the LOS of the observer (see section 1.2.1.4). Here the splitting
produce two strands of $\approx 5\ \rm{deg}$ each, that is about the typical
width of the HPS (see section 1.2.1.1 and Winterhalter et al. (1994)). Hence
WISPR may be the first heliospheric imager to unveil the thin and dense HPS
that is confined within the core of streamers. A visual inspection of the PFSS
reconstruction shown in Figure 1.8(a) reveals a HPS that is slightly warped on
the right-hand side of the figure, and that may produce the splitting of the
streamer rays observed by WISPR. To examine the origin of this feature in
detail, we exploit in the next sections additional inputs from high-resolution
3-D MHD simulations.

On November 1st and 2nd, large-scale fluctuations of the northern streamer
have been detected by WISPR-I. They were found to be associated to the
deflection of the streamer induced by the passage of two coronal mass
ejections (CMEs). These CMEs were slow and dragged along in the slow solar
wind (Hess et al., 2020; Rouillard et al., 2020c). I contributed significantly
to the Rouillard et al. (2020c) paper in which I tracked the slow CME imaged
by WISPR on November, 1-2 2018 as it propagated throughout the corona and
heliosphere, using both direct triangulation methods and a physical based
model.

Corrugations at the edge of the northern streamer can also be seen while being
much smaller at other times in the WISPR-I map, especially on November 3rd. It
is important to notice that PSP was in a quasi-corotation phase during this
period and therefore these perturbations should be transient structures
propagating along with the slow solar wind. Some of them have been associated
to density fluctuations expelled along the edge of streamers (Rouillard et
al., 2020a) that have been measured later on at PSP in situ. The presence of
such perturbations may be the signature of magnetic reconnection processes
occurring at lower heights and which generate density fluctuations that are
dragged in the slow solar wind, as stated in the dynamic theory introduced in
section 1.4.2 and that is further discussed in section 4.4 in lights of the
recent WISPR observations.

#### 4.3.4 Numerical setup

To interpret WISPR-I observations, I exploit the forward modeling method
already discussed together with a high-resolution simulation of the solar
corona and of the solar wind using the MULTI-VP MHD model introduced in
section 3.3.3 (see also Pinto & Rouillard, 2017). For this purpose, MULTI-VP
was run on thousands of magnetic flux tubes to simulate the entire solar wind
escaping the solar atmosphere. The inner boundary of the simulation domain is
at the photosphere and extends typically to about $30\ R_{\odot}$. For the
purpose of this study, the outer boundary of the MULTI-VP simulation was set
to $90\ R_{\odot}$ in order to include the brightness contribution of
electrons situated far behind the Thomson sphere.

The MULTI-VP model aims at simulating the solar wind as it forms along an
expanding magnetic flux tube. Therefore the heating flux $F_{h}$ that appears
in the energy equation (eq 3.4c) is a function dependent of the flux tube’s
cross sectional area $A(s)$ and of the magnetic field amplitude at the base of
the flux tube $|B_{\odot}|$ (Pinto & Rouillard, 2017), with a formulation
inspired from Withbroe (1988):

$F_{h}=12\times
10^{5}|B_{\odot}|\left(\frac{A_{\odot}}{A}\right)exp\left[-\frac{s-R_{\odot}}{H_{f}}\right]\
\ erg.cm^{-2}.s^{-1}\ (\text{with }|B_{\odot}|\text{ in }G)$ (4.2)

where $H_{f}=5f_{ss}^{-1.1}R_{\odot}$ is the heating scale height. The
expansion factor $f_{ss}=A_{ss}/A_{\odot}(r_{\odot}/r_{ss})^{2}$ is computed
between the base of the flux tube at $R=R_{\odot}$ and the height $R=R_{ss}$
in the corona from which the magnetic field becomes purely radial. In the case
where PFSS reconstructions are used to specify the magnetic field topology to
MULTI-VP, this height is the height of the PFSS model’s outer boundary called
the source surface (see section 3.3.1).

The MULTI-VP solar wind model runs on coronal magnetic fields that can be
derived by potential field source-surface (PFSS, see section 3.3.1)
extrapolations of magnetograms measured by different observatories. In this
study, we used photospheric magnetic field maps from WSO (see section 3.1.4.1)
and those computed by the ADAPT model (see section 3.1.4.3) based on GONG
magnetograms (see section 3.1.4.2). Hereafter we will refer to these two sets
of simulations as MVP-WSO and MVP-ADAPT.

The MULTI-VP simulations have a grid adapted to the input magnetogram
resolution. The MVP-WSO run has a grid of 5∘ resolution in both latitude and
longitude, and the MVP-ADAPT run is at a higher grid resolution of 2∘. The
MVP-ADAPT simulation provides 2.5 times finer resolution compared with the
MVP-WSO run; this provides a significant impact on the synthetic images which
will be discussed later on.

#### 4.3.5 WISPR-I synthetic images

MULTI-VP provides densities in all regions of space occupied by open magnetic
fields. Using those cubes, we produce synthetic WL imagery by applying the
theory of Thomson scattering (see Howard & Tappin, 2009) introduced in section
4.1 and the method described in section 4.2. These synthetic images are
compared with WISPR observations, providing a baseline to interpret the imaged
structures.

Figure 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.

Figure 4.5 presents examples of synthetic WL images produced from the MVP-WSO
(panel a) and MVP-ADAPT (panel b) runs. Panel (c) shows the corresponding
WISPR-I image on 2018 November 3 at 06:55 UT for comparison with the simulated
images. For illustration purposes, I rescaled the intensity for both synthetic
WL images to enhance the visibility of the streamer rays. As a consequence, in
the following sections I only perform a qualitative comparison of the features
aspect and position between synthetic products and real observations.

Comparing the synthetic images between the two runs, I find, as explained
earlier, similarities in the features observed near the equator, but there are
also some striking differences that are presented in detail in Poirier et al.
(2020).

Overall, the high-resolution MVP-ADAPT simulation produces a more detailed
view of the streamers and reveals substructures that are absent in the MVP-WSO
simulation. The streamer rays (orange arrows) are reproduced by both MVP-WSO
and MVP-ADAPT. This is also the case for the pseudo-streamer identified in
Figure 1.8(b) and here pointed with the green arrows.

The splitting of the northern streamer into two lanes observed in the WISPR-I
map (orange and red arrows in Figure 4.4) is also well reproduced in both
simulations but it appears at a slightly different latitude compared to the
observations. It is probably related to the inherent uncertainties in polar
field measurement. At first sight, it can be hard to identify the two-lane
splitting within the multiple adjacent thin rays visible in the MVP-ADAPT
image. Section 4.3.6 will remove this ambiguity and confirm the identification
made here.

Most of the differences between the two synthetic WL images of Figure 4.5
appear in the northern streamer, where the higher-resolution MVP-ADAPT
simulation shows an additional subdivision of the northern streamer into at
least two separate rays (orange and blue arrows). However, this subdivision is
not discernible in the WISPR image (panel c). The more diffuse streamer rays
that appear in the MVP-WSO image could result from multiple rays unresolved in
this lower resolution simulation but clearly seen in the higher-resolution
MVP-ADAPT run (panel b). Overall, the MVP-ADAPT simulation captures better the
finer-scale structures of coronal rays that are also observed by WISPR-I in
the streamer and pseudo-streamer. However, the MVP-ADAPT synthetic image tends
to show additional streamer rays in the northern streamer that are not seen
clearly in the WISPR-I image, and that we discuss in more detail in the next
section.

#### 4.3.6 Interpreting the small-scale structures

In the following, we further exploit the simulations to interpret the
properties of coronal rays observed by WISPR-I. The analysis of all the
features appearing in the real/synthetic images and maps has proven to be a
complex task. Mainly, the LOS effects make it difficult when we need to
identify the source regions responsible for the different rays visible in the
synthetic maps or images.

To get a better insight of where the bright rays are situated relative to the
Thomson sphere, I recomputed the synthetic images by splitting the integration
path along each LOS in two separate domains. The first domain covers only the
region from the observer (PSP) up to the Thomson sphere (“foreground” region),
while the second extends far out and beyond the Thomson sphere (“background”
region). In Figure 4.6, the initial WL synthetic image from the MVP-ADAPT run
(Figure 4.5(b)) is again plotted in panel (a) along with its associate
foreground (panel b) and background (panel c) subimages. The foreground
subimage (panel b) looks very similar to the full image (panel a) and
contributes to most of the diffuse brightness of the broad northern and
southern rays (annotated by the orange and green arrows). In contrast, the
background subimage (panel c) only reveals the thin and bright central ray at
a few degrees south (marked by the red arrow), which is not visible in the
foreground subimage. This is a clear hint that the full synthetic images
consist of rays located over an extended region in front of and beyond the
Thomson sphere. As already discussed in section 4.1, there are indeed non-
negligible contributions to the total brightness on both sides of the Thomson
surface that are included in the LOS integration domain.

Figure 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.  Figure 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.

In order to understand these subimages in more detail, I investigated further
the 3-D topology of the corona. For that we use the density in the simulations
as a proxy to visualize in 3-D these bright structures. Figure 4.7(a) shows a
Carrington map of the simulated density at $15\ R_{\odot}$ for the MVP-ADAPT
run. The white dashed line represents the PSP projected trajectory, and the
cyan square is the PSP position on 2018 November 3 at 06:55 UT. The magenta
line separates the foreground from the background integration domain. Figure
4.7(b) shows the Carrington map of the magnetic polarity from the MVP-ADAPT
run. Comparing both Carrington maps, we can see that the densest solar wind
forms in this simulation around the HCS and likely constitutes the HPS as
introduced in section 1.2.1.1. Dense wind also forms along arcs that connect
different parts of the HCS; these correspond to the cusp of pseudo-streamers
(see section 1.2.1.3). There are also patches of slightly more tenuous slow
winds extending away from the HCS. These are likely the product of
intermediate expansion factors of the coronal magnetic field that lead to
winds with intermediate speeds according to the quasi-stationnary theory
introduced in section 1.4.1.

One can see that the foreground domain is dominated by an intense and extended
density enhancement associated with the northern streamer. This is consistent
with the foreground subimage (Figure 4.6(b)) as PSP is close in space to this
high-density region. The instrument records a significant increase in
brightness over a broad region extending northwards from near the equator. PSP
is therefore imaging different regions of the streamer from a vantage point
that is just below the HPS.

We confirm that the pseudo-streamer rays (green arrow in Figure 4.5) originate
near the unipolar cusp identified in Figure 1.8. This region is located south
of the HCS and well in front of the Thomson sphere. Figure 4.7(a) reveals that
the wind forming in that region is not as dense as the simulated streamer
flows and would indeed appear less bright in the images. Consequently, this
less dense region appears in the foreground subimage as a much fainter diffuse
region in the lower half of the image (see the green arrow in Figure 4.6 panel
b).

On the contrary, the background integration domain covers a region of much
lower density with a thin and flat layer of local density enhancement
associated with the HPS. Imaging this east-west oriented structure from a
larger distance explains why the streamer appears this time as a very thin and
bright streamer ray visible in the background subimage (indicated by a red
arrow in Figure 4.6(c)).

Therefore, the full synthetic image (Figure 4.6(a)) shows both the broad and
diffuse light scattering emission of the foreground as well as the thin ray of
the same northern streamer that flattens at lower latitudes behind the Thomson
sphere. We must conclude that comparing WISPR-I images by simply taking slices
of simulations near the Thomson sphere is inadequate, and a complete analysis
of WISPR-I images requires an analysis that integrates foreground and
background features.

Figure 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.

Similarly to the WISPR-I map shown in Figure 4.4, we can build synoptic maps
from individual synthetic images generated from the MVP-ADAPT simulation. That
is shown in Figure 4.8(a) where the associated contributions from the
foreground and background are also given. Similarly to the analysis done on
the WL synthetic subimages, the foreground submap (panel b) contributes to
most of the bright structures seen in the full map (panel a). An exception is
the thin southern bright stripe already mentioned and located in the
background submap (red arrow in panel c). These submaps give us the last hint
to understand the origin of the apparent two-lane splitting of the northern
streamer ray that we identified earlier, which is annotated by the orange and
red arrows and visible from 2018 November 3 to 9 in the WISPR-I map (Figure
4.4).

From the decomposition of an MVP-ADAPT WL synthetic image into two subimages
(foreground and background as shown in Figure 4.6) as well as the analysis of
the polarity and density slices of the MVP-ADAPT run, we interpret this
splitting as the result of an LOS integration effect from two very distinct
regions. The initial northern streamer, slightly folded, visible in the
foreground and located at a few degrees above the equator flattens further in
the background to remain flat at a few degrees below the equator. This flat
part of the streamer in the background is highly visible in the polarity slice
(see Figure 4.7(b)) from $\sim$30∘ to $\sim$100∘ Carrington longitude.
Therefore, from the modeling, we can identify the apparent streamer splitting
in the WISPR-I map to be the result of an LOS integration effect of a slightly
folded HPS extending at and beyond the Thomson sphere.

The density and polarity maps (Figure 4.7(a) and (b)) also reveal that smaller
folds of the HCS situated in the foreground could create the main subdivision
of the northern streamer seen in the MVP-ADAPT synthetic image (see e.g. the
blue arrow in Figure 4.5(b)). This may also include the rays located at the
highest latitudes, which drift towards higher latitudes as the effect of PSP
plunging into these structures. Such features are hardly distinguishable in
the WISPR-I map shown in Figure 4.4, but they appear much more clearly in the
recent encounters (see Figure 4.9).

Several dense flux tubes can also be observed in the southern region in the
density map between Carrington longitudes 320∘ and 340∘. They are isolated
flux tubes with higher densities that produce additional thin rays in the
southern part of the synthetic foreground image (Figure 4.6). Others similar
patches can be seen all over the Sun but they are hardly distinguishable in
the dense northern streamer due to the saturation of the color scale. These
dense flux tubes are likely the product of different magnetic conditions at
the base of the flux tubes and that in turn affect the mass flux of the winds
simulated in MULTI-VP. This is an innate characteristic of MULTI-VP that we
already discussed at several occasions and that is built in the heating
prescription adopted in MULTI-VP. Therefore the fine pattern of the
photospheric magnetic map offered by the GONG-ADAPT magnetogram is reflected
in MULTI-VP by a myriad of winds with various mass fluxes. These individual
slow solar wind streams are, nonetheless, not clearly visible in the WISPR-I
observations and may be mixed up with the rays induced by the small warps of
the HPS discussed in the previous paragraph. Still, we shall point out that
similar fine-scale striations of the streamers can be seen in the more recent
encounters where the PSP-WISPR "microscope" effect was enhanced by taking
images from a vantage point that was well inside the solar corona. Whether or
not these individual slow solar wind streams predicted by the quasi-
stationnary theory and simulated by MULTI-VP are realistic or not will have to
be investigated in detail in lights of the more recent WISPR-I observations,
that is left for future studies.

#### 4.3.7 Conclusion

The previous section was rich in details, therefore we shall give a summary of
the main outcomes and limitations from this study and how these results may
affect our understanding of the slow solar wind. The reader is invited to
refer to Poirier et al. (2020) for an extensive discussion on the limitations
of this study.

Modeling of the solar wind and corona has been extensively used in this study
not to perform direct comparisons with WISPR-I images but to help us
understand the origin of the different coronal rays observed by WISPR-I during
the first encounter of PSP. We showed the need of having a fine-enough
simulation (e.g. MVP-ADAPT) in order to reproduce even the smallest features
observed by WISPR-I.

The high-resolution MVP-ADAPT simulation (with 2∘ angle resolution) allowed us
to give further context and potentially explain the apparent splitting of the
brightest streamer rays seen by WISPR-I. Our results suggest that this
originates from the LOS integration along an extended region where the HPS
undergoes a latitude change. Our model shows that the HPS latitude changes
from $\sim$10∘ to $\sim$-5∘ over a $\sim$60∘ Carrington longitude span at the
region where WISPR-I observations were made. The effect of such folds in the
HPS have been known since LASCO observations to produce separated streamer
rays as discussed in section 1.2.1.4 (see also Sheeley et al., 1997; Wang et
al., 1998). The novelty in WISPR-I observations is to act as a microscope to
catch even small latitudinal changes in the HPS, allowing a more detailed
evaluation of current coronal models.

This microscope effect manifests in WISPR-I images by revealing very thin
streamer rays of about $3-5^{\circ}$ in angular width. That is about the
typical width of the HPS measured in situ (Winterhalter et al., 1994), and
hence WISPR-I may be the first WL heliospheric imager to unveil the thin HPS.
As we shall see in section 4.4, that is even more remarkable in the recent
observations performed by WISPR-I.

### 4.4 A quick tour of subsequent WISPR-I observations

Figure 4.9:  WISPR-I WL synoptic maps for encounters 1, 2, 4, 5, 6, 8 and 9.

In figure 4.9 we show at a glance many WISPR-I observations represented as
latitude versus time maps for seven of the first ninth passages of PSP to the
Sun (same format as for Figure 4.4). A general observation is that the
streamer belt remained more or less in the equatorial plane for the first
seventh orbits, that is typical of a period with low solar activity. This was
also noted in the plasma measurements taken in situ at PSP which show a high
occurrence of slow and dense solar winds coming primarily from the edge of
streamers, as discussed in section 5.1 for encounter 2.

Encounters 8 and 9 reveal a much more structured corona with multiple
striations in the coronal rays. That may be the result of moving towards the
ascending phase of the solar cycle with a HPS that becomes significantly
warped. Whether all these striations are due to folding in the HPS is a
question that still remains to be elucidated and that we partially addressed
in section 4.3.6. In lights of the forward modeling that we performed for the
1st encounter, we also suggest that some of these striations may correspond to
individual slow wind streams of distinct origins and hence to winds that have
a different history in their formation (e.g. for the deposit of energy). We
will see in section 5.3.2 that at high solar activity, the solar atmosphere
does not solely consists of a HPS but rather forms a complex network of dense
layers that are associated with pseudo-streamers and more generally the QSLs
(S-web, see section 1.4.2.4). Therefore, WISPR-I may capture some of these
QSLs that complicates the interpretation of the observations. A modeling of
encounter 9 has been attempted in section 5.3.2 using a 3-D MHD simulation
from the WindPredict-AW model. We note that a fine level tuning of the model
would be required to match the actual WISPR-I observations. The exploitation
of WISPR-I images together with the use of synthetic imagery methods
constitute a baseline to improve the accuracy of the current models of the
solar atmosphere to an unprecedented level of detail.

Furthermore, one can notice a feature that is present in almost all WISPR-I
maps shown in Figure 4.9. I refer to the dark bands that split the bright
streamers in two separate rays. That feature generally results from a slight
inclination of the streamer belt (or HPS) with respect to the line-of-sight of
the observer, that has been explained in section 1.2.1.4 and examined in
detail during the 1st encounter in section 4.3.

One can also see rays that drift towards high latitudes. Such drifting rays
were poorly visible during the 1st encounter (Figure 4.4) but they become well
visible from encounter 6th, and is a 3-D perspective effect of PSP plunging
directly into the streamers (see also Liewer et al., 2019). Furthermore, this
feature will be more visible in the synthetic WISPR-I imagery performed in
section 5.3.2, from two 3-D MHD simulations of encounter 5 and 9 based on the
WindPredict-AW model.

Occasionally, one can see some corrugations in the streamer belt. As noted for
the 1st encounter (see section 4.3.3), some are related to the deflection of
streamers induced by the passage of CMEs. Other corrugations in the streamers
seen in WISPR have been associated to density fluctuations expelled along the
edge of streamers and measured in situ at PSP (Rouillard et al., 2020a), and
that present flux rope signatures as discussed in section 5.2. These small-
scale perturbations of the streamer become more and more preponderant in the
subsequent encounters as PSP gets closer to the Sun and as the microscope
effect of WISPR accentuates. That favors a theory of the formation of the slow
solar wind that is ruled by systematic magnetic reconnection events as
formulated in the dynamic theory introduced in section 1.4.2 and that is
discussed in more detail in chapter 5 in light of the recent PSP observations.

### 4.5 Conclusion

In the present chapter, novel remote-sensing observations from PSP-WISPR were
exploited showing a structuring of the solar wind at much smaller scales than
what has typically been achieved from 1 AU observatories. Because of the
peculiarity of PSP that dives deeply in the corona with unprecedented speeds,
I had to develop new methods to interpret the WL images taken by WISPR. At
several occasions I pointed out the importance of considering the Thomson
scattering theory (introduced in section 4.1) to interpret the nature of the
WL emissions captured by WISPR. I showed that even for WISPR that images at
the corona up close, it remains difficult to locate precisely the regions
where WL emissions occur. Fortunately, the closer-up PSP approaches the Sun,
the thinner is its area of sensitivity. The microscope effect of WISPR was
even more remarkable in the recent observations presented in section 4.4, that
reveal a fine-striation of the SSW at even smaller scales compared to the
first observations that we analysed in detail in Poirier et al. (2020)
(section 4.3).

I developed a new method to facilitate the interpretation of WISPR images. By
stacking images over time while accounting for PSP’s position and WISPR’s
pointing, I built synoptic maps that show at a glance the structure of the
corona imaged by WISPR. By identifying phases where PSP is in quasi-corotation
with the rotating solar corona, we showed that the time-dependent and spatial-
dependent components of the SSW can be disentangled. That way I could focus in
this chapter on the analysis of the fine-structure of a likely quasi-
stationnary SSW component imaged by WISPR, while the more dynamic component
SSW is discussed in detail in chapter 5.

Using a high-resolution MHD simulation from the MULTI-VP model, patches of
dense solar winds have been found to form over the entire source region of the
slow solar wind roughly below $40^{\circ}$ of heliographic latitude. This
points towards a "texture" of the solar wind that finds its root in the highly
structured nature of the coronal magnetic field. That was achieved following
the quasi-stationnary theory (section 1.4.1) where the energy source that
drives the solar wind is controlled by the topology of the coronal magnetic
field. By employing my forward modeling technique, I produced synthetic WISPR
images where these patches reflected in a fine striation of the streamer rays.

At the largest scales, this was interpreted as a small latitudinal shift of
the location of the HPS between the foreground and background. Although this
effect has long been known since the early stage of SoHO which orbits at near
1 AU, WISPR is able to capture irregularities in the shape of the HPS with
unprecendented details.

However, synthetic WISPR images built from the MULTI-VP simulations showed a
fine-structuring of the SSW at even smaller scales. We concluded that this is
likely the result of two distinct contributions: sub-foldings of the HPS at
smaller scales ($\lesssim 5^{\circ}$), and individual denser slow wind streams
that take root in regions of distinct origins. The presence of such fine rays
has been shown in highly processed eclipse images (see e.g. Figure 1.5 or also
November & Koutchmy (1996); Druckmüller et al. (2014)) and dominates
significantly the recent images taken by WISPR (see section 4.4). More
generally, this fine striation could be the manifestation of a SSW that forms
along a highly-structured S-web (section 1.4.2.4), where WL emissions are
expected not only from the HPS but from the QSLs at a much larger extent. The
last observation carried on by WISPR will have to be investigated in detail to
identify precisely the sources of the observed fine-structuring of the SSW,
that is left to future studies.

## Chapter 5 Variability of the slow solar wind

In this chapter we will first show in section 5.1 how the recent in situ
measurements taken by PSP support both the quasi-stationnary and dynamic
theories of the SSW formation introduced in section 1.4.1 and 1.4.2
respectively. Then we inspect in section 5.2 the intermittent nature of the
SSW (see section 1.2.2.1) from the recent close-up perspective offered by
WISPR. The WindPredict-AW MHD model introduced in section 3.3.2 is exploited
in section 5.3 to investigate the formation process of density perturbations
in the SSW through dynamic reconnection processes at the HCS, within the
observational context of WISPR. Finally, we show in section 5.4 that other
transient processes occurring lower in the solar corona are likely to generate
density fluctuations at higher frequencies.

### 5.1 Variability of the SSW as measured in situ at PSP

Figure 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).

During its first solar encounters, the Parker Solar Probe mostly measured SSW
with speeds less than $500\ km/s$. The examination of streamer rays in WL
observations by Rouillard et al. (2020a) during the second orbit of PSP
revealed that PSP mainly scanned dense and SSW plasma that originated from the
edge of streamers. However near perihelion, PSP exited temporarily the dense
streamer flows to sample a more tenuous SSW that likely emerged from deeper
inside a coronal hole. A thorough study of this same time interval by Griton
et al. (2021) revealed that the SSW measured in situ by PSP comes from two
different sources that have a distinct magnetic topology. At closest approach
PSP temporary exited the dense streamer flows to intercept magnetic field
lines that are rooted in the center of a small equatorial coronal hole. This
configuration is illustrated in Figure 5.1 that I produced for the Griton et
al. (2021) paper. The small equatorial coronal hole is visible as a small
darker region in the EUV synoptic map (top and middle panels).

My contribution to this paper consisted in providing a detailed analysis of
the PSP connectivity to the Sun. For this purpose I basically followed the
method presented in section 3.4.1 to select the most realistic PFSS
reconstruction of the coronal magnetic field, by comparing the modelled HCS
(or neutral line) with the WL observations of the streamer belt derived from
SOHO LASCO-C2 coronagraphic images. An optimal source surface height was found
at $2.1\ R_{\odot}$ that captured both the streamer belt structure and the
small equatorial coronal hole. This value is however not optimal to reproduce
at best the streamer belt shape alone. This dichotomy between the source
surface height optimized from EUV or WL observations has already been
discussed in section 3.4.2 and is a shortcoming of PFSS models that lack some
physics compared to more realistic MHD models. The connectivity of PSP to the
Sun was then estimated following the method of the Magnetic Connectivity Tool
described in section 3.2.1 and assuming that PSP connects to the source
surface with a Parker spiral (ballistic model).

Around closest approach, from March 28th to April 13th 2019, Griton et al.
(2021) identified two intervals of SSW that appeared to exhibit two distinct
states that differed by their densities, magnetic field properties and plasma
beta (the ratio between thermal to magnetic pressure, see section 1.3.1).
State 1 was the SSW interval associated to the edge of the streamer and was
found to be denser with a higher thermal pressure than the plasma coming from
deeper inside the small equatorial coronal hole, or state 2. The magnitude of
the magnetic field is also smaller near the streamer than in the center of the
coronal hole that results in a higher beta parameter in the streamer flow
(state 1). That is illustrated in the middle panel of Figure 5.1 where the
electron density has been plotted on top of a WL map of the streamer belt. In
this plot, the state 1 (and state 2) SSW that has a high (and low) electron
density can be identified to the greenish/yellowish (and blueish) colors.
Griton et al. (2021) showed using the MULTI-VP model that the quasi-
stationnary theory is able to reproduce the bulk properties of these two SSW
states, this is discussed in more detail below.

The study focused on time intervals of very calm solar winds that were
detected in _both_ state 1 and 2. While the SSW appeared overall much more
perturbed in the streamer flows than from the coronal hole flow, some very
quiet intervals were also measured inside the streamer flows (corresponding to
state 1) meaning that a transient origin of these streamer flows may not
always be necessary.

Figure 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).

The MULTI-VP model (introduced in section 3.3.3) was exploited to check
whether the quasi-stationnary theory could reproduce the two quiet SSW states
measured at PSP, which may result from the difference in the magnetic geometry
of the two distinct source regions. As introduced in section 1.4.1, the
various solar wind speeds produced by MULTI-VP directly reflect the
formulation that is adopted for the coronal heating (see section 4.3.4) where
the expansion factor controls the height at which the heat is deposited in the
corona and hence affects the density and speed of the solar wind (Pinto &
Rouillard, 2017). MULTI-VP has been run on two sets of magnetic flux tubes
associated with the SSW states 1 and 2 defined in Griton et al. (2021), of
which the solutions are given in Figure 5.2 where the red (and blue) color
denotes the SSW state 1 (and 2). The corresponding topology of the coronal
magnetic field is traced in the left-hand side panel of Figure 5.2 that I
produced for the Griton et al. (2021) paper. One can notice a strong expansion
of the flux tubes that are contiguous to the streamer (red lines) which have
the effect in MULTI-VP to increase the amount of energy that is deposited low
in the corona. As introduced in section 1.3.1, this pushes the transition
region downward at a height where the chromosphere can balance the enhanced
thermal pressure of the low corona. This results in the evaporation of
chromospheric material into the corona and solar wind. The outcome is a slow
and dense solar wind analogous to SSW state 1 measured by PSP at the streamer
edge. In contrast the magnetic field expansion is less strong along the flux
tubes that are rooted close to the center of equatorial coronal holes
producing a more tenuous and slightly more rapid SSW (state 2). The latter
remains nonetheless much slower than the typical fast winds that forms above
polar coronal holes. Because the equatorial hole considered here is much
smaller than typical polar coronal holes, and because it is enclosed by two
dominating closed-field regions (north: a streamer and south: a pseudo-
streamer) this configuration induces high-expansion factors along the field
lines associated with both states of SSW.

These recent PSP measurements illustrate nicely that the SSW can take
different forms characterised by different densities, plasma beta as well as
magnetic fields. While the SSW from streamers appears overall to be denser and
much more variable (as introduced in section 1.2.2.3) than the SSW released
from closer to the center of coronal holes, a quiet SSW appears to also form
inside streamers. Numerical modelling with MULTI-VP shows that the different
densities of the SSW from streamers and coronal holes could result from
distinct heating depositions with height that are controlled by the different
expansion factors. A theory explaining the properties of streamer flows must
therefore explain why it can be at times very calm and at other times very
variable. We investigate in section 5.3 and 5.4 using numerical modelling how
variability can be produced inside streamer flows, and what that variability
should look like in coronal imagery and from the WISPR perspective.

### 5.2 Variability of the SSW as observed by PSP-WISPR

The first observations provided by the heliospheric WL imager WISPR on board
PSP (see section 3.1.3) have been rich in structures that were often
unresolved from typical 1 AU observatories. In addition to revealing the fine-
structure of streamers as discussed in chapter 4, WISPR have also detected
many density perturbations propagating along the streamers of which, some of
them are shown in Figure 5.3. In overall we detect two different topologies in
the WL signatures that suggest either structures of different shapes or
different viewing conditions.

Figure 5.3:  A sample of several density fluctuations observed by WISPR-I.

The left-hand and right-hand side panels of Figure 5.3 depict two large-scale
structures that are likely situated close to PSP. They look very similar to
the signature of a flux rope seen edge-on, as in other events of pristine slow
CMEs imaged in great detail by WISPR (Hess et al., 2020; Korreck et al., 2020)
and for which I had tracked the flux rope associated to these CMEs throughout
the FOV of WISPR and other space-based WL imagers (Korreck et al., 2020;
Kouloumvakos et al., 2020b; Rouillard et al., 2020c). These flux rope
signatures often appear as a circular annulus of increased density that marks
the envelope of their cross-section, that is more or less deformed depending
on the viewing angle. Their trailing edge is particularly well visible as a
bright "u-shape" or "v-shape" even from 1 AU observatories (Rouillard et al.,
2009b). When seen face-on, the global shape of flux ropes is generally
identified as long arches which have long been observed from 1 AU (see e.g.
Sheeley & Rouillard, 2010).

The middle panel of Figure 5.3 presents a track of three small-scale blobs.
Similarly to the two other structures discussed in the previous paragraph,
these blobs appear as bright annulus that suggest a flux rope structure seen
edge-on. In this case however, the WL signature is significantly smaller and
smeared out. At this point it is hard to say whether these structures are
actually small-scale flux ropes or if that is because they are located far
away from WISPR.

The WISPR-I synoptic maps that I produced have been found very convenient to
track both small-scale and large-scale density perturbations over long time
intervals, as discussed in section 4.4 and shown in Figure 4.9. The large-
scale flux rope identified in Figure 5.3 (green color) is also visible in the
associated WISPR-I synoptic map shown in Figure 5.4. That is also true for the
trail made of three small-scale flux ropes (pink color), which show a
periodicity of $\approx 3-4\rm{hr}$.

Figure 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.

To summarize, the two large-scale flux ropes shown in Figure 5.3 might belong
to the "streamer blob" family that we discussed in section 1.2.2.1 and that is
represented in Figure 1.13. In contrast, the trail of small-scale flux ropes
shown in the middle panel of Figure 5.3 might be associated to quasi-periodic
density structures with a higher periodicity, which have already been measured
in situ at PSP but at another epoch (Rouillard et al., 2020a). Fluctuations at
even smaller scales can be seen in the WISPR-I movies as corrugations in the
streamers but would be hardly distinguishable in individual images.
Fortunately, the WISPR-I synoptic maps that I produced provide a nice view of
these corrugations as presented in Figure 4.9. They might fall in the small-
scale fluctuations unveiled from 1 AU by DeForest et al. (2018) (see also
Figure 1.13) or even correspond to structures of smaller sizes.

In this section we only presented a reduced sample of the myriad of blobs
observed by WISPR. A future study that assembles a large catalogue of these
structures based on their shape and periodicity would be highly constructive.
In the following sections we employ synthetic imagery methods combined with
global MHD simulations to get additional insights on the nature of the
perturbations observed by WISPR.

### 5.3 Quasi-periodic density structures generated by magnetic reconnection
at the HCS

We examine in this section the release of density perturbations from the tip
of streamers, and more specifically through the process of magnetic
reconnection at the HCS (see section 1.4.2.1), including the generation of
magnetic flux ropes (see section 1.4.2.3). To investigate the nature and
origin of these transient features observed in WL by WISPR (see section 5.2),
we exploit the 3-D MHD Alfvén-wave driven solar wind model WindPredict-AW
already introduced in section 3.3.2 in two different setups. We first use in
section 5.3.1 an idealized 2.5-D high-resolution resistive simulation that is
similar to the one presented in Réville et al. (2020a) and that simulates the
development of magnetic reconnection at the HCS through the tearing
instability, and consequently of the formation of quasi-periodic density
structures in the streamers. We then test in section 5.3.2 how this process
may hold in two realistic 3-D simulations of the corona and of the solar wind,
of which one has already been proven to generate 3-D magnetic flux ropes in
similar regions of the solar atmosphere as those observed in situ by the
Parker Solar Probe and Solar Orbiter (Réville et al., 2022).

#### 5.3.1 Idealized 2.5-D setup: the tearing instability

We exploit in this section an idealized 2.5-D version of the WindPredic-AW
model introduced in section 3.3.2 and that is described in detail in Réville
et al. (2020a). The motivation for this idealized setup is to analyse the
formation of magnetoplasma perturbations released from the tip of streamers as
suggested by the early observations of Sheeley et al. (1997) (see section
1.4.2.1). Réville et al. (2020a) showed with this type of simulation that the
generation of "streamer blobs" is two-fold, first a pressure instability
forces the outward stretching and expansion of coronal loops that begin to
thin out along what will become the heliospheric plasma sheet. This results
eventually into the onset of a tearing-mode instability that triggers magnetic
reconnection at the HCS with an associated release of a 2-D plasmoids
separated by density perturbations that propagate along the streamer.

Réville et al. (2020a) have identified two different periodicities in the
generated structures. They show that the stretching of coronal loops due to
coronal heating is further enhanced by the SSW that is accelerated along the
adjacent open field lines that form the cusp of the streamer. Overall this
stretching process occurs over a $\simeq 30\ \rm{hr}$ period and, as already
mentioned, is interrupted by the onset of the tearing-mode. In 2.5-D MHD this
process ejects plasmoids along the heliospheric plasma sheet, in 3-D MHD
large-scale magnetic flux ropes can be released through this process as
discussed in the next section. This $\simeq 30\ \rm{hr}$ periodicity roughly
corresponds to the 16-20 hour periodicity derived from STEREO HI-1
heliospheric imagery by Sanchez-Diaz et al. (2017a). The region situated by
subsequent plasmoid/flux ropes where the tearing instability develops marks
the onset of smaller reconnection events that form smaller scale quasi-
periodic structures (see also Figure 1.27). The periodicities of these
additional quasi-periodic structures were found to be consistent with the
hourly time scale of those observed in WL coronal and heliospheric imagers
(Viall et al., 2010; Viall & Vourlidas, 2015) and detected in situ (Kepko et
al., 2016).

Réville et al. (2020a, 2022) argue that the 30-hour cycles correspond to the
time for the streamer loops to refill in heated coronal material and recover
from its exhaust phase induced by the reconnection at the HCS. In essence the
recovery time of the streamer is linked to the processes that heat up the
coronal part of the loops that constitute the streamer. Streamers can become
unstable (Higginson & Lynch, 2018) and grow if their internal pressure exceeds
the confining force induced by the magnetic field. For the largest flux ropes
produced this way in the streamers, a toroidal geometry (with a poloidal and
toroidal field) could develop during the magnetic reconnection process. In
this case additional toroidal forces could also contribute to the outward
acceleration of these magnetic flux ropes as studied in the Rouillard et al.
(2020c) paper and for which I contributed significantly. For this paper I
integrated to a toy flux rope model a fully 3-D geometry as well as a
calculation of the shifting of flux surfaces known as the Shafranov shift.

During their replenishing growth phase, coronal loops stretch until they break
up by reconnection, this is accompanied by a significant transfer of matter
where the plasma that was initially confined in coronal loops gets expelled
into the SSW. Throughout this process, the slow wind should then be fueled
with low-FIP elements that have been fractionated from other elements
beforehand in the coronal loops following processes that have been discussed
in section 1.3.4. As the tearing instability develops the matter gets
concentrated primarily at the interstices between the generated fast quasi-
periodic structures mentioned above, where the resulting density enhancement
actually correspond to the "streamer blobs" observed in WL. That may also
explain the variability of the composition measured in situ in the SSW
precisely in these quasi-periodic structures (see e.g. Kepko et al., 2016).
The magnetic reconnection that develops near the tip of streamers as produced
in the WindPredict-AW simulation provides an attractive explanation for the
intermittency of the streamer flows (see section 1.2.2.1), its peculiar
composition typical of coronal loops (see section 1.2.2.2), and the
variability of its composition (see section 1.2.2.3).

Figure 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.

To favor an environment where the tearing instability can fully occur, the
choice is made of a 2.5-D setup that permits sufficiently high numerical
resolution near the HCS to model accurately the development of the tearing
mode. By doing so the magnetic reconnection at the HCS can be computed through
an explicit resistive term in the MHD equations. Additionally, the model
exploits a Harten–Lax–van Leer Discontinuities (HLLD) Riemann solver Miyoshi &
Kusano (2005) that can handle sharp gradients near the HCS. This setup allows
high values of the Lundquist number $S=Lv_{A}/\eta\geq 10^{4}$, an indicator
of the ability of the medium to trigger the tearing-mode instability. Although
this value of $S$ remains much lower than what is actually measured in situ in
the HCS ($S\approx 10^{14}$), Réville et al. (2020a) show that periodic
structures still arise and propagate with timescales coherent with those
typically measured in situ in the SSW. This is illustrated in Figure 5.5 for
two different Lundquist numbers, where the top panel depicts a situation that
is more favorable to the full development of the tearing instability thanks to
a Lundquist number that is higher than in the bottom panel.

#### 5.3.2 Realistic 3-D setup: comparison with PSP-WISPR observations

The idealized 2.5-D setup presented in section 5.3.1 allows us to generate
plasmoïd-like structures that propagate in the streamer. In reality, these
structures may consist of 3-D global structures in the form of magnetic flux
ropes as measured in situ in the SSW (Sanchez-Diaz et al., 2019). In this
sense, the two reconnection processes suggested by Sheeley et al. (1997) and
Gosling et al. (1995) and that are depicted in the panels b and c of Figure
1.25 may be interpreted as two manifestations of the same reconnection process
at the tip of streamers, where the panel c gives a more precise 3-D
description of the topology of the magnetic field. As introduced in section
1.4.2.3, such scenario would lead to the generation of successive 3-D flux
rope structures that are punctuated by streamer blobs at their interstice as
illustrated in Figure 1.27.

This motivated a subsequent study from Réville et al. (2022) where global 3-D
simulations of the solar wind using the WindPredict-AW model introduced in
section 3.3.2, have been exploited to study the generation and dynamics of
flux ropes propagating in the streamer and along the HCS. By a qualitative
comparison with real PSP and SolO in situ measurements taken on June 2020,
Réville et al. (2022) produced flux rope releases in roughly the same
heliospheric region as the flux ropes detected in situ. I contributed to this
study by helping to select an optimal photospheric magnetic map to be given as
an inner boundary condition to the WindPredict-AW model, using the WL remote-
sensing observations and following the methodology presented in section 3.4.1.
In the same vein as the extended benchmarking framework presented in section
3.4.2, Réville et al. (2022) performed an additional selection of the
photospheric magnetic map based on in situ measurements of the magnetic
polarity to ensure a correct modeling of the magnetic sectors at PSP and SolO.

In contrast to the 2.5-D setup, the 3-D setup cannot afford such a level of
refinement and therefore it is the numerical diffusion induced by the
numerical scheme itself that allows for reconnection to happen. This numerical
diffusion can be of several order of magnitudes larger than what is actually
permitted by the physics. Therefore, the magnetic reconnection process is
bound in the 3-D setup by the actual numerical size of the mesh near the HCS.
That constrains the 3-D setup with a lower Lundquist number $S\approx
500-1000$ that is not optimal for the full development of the tearing
instability as in the 2.5-D simulation. Despite this limitation, the 3-D
simulation does produce streamer blobs or flux ropes but only at the longest
periodicity of $\simeq 20\ \rm{h}$ (as those measured in situ by Sanchez-Diaz
et al. (2019)) and not the fast quasi-periodic structures of about hour-long
timescales achieved in the 2.5-D simulation (and which have been detected both
remotely (Viall et al., 2010; Viall & Vourlidas, 2015) and in situ (Kepko et
al., 2016)). An example of a magnetic flux rope structure is illustrated in
the right panel of Figure 5.6, after the main reconnection event at the HCS.

Figure 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).

Following the Thomson scattering theory (see section 4.1) and the same method
as in section 4.2, I here build WISPR-I synthetic images from the 3-D
datacubes of plasma density simulated by the WindPredict-AW model. In
particular, I exploit the June 14, 2020 simulation presented in Réville et al.
(2022) for a subset of the simulation snapshots between time $t=60\ \rm{h}$
and $t=180\ \rm{h}$ that were found to contain the flux ropes structures.
However, the cadence of the simulation outputs (i.e. $\simeq 2.2\ \rm{h}$) is
not as high as for WISPR-I (i.e. $\simeq 16\ \rm{min}$), hence a simulation
snapshot is used several times ($7.5$ in average) before using the next one.
In another 3-D WindPredict-AW simulation that we present later in this
section, we directly fixed the cadence of the simulation outputs to be the
same as WISPR-I so that the tracking of propagating structures in the WISPR
FOV is more accurate. We stress out, nonetheless, that the FOV of WISPR-I
remains almost unchanged over a time period of two hours during which PSP
moves only about $1.5^{\circ}$ in Carrington longitude with almost no
variation in distance to the Sun.

Furthermore, we need to calibrate the timeline of the simulation snapshots
with the observation times of WISPR-I. In Réville et al. (2022), a post-event
photospheric map (of June 14) has been used as the inner boundary to the
WindPredict-AW model for the reasons mentioned at the beginning of this
section. This particular map was updated to account for an active region that
emerged on the far-side of the Sun and hence was not captured by ground-based
observatories during the WISPR-I observation window (June 4-10). The timeline
is hence shifted so that the 5 days ($120\ \rm{h}$) of coverage of the
WindPredict-AW simulation are centered at the 5th perihelion of PSP, leading
to a simulation that spans a physical time interval from June 5 00:00 to June
9 23:59 with a cadence of $\simeq 2.2\ \rm{h}$. For the rest of the WISPR-I
observing window (June 4 and June 10), the WISPR-I synthetic images are
produced using either the $t=60\ \rm{h}$ (or $t=180\ \rm{h}$) simulation
snapshot.

Synthetic WISPR-I images of the formation and propagation of a streamer blob
as generated in the WindPredict-AW 3-D simulation are shown in Figure 5.7,
where subsequent images are subtracted following the running difference
technique to enhance the visibility of the WL signature. We do not find clear
similarities with the subset of flux ropes that we have identified in section
5.2 from WISPR-I images. That may be clarified by constructing an extended
observational dataset of blobs seen by WISPR and that is left for future
studies. Looking at Figure 8 in Réville et al. (2022), this flux rope likely
originates from a portion of the HCS that is within $150-250^{\circ}$ of
Carrington longitude among four others that are seen in the same region. I
only show one of the five flux ropes in Figure 5.7 but I could distinctly
identify all of them in the WISPR-I synthetic images. These flux ropes are
located far away from PSP which was located between $\simeq 15-60^{\circ}$ of
Carrington longitudes as illustrated in Figure 5.8. Therefore the structure of
these flux ropes is not well captured compared to the case shown later on in
Figure 5.10.

Figure 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).  Figure 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.

Similarly to the first observations of WISPR-I that we analyse in detail in
section 4.3, we can build a synthetic WISPR-I synoptic map that stacks
synthetic WISPR-I images over time (see section 4.3.2 for a description of the
method). The result of this approach is shown in Figure 5.9 where the
corresponding observed WISPR-I synoptic map was already presented in Figure
4.9. We can draw a similar conclusion to that of section 4.3 where the
apparent splitting of the streamer rays in two bands is again the result of
the HPS changing latitude between the foreground and background portion of the
LOS of WISPR-I. This effect is depicted in Figure 5.8 by the two dashed red
lines between $85-130^{\circ}$ and $140-180^{\circ}$ of Carrington longitude
that represent two bundles of LOSs along which a large portion of the dense
HPS is integrated and produces the two bright stripes visible in both the real
and synthetic WISPR-I maps. As discussed in section 4.4, one can also see a
clear drifting of the southern streamer ray away from the equator and that is
an effect of PSP plunging into the streamer. That corresponds to a portion of
the HPS that is in the direct vicinity of WISPR-I (red dashed line between
$50-85^{\circ}$ of Carrington longitude) at a time where PSP crosses the HCS
as shown in Figure 5.8. In addition, one can see clear signatures of five
distinct features in the streamer ray that appear every $\approx 20\ \rm{h}$
and which correspond to the passage of the five flux ropes mentioned above
across the FOV of WISPR-I. Finally, the flux ropes appear quite fragmented in
the synthetic WISPR-I map along the time axis (abscissa axis). That is the
effect of updating the simulated density every $\simeq 2.2\ \rm{h}$ whereas
WISPR-I images are cadenced at $\simeq 16\ \rm{min}$.

Figure 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 $12R_{\odot}$.

We repeat the analysis for the ninth PSP solar encounter where WISPR-I
observed a HCS that was much more warped. Since streamer flux ropes are
usually aligned with the streamer belt (Sheeley & Rouillard, 2010; Rouillard
et al., 2011), a warped streamer means that WISPR will observe different WL
signatures of these flux ropes that will either appear edge-on or with a high
inclination angle. The WindPredict-AW setup is similar to the previous one but
here the inner boundary is set with the GONG-ADAPT (11th realization)
magnetogram of August 14, 2021 00:00. Just as for the simulation of June 14
2020, the magnetogram was selected among many different sources and dates to
best match the observations. The latter included WL streamer belt observations
taken from $1\ AU$ (using the method presented in section 3.4.1) complemented
with the magnetic sectors and timing of HCS crossings measured in situ by PSP.
Once the simulation reached a permanent regime, outputs of the entire 3-D
simulated domain have been extracted every $\simeq 13\ \rm{min}$ to match the
cadence of WISPR-I. For the sake of memory space, we limit ourselves to
$\approx 100$ outputs nonetheless, that cover a time interval of $\simeq 22\
\rm{hr}$ around perihelion.

Following the same procedure as before, we show in Figure 5.10 synthetic
WISPR-I running-difference images of a flux rope produced by reconnection at
the HCS in the August 14 2021 simulation. The overall shape of the solar
corona (from the same simulation) is given in Figure 5.11 and appears, as
expected for the higher level of activity, much more structured and complex
than in the previous simulation. The flux rope seen in the synthetic WISPR-I
images (see Figure 5.10) comes from a portion of the HCS that is within
$\approx 70^{\circ}$ and $\approx 100^{\circ}$ of Carrington longitude. The WL
signature of the flux rope appears much more complex than in the previous case
(figure 5.7) and is very similar to actual flux ropes observed by WISPR-I (see
section 5.2) where WISPR-I is very close to flux ropes that are mostly seen
edge-on. The leading edge of the flux rope appears as arches (orange arrows)
whereas the trailing edge has a peculiar "v-shape" (green arrows). The growth
and expansion of the flux rope compresses the solar wind plasma that stacks on
the front and hence produces the bright arches. As discussed in section 5.2,
the "v-shape" could correspond to a LOS effect, but here this also likely a
remnant feature of the magnetic reconnection process itself which disconnected
the coronal loops as depicted in Figure 1.25(c). At the trailing edge the
reconnection accelerates and pushes the coronal loop plasma to the back of the
flux rope, that is also accompanied by an injection of poloidal flux into the
flux rope as we discussed in Rouillard et al. (2020c).

Figure 5.10:  Same as figure 5.7 but for a flux rope generated in the
simulation of August 14 2021.  Figure 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.

The HCS (see the dashed grey line in Figure 5.11) is significantly warped,
this is a consequence of a much more active Sun compared to the June 14 2020
simulation. This complicates the interpretation of the WL signature in recent
WISPR-I observations. The WISPR-I synoptic map shown in Figure 4.9 for the
ninth encounter (E9) now appears much more striated by streamer rays. As
discussed in section 4.4, these fine structures can be the result of multiple
folds in the HPS and other high-density plasma layers formed by pseudo-
streamers or more generally the complex network of QSLs (see section 1.4.2.4).
For illustration purposes we show in Figure 5.12 the synthetic WISPR-I map
built from the August 14 2021 simulation that looks very different to the real
map at first sight. At this stage we do not aim at reproducing the actual
position of each ray observed by WISPR-I because this would require further
iterations of model optimisation that is left for future study. However, this
simulation provides additional insight on the interpretation of WL signatures
in a highly-structured solar atmosphere. For that purpose we denoted several
bright rays in Figure 5.12 (dashed red lines) for which we could identify an
associated source region in the density map shown in Figure 5.11. The plunging
effect of WISPR-I into the corona is even more pronounced (ray 1) than in the
previous simulation, this orbital effect is a recurring feature of the latest
WISPR-I observations (see Figure 4.9). The multiple source regions identified
in Figure 5.11 and 5.12 further strengthen our previous observation that very
distant regions can contribute equally to the brightness received by WISPR-I
and that a significant part of the WL signal comes from high-density regions
that are not always associated with the bipolar streamer. This should be
investigated further in future studies, WISPR is here likely providing
detailed information on the network of pseudo-streamers and perhaps more
generally the S-web (see section 1.4.2.4).

Figure 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.

Both the leading and trailing edge of the flux rope that we have tracked in
the running difference images (see Figure 5.10) can also be seen distinctly in
the synthetic WISPR-I map and produce highly inclined stripes that spread over
a wide range of latitudes. The trailing edge produces vertical stripes (orange
arrows) that are similar to that of an actual flux rope observed by WISPR-I
and that we identified in Figure 5.4 by a green ellipse. Unfortunately the
leading edge was not clearly visible in this particular event, and the other
flux rope shown in the right-hand side panel of Figure 5.3 is too faint to be
properly tracked in a WISPR-I synoptic map. That should be clarified in
subsequent observations as WISPR will likely reveal more events of this kind
by getting closer and closer to the observed flux rope structures.

### 5.4 Propagating density structures triggered by transient heating at the
coronal base

(This section contains some material from the Griton et al. (2020) paper.)

We have already highlighted the successes of MULTI-VP to model the fine-
structure of the upper corona and nascent solar wind imaged by WISPR-I (see
section 4.3), including the SSW variability measured in situ at PSP (see
section 5.1). We see in this section that the MULTI-VP model can be also
exploited to interpret the origin of the transient structures generally
observed in WL coronagraphs.

The main motivation of the Griton et al. (2020) paper to which I contributed
significantly, was to understand the origin of some of the density
fluctuations observed by DeForest et al. (2018) during a recent (deep-field)
high-cadence campaign of the STEREO-A COR-2 coronagraph. We already introduced
their observations in section 1.2.2.1 and in Figure 1.12. This figure depicts
a highly structured corona typical of high solar activity where the HPS is
largely inclined with respect to the equator, this is illustrated in Figure
3.8 which I produced for the Griton et al. (2020) paper. The largest
fluctuations observed by DeForest et al. (2018) likely belong to the "streamer
blobs" family observed earlier on by Sheeley et al. (1997), which likely
originate from magnetic reconnection at the tip of streamers in the form of
magnetic flux ropes (as shown in the previous section). However, most
fluctuations observed by DeForest et al. (2018) are likely to be smaller
transient structures. Their presence at all position angles in the STEREO
images could be the result of the streamers being highly warped during that
period, meaning that the density fluctuations are released at high latitudes
as well. The origin of these small-scale density fluctuations is still debated
and they could come also from other coronal regions than just the streamers.
We have found no evidence from the dynamic WindPredict-AW simulations that
they could originate in magnetic reconnection from streamer tops. Therefore in
Griton et al. (2020) we investigated instead whether these density
fluctuations could originate from lower heights in the solar corona during
impulsive heating events.

In particular we wanted to check whether these density fluctuations are
generated low in the corona during transient heating events detected as local
brightenings in EUV and X-rays, known as coronal bright points (CBPs)
(Madjarska, 2019). These CBPs appear as hot loops (log$T$[K]$\simeq 6.2$) that
have received a significant amount of energy through what are likely to be
localised magnetic reconnection events (Kwon et al., 2012). The combined
analysis of intense CBPs observed in EUV and WL images of the upper corona
have revealed that outflowing density structures known as "coronal jets" often
originate as CBPs low in the corona (Wang et al., 1998). Recent numerical
simulations also corroborate this view (e.g. Singh et al., 2019). These
coronal jets become part of the outflowing solar wind outstreaming from
coronal holes (Wang et al., 1998). Hence an association exists between the
occurrence of density structures in the outflowing solar wind and the
occurrence of CBPs low in the corona. Furthermore, based on a statistical
analysis of the occurrence of CBPs carried out by Alipour & Safari (2015),
Griton et al. (2020) suggest that on average 1.6 CBPs should be expected for
each $1^{\circ}$ band of elevation angle in Figure 1.12, and producing
brightness fluctuations in WL every $\simeq 15$ hours.

An alternative setup of the MULTI-VP model introduced in section 3.3.3 was
developed to simulate the outward propagation of density fluctuations
originating in impulsive heating events at CBPs. For that an additional
transient heating term is added to the background coronal heating (eq 4.2) so
that the total heating rate $Q_{h}=-\nabla_{\parallel}\cdot F_{h}$ (in
$erg.cm^{-3}.s^{-1}$) is (Griton et al., 2020):

$Q_{h}=-12\times
10^{5}|B_{\odot}|\left(\frac{A_{\odot}}{A}\right)\left(\frac{1}{H_{f}}exp\left[-\frac{s-R_{\odot}}{H_{f}}\right]+\frac{1}{r_{p}}\mathcal{H}(t)a_{p}exp\left[-\frac{(s-R_{p})^{2}}{r_{p}^{2}}\right]\right)$
(5.1)

This perturbed component in $Q_{h}$ is a Gaussian function whose center, width
and amplitude is parametrized by $R_{p}$, $r_{p}$ and $a_{p}$. The time
dependency of the impulsive heating is introduced with the time dependent
$\mathcal{H}(t)$ factor.

A series of MULTI-VP runs was performed with this setup, first on a simple
idealized geometry and then a more realistic three-dimensional version, in
order to test the effect of each of the heating parameters on the generated
density fluctuations. The parameters assumed for the spatial distribition
(heliocentric latitude/longitude), frequency of occurrence, amplitude and
height deposition of the energy to the plasma were taken from observations of
CBPs (Alipour & Safari, 2015; Madjarska, 2019). These transient heating events
induce perturbations that have a significant impact on both the simulated
background solar wind and the generated density fluctuations (see Griton et
al., 2020). In a nutshell, the sudden deposition of energy in the low corona
triggers slow-mode (compressive) waves that propagate outwards at the local
sound speed in the simulated solar wind stream. The associated compression
fronts of these waves produce density enhancements that are found sufficiently
strong to be clearly visible in synthetic imagery and with intensity
variations that are comparable with STEREO-A COR-2 observations. The reader is
invited to refer to the Griton et al. (2020) paper for more details.

For this paper I performed WL synthetic imagery from the realistic three-
dimensional case mentioned above and which includes randomly distributed
heating events based on the occurrence rate of CBPs (simulation 9 in Griton et
al. (2020)). I followed the same method as the one described in section 4.2
based on the Thomson scattering theory introduced in section 4.1. Figure 5.13
shows a result from this process where the brightness is multiplied by $r^{3}$
(where $r$ is the heliocentric distance) to enhance the visibility of the
fluctuations as done in DeForest et al. (2018). To allow for a visual
comparison with their figure (Figure 1.12), I then employed a base-difference
method where a background synthetic image representative of the mean solar
wind flow (panel a) is subtracted from a raw synthetic image (panel b) to
enhance the visibility of brightness fluctuations (panel c). The color scale
is similar to the one used in Figure 1.12 so that the reader can directly
compare our simulated density fluctuations with the ones observed by DeForest
et al. (2018) from STEREO-A COR-2 images.

Figure 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.

Figure 5.13(c) shows several propagating structures characterised by increases
in total brightness $\lesssim 4\times 10^{-10}\ B_{\odot}R_{\odot}^{3}$ or
alternatively $\sim 11\%$ with respect to the mean background image. The shape
and order of magnitude are consistent with the ubiquitous brightness
fluctuations observed by STEREO COR2 and analyzed in DeForest et al. (2018).
Synthetic brightness fluctuations show different shapes, scales and
intensities. Regarding shapes, one can see patches (or blobs) alongside
elongated features. Elongated features are mainly visible due to the
saturation of the color scheme, where neighboring fluctuations coalesce. They
may also result from the superposition of background and foreground
contributions.

In conclusion, this study reveals that at least a fraction of the brightness
fluctuations observed in the corona could be caused by heating events due to
CBP occurring below, near the coronal base. Since CBPs occur over the entire
solar surface, the formation and outward propagation of these density
fluctuations is not limited to the streamer belt and can appear both in the
slow and fast solar wind regions. A process that we neglected and is known to
occur during CBPs is magnetic reconnection. In particular CBPs likely result
from reconnection between coronal magnetic fields. At the source of the solar
wind, interchange reconnection between the pre-existing open-field of coronal
holes and the highly dynamic closed-field of small-scale loops is likely to
occur and generate complex, probably twisted, magnetic fields that are
released as jets or even microjets. As discussed in section 1.4.2.2 and
1.4.2.4, many mechanisms can favor such interchange reconnection in coronal
holes, of which the emergence of magnetic flux, the convective motions at the
photosphere, and the differential versus rigid rotation. We have seen in this
section that, given a source of energy provided by dynamic processes, the
quasi-stationnary theory can explain part of the fluctuations observed in the
solar wind. In the Griton et al. (2020) study the simulated fluctuations were
the most visible in the high speed and tenuous solar wind streams that
originate far from the streamer, but they were also seen in the dense slow
wind that typically forms further away from the center of coronal holes or
near the center of small equatorial coronal holes (such as the one discussed
in section 5.1).

### 5.5 Conclusion

We showed in the previous chapter 4 that coronal and solar wind imagery can be
used to investigate the SSW’s spatial and temporal variability with greater
ease than with the in situ measurements considered in this chapter (see
section 5.1). This is because time-dependent variability tends to manifest
itself as the outward propagation of bright (high density) structures in the
images whereas spatial variability induces brightness variations that evolve
more slowly in the images even for the rapidly-moving Parker Solar Probe. In
chapter 4 we depicted SSW source regions that are highly structured spatially
in the form of highly striated streamer rays observed in great detail by
WISPR.

In the present chapter, we looked at whether this spatial variability could to
some extent be isolated in in situ measurements thanks to the PSP’s novel
orbit. We therefore carried out a more in-depth analysis (section 5.1) of how
the variability of the SSW measured in situ by PSP results from both spatial
and time-dependent effects on a broad range of scales. We focused first on a
time period when PSP transitioned from what was interpreted as streamer flows
to coronal hole flows. While streamer belt flows appear much more variable,
exhibiting strong variations in plasma and magnetic field parameters (Réville
et al., 2022), we found that a quiet solar wind component also exists in this
streamer flow (Griton et al., 2021) and that its mean properties differ from
those measured in the quiet component of the coronal hole flow.

We interpreted these differences as a _spatial-dependent_ effect of the probe
moving across distinct SSW that form along magnetic field lines with different
magnetic topologies at their source. The quasi-stationary theory was found to
provide a quantitative assessment of what we categorised as a spatial
variability in the SSW. Just as we did in chapter 4 to interpret the fine
structure of streamer rays observed in PSP-WISPR, we used the MULTI-VP MHD
model to study quantitatively the differences between the quiet solar wind
measured in streamers and coronal hole flows. To do that we kept the coronal
heating term (eq 4.2) dependent on the expansion factor of flux tubes
channelling the solar wind. A transition of what appears to be two states of
the SSW measured in situ by PSP during its second encounter is naturally
interpreted as the transition between field lines that have different
expansion factors at their source. This topological change in magnetic field
is related to PSP’s magnetic connectivity shifting from the edge of the
streamer belt to the center of a small equatorial coronal hole. If the first
in situ measurements taken by PSP were mostly constituted of streamer-like
SSW, PSP has now collected over the recent orbits a wealth of in situ data of
solar wind plasmas that originate from a diversity of source regions. A
systematic comparison between the magnetic topology of the source regions and
the bulk properties of the SSW measured in situ at PSP could provide further
tests to the quasi-stationary theory of the SSW.

We then examined how the various physical processes proposed in the dynamic
theory of the SSW could explain the _time-dependent_ variability of the SSW.
For that we analysed and interpreted further WISPR observations by exploiting
advanced 3-D time-dependent numerical modelling. As already noted in chapter
4, WL emissions from the stationary part of the corona detected in the
foreground and background of WISPR-I images can complicate their
interpretation. However we show that WISPR-I can capture perturbations in the
SSW with fine details that are otherwise smeared out at further distances to
the Sun, this includes what appear to be very small plasmoids outflowing along
streamer rays. I exploited the WindPredict-AW MHD simulations performed by
Victor Réville and described in Réville et al. (2020a, 2022) to examine how
well they can explain the transients observed in WISPR. Because this model
solves the coupled transport of the solar wind plasma with the magnetic field,
dynamic processes such as the generation of periodic structures from the tip
of streamers could be investigated in detail. The large plasmoids and flux
ropes released from the streamers result from the extension and thinning of
streamer loops. This process is driven by pressure imbalance between the loops
and the surrounding medium and that then provides favorable conditions for
magnetic reconnection to occur at the HCS through the tearing instability. The
reconnection between coronal loops that have been stretched during this
process leads to the formation of flux ropes that propagate along with the
SSW, whose signatures have been detected in situ in past studies (Rouillard et
al., 2009a; Sanchez-Diaz et al., 2019).

The 2.5-D MHD numerical setup of WindPredict-AW that was introduced in section
5.3.1, reproduces nicely the density structures that occur with periodicities
consistent with past in situ and remote-sensing observations. In order to
fully address the spatial and temporal variability in these images I had to
exploit the 3-D setup of WindPredict-AW. The 3-D MHD setup cannot resolve the
scales necessary to model accurately the tearing mode instability but it does
produce (streamer-induced) outflowing 3-D flux ropes, albeit with greater size
and slightly longer periodicities that deduced from both in situ measurements
and remote-sensing observations by SolO and PSP (Réville et al., 2022). I used
my forward modeling technique to simulate WISPR images of the quasi-periodic
structures generated in WindPredict-AW. That includes a close-up view of a
flux rope simulated in WindPredict-AW that looks similar to other flux ropes
imaged by WISPR-I and which have been presented in section 5.2. Future studies
should extend this type of analysis by searching and cataloguing more small
flux rope like structures observed in WISPR images in order to include many
more cases.

Section 5.4 extends the analysis made in section 5.3 to address the origin of
even smaller density structures that appear ubiquitous in coronal imagery from
STEREO. In contrast to the formation process discussed in section 5.3, we
examine in section 5.4 the generation of density perturbations at coronal
heights situated well below the tip of streamers, at the base of the corona.
The MULTI-VP model was exploited to simulate how the quasi-stationnary solar
wind may be altered when subjected to a varying input of energy. As discussed
in section 1.4.2.2 and 1.4.2.4, magnetic reconnection is thought to be
ubiquitous in the low corona. The heating associated with these reconnection
events was considered in MULTI-VP by an additional impulsive heating component
that triggered periodic pressure pulses that induced high-density structures
rising up in the corona. By supposing that these heating events follow the
statistics of CBPs which are interpreted as electromagnetic signatures of
magnetic reconnection events happening in the low corona, we showed that
periodic heating can generate density perturbations that fill out a
significant portion of the corona typically observed by WL coronagraphs.
However, it remains to assess how much of these perturbations relate to those
actually observed by WISPR, that is left for a future study.

## Chapter 6 The Irap Solar Atmosphere Model (ISAM)

ISAM (for Irap Solar Atmosphere Model) is a multi-specie kinetic-fluid model
that solves for the transport of neutrals and charged particles along a
magnetic field line and from the chromosphere to the corona. The model assumes
gyrotropy of the velocity distribution function (VDF) and solves a set of
transport equations for 16 moments of the VDF ( $n,\ u,\ T^{\parallel},\
T^{\perp},\ q^{\parallel},\ q^{\perp}$ ) (see section 6.2 and 6.3). The
transport equations are solved for neutral atoms, ions and electrons. ISAM
solves for transport equations that are concretely in 1-D but include the
effects of pressure anisotropy. The predominant ionization and recombination
processes are accounted for in order to model partial ionization in the
chromosphere (see section 6.4), this is essential to address the FIP effect.
Thermal and momentum exchanges between species through collisions are also
accounted for (see section 6.5). The solution of the model will depend
strongly on the chosen heating profile which can be either prescribed with an
ad-hoc heating function or through an explicit calculation of the dissipation
of Alfvén waves (see section 6.6). Near the top of the chromosphere and at the
transition region, the energy balance includes cooling/heating terms that
account for the radiative emissions/absorptions in the optically thick and
thin regions (see section 6.7). This comprehensive set of physical processes
solved by ISAM allows us to compute dynamically the different layers of the
solar atmosphere: chromosphere, transition region and corona.

### 6.1 Legacy

The structure of ISAM is based on an ionospheric code called IPIM (Blelly &
Schunk, 1993; Marchaudon & Blelly, 2015). The model has been adapted to the
solar atmosphere by my colleague Michael Lavarra who included the coupling of
neutrals to charged particles and exploited the model to study the properties
of the solar wind in open magnetic field geometries (Lavarra et al., 2022). In
parallel I developed a different setup to simulate closed-field plasmas
confined in coronal loops.

ISAM is a model that follows the footsteps of previous implementations of
high-order moment approaches (Schunk, 1977; Demars & Schunk, 1979; Blelly &
Schunk, 1993; Lie-Svendsen et al., 2001; Killie et al., 2004; Janse et al.,
2006). They all aim at describing in detail the transport of matter and energy
with a self-consistent resolution of the heat flux that seeks to be valid for
both collision-dominated and collisionless plasmas.

### 6.2 Origin: the Boltzmann equation

High-order approaches all start from a kinetic description of the plasma.
Because a peculiar description of each particle that constitutes the plasma is
computationally prohibitive for typical densities around $10^{10}-10^{20}$ in
the solar atmosphere, therefore a statistical description using the velocity
distribution function $f(\mathbf{r},\mathbf{v},t)$ (VDF) is more appropriated.
In other words, $f(\mathbf{r},\mathbf{v},t)$ is the particle density at time
$t$ in the volume defined by
$\mathbf{r}-d\mathbf{r}\leq\mathbf{r}\leq\mathbf{r}+d\mathbf{r}$ with a
peculiar velocity
$\mathbf{v}-d\mathbf{v}\leq\mathbf{v}\leq\mathbf{v}+d\mathbf{v}$.

The dynamical evolution of such a system that is out of balance due to
external interactions is described by the Boltzmann equation. For an ensemble
$s$ of particles (species) with mass $m_{s}$ and VDF $f_{s}$ undergoing a
force $F_{s}$, the Boltzmann equation reads:

$\frac{D}{Dt}f_{s}=\frac{\partial}{\partial
t}f_{s}+\mathbf{v_{s}}\cdot\mathbf{\nabla}_{r}f_{s}+\frac{1}{m_{s}}\mathbf{F_{s}}\cdot\mathbf{\nabla}_{v}f_{s}=\frac{\delta
f_{s}}{\delta t}$ (6.1)

where $\mathbf{\nabla}_{r}$ and $\mathbf{\nabla}_{v}$ are the gradients in the
coordinate and velocity space. Collisions of a specie $s$ with other species
will affect the VDF throughout the right-hand side collision term
$\frac{\delta f_{s}}{\delta t}$. Solving directly for the Boltzmann equation
is usually not appropriated to describe the macroscopic state of a system.
Only in rare occasions the Boltzmann equation has been fully solved
numerically, as in the work of Spitzer & Härm (1953) who gave their names to
the well-known Spiter-Härm classical heat flux prescription that is widely
used by the scientific community to describe heat flux in collisional plasmas.
Such approaches are usually referred as the classical theories.

Since we are interested in the macroscopic properties of a plasma, it is more
convenient to represent its statistical state through the moments of the VDF,
assuming that the macroscopic scales are sufficiently large for the notion of
density and temperature to be meaningful. For plasmas typically the particles
exhibit collective behaviour mediated by collisions and/or the electromagnetic
field. In a gas composed of charged particles such as a plasma, a convenient
scale can be introduced above which the electric field induced by for instance
an ion in the gas will be screened by the surrounding electrons, the time
required for this screening to take place in a weakly collisional plasma is
related to the transit time of a thermal particle to cross the Debye length:

$\lambda_{D}=\sqrt{\frac{\epsilon_{0}k_{b}T_{e}}{e^{2}n_{e}}}\frac{1}{\sqrt{1+\sum_{j}Z_{j}^{2}\frac{n_{j}}{n_{e}}\frac{T_{e}}{T_{j}}}}$
(6.2)

where $\epsilon_{0}$ is the vacuum permittivity, $k_{B}$ the Boltzmann
constant, $e$ the electron elementary charge, $Z_{j}$ the ion charge state,
$n_{e,j}$ and $T_{e,j}$ the electron and ion density and total temperature.
The Debye sphere of radius $D$ defines the region of maximum influence of a
particle in the plasma. The typical Debye length reaches macroscopic scales of
about $10^{-6}-10^{-2}\ \rm{m}$ in the solar chromosphere and corona, and
$\approx 10\ \rm{m}$ in the solar wind but in any case it remains much smaller
than the minimum grid cell size in ISAM (i.e. $15\ \rm{km}$ at the transition
region for the runs presented in chapter 7).

The first 16 moments of the VDF solved in ISAM are obtained by subsequent
averaging of the VDF:

$\displaystyle n_{s}$ $\displaystyle=\int f_{s}d\mathbf{v_{s}}\ \ $ density
(6.3a) $\displaystyle\mathbf{u_{s}}$
$\displaystyle=\left<\mathbf{v_{s}}-\mathbf{t_{s}}\right>=\left<\mathbf{v_{s}}\right>\
\ $ drift velocity (6.3b) $\displaystyle T_{s}^{\parallel}$
$\displaystyle=m_{s}/k_{b}\left<\mathbf{t_{s}}^{\parallel 2}\right>\ \ $
parallel temperature (6.3c) $\displaystyle T_{s}^{\perp}$
$\displaystyle=m_{s}/(2k_{b})\left<\mathbf{t_{s}}^{\perp 2}\right>\ \ $
perpendicular temperature (6.3d) $\displaystyle\mathbf{q_{s}^{\parallel}}$
$\displaystyle=n_{s}m_{s}\left<\mathbf{t_{s}}^{\parallel
2}\mathbf{t_{s}}\right>\ \ $ heat flow vector for parallel energy (6.3e)
$\displaystyle\mathbf{q_{s}^{\perp}}$
$\displaystyle=n_{s}m_{s}/2\left<\mathbf{t_{s}}^{\perp
2}\mathbf{t_{s}}\right>\ \ $ heat flow vector for perpendicular energy (6.3f)
$\displaystyle\mathbf{P_{s}}$
$\displaystyle=n_{s}m_{s}\left<\mathbf{t_{s}}\mathbf{t_{s}}\right>\ \ $
pressure tensor (6.3g) $\displaystyle\mathbf{\tau_{s}}$
$\displaystyle=\mathbf{P_{s}}-p_{s}^{\perp}\mathbf{I}-(p_{s}^{\parallel}-p_{s}^{\perp})\mathbf{e_{3}}\mathbf{e_{3}}\
\ $ stress tensor (6.3h)

where the averaging operator is defined as:

$\left<\mathbf{A}(\mathbf{r},\mathbf{v_{s}},t)\right>=\frac{1}{n_{s}(\mathbf{r},t)}\int
f_{s}(\mathbf{r},\mathbf{v_{s}},t)\mathbf{A}(\mathbf{r},\mathbf{v_{s}},t)d\mathbf{v}_{s}$
(6.4)

and $\mathbf{t_{s}=v_{s}-u_{s}}$ is the random (thermal) speed,
$p_{s}^{\parallel}=n_{s}k_{b}T_{s}^{\parallel}$ and
$p_{s}^{\perp}=n_{s}k_{b}T_{s}^{\perp}$ the parallel and perpendicular thermal
pressures,
$\mathbf{I}=\mathbf{e_{1}}\mathbf{e_{1}}+\mathbf{e_{2}}\mathbf{e_{2}}+\mathbf{e_{3}}\mathbf{e_{3}}$
is the unit dyadic tensor (or identity matrix). Note that the $\parallel$ and
$\perp$ subscripts for the heat flow vectors indicate whether the parallel or
perpendicular energy is transported. In all other cases the $\parallel$ and
$\perp$ subscripts refer to the vector component parallel and perpendicular to
the magnetic field.

By forming the relevant moments listed in equation 6.3 from the Boltzmann
equation, one can derive a complex set of transport equations. This system of
equations needs to be closed though because the transport equation for the
$n^{th}$ moment introduces the moment of order $n+1$. A closed-set of
transport equations can be obtained by approximating the shape of the VDF.

An isotropic plasma in thermodynamic equilibrium can be described by a Maxwell
distribution function of the form:

$f_{s0}^{M}=n_{s}\left(\frac{\beta_{s}}{2\pi}\right)^{3/2}\exp\left(\frac{-\beta_{s}\mathbf{t_{s}}^{2}}{2}\right)$
(6.5)

with $\beta_{s}=m_{s}/(k_{b}T_{s})$. However such approximation is not
appropriated in a plasma where collisions are too scarce to allow an efficient
redistribution of the energy, as in the solar corona. A bi-Maxwellian is
therefore recommended to account for the plasma anisotropy in the direction
parallel $T_{s}^{\parallel}$ and perpendicular $T_{s}^{\perp}$ to the magnetic
field: