11institutetext: Rosseland Centre for Solar Physics, University of Oslo,
Postboks 1029 Blindern, N-0315 Oslo, Norway 22institutetext: Institute of
Theoretical Astrophysics, University of Oslo, Postboks 1029 Blindern, N-0315
Oslo, Norway 33institutetext: IRAP, Université Toulouse III - Paul Sabatier,
CNRS, CNES, Toulouse, France 44institutetext: The Johns Hopkins University
Applied Physics Laboratory, 11101 Johns Hopkins Road, Laurel, MD 20723, USA

# Variability of the slow solar wind: New insights from modelling and PSP-
WISPR observations.

Nicolas Poirier 1122 Victor Réville 33 Alexis P. Rouillard 33 Athanasios
Kouloumvakos 44 Emeline Valette 33

###### Abstract

Aims. We analyse the signature and origin of transient structures embedded in
the slow solar wind, and observed by the Wide-Field Imager for Parker Solar
Probe (WISPR) during its first ten passages close to the Sun. WISPR provides a
new in-depth vision on these structures, which have long been speculated to be
a remnant of the pinch-off magnetic reconnection occurring at the tip of
helmet streamers.

Methods. We pursued the previous modelling works of Réville et al. (2020,
2022) that simulate the dynamic release of quasi-periodic density structures
into the slow wind through a tearing-induced magnetic reconnection at the tip
of helmet streamers. Synthetic WISPR white-light (WL) images are produced
using a newly developed advanced forward modelling algorithm that includes an
adaptive grid refinement to resolve the smallest transient structures in the
simulations. We analysed the aspect and properties of the simulated WL
signatures in several case studies that are typical of solar minimum and near-
maximum configurations.

Results. Quasi-periodic density structures associated with small-scale
magnetic flux ropes are formed by tearing-induced magnetic reconnection at the
heliospheric current sheet and within $3-7\ \rm{R_{\odot}}$. Their appearance
in WL images is greatly affected by the shape of the streamer belt and the
presence of pseudo-streamers. The simulations show periodicities on $\simeq
90-180\ \rm{min}$, $\simeq 7-10\ \rm{hr,}$ and $\simeq 25-50\ \rm{hr}$
timescales, which are compatible with WISPR and past observations.

Conclusions. This work shows strong evidence for a tearing-induced magnetic
reconnection contributing to the long-observed high variability of the slow
solar wind.

###### Key Words.:

Sun: solar wind – Methods: numerical – Methods: observational –
Magnetohydrodynamics (MHD) – Instabilities – Magnetic reconnection

##  1 Introduction

In contrast to the fast solar wind, a mystery remains on the origin of the
slow solar wind (SSW) and of its high variability. This variability can be the
result of time-dependent and/or spatial-dependent effects.

The spatial-dependent variability often emerges in structured bundles of
bright rays in coronal white-light (WL) emissions, that have long been
observed from coronagraphs and heliospheric imagers such as the Solar and
Heliospheric Observatory (SoHO: Domingo et al., 1995) and the Solar
TErrestrial RElations Observatory (STEREO: Kaiser et al., 2008). Recently, the
Wide-Field Imager for Parker Solar Probe (WISPR: Vourlidas et al., 2016)
revealed a finer structuring of the slow solar wind at scales down to the thin
heliospheric plasma sheet (HPS) (Poirier et al., 2020; Liewer et al., 2022;
Howard et al., 2022).

On the other hand, it has been shown that the SSW is also highly time
dependent, by hosting up to $\approx 80\%$ of quasi-periodic structures (Viall
et al., 2008). This paper focus on the time-dependent variability of the slow
wind, as captured from the WISPR novel perspective and in light of state-of-
the-art modelling.

Density transient structures that propagate along with the SSW have long been
observed in white-light imagery, with a great variety of shapes, speed, and
origins. Among the most evident are coronal mass ejections (CMEs), which by
releasing tremendous amount of coronal material into the heliosphere, generate
significant brightness enhancements in both coronagraph and heliospheric
images (see e.g. the review by Webb & Howard, 2012). Some CME events that
undergo more moderate and progressive accelerations, called streamer blowouts,
have been particularly observed to deflect towards the cusp of helmet
streamers and further propagate within the SSW (see e.g. the recent WISPR
observations described in Hess et al., 2020; Korreck et al., 2020; Rouillard
et al., 2020c).

Since the beginnings of the SoHO-LASCO coronagraph, other CME-like flux rope
structures known as the Sheeley blobs (Sheeley et al., 1997) have been
observed to propagate along the bright rays associated with streamer stalks
where the densest slow wind originates. Early interpretations suggested that
these structures formed as a result of a pinch-off reconnection at the tip of
helmet streamers that would have been stretched out beforehand (Gosling et
al., 1995; Wang et al., 1998); the conditions leading to this stretching and
eventually to the reconnection are still unclear. On some occasions they also
appear as bright arches, which may be more or less ‘squashed’ depending on
their inclination with respect to the observer (see e.g. Sheeley & Rouillard,
2010).

Large loops from active regions (ARs) have also been observed to leave arch-
like signatures as they gradually expand into the corona (as seen in both
X-ray and WL emissions; see Uchida et al., 1992; Morgan et al., 2013). A
helmet streamer made of such expanding loops may then be prone to stretching,
and to the formation of streamer blobs via the pinch-off reconnection
scenario.

This picture has the advantage of also being consistent with observations of
plasma inflows in LASCO (Wang et al., 2000; Sheeley & Wang, 2002), which were
associated for the first time with outflowing blobs later on (Sanchez-Diaz et
al., 2017b; Lynch, 2020). The continuous tracking of blobs expelled from the
tip of helmet streamers all the way to points of in situ measurements reveals
that they transport helical magnetic fields (Rouillard et al., 2009b, 2011),
which is further supported by recent Parker Solar Probe (PSP) observations
(Lavraud et al., 2020; Rouillard et al., 2020a). More systematic statistical
analyses based on STEREO images, and of in situ measurements inside the HPS
revealed that the topology of blobs is consistent with magnetic flux ropes
(Sanchez-Diaz et al., 2019) that could form via magnetic reconnection at the
tip of helmet streamers (Sanchez-Diaz et al., 2017a).

The modelling of streamer instabilities in time-dependent magnetohydrodynamics
(MHD) simulations also supports this scenario (Chen et al., 2009; Lynch,
2020). Following in the footsteps of these models, Réville et al. (2020)
investigated in detail the tearing instability that occurs near the cusp of
streamers, in a high-resolution 2.5D simulation of the corona and using an
idealistic dipolar configuration of the solar magnetic field. Streamer blobs
were reproduced in addition to a plethora of quasi-periodic structures over a
wide range of frequencies. Past studies based on near 1 AU remote-sensing
and/or in situ observations also reveal the existence of quasi-periodic
structures with periodicities varying from $\approx 90-180\ \rm{min}$ to
$\approx 8-16\ \rm{hr}$ (Viall et al., 2010; Viall & Vourlidas, 2015; Kepko et
al., 2016; Sanchez-Diaz et al., 2017b), which were found in Helios (Di Matteo
et al., 2019) and in recent PSP observations as well (Rouillard et al.,
2020a). From a high-cadence campaign on the STEREO-A COR-2 coronagraph,
DeForest et al. (2018) recently revealed the ubiquitous presence of density
fluctuations at even smaller scales $\approx 20-40-60\ \rm{min}$.

In addition to the density and magnetic field, other plasma properties have
also been measured to vary during the passage of these transients. For
instance, Kepko et al. (2016) showed a similarity between the variability of
the charge state ratios measured in situ in the slow wind and the short hourly
timescale of the quasi-periodic structures observed remotely in WL streamers.
This SSW originating from streamers tends to exhibit high charge state ratios
typical of hot ARs, whereas the SSW that emerges farther away from streamers,
probably from deeper inside coronal holes is characterised by lower charge-
state ratios comparable to those measured in the fast wind (Neugebauer et al.,
2002; Liewer et al., 2004; Stakhiv et al., 2015, 2016). The streamer-like SSW
is also known to be more enriched in heavy ions having a low first ionisation
potential (FIP) (see e.g. von Steiger, 1996; Peter, 1998), a composition
typical of closed-field plasma from ARs (see e.g. Ko et al., 2002; Brooks &
Warren, 2011; Doschek & Warren, 2019). The streamer-like SSW, or at least its
dynamic component, could hence be conveniently interpreted as originating from
the pinch-off reconnection mechanism by offering a channel through which
closed-field material can be intermittently released into the slow wind.

This paper further investigates this scenario through a qualitative comparison
between the recent highly resolved WL observations taken by WISPR, and high-
resolution simulations of the solar corona and solar wind. We first analyse in
section 2 two events observed by WISPR that depict quasi-periodic structures.
We then present in Sect. 3 our modelling approach to reproduce such structures
through the tearing-induced reconnection at the tip of streamers. Synthetic
WISPR images are then produced and compared against observations in Sect. 4.
Limitations and future perspectives on this work are discussed in Sect. 5. We
finally conclude on the possible implications of this work for the
understanding of the slow solar wind in Sect. 6.

##  2 Observations

After its first 11 successful encounters WISPR has already provided a wealth
of images rich in structures that were often unresolved from typical 1 AU
observatories (see e.g. Howard et al., 2022, for an overview of the tenth
encounter). This is a direct benefit of bringing an imager so close to the
Sun, to a vantage point that is located inside the corona. By drastically
shortening the line-of-sight integration path, WISPR is able to resolve with
unprecedented detail the density structures that propagate within the slow
solar wind. WISPR consists of two WL heliospheric imagers that are mounted on
the ram side of PSP, and so the solar wind structures can be imaged prior to
their in situ measurement (Vourlidas et al., 2016). WISPR offers a large field
of view (FOV) thanks to its two telescopes that cover in elongation angle
($\epsilon$, angle away from the Sun) $13.5-53.0^{\circ}$ for the inner
telescope (WISPR-I) and $50.5-108.5^{\circ}$ for the outer telescope
(WISPR-O). At the closest approach to be reached by PSP in 2024 ($9.86\
\rm{R_{\odot}}$), WISPR-I will be able to observe the corona from only $2.3\
\rm{R_{\odot}}$. We exploit WISPR level 3 images,111data source:
https://wispr.nrl.navy.mil/wisprdata which have been calibrated (see Hess et
al., 2021) and where contributions to the white-light emissions by dust
particles (i.e. the F-corona) have been subtracted to reveal only the faint
K-corona made up of coronal electrons (see Howard et al., 2022, for more
details on the procedure).

###  2.1 First insights on the transients observed by WISPR

Figure 1: Two examples of large-scale density structures observed by WISPR-I
on April 26, 2021, 06:42 universal time (UT, top panel) and May 04, 2021,
11:27 UT (bottom panel). The helioprojective-radial coordinate grid is plotted
as solid white isolines for the position angle ($\psi$, i.e. the anticlockwise
angle from solar north) and dashed white isolines for the elongation angle
($\epsilon$), both in degrees.

WISPR has detected a wealth of fluctuations in the slow wind, whose signatures
can be very diverse. Among these fluctuations many have been associated with
magnetic flux ropes with a clear dark cavity, suggesting that these flux ropes
were likely observed edge-on or at a small inclination angle.

We present two examples of such signatures in Fig. 1, captured by the inner
telescope WISPR-I during the eighth PSP encounter. Two bright shells (green
arrows) can be seen, one circular (bottom panel) and the other more V-shaped
(top panel). Such shapes have already been observed from 1 AU, albeit to a
larger spatial extent. They have been especially captured in great detail in
events associated with pristine slow CMEs observed by WISPR. The V-shape was
interpreted either as a slight inclination of the flux rope with respect to
the line of sight (LOS) of the observer or as a byproduct of the reconnection
process itself that generates the flux rope (Thernisien & Howard, 2006;
Rouillard et al., 2009a, 2020c). The fact that the brightness enhancement is
often more marked at the back end of the flux rope supports the latter
scenario, by an accumulation of plasma from the reconnection exhaust. A closer
look at the bottom panel of Fig. 1 shows an inner circular structure (yellow
arrows) that has also been detected in other transients observed by WISPR (see
e.g. Hess et al., 2020; Rouillard et al., 2020c; Howard et al., 2022).

Most signatures of this scale have been found to be associated with sporadic
(slow) CME events, where flux ropes are already present low in the corona well
below the tip of streamers (see e.g. Hess et al., 2020; Korreck et al., 2020;
Rouillard et al., 2020c; Howard et al., 2022). In this paper we focus on flux
ropes that form on a regular basis just above the tip of helmet streamers,
which are presumed to be major contributors to the variability of the slow
wind.

Figure 2: Two examples of small-scale density structures observed by WISPR-I
on April 26, 2021, 22:57 UT (top) and March 01, 2022, 15:15 UT (bottom). The
helioprojective-radial coordinate grid is again represented in the same format
as in Fig. 1. The green lines represent the pixels extracted to build the
time–distance (J-)maps shown in Fig. 4.

Compared to past near 1 AU observations, the novelty of WISPR observations is
in imaging streamers from much closer in, providing clearer signatures of its
embedded transients and access to smaller scales. In figure 2 we show two
events that may be related to quasi-periodic structures captured by WISPR-I
during the 8th (top panel) and 11th (bottom panel) PSP encounters, hereafter
referred to as the April 2021 and March 2022 events, respectively. We were
able to identify a series of other similar events in WISPR-I images, but we
only selected here the most visible ones for illustrative purposes,
particularly in the most recent WISPR-I observations. Although more difficult
to interpret because of a higher solar activity, they reveal local features
that clearly stand out from the background signal.

The top panel unveils a track of three small-scale structures. Similarly to
the larger-scale events shown previously in Fig. 1, they also appear as bright
annuli suggesting flux ropes seen edge-on (yellow arrows). At this point it is
hard to say whether these structures are actual small-scale flux ropes or if
they are located far away from WISPR, which we discuss below. In contrast, the
bottom panel shows arch-like signatures (orange arrows). Similar signatures
have already been observed from 1 AU, and have been related to flux ropes seen
with a greater inclination angle or almost face-on (Sheeley & Rouillard, 2010;
Rouillard et al., 2011), or also to expanding AR loops (Morgan et al., 2013).
In the latter case though, the expansion of the loops is much slower than the
propagation speed of the transients captured by WISPR (see the fitting
performed in Sect. 2.3).

Similarly to the April 2021 event shown in the top panel, a close-up visual
inspection of the March 2022 event also reveals consecutive arches following
each other. Both these events show interesting periodic behaviour in their
spatial distribution, and hence they may be connected to the above-mentioned
$90-180\ \rm{min}$ quasi-periodic structures that have been previously
detected in the slow wind; this is discussed further in Sect. 2.3.

Finally, we can give a rough estimate of the brightness variation induced by
the passage of these transients. For this purpose, we examined the pixel
values as given in units of mean solar brightness ($B_{\odot}$) in the level 3
WISPR-I _.fits_ files. It is important to note that these data products are
not photometrically accurate because some of the K-corona emissions might be
removed during the background removal procedure.222See the disclaimer about
the level 3 (version 1) data at https://wispr.nrl.navy.mil/wisprdata We
averaged the emissions over representative areas that define the transients
and the background (host) streamers, and computed the relative difference
$(\bar{B}_{transient}-\bar{B}_{streamer})/\bar{B}_{streamer}$. We found
relative brightness increases of $\approx 80-95\%$ for the April 2021 (edge-on
case) event and $\approx 30-50\%$ for the March 2022 (face-on case) event,
which is brighter than what has been typically measured from 1 AU for the
Sheeley et al. (1997) blobs. As we show throughout this paper, the Sheeley
blobs and the quasi-periodic structures captured by WISPR can be related to
two different families of transients that result from a pinch-off reconnection
at the tip of helmet streamers.

###  2.2 Global context from near 1 AU observations

To get a better context for these events, we construct WL maps of the streamer
belt as observed from near 1 AU by LASCO-C2 over half a solar rotation. We
show these maps in Fig. 3 (panels b and d). Estimates of the heliospheric
current sheet (HCS) derived from potential field source surface (PFSS)
extrapolations are plotted (red dashed lines) to help us differentiate pseudo-
streamers (unipolar structures) from the main streamer belt (where the
magnetic polarity switches sign). Assuming that the above transients originate
from and propagate within the streamer belt, we identified two possible source
regions that have an inclination consistent with the April 2021 and March 2022
events observed by WISPR. Since the imaged transients significantly stand out
from the background streamers, they should be located quite close to the
Thomson sphere (see the magenta lines) where most of the WL emissions are
expected to originate (see Sect. 3.3). We identified two possible source
regions, a nearly aligned section of the streamer belt located within
$60-110^{\circ}$ (and $\approx 0^{\circ}$, see panel b) of Carrington
longitudes (and latitudes), and an inclined section of the streamer belt
located within $220-280^{\circ}$ (and $\operatorname{-}20-0^{\circ}$, see
panel d). In an independent study Liewer et al. (2023) determined a similar
source region for the April 2021 event using a more sophisticated tracking
technique, and hence supports the fact that this event was indeed propagating
within the main streamer belt.

The low coronal structures underlying the streamer belt, as observed in the
extreme ultraviolet (EUV) by the Atmospheric Imaging Assembly (AIA) on board
the Solar Dynamics Observatory (SDO) are shown in panels a and c of Fig. 3. In
panel c two major ARs can be clearly seen at longitudes $210^{\circ}$ and
$245^{\circ}$, which are located underneath the potential source streamer that
produced the March 2022 event. These ARs may prove important in the formation
of streamer transients. The hot plasma in such active closed-field regions can
be prone to expansion into the corona through thermal instability, bringing
the AR loops that are frozen into that plasma to higher coronal heights,
eventually up to the cusp of streamers. On the other hand, in panel a no
significant AR is visible beneath the streamer that potentially produced the
April 2021 event observed by WISPR. However, it has also been suggested that
the stretching of streamers can naturally occur simply as the result of the
magnetic field near the cusp being too weak to hold the thermal pressure
exerted by the underlying plasma Chen et al. (2009).

The two mechanisms could act together in the formation of streamer transients
through the pinch-off reconnection process, which is actually supported by
observations (see e.g. Morgan et al., 2013) and by simulations (Chen et al.,
2009; Réville et al., 2020). In particular, as we discuss in the results
section (4.1), that the pinch-off reconnection process can generate transients
with a low frequency that is quite variable and dependent on the local coronal
conditions beneath the streamer. Therefore, the presence and amount of ARs
beneath streamers could implicitly affect the rate at which these low-
frequency transients are produced. From past observations near 1 AU, these
transients were detected with periods varying from $\approx 8-16\ \rm{hr}$
(Sanchez-Diaz et al., 2017a, near solar maximum) to $0.5-2\ \rm{days}$
(Morgan, 2021, near solar minimum). A future statistical study that links
these heliospheric measurements to low atmospheric EUV observations would be
helpful to better assess the contribution of ARs on the release of streamer
transients.

Figure 3:  Contextual imagery from near 1 AU. Panels a and c: Synoptic maps of
the low solar atmosphere derived from SDO-AIA
(171$\mathrm{\SIUnitSymbolAngstrom}$) EUV observations. Panels b and d:
Synoptic maps of the WL corona as seen by the SoHO LASCO-C2 coronagraph. These
maps have been produced by combining on-disk (off-limb) images over a full
(half) solar rotation prior to the April 26, 2021 (panels a and b), and March
01, 2022 (panels c and d), events (see section 4.2 of Rouillard et al., 2020b,
for a description of the method). The dashed red line represents an estimate
of the shape of the HCS derived from PFSS extrapolations of the photospheric
magnetic field. Among the multitude of photospheric maps available we picked
two maps that match both the shape of the WL streamer belt (see Poirier et
al., 2021) and the timings of the HCS crossings determined from magnetic
sector measurements taken in situ at PSP. In the present case, the magnetic
maps from the Air Force Data Assimilative Photospheric flux Transport (ADAPT)
model (Arge et al., 2010, 2011, 2013) for April 16 12:00UT (8th realisation)
and March 01 12:00UT (11th realisation) were selected assuming a source
surface height of $2.0\ \rm{R_{\odot}}$ and $2.5\ \rm{R_{\odot}}$,
respectively. The vertical (radial) projection of PSP orbit onto the solar
disk is plotted in cyan; the stars indicate several dates at PSP. The magenta
surfaces represent the projections of the regions scanned by WISPR-I at the
time of the two images shown in Fig. 2, with a solid line indicating the
location of the Thomson sphere extracted at $\epsilon=22^{\circ}$.

###  2.3 Tracking transients in WISPR J-maps

The usual and efficient method to further characterise transient features and
have more insights into their possible generation mechanism is to measure
their periodicity. For this purpose, a method that has been widely used across
the community is to track transients in distance-time maps called J-maps (see
e.g. Sheeley et al., 1999, 2008; Rouillard et al., 2008, 2009b). The bright
features that appear in J-maps then provide insightful information on the
periodicity, propagation speed, and acceleration profiles of the transients
(albeit with some limitations, as discussed below).

These maps are commonly produced by extracting pixels along a fixed direction
(either at the ecliptic or at another position angle), and using the
elongation angle ($\epsilon$) as a measure of the angular distance away from
Sun centre.

Tracking transient features in heliospheric images has long been a delicate
task, and even more for a rapidly moving and up-close imager like WISPR. New
techniques have been developed to better track WISPR features, whether they
are static (e.g. coronal rays: Liewer et al., 2022) or dynamic (e.g. CMEs:
Liewer et al., 2020). These techniques include a number of corrections to
account for the effects of spacecraft motion, perspective, and orbit out of
the solar equatorial plane. For instance, a transient propagating radially
outwards from the Sun does not necessarily remain at a constant position angle
as it moves across the WISPR FOV, and its distance to WISPR may also vary.
This can affect their appearance in J-maps. For instance, curved signatures
were noted during a CME event that came close to the two heliospheric imagers
on board STEREO (Sheeley et al., 2008; Rouillard et al., 2008). Furthermore,
when a target moves away from the observer, hence leading to an apparent
slowing of its propagation speed, this can also produce curved signatures in
J-maps as we see later for the April 2021 event. Performing a precise fitting
of the transients observed by WISPR is beyond the scope of this study, and
hence circumvents the need of a complex tracking technique as developed by
Liewer et al. (2022).

Figure 4:  WISPR-I distance–time (J-)maps for the April 2021 (left) and March
2022 (right) events, and along the two slits shown in Fig. 2 (solid green
lines). The fitted profiles are plotted as dashed red lines, with constant
velocity of $160\ \rm{km/s}$ (left) and $415\ \rm{km/s}$ (right), i.e. about
the solar wind bulk velocity measured a few days later by PSP-SWEAP/SPC in
situ (see Fig. 5).

The J-maps associated with the March 2022 and April 2021 events introduced
earlier are shown in Fig. 4. The corresponding slits along which the J-maps
were constructed are plotted in Fig. 2 as green lines. These J-maps are
classical distance–time maps, except that instead of the typical elongation
angle that measures the angle away from the Sun centre, the radial distance to
the Sun is used. The implicit assumption to determine this parameter, however,
is that extracted pixels are projected onto the Thomson sphere, a reference
surface where WL emissions are expected to be strongest (see discussion in
Sect. 3.3). From these J-maps we were able to infer some insightful properties
of the transient structures captured by WISPR, which we describe in the
following paragraphs.

Propagation profiles
The J-maps allow us to perform a direct visual inspection of the speed profile
of the propagating structures. For this purpose we fitted profiles of constant
speed for several of the most visible stripes in Fig. 4. We note that several
sub-structures could also be seen between some of the brightest fronts, but
were too faint to be shown here. The March 2022 event (right panel) is fairly
well described by constant speed profiles at $\approx 415\ \rm{km/s}$. In
contrast, the April 2021 event (left panel) shows curved stripes that deviate
from constant speed profiles, and with a very low speed ($\approx 160\
\rm{km/s}$). Using a more sophisticated tracking technique on the April 2021
event, Liewer et al. (2023) obtained a similar low propagation speed of
$\approx 190\ \rm{km/s}$. Although such a low propagation speed could be due
to perspective effects (e.g. of the transients moving away from WISPR), we
show in Sect. 2.4 that here this is probably associated with a very slow and
dense slow solar wind flow within the HPS.

Curved signatures in J-maps have already been found in STEREO observations for
instance, (Sheeley et al., 2008; Rouillard et al., 2008) and also more
recently in WISPR (Howard et al., 2022). These apparent decelerations were in
fact associated with the effect of the imaged structures getting closer to or
farther away from the observer. Since PSP remained relatively ‘static’ (i.e.
co-rotating with the solar corona) at that time, that might suggest that the
structure itself was moving with respect to WISPR. This is also consistent
with the WL signatures that suggest that these flux ropes propagated along PSP
orbital plane (see the green dashed line in the top panel of Fig. 2), where
their legs may have come closer to or farther away from WISPR during their
expansion. The pinch-off reconnection mechanism at the tip of streamers indeed
predicts that such flux ropes develop large azimuthal extents, from their
generation and during their expansion in the solar wind (Sanchez-Diaz et al.,
2019).

Periodicities
The periodicities measured between the fitted stripes range from $\approx
110-120\ \rm{min}$ (Fig. 4, left panel) and $\approx 130-175\ \rm{min}$ (right
panel) between the fitted stripes. This falls well within the typical $\approx
90-180\ \rm{min}$ range previously detected from near 1 AU observations.
Furthermore, similar periodicities are also retrieved in an upcoming
statistical study from Viall & Vourlidas (2023) that includes a few PSP
encounters. Because of the rapidly varying viewing conditions of WISPR, quasi-
corotation phases do not last very long, and hence periodicities above
$\approx 10\ \rm{hr}$ cannot easily be detected. Therefore, it remains
difficult to check whether the longer $\approx 8-16\ \rm{hr}$ periods of
streamer blobs measured previously from slowly moving 1 AU observatories also
manifests in WISPR images.

As we discuss in the next section, these periodicities could be byproducts of
the pinch-off reconnection process occurring at the tip of streamers. More
precisely, they could be the manifestation of multiple modes associated with
the tearing instability that can develop at the HCS.

###  2.4 Insights from plasma measurements taken in situ at PSP

Figure 5:  Solar wind measurements taken in situ at PSP. Top panel: Bulk
velocity (left axis) and density of the proton solar wind (right axis)
measured by the Solar Probe Analyzer (SPAN-i), which is part of the Solar Wind
Electrons Alphas and Protons (SWEAP: Kasper et al., 2016) investigation.
Middle panel: Magnetic field (1 min average) measured by the magnetometer
(MAG) of the FIELDS instrument suite (Bale et al., 2016). Bottom panel: Radial
distance to the Sun and Carrington longitude of PSP. A few days after the
April 2021 event observed by WISPR-I, PSP sampled a very dense and slow solar
wind typical of HPS, with several magnetic polarity inversions suggesting
potential crossings of the HCS. Figure produced with the AMDA web tool
publicly available at (http://amda.irap.omp.eu/).

We check whether the low propagation speeds measured for the April 26, 2021,
event imaged by WISPR-I are realistic, by making a rough comparison with the
solar wind speeds measured in situ at PSP around that time, as shown in Fig.
5. This is possible because after that event, PSP entered in a pro-grade
phase, and hence could a few days later sample a solar wind channel that
probably hosted the transients captured by WISPR-I (see the PSP orbit plotted
in the top panel of Fig. 3).

In the top panel, we can see that PSP indeed measured very slow (purple dots)
and dense (emerald green dots) solar wind with $\approx 160-250\ \rm{km/s}$
and $\approx 1-8\times 10^{3}\ \rm{1/cc}$ at around $15\ \rm{R_{\odot}}$. Such
plasma flows are typical of coronal streamers at that distance (Cho et al.,
2018; Morgan & Cook, 2020), and more generally of HPS (see e.g. Winterhalter
et al., 1994; Sanchez-Diaz et al., 2019; Lavraud et al., 2020), which is also
supported by a few HCS crossings during this interval (here simply identified
by a global polarity inversion of the magnetic field, see middle panel). These
SSWs are hence potential hosts of blobs and quasi-periodic structures like
those observed by WISPR-I. Unfortunately, in this case the transients imaged
by WISPR-I on April 26 (located within $\approx 8-16\ \rm{R_{\odot}}$) already
moved far away before PSP could get inside that streamer belt starting from
April 29. Nevertheless, the solar wind velocities measured there (at $\approx
15\ \rm{R_{\odot}}$, see the top and bottom panels of Fig. 5) closely match
the $\approx 160\ \rm{km/s}$ fitted speed of the imaged transients.

##  3 Modelling: Method

Several main characteristics of transients in the slow wind have been
extracted from WISPR observations. We introduce in this section our modelling
approach to get more insights into the possible origin of these structures. We
test the idea that the pinch-off reconnection process at the tip of streamers
is responsible for the formation and release of the small transients observed
by WISPR.

This mechanism is tested first in an idealistic simulation of a very high-
resolution time-dependent 2.5D MHD dipolar corona (Sect. 3.1), and then a
lower resolution time-dependent 3D MHD simulation of the conditions
encountered by PSP during its ninth passage near the Sun (Sect. 3.2). We then
present in Sect. 3.3 our approach to building synthetic products that can be
compared with WISPR observations.

###  3.1 Idealistic simulation of a dipolar corona

Figure 6:  Simulated electron density from the idealistic dipolar set-up.
Colours are plotted in a logarithmic scale. The approximate FOV of WISPR-I
when PSP is located at $10\ \rm{R_{\odot}}$ and $35\ \rm{R_{\odot}}$ is
depicted by a dark and white dashed rectangle, respectively. An animated
version of this figure is available online at
https://doi.org/10.5281/zenodo.8135596.

Réville et al. (2020) describe in detail the pinch-off reconnection mechanism
induced by the tearing instability, in an idealistic 2.5D simulation of the
solar corona. To allow a fair comparison with actual observations from WISPR,
the Réville et al. (2020) simulation was run again with more outputs (one
every $\simeq 13\ \rm{min}$) to match the typical temporal cadence of WISPR.

In figure 6 we illustrate the main phases of the pinch-off reconnection
mechanism, with several snapshots extracted from the simulation and zoomed-in
views in the panels on the left side. Starting from a near equilibrium state
($t=0\ \rm{min}$, first row), the tip of the helmet streamer eventually
expands ($t=1540\ \rm{min}$, second row) due to pressure imbalance between the
closed-field plasma confined beneath (as in coronal loops) and the out-flowing
plasma from the adjacent open field. As the helmet streamer expands a thinning
also occurs at its back end, up to the point where the streamer gets
sufficiently thin locally for the tearing instability to trigger magnetic
reconnection ($t=2349.8\ \rm{min}$, third row), referred to as the ‘ballooning
mode’ in Réville et al. (2020). The ejecta of a large plasmoid of dense and
initially closed-field material follows ($t=2880.8\ \rm{min}$, fourth row).
The tearing instability further develops at smaller scales triggering
reconnection at multiple secondary sites, and is called the ‘tearing mode’.
Plasma material is pushed away from these reconnection sites and then
accumulates in small-scale and dense plasmoids. More precisely, this plasma
concentrates in shell-like structures where the magnetic field is mostly
poloidal, as in magnetic flux ropes. In contrast, the core of these structures
is less dense due to a dominant toroidal magnetic field component that is
directed across the figure plane (see animation associated with Fig. 6
available online,333https://doi.org/10.5281/zenodo.8135596 as well as those
presented in the original paper by Réville et al. 2020).

For the first time, the inner and outer WISPR telescopes combined can provide
an in-depth view of the transient structures that form from pinch-off
reconnection, right in their formation regions. As illustrated in figure 6,
WISPR may see different signatures according to its distance from the Sun,
where the dashed white and dark rectangles show the approximate WISPR-I FOV,
assuming that PSP is located respectively at $35\ \rm{R_{\odot}}$ and $10\
\rm{R_{\odot}}$ from the Sun ($10\ \rm{R_{\odot}}$ being an estimate of the
closest approach to be reached by 2024). For instance, it happens that some of
the simulated transients eventually merge together along their propagation to
form larger and/or denser plasmoids (see lower left panel of Fig. 6). Hence,
we pursue here the work of Réville et al. (2020) to examine how such simulated
structures may look in a white-light imager such as WISPR.

To produce synthetic white-light observables, we need first to extend the 2.5D
simulated domain into three dimensions. We perform an axisymmetric
demultiplication of the 2.5D simulation about the solar rotation axis, hence
producing a 3D corona with a flat streamer belt at the equator. By changing
the position of our virtual observer we can then test most situations
encountered by WISPR along its orbits and more generally throughout the solar
cycle, that is from a horizontal to vertical streamer belt configuration
typical of a solar minimum and maximum, respectively.

###  3.2 Case study simulation of the ninth PSP encounter

Because the 2.5D set-up presented in Réville et al. (2020) is highly
idealistic, an attempt has been made to extend this work to a fully fledged 3D
model that is called WindPredict-AW (see Parenti et al., 2022; Réville et al.,
2022, for a detailed description). In this modelling framework, the 2.5D
magnetic structures mentioned above translate into 3D magnetic flux ropes,
where their generation and propagation can now be studied in a self-consistent
manner. The disadvantage, however, is that the 3D set-up provides a lower
level of spatial resolution than the 2.5D set-up. Since magnetic reconnection
is allowed by numerical diffusion of the numerical scheme itself, it is bound
by the actual numerical size of the mesh near the HCS. The 3D set-up is hence
not optimal for the full development of the tearing instability, as is the
idealistic set-up (see Réville et al., 2022, for more details). Despite this
limitation, the 3D simulation still does produce transients but at low
frequency, which are the ballooning modes and only the quasi-periodic
structures with periods $\gtrsim 4\ \rm{hr}$. By applying a realistic
photospheric magnetic map at the inner boundary, Réville et al. (2022) was
able to reproduce the statistical occurrence of streamer flux ropes that
intersected both PSP and SolO during the joint observation campaign of June
2020.

We here pursue the work of Réville et al. (2022) with a similar 3D simulation
set-up, but applied to the ninth PSP encounter (August 2021). The inner
boundary is set with the GONG-ADAPT (11th realisation) magnetogram of August
14, 2021, 00:00 UT, and kept fixed over the entire simulated period. The
magnetogram was selected among many different sources and dates to best match
the observed shape and location of the streamer belt, as seen from 1 AU by
SoHO-LASCO over a full solar rotation. The selection process is based on the
method presented in Poirier et al. (2021). Another criteria was also the
correct prediction of both magnetic sectors and timing of HCS crossings
measured in situ by PSP. Once the simulation relaxed, outputs of the entire 3D
simulated domain were extracted every $\simeq 13\ \rm{min}$ to match the
actual cadence of WISPR. For the sake of computational time the simulation was
run until $100$ outputs were obtained, which covers a time interval of $\simeq
22\ \rm{hr}$ starting at perihelion. The simulation is kept fixed outside this
interval, and allows us to synthesise WISPR images over a longer period even
though the simulated solar wind remains static. The static phase is still
meaningful to differentiate the effect of the fast-moving probe from the
propagation of the solar wind structures within the synthesised images. The
procedure to produce WISPR synthetic images is described in Sect. 3.3.

The simulated streamer belt and density structures propagating within its core
are shown in Fig. 7, along with the FOV of both WISPR-I (in white) and WISPR-O
(in grey). At that time WISPR was imaging from a distance of $\approx 26\
\rm{R_{\odot}}$, a highly warped streamer belt typical of a high solar
activity. Throughout the region scanned by WISPR, the streamer belt undergoes
significant latitudinal shifts within $\approx\operatorname{-}15-25^{\circ}$
of Carrington latitude. A few flux rope structures have been identified in the
simulation (see the coloured arrows). All of them, except the farthest one
(cyan arrow) produce visible WL signatures in the synthetic WISPR images (see
Sect. 4.2). The flux ropes have different extents and widths that can be
explained by a different stage of their formation and/or evolution. We also
find that the spatial extent of these flux ropes within the streamer belt
varies, and that it is delimited by intersections of pseudo-streamers with the
main streamer belt (see also Réville et al., 2022). This makes up a complex
network that is inherently connected to the S-web (or web of separatrix and
quasi-separatrix layers; see Antiochos et al., 2011). Finally, each of these
flux ropes shows a different inclination. All of this will affect their
appearance from the WISPR perspective as we show in Sect. 4.2.

Figure 7:  Simulated density structures from the WindPredict-AW fully fledged
3D MHD model. Top two panels: Front and top views of the 3D density structures
propagating within the core of the (true) streamer belt (identified with a
$Br=0$ isosurface). Coordinates in the Carrington frame are also given, for
the longitudes (lighter colours, in degrees) and radial distances (darker
colours, in solar radii). Bottom panel: Synoptic (Carrington) map at $r=16.5\
\rm{R_{\odot}}$ with PSP orbit plotted in cyan. To identify the true streamer
belt from pseudo-streamer structures, a $Br=0$ contour is plotted as a red
dashed line. In both panels the FOV of WISPR-I and WISPR-O are shown in white
and grey, respectively. In both panels the logarithmic colour scale represents
the flux of plasma density scaled by its value at the inner boundary
($r=1.0002\ \rm{R_{\odot}}$). The coloured arrows point to flux rope
structures (see main text for dicussion). An animated version of this figure
is available online at https://doi.org/10.5281/zenodo.8135596.

###  3.3 Producing synthetic WISPR images

Synthetic WISPR images are produced similarly to what was done in Poirier et
al. (2020), except that in the present work we used time-dependent simulations
rather than static simulations.

Following the Thomson scattering theory (Howard & Tappin, 2009; Howard &
DeForest, 2012), the total intensity received by a pixel detector from
scattered electrons can be expressed as an integral along the path length $z$
along each LOS:

$\begin{aligned}
I^{tot}_{t}&=\int_{z=0}^{z\rightarrow+\infty}I_{t}dz=\int_{z=0}^{z\rightarrow+\infty}n_{e}z^{2}Gdz\
\quad\text{(in }\rm{W.m^{-2}.sr^{-1}}\text{)}\\\
G&=\frac{B_{\odot}\pi\sigma_{e}}{2z^{2}}\left(\underbrace{2}_{1}\underbrace{\left[(1-u)C+uD\right]}_{2}\underbrace{-\sin{\chi}^{2}}_{1}\underbrace{\left[(1-u)A+uB\right]}_{2}\right)\end{aligned}.$
(1)

Here $I_{t}$ refers to the total (and not polarised) intensity,
$B_{\odot}\simeq 2.3\times 10^{7}\ \mathrm{W.m^{-2}.sr^{-1}}$ the Sun’s mean
radiance (or surface brightness), and $\sigma_{e}=r_{e}^{2}\simeq 7.95\times
10^{-30}\ \mathrm{m^{2}}$ the electron cross-section. The electron density
$n_{e}$ is an input 3D datacube interpolated at each LOS point. The $G$
function here includes contributions from both pure-geometric scattering
(indicated by $1$) and the solar illumination function (indicated by $2$). For
far distances to the Sun, $G$ can be approximated as
$(R_{\odot}/r)^{2}(2-\sin{\chi}^{2})=(R_{\odot}/r)^{2}(1+\cos{\chi}^{2})$,
where $\chi$ is called the scattering angle between the scattering site and
the Sun-observer line, and $(R_{\odot}/r)^{2}$ represents the classical fall-
off of sunlight with heliocentric distance $r$ (see Howard & DeForest, 2012).
For an observer as close to the Sun as WISPR, additional effects should be
considered, such as the collimation of sunlight and limb-darkening, using for
instance the van de Hulst coefficients $A$, $B$, $C$, and $D$ defined in
Howard & Tappin (2009, Eqs. 25-28). A direct observation of Eq. (1) shows that
the integral is semi-infinite on the path length $z$. In practice, it is
possible to shrink this integral to a limited (finite) region that includes
most contributions to the total brightness (see discussion in Appendix A).

Theoretical works have shown that WL emissions produced from Thomson
scattering are expected to peak at a surface called the Thomson sphere (TS,
Vourlidas & Howard, 2006). This can be geometrically defined by a sphere with
its centre located halfway along the Sun–observer line, and with the length of
this line for diameter. However, Howard & Tappin (2009) and Howard & DeForest
(2012) have demonstrated that this peak at the Thomson sphere is very smeared
out. Therefore, a detector such as WISPR would not be sensitive to electrons
that are concentrated near the Thomson sphere, but rather to a much broader
region on either side of the Thomson sphere ($\approx\chi_{TS}\pm 45^{\circ}$)
that is called the Thomson plateau (TP, Howard & DeForest, 2012). An
illustration of this effect for WISPR is given in Poirier et al. (2020, figure
14).

Although there already are several numerical implementations of this theory
within the scientific community (e.g. in the FORWARD tool: Gibson et al.,
2016), we opted to develop a new algorithm that we can tailor to the specific
constraints of WISPR and the needs of this study. The procedure is summarised
below.

_Instrument definition:_ Given both ephemeris (positioning) and pointing
information for our virtual instrument, we build a 2D matrix of LOS
coordinates.

_Grid optimisation_ : A dynamic grid refinement algorithm adjusts the sampling
along each LOS, so as to capture at best the smallest physical structures in
the simulation box (see Appendix A). The sample points are distributed from
PSP, and pass through and beyond the Thomson sphere. For the WISPR images
synthesised in this work, this represents $241$ million sample points to be
optimised.

_Thomson scattering computation_ : Given a 3D simulated datacube of electron
density, the Thomson scattering formula (Eq. 1) is computed at each sample
point. This includes beforehand an interpolation step that can be very costly,
given the large number of sample points and the size of the input datacubes
used in this work ($(n_{r},n_{\theta},n_{\phi})=(768,384,360)$ and
$(256,160,320)$ for the idealistic and fully fledged 3D set-up, respectively).

_LOS integration_ : The synthetic image is finally obtained by summing up all
local contributions to the total brightness along each LOS.

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 in its distance to the Sun. The WISPR FOV must therefore be updated
very regularly, which is done by rerunning phase 1 and 2 for every single
image to be synthesised. As WISPR is also much closer to the imaged coronal
structures, it is critical to keep an accurate tracking of WISPR’s pointing by
using the World Coordinate System (WCS). For this purpose, phase 1 exploits
the IDL routines provided by the WISPR instrument team through the SolarSoft
library.

For the idealistic dipolar numerical set-up, however, we opted for a simple
user-defined FOV that we can easily control to test different scenarios in a
sandbox. This allows us to simulate various viewing conditions that WISPR-I
have encountered (or may in the future) at distinct phases of the solar cycle.
We define a FOV representative of WISPR-I in the helioprojective–Cartesian
frame with $HPLN=(10,50)^{\circ}$(azimuthal angle) and
$HPLT=(-20,20)^{\circ}$(elevation angle) where $(HPLN=0,HPLT=0)$ points
towards solar centre. We assume a null roll angle for simplicity. The
helioprojective frame is a sphere centred at the observer position, which
needs to be defined as well. We assumed a PSP-Sun distance of $35\
\rm{R_{\odot}}$ and $10\ \rm{R_{\odot}}$; the first is an average between the
March 2022 and April 2021 events presented earlier, and the second is intended
to represent the closest approach that PSP will ever reach in 2024. The
remaining parameter is the latitude of our virtual observer; the longitude
does not matter since the idealistic simulation is axisymmetric about the
solar rotation axis. Varying the latitude $\theta$ allows us to mimic
different inclinations of the streamer belt from the WISPR perspective, where
large (or small) $\theta$ (in absolute value) are intended to be
representative of a streamer belt seen face-on (or edge-on) during solar
maximum (or minimum) conditions. An inclination angle $\theta$ of $0^{\circ}$
and $\operatorname{-}40^{\circ}$ has been assumed, for comparison with the
April 2021 and March 2022 event, respectively.

##  4 Modelling: Results

###  4.1 Idealistic simulation of a dipolar corona

In Appendix B we gather the raw (absolute brightness) synthetic WISPR images
produced from the idealistic dipolar modelling set-up introduced in Sect. 3.1.
To enhance the visibility of transient structures, we follow the base
difference method where a background image (here computed as the average
brightness over the entire time interval) is subtracted from each individual
image. The resulting base-difference synthetic images are shown in Figs. 8-9,
where bright or dark colours respectively correspond to an enhancement or
depletion in electron density with respect to the background solar wind. These
base-difference images reveal faint brightness variations much more clearly,
and thanks to the new adaptive grid refinement method developed for this
paper, small-scale density structures are rendered with great precision. This
manifests as very smooth brightness variations across the LOS, where otherwise
sharpness would indicate an inappropriate sampling along the LOS. Some small
spurious features (especially in the $\theta=\operatorname{-}40^{\circ}$ case)
are sometimes seen. These are remnant artefacts from the adaptive grid
refinement method that need further adjustments (discussed in Sect. 5).

Figure 8:  Synthetic WISPR-I images. The images were produced from the
idealistic dipolar set-up introduced in Sect. 3.1, using a layout similar to
that in Fig. 6, assuming that PSP is located at $10\ \rm{R_{\odot}}$ (left
column) and $35\ \rm{R_{\odot}}$ (right column). This figure depicts the case
of a flat streamer seen edge-on by WISPR ($\theta=0^{\circ}$). The relative
brightness difference with the mean brightness computed over the full interval
is colour plotted using a symmetrical logarithmic scale (with linear scale
below a cut-off value of $10^{-2}$). Isolines of the position ($\psi$) and
elongation ($\epsilon$) angles (in degrees) are also plotted as solid and
dashed white lines respectively. The coloured arrows point to transient
structures (see main text for discussion). An animated version of this figure
is available online at https://doi.org/10.5281/zenodo.8135596.  Figure 9:
Same as Fig. 8, but for a flat streamer seen at a
$\theta=\operatorname{-}40^{\circ}$ inclination angle by WISPR. An animated
version of this figure is available online at
https://doi.org/10.5281/zenodo.8135596.

WL signatures
We focus first on the two right-hand side panels of Figs. 8-9, where the
distance of PSP is taken close to that of the April 2021 and March 2022 events
observed by WISPR (i.e. $35\ \rm{R_{\odot}}$). The four rows cover two full
cycles of the development of the tearing instability. The $t=2880.8\ \rm{min}$
snapshot (bottom rows) is the one that illustrates the different phases best.
We can group the synthetic WL signatures in two main families: bright diffuse
patches and bright and more concentrated emissions. They are described below.

The bright diffuse patches result from the main onsets of the tearing
instability (i.e. the ballooning mode; see the orange arrows). Due to their
rather large scale, they are likely the Sheeley blobs that have long been
observed from 1 AU, and first detected by SoHO-LASCO (Sheeley et al., 1997).
When seen edge-on ($\theta=0^{\circ}$, Fig. 8) they show quite significant
brightness enhancement of up to $\approx 35\%$; instead, when seen with some
inclination ($\theta=\operatorname{-}40^{\circ}$, Fig. 9) they appear slightly
dimmer with brightness enhancements below $\approx 10\%$. Because the
simulated transients have here an infinite extent in azimuth, they show
drifting signatures towards the FOV edges as they pass over WISPR location
(see second row of Fig. 8, right panel), which are similar to WL signatures
that were observed when WISPR approached and went through the streamer belt
(Howard et al. 2019; Poirier et al. 2020; see also the simulations by Liewer
et al. 2019). Nonetheless, having infinite azimuthal extents for such
transients is not realistic, as clearly shown by the March 2022 event, and we
show in Sect. 4.2 that this can be solved using the fully fledged 3D modelling
set-up.

The bright and more concentrated emissions exhibit quasi-periodic formation
(see the green and purple arrows). Similar spatial distributions and widths to
those of the April 2021 and March 2022 events observed by WISPR can be seen in
the lower right panels of Figs. 8 and 9, respectively. These quasi-periodic
structures develop at the back of the main onsets described above, and can be
connected to the long-observed hourly periodicities measured both remotely and
in situ in the slow solar wind (Viall et al., 2010; Viall & Vourlidas, 2015;
Kepko et al., 2016). These structures are smaller in size than the streamer
blobs discussed above, and hence we expect them to contribute less to the
total brightness integrated along the LOS. Conversely, they exhibit much
higher brightness enhancements because they contain a much higher
concentration of plasma. Their brightness variation ranges $\approx 20-100\%$
(edge-on case, $\theta=0^{\circ}$, Fig. 8) and $\approx 1-40\%$ (face-on case,
$\theta=\operatorname{-}40^{\circ}$, Fig. 9). Some of these structures
propagate faster, and as a result sometimes coalesce with their preceding
fellows or even merge with the main onset (see the purple arrows).

In terms of brightness variation, there is a fair agreement between the
simulated quasi-periodic structures and the transients observed by WISPR, that
is $\approx 80-95\%$ for the April 2021 event (edge-on case, to be compared
with $\theta=0^{\circ}$) and $\approx 30-50\%$ for the March 2022 event (face-
on case, to be compared with $\theta=\operatorname{-}40^{\circ}$). On the
other hand, the simulated ballooning modes (which can be associated with the
Sheeley blobs) may be more difficult to see from the WISPR perspective as
their signature is fainter and more diffuse. They are also less likely to be
detected by WISPR due to their long periodicity, as we describe below in ‘
_Periodicities_ ’.

Access to shorter heliocentric distances might help to better resolve both
structures as illustrated in the left panels of Figs. 8-9, assuming the
hypothetical $\simeq 10\ \rm{R_{\odot}}$ to be reached by PSP in 2024 at the
closest approach. There the shrinking of the Thomson plateau (i.e. of the
sensitive area of WISPR; see Sect. 3.3) should allow transients within
streamers to more easily stand out from the background emissions. WISPR will
also be able to observe these structures right in their formation region
($3-7\ \rm{R_{\odot}}$), and therefore might provide new clues about the
tearing instability occurring at the HCS.

Figure 10:  Time analysis of WL transients simulated from the idealistic
dipolar set-up introduced in Sect. 3.1. Top panel: Synthetic WISPR-I J-maps
built from the two-image sequences shown in Fig. 8 and extracted along the
solar equator ($\psi=270^{\circ}$). Bottom panel: Associated period
decomposition at $r=10\ \rm{R_{\odot}}$ using a 1D continuous wavelet
transform (Morse wavelet, Torrence & Compo, 1998).

Propagation velocities and acceleration profiles
We now look at the kinematics of the simulated transients, by making J-maps as
those shown in Fig. 4.

A synthetic J-map for the $\theta=0^{\circ}$ case is given in Fig. 10, where
the slit has been taken at the solar equator (i.e. along the streamer). The
$\theta=\operatorname{-}40^{\circ}$ case is not shown because it does not
change the rest of the analysis much. We retrieve here the two main families
of WL signatures identified in the previous section. First, the wide patches
associated with the main onsets of the tearing instability (i.e. the
ballooning mode; see orange arrows). Second, the more concentrated emissions
associated with quasi-periodic transients, not pinpointed here as they clearly
stand out as bright thin stripes from the rest.

The ballooning modes show quite different signatures in the J-map, with more
curvature and less inclination. Their lower inclination indicates that they
propagate at a slightly lower speed than the quasi-periodic structures, which
are then likely to merge together as discussed previously. Their curvature may
also be indicative of a more progressive acceleration until they reach their
terminal speed after $\approx 10-15\ \rm{R_{\odot}}$; in contrast, the fast
quasi-periodic structures show clear constant speed profiles. Here the
acceleration patterns are likely to be actual and not apparent accelerations
as our WISPR-I FOV remains static in this simulation set-up (see Sect. 3.3),
although LOS-integration effects could still contribute to these curvatures,
as already discussed in Sect. 2.3. In both cases, the simulated transients
reach a terminal speed of $\approx 250\ \rm{km/s}$, which corresponds to the
bulk speed of the very slow and dense wind at the core of the streamer belt in
the simulation.

Periodicities
The bottom panel of Fig. 10 shows the dominant periodicities over four full
cycles of the tearing instability. The time elapsed between each main onset of
the tearing instability (i.e. the ballooning modes) is quite variable,
reaching $\approx 25\ \rm{hr}$ between the first two and $\approx 50\ \rm{hr}$
for the others. Here, this is mostly driven by how fast helmet streamer loops
can grow due to pressure imbalance, and hence is highly sensitive to local
coronal conditions. This long periodicity is then likely to vary significantly
from one simulation to another, and more generally over solar longitudes and
along the solar cycle (see the discussion in Sect. 2.2). In comparison, a
shorter $\approx 8-16\ \rm{hr}$ period had been detected from past STEREO-A
observations (Sanchez-Diaz et al., 2017a), but this study focused on a few
specific events around the solar maximum of cycle 24. More recently, an
analysis of the observations taken by the STEREO-A COR-2 coronagraph near
solar minimum has shown density variations in the streamer belt on timescales
of 0.5-2 days (Morgan, 2021), which this time agree with our simulation. Due
to a fast and highly elliptical orbit, WISPR is not appropriate to detect such
long periodicities. Therefore, the legacy 1 AU observatories remain valuable
assets, which in complement to the recently launched Solar Orbiter might
finally allow us to better parametrise such events.

Regarding the quasi-periodic structures generated between each main onset, a
wavelet power spectrum (Torrence & Compo, 1998) reveals dominant periods
around $\approx 2-3\ \rm{hr}=120-180\ \rm{min}$ and $\approx 7-10\ \rm{hr}$.
In addition, we note that the simulation exhibits some periodicities as low as
$\approx 1.5\ \rm{hr}=90\ \rm{min}$ (at $t=100-105\ \rm{hr}$). These results
are in good agreement with the $\approx 90-180\ \rm{min}$ to $\approx 8-16\
\rm{hr}$ periods that have been typically detected from 1 AU (Viall et al.,
2010; Viall & Vourlidas, 2015; Kepko et al., 2016; Sanchez-Diaz et al.,
2017b). More specifically, they also agree with those measured during the
April 2021 ($130-175\ \rm{min}$) and March 2022 ($110-120\ \rm{min}$) events
observed by WISPR.

###  4.2 Case study simulation of the ninth PSP encounter

We now exploit the fully fledged 3D simulation set-up introduced in Sect. 3.2
and detailed in Réville et al. (2022). Starting from April 2021 (eighth
encounter), interpreting WISPR observations has become highly challenging even
with the use of such state-of-the-art modelling. This is primarily due to PSP
diving much deeper inside the nascent solar wind and to an increase in the
solar activity resulting in a much more structured corona. Tremendous efforts
in tuning-up the model parameters would be required for a fair one-to-one
comparison with the actual WISPR observations taken during the ninth encounter
(August 2021). A dynamic update of the magnetic map at the inner boundary
would also be essential to reach this goal. We leave this for future works,
but there is still a valuable set of information that we can extract from this
simulation to feed the current discussion.

We present in figure 11 the result of our forward modelling method applied to
this simulation. Similarly to Figs. 8-9, a difference method is used to better
visualise brightness fluctuations due to transient propagating structures.
However, computing a mean background over the entire time interval is no
longer appropriate since our virtual WISPR observer is no longer static. We
then follow the well-known running difference method here, where the mean
background image is computed over a sliding temporal window. For completeness
and a more realistic impression, the raw synthetic products in absolute
brightness are also shown in Fig. 14.

In contrast to the idealistic modelling set-up, a wealth of signatures are
produced here with a rich diversity of shapes, extents, and locations. They
are indicated in Fig. 11 by coloured arrows, which can be directly connected
to the flux rope structures present in the simulation snapshot shown in Fig.
7. Transients can be seen throughout the WISPR-I/O FOVs. Overall they show
great topological similarity with the April 2021 and March 2022 events
observed by WISPR-I (see Fig. 2), namely arch-like (top panel) and blob-like
(bottom panel) signatures representative of flux ropes seen face-on and edge-
on, respectively. This time, the fully fledged 3D set-up is more realistic
with arch signatures of finite (and not infinite) spatial extent because it
includes secondary (or pseudo) streamer structures that were missing in the
idealistic dipolar set-up (see Sect. 3.2).

Furthermore, using Fig. 11 we can estimate that the simulated transients
induce a similar increase in relative brightness of $\approx 1-4\%$ during
their passage, although they are located at different distances from WISPR.
These values are quite low compared to the brightness variations estimated
above from WISPR observations and the dipolar corona set-up because the
background field here is computed over a much shorter temporal window, which
is not representative of the emissions from the background streamers. For this
purpose we should use instead the raw synthetic images (in absolute
brightness) shown in Fig. 14, and compare the absolute brightness inside and
just outside the transients (similarly to what we did for the real WISPR
observations). By doing so we obtain more reasonable brightness variations of
$\approx 8-17\%$.

Many of the brightness variations visible in Fig. 11 do not result from a
propagating transient feature, but from a change in the viewing conditions of
WISPR as PSP flies rapidly along its orbit. This has been seen many times, for
instance from the drift of streamer rays towards the FOV edges (Liewer et al.,
2019; Howard et al., 2019; Poirier et al., 2020). This effect can be
visualised in a supplementary movie provided with Fig. 11. Starting from
August 09, 2021, 18:37 UT, the movie shows the effect of PSP moving throughout
the streamer belt, for the first time with a fully dynamic 3D modelling set-up
that includes the self-generation of streamer transient structures. As WISPR
moves closer to (and probably into) these imaged transients, their morphology
changes quite significantly, which is the consequence of a change in the
perspective and of a change in the sensitivity area of WISPR (see Sect. 3.3).
Our comparison basis with actual observations can only be qualitative here for
all the reasons mentioned above, such as the fact that the simulation does not
cover the entire WISPR interval studied here (see Sect. 3.2). Even so, we were
able to identify many similarities between this synthetic movie and the newest
WISPR observations starting from August 2021 (official movies for all WISPR
observations can be found online444https://wispr.nrl.navy.mil/encounter-
summaries).

Figure 11:  Synthetic images of inner WISPR-I and outer WISPR-O telescopes
combined. The images were produced with the fully fledged 3D simulation set-up
introduced in Sect. 3.2. A mean background was subtracted from each image
using a sliding temporal window ranging $\approx 5-10\ \rm{hours}$ (running
difference method). The relative brightness difference to this mean background
is colour plotted with a symmetrical logarithmic scale (with linear scale
below a cut-off value of $10^{-2}$). As the simulation is $\approx 22\
\rm{hours}$ long, only a short temporal window around perihelion could be
treated in a self-consistent manner (dynamic mode). Outside this dynamic
interval, images were then produced with either the first or last simulation
snapshot (static mode). An animated version of this figure is available online
at https://doi.org/10.5281/zenodo.8135596.

##  5 Discussion

In this study we focused only on a few transient events observed by WISPR that
we considered as promising candidates produced by the streamer pinch-off
reconnection mechanism. The April 26 and March 01 events were picked
especially for their great visibility, but do not represent the full set of
observations. WISPR observations show a plethora of transient features, in
particular in the late PSP encounters (from September 2020 and on). Further
studies are required to infer those transient properties in a statistical
manner (see e.g. the upcoming paper from Viall & Vourlidas, 2023).

Although the two modelling set-ups presented in this work show a great
potential for interpreting some of the transient nature of WISPR images, there
are still many improvements that are needed. Significant advances have been
made in the forward modelling procedure compared to our previous study
(Poirier et al., 2020), where most of the previous difficulties have been
resolved. The adaptive grid refinement method allows for a much more accurate
sampling of the LOS, which results in a very smooth rendering of small-scale
density structures. Despite the high precision achieved, we still note some
remaining artefacts of minor importance in the difference-based synthetic
products. For the sake of computational tractability, for each individual
image we had to keep the number of optimisation steps for the LOS sampling to
a maximum of 30 in our case. This implies that the optimisation procedure may
not fully converge for all LOS, at times generating some spurious features.

These are minor inconveniences, and now the performance of our white-light
rendering code is primarily limited by the quality of the input simulation.
Future efforts should then concentrate on pushing existing solar coronal or
wind models beyond their current capabilities in order to allow for a thorough
understanding of the latest WISPR observations. In light of this work, we
present below some areas of progress that could be addressed.

_Coronal structure fidelity:_ Current comparisons against WISPR observations
greatly suffer from a misplacement of the main streamer belt and pseudo-
streamer structures. This highlights the unique capability of WISPR to provide
more stringent constraints to current coronal models. For instance the
magnetic map set at the inner boundary is known to have a critical impact on
the performance of existing MHD models. Having a magnetic map that is updated
dynamically over time will be necessary improve comparisons with recent WISPR
observations. On-going efforts have also been carried on towards a systematic
benchmarking of MHD models against observations (see e.g. Badman et al.,
2022), and WISPR observations could both greatly benefit from and support such
works.

_Spatial resolution:_ Although the fully fledged 3D simulation set-up depicts
a much more realistic solar coronal structure, it only permits a partial
development of the tearing instability compared to the idealistic but much
more (spatially) resolved 2.5D set-up. Future works will have to refine the
spatial resolution around the HCS further, but that is extremely challenging
in such global 3D MHD models while maintaining computational tractability.

_Temporal cadence and duration:_ Together with a sufficiently spatially
resolved model, temporal cadence is also important to track these streamer
transients as they rapidly propagate throughout the WISPR FOV. This will be
important especially in the years to come when PSP will be able to sample
these transients right in their formation region (i.e. the $10\
\rm{R_{\odot}}$ case treated with the 2.5D set-up). Having simulation
snapshots at a higher cadence than WISPR could also help to get a better match
by allowing the synthetic images to mimic the ‘blurring’ effect of exposure
time. Furthermore, simulation duration of more than one day would be
preferable to maximise the scientific output. This can easily be achieved in a
2.5D context, but implies large amounts of data in full 3D modelling set-ups,
a challenge to reach with modern computational facilities.

##  6 Conclusion

The variability of the slow wind that originates from streamers has been
analysed in light of the latest observations taken by WISPR. A few transient
events have been identified with periodicities that are consistent with the
previous $90-180\ \rm{min}$ range detected from near 1 AU observations. The
pinch-off reconnection mechanism occurring at the tip of helmet streamers has
long been predicted as a potential source mechanism of these quasi-periodic
structures.

For the first time this work provides strong evidence to support this scenario
using two advanced MHD models of the solar wind and corona, each with its own
pros and cons. Both give rise to the same fundamental process, however. First
there is a pressure instability of the coronal loops that are lodged beneath
the helmet streamers that allows them to rise in the corona. They stretch to a
point where the current sheet that develops at their back becomes so thin that
magnetic reconnection eventually occurs via the tearing instability. A large
flux rope made of streamer material is then released (i.e. the main onset or
ballooning mode), corresponding to the Sheeley blobs that were first detected
in SoHO-LASCO. Behind this main ejecta follows the further development of the
tearing instability at the HCS, that generates a myriad of quasi-periodic
smaller-scale structures. We show that these quasi-periodic structures exhibit
local density enhancements that are strong enough to be detected by WISPR, and
that they actually show great topological similarities with two real events
captured by WISPR. In addition, the simulated quasi-periodic structures have
periodicities that agree well with these events, and also more generally with
the $90-180\ \rm{min}$ range detected in past observations.

These quasi-periodic structures could be reproduced in an idealistic dipolar
set-up thanks to a very high spatial resolution at the HCS. However, this set-
up lacked some consistency to simulate properly the actual WL signatures
observed by WISPR. A global fully fledged 3D MHD model was then necessary to
simulate the aspect of these structures in a self-consistent manner. However,
the coarser spatial resolution did not allow the full development of the
tearing instability, and thus of the quasi-periodic structures where only the
$\gtrsim 4\ \rm{hours}$ long periodicities could be reproduced.

This work highlights the importance of the tearing instability occurring at
the tip of streamers to fuel the long-observed high variability of the slow
solar wind. Furthermore, we discuss how extremely challenging the latest (and
upcoming) WISPR observations are to interpret, even just for the quasi-steady
component of the slow wind because PSP is diving deeper and deeper within a
solar corona that becomes highly structured with the rising phase of the solar
cycle. Therefore, WISPR offers new stringent constraints to push existing
models of the solar corona and wind beyond their current capability, which in
turn should help in better understanding WISPR observations.

## Acknowledgements

The authors are indebted to an anonymous referee whose valuable suggestions
greatly improved this work.

This research has been funded by the ERC SLOW_SOURCE (DLV-819189) and NRC ORCS
(324523) projects. A.K. was supported by NASA’s Parker Solar Probe mission
under contract NNN06AA01C.

The authors are grateful to Nicholeen Viall and Angelos Vourlidas for
insightful discussions, and the WISPR team for providing the data. The Wide-
Field Imager for Parker Solar Probe (WISPR) instrument was designed, built,
and is now operated by the US Naval Research Laboratory in collaboration with
Johns Hopkins University/Applied Physics Laboratory, California Institute of
Technology/Jet Propulsion Laboratory, University of Gottingen, Germany, Centre
Spatial de Liege, Belgium and University of Toulouse/Research Institute in
Astrophysics and Planetology. Parker Solar Probe was designed, built, and is
now operated by the Johns Hopkins Applied Physics Laboratory as part of NASA’s
Living with a Star (LWS) program.

The authors also thanks A. Mignone and the PLUTO development team, on which
WindPredict-AW is based. The 2.5D and 3D WindPredict-AW simulations were
performed on the Jean-Zay supercomputer (IDRIS), through the GENCI HPC
allocation grant A0130410293.

The photospheric magnetic maps used in this work are produced collaboratively
by AFRL/ADAPT and NSO/NISP. The SOHO/LASCO data are produced by a consortium
of the Naval Research Laboratory (USA), Max-Planck-Institut für Aeronomie
(Germany), Laboratoire d’Astronomie (France), and the University of Birmingham
(UK). SOHO is a project of international cooperation between ESA and NASA.

This work made use of the data mining tools AMDA555http://amda.irap.omp.eu/
developed by the Centre de Données de la Physique des Plasmas (CDPP) and with
financial support from the Centre National des Études Spatiales (CNES). This
study also used the NASA Astrophysics Data System
(ADS666https://ui.adsabs.harvard.edu/), the open-source GNU Image Manipulation
Program (GIMP777https://www.gimp.org/) and the
ImageJ888http://imagej.nih.gov/ij image processing tool developed by Wayne
Rasband and contributors at the National Institutes of Health, USA.

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##  Appendix A Computational challenges and method

Having underresolved LOSs has been shown to have significant consequences on
the synthetic images produced (e.g. artefacts, missing density structures; see
Poirier et al. 2020), and as such $121$ points appeared to be sufficient to
resolve the features studied in that previous work. However, the idealistic
dipolar set-up introduced in Sect. 3.1 requires resolving spatial structures
as small as $0.1\ \rm{R_{\odot}}$. As a consequence, running again our
synthesising script with $121$ points only was no longer adequate. A
significant improvement compared to our previous work (Poirier et al. 2020)
has then been to push the LOSs resolution from $121$ to $241$ points,
resulting in several computational challenges to tackle, which are discussed
below.

Synthesising a white-light image of $1000$-by-$1000$ pixels implies computing
multiple 3D matrices with up to $241$ million elements each, for LOSs resolved
with $241$ points. In typical $32GB$ memory systems this quickly leads to a
memory overflow. One main challenge has been to optimise the code so as to
minimise the memory usage, and at the same time maximise the workload on CPUs.
A prior step before actual computation is to break the image down into smaller
sections, where the number of sections is adjusted automatically to ensure
maximised performances. Each sub-section is then computed in parallel, hence
using at most the capacity of current multi-core systems.

In theory the LOSs could be better resolved than $241$ points, resulting in
more sub-sections to compute, and hence to a longer computational time.
However, we restrained ourselves to $241$ points instead and worked on
optimising the point distribution along each LOS. To do so one needs to have a
prior idea of which portions of the LOSs need to be better resolved than
others. The Thomson scattering (introduced in Sect. 3.3) served as a basis to
optimise the point distribution, following a two-step procedure described
below.

We start by defining a uniform grid in scattering angle $\chi$ that covers
both the foreground ($\chi=90^{\circ}\rightarrow\chi=\chi_{max}$) and
background ($\chi=\chi_{min}\rightarrow\chi=90^{\circ}$) with respect to the
Thomson sphere ($\chi=90^{\circ}$), where $\chi_{min}=0^{\circ}$ and
$\chi_{max}=180^{\circ}-\alpha$ are the asymptotic limits to the acceptable
range of $\chi$ angles ($\alpha$ is the central angle between the LOS and the
observer–Sun line). Using a $\chi$-defined uniform grid is convenient as it
naturally produces a non-uniform grid in the path length $z$ (i.e. the
distance along a LOS from the observer’s position), with a minimum spatial
step near the Thomson sphere. We make a first computation of the total
brightness on this uniform grid. Before proceeding to the grid optimisation
described afterwards, the spatial extent of each LOS is reduced to a region
that accounts for most of the total integrated brightness; we used $99\%$ in
this paper. The upper $\chi_{u}$ and lower $\chi_{l}$ limit to the integral of
the total brightness (Eq. (1)) are determined when the ratio

$\mathcal{R}=\frac{\sum_{\chi=90^{\circ}}^{\chi_{l,u}}n_{e}z^{2}Gdz}{\sum_{\chi=90^{\circ}}^{\chi_{min,max}}n_{e}z^{2}Gdz}$
(2)

reaches $0.99$ in both the foreground and the background. This allows us to
save more grid points in needed areas and to maximise the efficiency of the
grid refinement step described in the next paragraph.

In the second step we implemented an adaptive grid method following a similar
approach to that used by Dorfi & Drury (1987), where the spatial refinement
adapts dynamically according to the physical structures to be resolved, which
are here the density structures along each LOS. For this purpose we define the
grid point density by a function that includes the local distribution of the
total (i.e. not polarised) WL intensity along each LOS

$\begin{aligned}
R^{k+1}_{i}&=\frac{1}{c}\sqrt{w_{1}\left(\frac{\sum_{i=1}^{i=nlos}\Delta\chi^{k}_{i}}{\chi_{u}-\chi_{l}}\right)^{2}+w_{2}\left(\frac{I^{k}_{t,i}/\Delta
z^{k}_{i}}{mean(I^{k}_{t,i}/\Delta z^{k}_{i})}\right)^{2}}\\\
c&=\sqrt{w_{1}+w_{2}}\end{aligned},$ (3)

where the lower $i$ and upper $k$ sub-scripts refer to the spatial index
(position along the LOS) and the optimisation iteration number, respectively.
This formulation allows us to apply multiple optimisation criteria pondered by
their respective weights $w_{*}$, all chosen as equal to one in this paper
(after performing several tests). The first criterion (left term) allows us to
constrain the extent of the optimised grid near the previously determined
$99\%$ range of interest. The second criterion purely depends on the studied
physical system where more points are set where the local intensity is
greater. One should make sure to use normalised quantities to get a proper
balance between each criterion. Using the mean value of the integrated
intensity $mean(I^{k}_{t,i}/\Delta z^{k}_{i})$ along each LOS appears to work
best. We note that $w_{2}=0$ would lead to a uniform grid in $\chi$. The
actual spatial step in $\chi$ angle at the next optimisation step is then
determined by

$\Delta\chi^{k+1}_{i}=\Delta\chi^{k}_{i}\frac{R^{k}_{i}}{R^{k+1}_{i}}.$ (4)

This procedure is applied to each LOS and is repeated iteratively until the
brightness integrated along each LOS converges to a stable value, which is
defined as $<1\%$ variation with previous iteration in this work. To
accelerate the optimisation process, the optimised grid is re-used from one
image to another as a new initialisation instead of the $\chi$-defined uniform
grid.

##  Appendix B Raw synthetic products, absolute brightness

In terms of absolute brightness most of the transient signatures remain
relatively faint over the background solar wind (see the coloured arrows in
Figs. 12-13), even though our virtual WISPR observer is imaging them from a
very close distance. The idealistic case of an inclined streamer belt
($\theta=\operatorname{-}40^{\circ}$, Fig. 13) is even dimmer compared to the
LOS-aligned streamer belt case ($\theta=0^{\circ}$, Fig. 12), as a much
smaller portion of these transients is integrated along the LOS.

A similar comment can be made concerning the absolute brightness images
synthesised from the fully fledged 3D modelling set-up (see Fig. 14), for
which the results are discussed in Sect. 4.2. Here only one flux rope
structure barely stands out from the background. For a better visualisation of
these transients, we decided to work primarily with difference images as those
shown in the core text.

Figure 12:  Same as Fig. 8, but for absolute brightness, plotted in units of
mean solar brightness $B_{\odot}=2.3e7\ \rm{W.m^{-2}.sr^{-1}}$ and in
logarithmic scale. An animated version of this figure is available online at
https://doi.org/10.5281/zenodo.8135596.  Figure 13:  Same as Fig. 9, but for
absolute brightness, plotted in units of mean solar brightness
$B_{\odot}=2.3e7\ \rm{W.m^{-2}.sr^{-1}}$ and in logarithmic scale. An animated
version of this figure is available online at
https://doi.org/10.5281/zenodo.8135596.  Figure 14:  Same as Fig. 11, but for
absolute brightness, plotted in units of mean solar brightness
$B_{\odot}=2.3e7\ \rm{W.m^{-2}.sr^{-1}}$ scaled with the square of the radial
distance $B_{\odot}(R/R_{\odot})^{2}$, and in logarithmic scale. An animated
version of this figure is available online at
https://doi.org/10.5281/zenodo.8135596.