11institutetext:  Département d’Astrophysique/AIM, CEA/IRFU, CNRS/INSU, Univ.
Paris-Saclay & Univ. de Paris, 91191 Gif-sur-Yvette, France
22institutetext: IRAP, Université de Toulouse; UPS-OMP, CNRS; 9 Av. colonel
Roche, BP 44346, F-31028 Toulouse cedex 4, France
22email<EMAIL_ADDRESS>
33institutetext: Laboratoire d’Astrophysique de Bordeaux, Univ. Bordeaux,
CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France

# Solar wind rotation rate and shear at coronal hole boundaries, possible
consequences for magnetic field inversions.

R. F. Pinto 1122 N. Poirier 22 A. P. Rouillard 22 A. Kouloumvakos 22 L. Griton
22 N. Fargette 22
R. Kieokaew 22 B. Lavraud 2233 A. S. Brun 11

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. In situ measurements by several spacecraft have revealed that the
solar wind is frequently perturbed by transient structures that have been
interpreted as magnetic folds, jets, waves and flux-ropes that propagate
rapidly away from the Sun over a large range of heliocentric distances. Parker
Solar Probe (PSP), in particular, has detected very frequent rotations of the
magnetic field vector at small heliocentric radial distances, accompanied by
surprisingly large solar wind rotation rates. The physical origin of such
magnetic field bends and switchbacks, the conditions for their survival across
the interplanetary space, and their relation to solar wind rotation are yet to
be clearly understood.

Aims.  We aim at characterising the global properties of the solar wind flows
crossed by PSP, at relating those flows to the rotational state of the low
solar corona, and at identifying regions of the solar surface and corona that
are likely to be sources of switchbacks and bends.

Methods.  We traced measured solar wind flows from the spacecraft position
down to the surface of the Sun to identify their potential source regions, and
used a global magneto-hydrodynamic (MHD) model of the corona and solar wind to
analyse the dynamical properties of those regions. We identify regions of the
solar corona for which solar wind speed and rotational shear are important and
long-lived, that can be favourable to the development of magnetic deflections
and to their propagation across extended heights in the solar wind.

Results.  We show that coronal rotation is highly structured, and that
enhanced flow shear and magnetic field gradients develop near the boundaries
between coronal holes and streamers, around and above pseudo-streamers, even
when such boundaries are aligned with the direction of solar rotation. The
exact properties and amplitudes of the shears are a combined effect of the
forcing exerted by the rotation of the corona and of its magnetic topology. A
large fraction of the switchbacks identified by PSP map back to these regions,
both in terms of instantaneous magnetic field connectivity and of the
trajectories of wind streams that reach the spacecraft.

Conclusions.  We conclude that these regions of strong shears are likely to
leave an imprint on the solar wind over large distances and to increase the
transverse speed variability in the slow solar wind. The simulations and
connectivity analysis suggest they can be a source of the switchbacks and
spikes observed by Parker Solar Probe.

###### Key Words.:

Sun – Solar wind

##  1 Introduction

Measurements made by Parker Solar Probe (PSP; Fox et al., 2016) during its
first set of orbits have revealed several intriguing properties of the solar
wind at heliocentric distances that had never been probed before. PSP has
shown that strong magnetic perturbations in the form of localised magnetic
field-line bends, the most intense of which are termed switchbacks (SB), are
omnipresent in the pristine solar wind measured near the Sun (Bale et al.,
2019). PSP observations have also shown that the magnitude of the transverse
(i.e, rotational) speeds of the solar wind increases rapidly in the corona
with increasing proximity to the solar surface, up to amplitudes that were not
expected considering its general radial trend higher up in the heliosphere
(Kasper et al., 2019). However, current knowledge of the exact way that the
solar rotation propagates into and establishes in the highly magnetised solar
corona is insufficient to interpret adequately the rotation state of the
observed solar wind flows. Physical links between these coincidental phenomena
are yet to be identified, but possibly relate to the structure of the solar
wind and magnetic field at large scales, and to how their interplay generates
regions with contrasting magnetic field directions and flows. Interchange
reconnection has been pointed out as a potential candidate mechanism (Fisk &
Kasper, 2020), as it links large-scale rotation to magnetic reconnection.
Interchange can occur at a variety of scales in the solar corona, from polar
plumes above small magnetic bipoles inside coronal holes (Wang et al., 2004,
2012; Owens & Forsyth, 2013) to streamer/coronal hole boundaries.

Sudden reversals of the magnetic field have been observed for several decades.
Early examples can be spotted in observations from the Helios mission
(Behannon & Burlaga, 1981). These reversals have since been studied at various
distances from the Sun and related to different photospheric, coronal or
heliospheric phenomena (Kahler et al., 1996; Ballegooijen et al., 1998; Balogh
et al., 1999; Yamauchi et al., 2004b, a; Velli et al., 2011; Neugebauer, 2012;
Wang et al., 2004, 2012; Owens & Forsyth, 2013; Neugebauer & Goldstein, 2013;
Matteini et al., 2014; Borovsky, 2016; Horbury et al., 2018; Sterling & Moore,
2020). SB’s observed by PSP display a high degree of alfvenicity (cross-
helicity), indicating that they propagate outwards mostly in the form of
incompressible MHD waves (Bale et al., 2019), that they correspond to a
rotation of the magnetic field vector without change to its absolute value
(Mozer et al., 2020), and that equipartition between the transverse velocity
($\delta v_{\perp}$) and magnetic field ($\delta b_{\perp}/\sqrt{\rho}$)
perturbations is observed. The physical origin and evolution of magnetic SBs
across vast heliocentric distances remains elusive to this date. Currently
debated hypothesis interpret them as products of surface and coronal dynamics
(reconnection, jets, plumes) or, inversely, claim them to be formed in-situ in
the heliosphere (turbulence, large-scale wind shear). Numerical simulations of
alfvénic perturbations by Tenerani et al. (2020) suggest that switchbacks
originating in the lower corona may survive out to PSP distances on their own,
but only as long as they propagate across a sufficiently unperturbed
background solar wind (free of significant density fluctuations, flow and
magnetic shears that could destabilize them, e.g., the parametric
instability). Owens et al. (2020), however, argue that solar wind speed shear
is an essential factor for the survival of heliospheric magnetic field
inversions (switchbacks) produced close to the Sun up to $1\ \mathrm{AU}$, as
these should not last long enough without being amplified by solar wind speed
shear along their propagation path. Macneil et al. (2020) have furthermore
shown that these magnetic inversions grow in amplitude and in frequency with
altitude in HELIOS data, favouring the idea that they are either created or
amplified by favourable wind shear in the heliophere. From a different
perspective, Squire et al. (2020) propose that magnetic SBs are a natural
result of solar wind turbulence, and should therefore be produced in loco
throughout the heliosphere. Ruffolo et al. (2020) furthermore suggest that
shear-driven MHD turbulence is capable of producing magnetic SBs, especially
on the regions just above the Alfvén surface, where the observed solar wind
flow density transitions from a striated to a flocculated pattern (_cf._
DeForest et al., 2016).

Large-scale solar wind shear builds up naturally from two different
components: gradients in wind speed in the meridional plane (transitions
between slow and fast wind streams, between open and closed field regions),
and gradients in azimuthal wind speed (rotation). If the role of the former is
reasonably easy to identify on large scales, that of the latter is much less
well understood. The solar photosphere is known to rotate with a well-defined
differential latitudinal profile (with the equator having a higher rotation
rate than the poles). The corona above, albeit magnetically rooted in the
photosphere, seems to exhibit a different rotation pattern, with regions that
often appear to be rotating rigidly (Antonucci & Svalgaard, 1974; Fisher &
Sime, 1984). It has been suggested that this rigidity could be the result of
the interplay between emerging magnetic flux and the global field, involving
sustained magnetic reconnection (Wang et al., 1988; Nash et al., 1988).
Overall, the coronal plasma seems to rotate with a more solid-body like
pattern than the photosphere, and the more so at mid and high coronal
altitudes (Insley et al., 1995; Bagashvili et al., 2017). The rotation profile
of the corona also seems to evolve in time, and to be linked to the solar
cycle phase or, at least, to the specific coronal magnetic field configuration
at a given moment (Badalyan et al., 2006; Badalyan, 2010). Observations by
Giordano & Mancuso (2008) using SoHO/UVCS have shown that a number of features
superpose to the average large-scale latitudinal trends of the coronal
rotation during solar minimum. They observed zones displaying particularly low
rotation rates (or high rotation periods) that are likely to be located near
the coronal hole/streamer boundaries (_cf._ their Figs. 5 and 6). Similar
observations performed during solar maximum (Mancuso & Giordano, 2011)
indicate a flatter (more solid-body like) rotation profile, albeit with sub-
structures that are harder to link clearly to the coronal topology.s

In any case, a detailed picture of how the solar rotation establishes
throughout the corona, including coronal holes and streamers, and whether it
can produce sustained wind shears is still lacking. Closed magnetic loops will
tend toward uniform rotation rates all along them (thus opposing rotation-
induced shearing; Grappin et al., 2008), while open field lines will develop a
wind flow that will see its azimuthal speed decreasing with distance from the
Sun (in order to conserve angular momentum, as long as the magnetic tension
exerted by the background field becomes weak enough). Streamers are systems of
closed magnetic loops, and as such should acquire a shape and rotation pattern
that depend on the specific range of solar latitudes that they are
magnetically rooted at. Large streamers can encompass a large range of
magnetic loop sizes and footpoint rotational speeds, and may develop an
internal differential rotation structure. Open field lines will follow either
a vertical path or one with strong inflexions around streamers depending on
where they are rooted at the surface of the Sun, thus having an impact on the
transport of angular momentum from the surface up to the high corona. Thus,
rotation shearing layers can develop at specific places of the solar corona,
such as at the interfaces between coronal holes and streamers. Such shear
layers can be off importance to the formation of magnetic field reversals, if
ever they can develop MHD instabilities that allow for the transport of mass,
vorticy and helicity across different topological regions (via shearing,
resistive or Kelvin-Helmhöltz instabilities; cf. e.g. Dahlburg & Einaudi,
2003; Ruffolo et al., 2020). Additionally, the shear patterns formed can be of
importance to the transport and amplification of such magnetic structures (or
at least a fraction of them).

The first four PSP perihelia occurred during solar minimum, between about
November 2018 and February 2020. During these close passes, the spacecraft
remained within 5 degrees of the solar equator during that time, and also
close to the heliospheric current sheet (HCS). Solar minimum conditions
translated into a corona displaying a large equatorial streamer and two polar
coronal holes, essentially in an equator-symmetric configuration, except for
the occurrence of a small low-latitude CH (visible during the first and second
orbits) and a few equatorward polar CH extensions and small equatorial CHs.

This manuscript focuses on investigating the response of the coronal magnetic
field and solar wind flows to photospheric rotation, and on pointing out
consequences to the interpretation of recent measurements made by PSP. We used
an MHD numerical model of the solar wind and corona, and estimations of the
sun-to-spacecraft connectivity using the IRAP’s connectivity tool to determine
the coronal context of the solar wind flows probed by PSP during its first few
solar encounters (focusing on the first, second and fourth encounters, due to
the lack of continued good quality solar wind data for the third encounter).
We suggest that the global dynamics of the rotating solar corona can impact
the conditions for the formation of magnetic disturbances such as SBs (among
others).

##  2 Numerical model of the rotating corona and wind

Figure 1:  Coronal magnetic field, wind velocity and rotation rate at three
illustrative moments of the solar cycle (from left to right: solar minimum,
maximum and decay phase, corresponding to instants $t=0$, $3.8$, and $4.\
\mathrm{yr}$ in Fig. 3 of (Pinto et al., 2011)). The left halves of the panels
show meridional slices of the wind speed (sonic Mach number, from $0$ in solid
blue to $2$ in solid yellow, and with the yellow/blue boundary representing
the sonic surface), and the right halves show slices of the rotation rate of
the solar corona (from $0$ in black to $14\ \mathrm{{}^{\circ}/day}$ in light
orange). Top and bottom rows correspond to cases with a standard differential
rotation and to solid-body rotation at the lower boundary (the lower boundary
of the corona is coloured with the same $\Omega$ colour scale as the slices on
the right side of the images). Blue lines represent magnetic field lines. The
frame of reference is inertial (not comoving with the Sun), hence $\Omega$ is
positive everywhere.

Figure 2:  Rotation period as a function of latitude at different solar radii
($r=1.03,\ 2.0,\ 4.0,\ 8.0$ and $16\ R_{\odot}$, from darker to lighter green
lines) for the runs represented in Fig. 1 (from left to right: solar minimum,
maximum and decay phase; differential surface rotation on the top row, solid-
body rotation on the bottom row). The red curves indicate the rotation period
imposed at the surface (eq. 6, with $\Omega_{b}$ and $\Omega_{c}$ equal to $0$
in the solid rotation case). Rotation periods peak just outside CH/streamer
boundaries, as in the UVCS observations by Giordano & Mancuso (2008). Global
(resonant) oscillations of closed loops within the streamers are visible
within these main peaks, especially in the case with differential rotation.
Maximum rotational shearing occurs at mid-altitudes (below maximum streamer
height). Wind shear is transmitted upwards, well above the streamer heights,
at the vicinity of HCSs. At higher heights, the corona tends to progressively
approach solid-body rotation with height within coronal holes (note that the
case in the third column has a polar pseudo-streamer).

Figure 3:  Flowlines for the non-rotating (left) and rotating (right) cases,
rendered from different perspectives (from top to bottom: front, side, and
pole-on views). For visual clarity, only the streamlines crossing the northern
hemisphere are shown in the front view. The length of each streamline
indicates the total displacement of a given plasma element during the same
period of time. Streamlines that correspond to plasma lying well within the
streamer make an arc of a circle around the Sun (CCW on the pole-on view),
while those connected to solar wind flows extend outward.

###  2.1 Numerical code and setup

We used the numerical code DIP to model a 2.5 D axi-symmetric solar corona,
setup in a similar way as in Pinto et al. (2016) and Pinto et al. (2011),
although with a higher spatial resolution ($768\times 768$ grid, non-uniform
in radius and uniform in latitude) and including rotation. The code solves a
system of MHD equations that describes a one-fluid, isothermal, fully ionised
and compressible plasma:

$\displaystyle\partial_{t}\rho$ $\displaystyle+$
$\displaystyle\nabla\cdot{\rho\mathbf{u}}=0\ ,$ (1) $\displaystyle P$
$\displaystyle=$ $\displaystyle\frac{2}{m_{H}}\rho k_{B}T\ ,$ (2)
$\displaystyle\partial_{t}\mathbf{u}$ $\displaystyle+$
$\displaystyle\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}=-\frac{\nabla
P}{\rho}+\frac{\mathbf{J}\times\mathbf{B}}{\mu_{0}\rho}-\mathbf{g}+\nu\nabla^{2}\mathbf{u}\
,$ (3) $\displaystyle\partial_{t}\mathbf{B}$ $\displaystyle=$
$\displaystyle\nabla\times{\left(\mathbf{u}\times\mathbf{B}\right)}+\eta\nabla^{2}\mathbf{B}\
.$ (4)

The model assumes the corona and the solar wind to be isothermal with a
uniform coronal temperature $T_{0}=1.3\ \mathrm{MK}$ and a specific heat ratio
$\gamma=1$. The magnetic field $\mathbf{B}$ separates into a time-independent
external component $\mathbf{B^{0}}$ (a potential field resulting from the
internal dynamics of the Sun) and into an induced field $\mathbf{b}$. We
adopted several configurations for $\mathbf{B^{0}}$ in order to simulate
different moments of the solar activity cycle. The equations are integrated
using a high-order compact finite difference scheme (Lele, 1992) with third-
order Runge-Kutta time-stepping (cf. Grappin et al., 2000). The diffusive
terms are adapted to the local grid scale ($\Delta l$), that is non-uniform in
the radial direction ($\Delta l$ is minimal close to the lower boundary). The
kinematic viscosity is defined as $\nu=\nu_{0}\left(\Delta l/\Delta
l_{0}\right)^{2}$, typically with $\nu_{0}=2\times 10^{14}\
\mathrm{cm^{2}\cdot s^{-1}}$ and $0.01\lesssim\left(\Delta l/\Delta
l_{0}\right)^{2}\lesssim 10$. The magnetic diffusivity $\eta$ is scaled
similarly.

The boundary conditions are formulated in terms of the MHD characteristics by
imposing the amplitudes of the MHD characteristics propagating into the
numerical domain (the outgoing ones being already completely determined by the
dynamics of the system). The upper boundary is placed at $r=15\ R_{\odot}$ and
is fully transparent (open to flows and transparent to waves). The lower
boundary is placed at $r=1.01\ R_{\odot}$ and is semi-reflective with respect
to the Alfvén mode (but transparent with respect to all others), and is also
open to flows. We treat the chromosphere and the transition region layers as
an interface (or rather a discontinuity), and define the chromospheric
reflectivity $a$ in terms of the ratio $\epsilon$ of Alfvén wave speeds above
and below:

$a=\frac{\epsilon-1}{\epsilon+1}\ ,\ \mathrm{with}\
\epsilon=\frac{C_{A}^{photosph}}{C_{A}^{corona}}\ ,$ (5)

$C_{A}$ representing the Alfvén speed $B/\left(\mu_{0}\rho\right)^{1/2}$. This
approximation is valid for perturbations whose characteristic wavelength is
much larger than the thickness of the chromosphere, which is the case for the
quasi null-frequency (non-oscillating) rotational forcing that we apply here
at the lower boundary.

In order to establish coronal rotation, we first let a non-rotating solar wind
solution fully develop in the whole numerical domain, and then apply a torque
at the lower boundary which accelerates it progressively to the following
rotation rate profile

$\Omega\left(\theta\right)=\Omega_{a}+\Omega_{b}\sin^{2}\theta+\Omega_{c}\sin^{4}\theta,\
$ (6)

with $\theta$ being the latitude, $\Omega_{a}=14.713\
\mathrm{{}^{\circ}/day}$, $\Omega_{b}=-2.396\ \mathrm{{}^{\circ}/day}$ and
$\Omega_{c}=-1.787\ \mathrm{{}^{\circ}/day}$, following Snodgrass & Ulrich
(1990). We also tested solid-body solar rotation profiles, that were achieved
by setting $\Omega_{b}$ and $\Omega_{c}$ to $0$ in eq. (6). The duration of
the initial acceleration period was defined to be $\sim 1/4$ of the average
(final) rotation period at the surface, and we let the system relax for at
least $10$ Alfvén crossing times of the whole domain (lower to upper
boundary). The initial transient propagates upwards (after crossing the
idealised chromospheric interface) predominantly as an Alfvénic wavefront
(with little power on the non-alfvén charateristics), accelerating the open
field plasma in the azimuthal direction and exciting a few global oscillations
in the closed field regions. After a few Alfvén transit times, and for a small
enough $\epsilon$, the corona and wind settle down into a quasi-steady state.
For $\epsilon\sim 1$, the highly transparent surface spins down very quickly,
as this configuration corresponds to a lower boundary that cannot oppose the
braking torque that results from net outward angular momentum flux carried
away by the solar wind. For $\epsilon\sim 0$, the lower coronal boundary
maintains the imposed rotation, and the magnetic field is line-tied to it.
Smaller closed loops keep oscillating resonantly for a long time and the
larger ones are sheared indefinitely (as their foot-points suffer a larger
range of azimuthal speeds due to differential surface rotation). Intermediate
(more solar-like) values of $\epsilon$ allow for the surface rotation to be
maintained in the open-flux regions while the minimal required amount of
footpoint leakage is allowed for the closed-flux regions to stabilise (_cf._
Grappin et al., 2008). The streamers gradually evolve towards a nearly solid-
body rotation profile, with a rotation rate determined by that of its magnetic
foot-points. Large streamers encompass a wide latitudinal range, and therefore
a wide range of footpoint rotation rates. As a result, such streamers develop
a more distinguishable differential rotation pattern than the smaller ones.
The open field regions (coronal holes) develop permanent azimuthal velocity
and magnetic field components, with a reasonably complex spatial distribution
in the low corona converging into what can be thought of as the beginning of
the Parker spiral on the outer part of the domain. The finite magnetic
resistivity affects the width of these regions, but has a negligible effect on
the overall rotation rates. We note that in order to achieve long-lasting
(_i.e_ , stable) coronal rotation profiles we had to resort to values of
$\epsilon$ of about $10^{-3}$, rather than to a more realistic
$\epsilon=10^{-2}$. The reason for this is that the convective dynamics of the
surface and sub-surface layers of the Sun are absent from our model, and
therefore so are the resulting torques that would counter-balance the weak
(but finite) braking torque exerted on the lower boundary by the rotating
solar wind.

s

###  2.2 Rotating corona and solar wind: overview

We ran a series of numerical MHD simulations of the solar corona and wind on
which axisymmetric streamers and coronal holes are set under rotation
following the methods described in Sect. 2. Figure 1 shows three-dimensional
renderings of different simulation runs, corresponding to different moments of
the solar cycle (from left to right: activity minimum, maximum and decay
phase) and to different surface rotation profiles (from top to bottom:
differential and solid-body). The blue lines are magnetic field lines. The
left halves of images display meridional cuts of the solar wind speed (in
units of sonic Mach number, from $0$ in solid blue to $2$ in solid yellow),
while the right side halves show the rotation rate $\Omega$ (from $0$ in black
to $14\ \mathrm{{}^{\circ}/day}$ in light orange, defined with respect to the
inertial reference frame). The solar surface is coloured with the same colour
scheme as the right side of the images, and hence show the rotation pattern at
the lower coronal boundary. It is immediately clear from the figure that the
solar corona assumes a rotation state that is highly structured and that
reflects the large-scale topology of the magnetic field. As described above,
the streamers tend to set themselves into solid body rotation, with the
largest ones developing a more complex internal rotation structure (with some
of its larger inner loops undergoing global resonant transverse oscillations
for a long period of time). Open field lines that pass well within coronal
holes (faraway from CH/streamer boundaries) progressively develop a backward
bend (opposed to the direction of rotation).The open field-lines that pass
close to the coronal hole boundaries suffer stronger expansions and deviations
from the radial direction, drive the slowest wind flows, and acquire the
lowest rotation speeds at mid coronal heights (not very far from the streamer
tips). The rotation rates remain high at and right above the streamers tips
(close to the HCS/HPS), nevertheless, as these tend to assume a rotation
profile more closely linked with that of the surface (and of the corresponding
streamers). As a result, the strongest spatial contrasts in wind speed and
rotation rate $\Omega$ are usually found across these regions of the corona.

Wind speed gradient $\left|\nabla u\right|$

Rotation rate gradient $\left|\nabla\Omega\right|$

Figure 4:  Zoomed-in view of the absolute magnitudes of the gradients of flow
speed (top) and of the rotation rate $\Omega$ (bottom) close to the equatorial
streamer on the solar minimum (left) and maximum (right) configurations. For
simplicity, only differentially-rotating cases are shown and both quantities
are displayed in normalised units (in units of $6.27\times 10^{-4}\
\mathrm{s^{-1}}$ and $9.0\times 10^{-13}\ \mathrm{rad\ s^{-1}m^{-1}}$,
respectively). The axis indicates distances in solar radii. The green lines
are magnetic field-lines. The coronal hole/streamer boundaries systematically
develop flow shears due to gradients in the solar wind flow along the magnetic
field and to gradients in the rotation pattern (flow across the field)
altogether. These shearing regions reach well into the coronal holes, and into
the wind flow well past the streamer top height.

###  2.3 Varying surface rotation profiles and cycle phase

Different surface rotation profiles lead to qualitatively similar results,
with differences between the differentially-rotating and the solid-body cases
being more pronounced on large coronal loop systems. Streamers that are fully
rooted at mid or high-latitude (i.e, not equator-symmetric) are subject to
footpoint shearing (albeit shallow in amplitude), and acquire an orientation
oblique in respect to the meridional plane (which does not happen with solid-
body rotation). The positions and amplitudes of the fast and slow wind streams
do not change when switching between the two types of rotation. The positions
of the higher and lower rotation rates regions are also maintained, but their
amplitudes and substructure can differ sensibly.

Figure 2 shows the rotation period (in days) as a function of latitude at
different radii ($r=1.03,\ 2.0,\ 4.0,\ 8.0$ and $16\ R_{\odot}$, from darker
to lighter green lines) for the same runs as those represented in Fig. 1 (from
left to right: solar minimum, maximum and decay phase; differential surface
rotation on the top row, solid-body rotation on the bottom row). The red
curves indicate the rotation period imposed at the surface (eq. 6, with
$\Omega_{b}$ and $\Omega_{c}$ equal to $0$ in the solid-body rotation case).
The rotation regimes of coronal holes and closed-field regions are clearly
distinct, with the former tending to progressively approach solid-body
rotation with increasing altitude (note that the case in the third column has
pseudo-streamers at the north and south poles), and the latter showing a more
complex rotation structure. The direct imprint of the imposed surface profile
is only observed within the first few solar radii, with the overall
latitudinal trend of the rotation period (equator to poles) even reversing in
the high corona in some cases (e.g., in the first and third columns of the
figure). Remarkably, rotation periods peak strongly just outside CH/streamer
boundaries. These peaks are present along the whole extension of the
boundaries (note how they get closer together with increasing altitude,
especially for the cases with large streamers), and even beyond. They
correspond spatially to the slowing down of the rotation rate visible as dark
patches in the right halves of the panels in Fig. 1. These features are
particularly prominent in the solar minimum configuration, with a very large
equatorial streamer, and strikingly consistent with the solar coronal rotation
periods measured below $2\ R_{\odot}$ by Giordano & Mancuso (2008) using
SoHO/UVCS data (during activity minimum, May 1996 to May 1997). Their
observations highlight that, as in our simulation, the larger gradients of
rotation rate are found at the boundaries between open and closed magnetic
field lines. Our simulations suggest furthermore that this behaviour is
universal across the activity cycle, although less easy to distinguish during
solar maximum (cf. second column in Fig. 2), due to a more intricate mixture
of smaller streamers/pseudo-streamers and of thinner coronal holes. This solar
minimum to maximum variation is also consistent with the results of Mancuso &
Giordano (2011), who performed a similar analysis with UVCS data during solar
maximum (March 1999 to December 2002). Global (resonant) oscillations of
closed loops within the streamers are visible in the interval between these
main peaks in rotation period (that is, within streamers), especially in the
cases with differential rotation. While latitudinal gradients in rotation rate
tend to be maximal at mid-altitudes (below maximum streamer height), they
remain significant much beyond the height of the largest streamers, at the
vicinities of HCSs. The gradients in rotation period are overall steeper in
the cases under differential surface rotation than on the cases with solid-
body surface rotation.

$u_{\phi}$ to $u_{\left(r,\theta\right)}$ ratio

$B_{\phi}$ to $B_{\left(r,\theta\right)}$ ratio

Figure 5:  Ratio of azimuthal to meridional vector components of the velocity
(top) and magnetic field (bottom) for the solar minimum (left) and solar
maximum (right) configurations. Distances are in solar radii, and green lines
are magnetic field-lines. For simplicity, only differentially-rotating cases
are shown. The core of closed-field regions display high
$u_{\phi}/u_{\left(r,\theta\right)}$ ratios and
$B_{\phi}/B_{\left(r,\theta\right)}\approx 0$ due to them being in solid-body
rotation. Open-field regions develop a $u_{\phi}/u_{\left(r,\theta\right)}$
profile that decays with altitude while $B_{\phi}/B_{\left(r,\theta\right)}$
grows. Coronal hole/streamer boundaries show stark contrasts in velocity and
magnetic field pitch angles, and also extend coronal regions above the mid-
latitude pseudo-streamers on the right panels.

###  2.4 Shear flow morphology

In order to better show the form and amplitude of the shearing flows imposed
by the global coronal rotation, Figure 3 displays a series of velocity
streamlines (or flowlines) corresponding to flows passing above and below the
CH/streamer interface (from streamer mid-height to top). The plots to the left
show, for comparison, a non-rotating solar wind solution, while the plots to
the right show the solar minimum configuration with solar-like differential
rotation. Three perspectives are presented: a front view, a side view, and a
pole-on view (from the north pole), all in the inertial (not rotating)
reference frame. All streamlines are integrated for the same physical time
interval, such that the length of each streamline indicates the total
displacement of a given fluid element during that period of time. Streamlines
that correspond to plasma lying well within the streamer make an arc of a
circle around the Sun (CCW on the pole-on view), while those connected to
solar wind streams extend outward. Among the latter, those that are farther
away from the CH/streamer boundary cross the domain faster and follow a
straighter path, while those that pass closer to it also accelerate slower and
suffer a larger azimuthal deviation before they join the bulk of the wind flow
above. Some flows transition between the two regions, especially those passing
very close to the interface for which diffusive processes favour that
transition. The side-view (second row) also shows how coronal rotation makes
the equatorial streamer slightly taller and with a sharper transition to the
neighboring coronal holes, on the region that develops the strongest magnetic
and flow shear.

###  2.5 Wind shear near the coronal hole boundaries and on the extended
corona

Flow vorticity $\left|\nabla\times\mathbf{u}\right|$

Ratio of poloidal (rotation-induced) to azimuthal (parallel flow induced)
vorcity

Figure 6:  Unsigned flow vorticity (top) and ratio of poloidal to azimuthal
vorticity components (bottom) in the low-latitude regions for the solar
minimum and solar maximum configurations. For simplicity, only differentially-
rotating cases are shown and flow vorticity is displayed in normalised units
($9.0\times 10^{-13}\ \mathrm{rad\ s^{-1}m^{-1}}$). The axes indicates
distances in solar radii. Green lines are magnetic field-lines. The
combination of the shearing flows in Fig. 4 are a source of vorticity that
extend along the CH/steamer boundaries and above. The yellow regions in the
bottom plots indicate zones where the poloidal component (the only one that
gives rise to field-aligned vorticity, due to gradients of $v_{\phi}$ produced
by coronal rotation) is predominant. Pink areas are dominated by wind speed
shear (rather than rotational shear). Streamers have a strong rotational
signature. Pseudo-streamer axis display elongated rotational shearing zones.

The CH/streamer interface region combines two mechanisms that generate
shearing flows: one that acts on the meridional plane (due to the transition
from the inner streamer stagnant flow, to the slow wind region and finally to
the fast wind), and another one that translates into changes in the azimuthal
component of the flow velocity (due to the different rotation regimes on each
side of the boundary). Figure 4 shows the absolute values of the gradients of
the wind speed (top) and $\Omega$ (bottom) in the low corona during solar
minimum (left) and solar maximum (right), focusing on the low latitude
regions. In absolute terms, flow shear is maximum on broad regions around the
CH/streamer interfaces, that extend partially into the close-field regions and
into the coronal holes. These broad regions of strong shear clearly extend
upwards, surrounding the heliospheric currents sheets (HCSs). The closed-field
side of the shear region encloses the transition from the solidly-rotating but
windless part of the domain to the slow wind zone, while the open-field side
covers the transition from slow to fast solar wind flow (with decreasing
rotation rates). These shearing layers are longer and thicker on the solar
minimum configuration, which comprises a large equatorial streamer. The
contribution of coronal rotation to the overall shear is more clearly visible
on the bottom row of the figure. Strong gradients of $\Omega$ are found to be
more tightly concentrated around the interface layers. However, these
gradients extend sideways well into the coronal holes, and upwards asides HCSs
and also along pseudo-streamer axis (cf. the two mid-latitude pseudo-streamer
structures on the panel to the right of the figure). In addition, current
density ($\sim\nabla\times\mathbf{B}$) also accumulates at the coronal
hole/streamer boundaries, at the streamer tip and along the HCS. Plasma
$\beta$ increases with height along the interface, eventually becoming larger
than 1 close to the streamer tip (and base of the HCS).

Angular deviations of magnetic field lines relative to the meridional plane
(due to rotation) are small at these heights, and therefore hard to visualize
directly. But they are present nonetheless, and some of the plotted fieldlines
can be seen to go in and out of the meridional planes represented in Fig. 1.
Open field lines bend backwards, as expected, and more strongly in the slow
wind parts, while the closed loops on the other side of the interface are
close to solid-body rotation. From the high corona upward, the transverse
(azimuthal) component of the magnetic field grows in amplitude with growing
distance from the Sun in response to solar rotation, as expected from classic
wind theory. The azimuthal component of the velocity field, conversely, can be
significant in the low corona, but decreases asymptotically in the high
corona. Fig. 5 shows the spatial distribution of the ratios between the
azimuthal and the poloidal components of the velocity and of the magnetic
field vectors ($u_{\phi}/u_{\left(r,\theta\right)}$ and
$B_{\phi}/B_{\left(r,\theta\right)}$, in absolute value). High values
correspond to high pitch angles, defined as the angle between the vector and
the meridional plane, or the angular deviation in respect to a purely poloidal
flow (in $r$ and $\theta$). A non-rotating corona displays null values
everywhere on a non-rotating corona. A rotating corona will contain closed-
field region in (quasi) solid-body rotation with very high
$u_{\phi}/u_{\left(r,\theta\right)}$ and
$B_{\phi}/B_{\left(r,\theta\right)}\approx 0$, and open-field regions for
which $u_{\phi}/u_{\left(r,\theta\right)}$ decreases and
$B_{\phi}/B_{\left(r,\theta\right)}$ increases with altitude. Magnetic field-
lines are represented as green lines. Some magnetic features, such as the
pseudo-streamers on the case represented on the right panels, generate
localized enhancements of the transverse (azimuthal) fields that are felt
across the radial extent of the numerical domain.

The combination of flow shears in the meridional and azimuthal directions act
as a persistent source of flow vorticity (defined as
$\nabla\times\mathbf{u}$), also with multiple components. Pure wind speed
shear, due to variations in the radial and latitudinal components of the wind
speed gradients, translates into an azimuthal vorticity vector that represents
vortical motions contained within the $\left(r,\theta\right)$ plane. On the
other hand, spatial variations in the azimuthal speed (due to rotation)
produce a vorticity vector with only radial and latitudinal components
(poloidal vorticity), corresponding to vortical flows that develop in the
$\left(\theta,\phi\right)$ or in the $\left(r,\phi\right)$ planes. Figure 6
shows the spatial distributions of the absolute values of flow vorticity (top)
and the ratio of the poloidal to azimuthal vorticity (i.e, ratio of rotation-
induced to wind-speed induced vorticities; bottom plots). Left and right
columns show the differentially-rotating cases at solar minimum and at solar
maximum, as in the previous figures. Flow vorticity is maximal at the
interfaces between closed and open magnetic field, as can be guessed from
Figs. 4 and 5, and extends outward around streamer and pseudo-streamer stalks.
The yellow regions in the bottom plots indicate zones where the poloidal
component – the one that is rotation-induced and can give rise to field-
aligned vorticity – is predominant. The pink areas are, conversely, dominated
by wind speed shear caused by the spatial distributions of slow and fast wind
flows and of wind-less closed-field regions. Streamers (and generally closed-
field regions) have a strong rotational signature that is present up to their
boundaries. The accelerating wind flows that surround them rapidly reach
speeds high enough to overcome the azimuthal (rotational) speeds, and most of
the coronal hole vorticity becomes dominated by wind speed shear. However,
some coronal regions develop a strong and extended rotation-induced vorticity
signature. The most remarkable examples are pseudo-streamers located at mid-
latitudes, which develop elongated rotational shearing zones along their
magnetic axis, as shown in the bottom right panel of Fig. 6. These structures
separate regions of the corona filled with magnetic field rooted at very
different latitudes (polar and quasi-equatorial CHs) with rather different
surface rotation rates. Flow shear remains predominantly forced by global
rotation through large distances on these layers, and thus developing a mainly
field-aligned vorticity (corresponding to vortical motions orthogonal to the
main magnetic field orientation).

##  3 Sun – Parker Solar Probe connectivity context

Figure 7:  Sun to PSP connectivity maps for the first (Oct. – Nov. 2018),
second (Mar. – Apr. 2019) and fourth (Jan. – Feb. 2020) encounters (top,
middle and bottom rows, respectively). Connectivity is computed by projecting
the solar wind speed measured at PSP backwards to the surface of the Sun,
considering the most likely Parker spiral, accounting for solar wind travel
time and connecting to coronal field reconstructions based on magnetograms
corresponding to the expected solar wind release times. The left panels
display Carrington maps of EUV emission (SDO/AIA $193\AA$) overlaid with the
most probable footpoints (solar wind source positions, coloured circles) at
the surface of the Sun for the dates indicated, and the centroid of the
connectivity probability distribution at $5\ R_{\odot}$ (coloured stars). The
cyan line indicates the trajectories of the $2.5\ R_{\odot}$ connectivity
point. The coloured dotted lines are visual aids to connect source-surface to
surface positions (but do not trace the actual field-line trajectories). Date
labels indicate in-situ solar wind measurement times. The right panels show
similar maps of white light emission at $5\ R_{\odot}$ (SoHO/LASCO C3),
overlaid with similar markers (at $5\ R_{\odot}$). The red dashed-line
represents the neutral line (base of the HCS) obtained from PFSS extrapolation
of ADAPT/GONG maps. PSP spent a significant fraction of its first passages
sampling solar wind streams that developed at the vicinity of predominantly
azimuthaly-aligned CH-streamer boundaries, and that propagated through the
heliospheric plasma sheet (in close proximity to the HCS, within the bright
streamer belt region).

Figure 8:  Three-dimensional rendering of the Sun-to-PSP connectivity paths on
Mar 23th 2019 at 00:00 UT, during PSP’s second encounter (_cf._ middle panels
of Fig. 7). The two panels show side and top views, as for the simulations in
Fig. 3 (middle and bottom panels). The radial component of the surface
magnetic field is represented in green (positive polarity) and red (negative
polarity) tones, retrieved from the same ADAPT/GONG magnetograms as in the
previous figure. The transparent grey surface represents the boundary between
the large equatorial streamer and the polar coronal holes, and the violet
ribbon indicates the base of the HCS. The light yellow lines cover the whole
uncertainty ellipse taken into account by the Connectivity Tool. A fraction of
these lines map towards the northern hemisphere, but can be discarded because
they correspond to a magnetic polarity inverse of what PSP measured in-situ
(although they fall into topologically and dynamically equivalent regions of
the corona). The dark red lines indicate the connectivity paths ranked with
the highest connectivity probability. The blue lines trace a few additional
open field lines rooted inside coronal holes. PSP is connected to field lines
(and wind streams) that closely delineate azimuthaly-aligned CH-streamer
boundaries, in concordance with the MHD simulations discussed in Sect. 2. Note
how the lines (than sample the whole uncertainty ellipse) clearly fall into
compact regions that correspond to the broad shearing zones in the top-left
panels of Figs. 4 and 6.

In order to establish links between the results of our numerical simulations
and measurements made by Parer Solar Probe, we attempted to determine the
source regions of the wind streams detected in-situ, together with their
trajectories across the solar corona. This let us verify the applicability of
the model to the coronal context at play, and hence to determine the physical
conditions most likely experienced by those solar wind flows.

###  3.1 The Connectivity Tool

We have used the IRAP’s Connectivity Tool (http://connect-tool.irap.omp.eu/)
to determine the most probable paths that the solar wind streams took all the
way from the solar surface to the position of PSP during its first four
perihelia. The connectivity tool calculates continuously the magnetic and
solar wind connectivity between the Sun and different spacecraft (or Earth),
so as to establish physical links between them all along their orbits
(Rouillard et al., 2020b). The tool offers different possible sources of input
magnetogram data, extrapolation methods and wind propagation models, and lets
the user assess uncertainties and inconsistencies related to the measurements
and models used. We chose to setup the connectivity tool with standard
Potential Field Source-Surface (PFSS) extrapolations of Air Force Data
Assimilative Photospheric Flux Transport (ADAPT) magnetograms (Arge & Pizzo,
2000; Arge et al., 2003). It also provides an evaluation of the Potential
Field Source-Surface (PFSS) reconstructions by comparing the topology of the
neutral line with that of streamers (Poirier et al. 2021, _submitted_). The
choice of the ADAPT magnetograms was based on this evaluation procedure.
Propagation paths and temporal delays were determined by adjusting a solar
wind profile to the wind speeds measured at the position of PSP at each moment
of its trajectory. The wind velocities were obtained from the SWEAP instrument
suite (Kasper et al., 2016), and particularly from the plasma moments from the
Solar Probe Cup (SPC; Case et al., 2020). The solar wind mapping from the
spacecraft position to the low corona, and from there to the surface, can be
affected by different sources of error. Uncertainties related to the exact
wind acceleration profile can lead to different solar wind propagation paths,
and hence to deviations in longitude in the high corona, and total wind travel
time. PFSS extrapolations from magnetic fields measured at the surface of the
Sun are furthermore occasionally affected by positional errors that translate
into few-degree deviations of the coronal structures in latitude. In order to
cope with these issues, the Connectivity Tool determines the points at the
surface of the Sun that connect to an uncertainty ellipse around the orbital
position of the target spacecraft (covering the expected latitudinal and
longitudinal uncertainties). For any given time, the tool hence provides a
list of surface foot-points with different associated probabilities, rather
than unique positions. The time period covered by this manuscript corresponds
to solar minimum configuration, with a close to axi-symmetric corona and a
rather flat HCS. As a result, the dispersion of the foot point positions
predicted for any given date and time leads to region very elongated in the
azimuthal direction, with the corresponding errors rarely leading to
topologically different regions of the Sun. This holds also for moments when
the spacecraft is in close proximity to the HCS and the expected footpoints
split into both solar hemispheres (northern and southern end-regions are
topologically similar). We also made extra runs with lower and higher solar
wind speeds for the duration of the time-periods analysed, and found that the
properties of the connected regions do not change significantly. Our aim here
is to identify the types of solar wind source regions (in the topological
sense) rather than exact surface footpoint coordinates. Therefore the error
sources described above do not translate into variations of the dynamical and
topological properties of the regions crossed by the solar wind, and therefore
should not affect our analysis.

###  3.2 Sun to spacecraft connectivity, solar wind trajectories

Figure 7 shows the trajectory of PSP during perihelia 1,2 and 4 (thick black
line on the left, green on the right panels) plotted over Carrington maps of
coronal EUV emission (SDO/AIA 193 Å; first column) and white-light emission at
$5\ \mathrm{\ R_{\odot}}$ (SoHO/LASCO C3; second column). The corresponding
time periods are during Oct. – Nov. 2018 (top), Mar. – Apr. 2019 (middle) and
Jan. – Feb. 2020 (bottom). Pale blue and red layers cover the base of coronal
holes with positive and negative polarity, respectively. The colored symbols
represent the positions of the solar wind plasma that reached PSP computed
using the IRAP’s connectivity tool, with in-situ measurement dates being
labelled in corresponding colours. Stars indicate solar wind plasma position
at the PFSS source-surface altitude (only the centroid of the uncertainty
ellipse is shown, for simplicity), and circles indicate the most likely wind
source positions at the solar surface. The dotted lines that connect them are
visual aids and, for simplicity, do not trace the actual field-line
trajectories. Following the description in Sect. 3.1, each star corresponds to
several circles that indicate the scatter due to mapping uncertainties. The
red dashed-line represents the neutral line (base of the HCS) obtained from
PFSS extrapolation of ADAPT/GONG maps. We have excluded the third PSP
encounter due to the unavailability of continued good quality solar wind data
during the corresponding time interval.

The surface footpoint charts overlaid on the EUV 193 Å synoptic maps indicate
that PSP spent a significant fraction of its first passages sampling solar
wind streams that developed at the vicinity of predominantly azimuthaly-
aligned CH-streamer boundaries. These regions correspond topologically to the
boundary shear layers illustrated in the top-left panel of Fig. 4 (for our
simulated solar minimum case). The mappings at $5\ R_{\odot}$ on the right
panels (white-light maps) show that the streams detected by PSP propagated
through the heliospheric plasma sheet (in close proximity to the HCS), and
through the bright streamer-belt region. These region corresponds to the
equatorial regions in the high coronain our solar minimum simulations, just
above the tip of the large equatorial streamer.

Solar wind connectivity across this range of heliocentric distances ($1$ to
$5\ R_{\odot}$) is better shown in Figure 8 for one selected instant (during
the second PSP encounter). The figure shows a three-dimensional rendering of
the global magnetic field structure of the corona on March 23rd 2019 (_cf._
Fig. 7) and the magnetic field lines through which the wind plasma that
reached PSP at the instant represented is likely to have escaped from
beforehand. The two panels show side and top views, as in Fig. 3 (middle and
bottom panels) for our solar minimum MHD simulations. The ADAPT/GONG
magnetogram used on our coronal field reconstruction is plotted over the
surface of the Sun, with green and red shades representing positive and
negative polarities. The transparent grey surface represents the boundary
between the large equatorial streamer and the polar coronal holes, and the
violet ribbon indicates the base of the HCS (polarity inversion line). Light
yellow lines represent a series of magnetic field lines that sample the whole
uncertainty ellipse taken into account by the Connectivity Tool. A fraction of
these lines map towards base of the coronal hole on the northern hemisphere,
but are discarded because they correspond to a magnetic polarity inverse of
that measured in-situ by PSP. The dark red lines indicate the connectivity
paths ranked with the highest connectivity probability. The blue lines trace a
few additional open field lines rooted inside coronal holes. It is clear from
the figure that the wind streams sampled by PSP at this instant developed
along paths that closely delineate azimuthaly-aligned CH-streamer boundaries,
in concordance with the MHD simulations discussed in Sect. 2. In view of the
temporal sequence of Sun-to-PSP solar wind connectivity displayed in Fig. 7,
this configuration was the most common one throughout the periods analysed.

The most probable Sun–spacecraft solar wind propagation paths lie, for the
most, right along the boundary between the large equatorial streamer and the
polar coronal holes. As PSP proceeded on its orbit, the source regions at the
surface scanned this boundary continuously, unless during brief periods of
connection to low latitude coronal holes or to deep equatorward polar CH
extensions. PSP spent a large fraction of its first few encounters probing
solar wind streams that formed in the vicinity of azimuthally aligned
CH/streamer boundaries, and especially so during the 2nd and 4th encounters,
with occasional connections to small low latitude coronal holes or equatorward
polar coronal hole extensions (Griton, et al, 2020; in press). The solar
corona retained a high degree of axial symmetry throughout the time periods
involved in this analysis. The HCS (and HPS) maintained a predominant E-W
orientation in the solar regions connected to PSP. These reasons justify the
use of the MHD simulations represented in Figs. 1 to 6.This is especially true
for the solar minimum case (with an axi-symmetric large equatorial streamer).
Deviations to this configuration, such as the coronal hole extensions and
equatorial coronal holes visible near Carrington longitudes $280$ and $80$ in
Fig. 7, are in principle associated with pseudo-streamers such as those in our
simulations for the cycle decay phase (right panels of Figs. 4 to 6, at mid-
latitudes), although perhaps with different sizes and orientations. We
consider that the regions rooted at about $\pm 60\ \mathrm{deg}$ in our
simulations for the solar minimum case (_cf._ Figs. 4 to 6, left) are the most
representative of the source regions of the wind flows detected by PSP. These
are coronal hole boundary regions oriented in the east-west direction
(parallel to the direction of solar rotation). Solar wind streams originating
from these regions are accelerated through an environment with significant and
spatially extended solar wind speed and rotation shear, that also correspond
to the peaks in rotation period in the left panels of Fig. 2.

The dynamical properties of these zones of the corona should have an impact on
the properties of the wind measured _in-situ_ , if not being being responsible
for some of their characteristics. On PSP data, transitions from streamer
(i.e, boundary layer) to non-streamer (core of coronal hole) wind flows were
accompanied by a clear decrease in the variablity of the wind (Rouillard et
al., 2020a), both in frequency and amplitude of magnetic SBs and on the
occurrence of strong density fluctuations. This is suggestive that the
physical conditions associated with coronal hole boundaries are favourable to
the development of such perturbations. Interchange reconnection, often invoked
as a possible SB generation mechanism (Fisk & Kasper, 2020), relies on the
forcing of these boundary regions by the large-scale rotation of the corona.
However, it is expected to be enhanced (or more efficiently driven) at
CH/streamer boundaries that are orthogonal or inclined with respect to the
direction of rotation as, e.g, a streamer pushes into neighbouring CHs (see
e.g. Lionello et al., 2005), and reduced on azimuthally aligned CH/streamer
boundaries. This is in contrast with the general spatial orientation of the
equatorial streamer observed during this period and that of our simulations,
on which streamer/CH boundaries are parallel to the direction of rotation.
While less favourable to interchange reconnection, this spatial orientation
does not hamper the formation of the shear flows in the solar wind shown in
Figs. 3 to 6. Furthermore, these shears remain visible up to large
heliocentric distances, which could favour the propagation (or even
amplification) of magnetic perturbations formed in the low corona, allowing
them to survive more easily up to the altitude of detection (Owens et al.,
2020; Macneil et al., 2020). Beyond this phenomenology, the boundaries of
polar coronal holes are also known to be highly dynamic and undergo
significant reconfiguration over the roughly 24-hour timescale of
supergranules (Wang et al., 2010). These effects are not modelled in this
paper, but could induce additional variability that could be felt by PSP. The
existence of neighbouring solar wind streams with different rotation rates (as
those in the streamer stalk regions on our simulations) should furthermore
contribute to increasing the variability of the transverse (rotational)
velocities measured by PSP, especially as the HCS is slightly warped (_cf._
Fig. 7).

##  4 Conclusion

###  4.1 Summary

We investigate the development of spatially extended solar wind shear regions
induced by solar rotation and by variations in solar wind speed in the light
of recent Parker Solar Probe findings. Our analysis combines simulations made
using a MHD numerical model of the solar wind and corona, and estimations of
the sun-to-spacecraft connectivity during the first 4 PSP solar encounters to
aid in associating model results to s/c data. Our main findings are that:

  1. 1.

Solar wind flows that develop in the vicinity of coronal hole boundaries are
subject to persistent and spatially extended shearing. There are two
components to this shearing: a wind speed shear due to the transition from
closed-field (no-wind) to the slow and fast wind regions, and a rotational
shear due to the way coronal rotation settles in response to the rotation of
the solar surface. The most significant shearing occurs in thin layers that
lie along CH/streamer (or pseudo-streamer) boundaries, and that stretch
outwards in the vicinity of heliospheric current sheets and pseudo-streamer
stalks. Wind speed shear generates a spatially broader shearing signature
associated with a large-scale vorticity vector oriented in the azimuthal
direction Rotational shearing produces shearing patterns with field-aligned
vorticity that become predominant in elongated regions above pseudo-streamer
stalks that extend to great distances from the Sun.

  2. 2.

The solar corona acquires a complex rotation pattern that differs
significantly from that of the surface rotation that drives it. Closed-field
regions (streamers, pseudo-streamers) tend to set themselves into solid body
rotation, with a rate consistent to that of the surface regions at which they
are rooted. Open field lines show a variety of rotation rates, with those that
pass near coronal hole boundaries acquiring the lowest rotation rates at mid
coronal heights, in stark contrast with the closed-field regions across the
boundary. This results in clear increases of rotation period adjacent to
streamers (especially visible in large streamers), in agreement with SoHO/UVCS
observations. Streamer stalks (and the vicinity to HCS/HPS) can contain a
mixture of wind streams at different rotation rates (slow rotating flows
coming from the CH boundaries, faster wind flows coming from the streamer
tips).

  3. 3.

Solar wind flows probed by Parker Solar Probe during its first four orbits
form and propagate away from the Sun through regions of enhanced wind speed
and rotational shear. Our Sun-to-spacecraft connectivity analysis shows that
such solar wind flows originated mostly at the boundaries of quasi-axisymetric
polar coronal holes, with occasional crossings of low-latitude coronal holes.
The measured wind flows showed a strong and complex rotational signature
permeated by pervasive magnetic perturbations such as switchbacks (among
others). Our results suggest that the slow wind flows detected by PSP should
be experience persistent shears across their formation and acceleration
regions, supporting the idea that these should have an impact on the formation
localised magnetic field reversals and be favourable to their survival across
the heliosphere.

###  4.2 Discussion and perspectives

Unlike its photospheric counterpart, current knowledge of the rotational state
of the solar corona is very limited. Different observation campaigns (relying
on different measurement methods) have suggested that the corona rotates in a
manner that is not in direct correspondence with the surface rotation, and
that it depends to some degree on the solar cycle phase (that is, on the
global magnetic topology). More recently, Parker Solar Probe unveiled the
prevalence of strong rotational flows that increase rapidly in amplitude as
the spacecraft approached the Sun, at least in the regions probed during its
first few close passes (Kasper et al., 2019). Our MHD simulations show that
surface rotation is transmitted to the corona in a complex manner that depends
intrinsically on the organisation of the large-scale magnetic field at any
given moment. Coronal rotation is highly structured at low coronal altitudes,
with a clear signature of slowly rotating flows that follow the CH/streamer
boundaries, in agreement with the observations by Giordano & Mancuso (2008).
Some of these strong gradients in rotation rate produce an imprint that
extends faraway into the high corona (up to the upper boundary of the
numerical model). These non-uniformities in rotation rate translate into solar
flow shear with a vorticity component oriented along the magnetic field and
solar wind propagation direction, that adds up to the shear caused by the
spatial distribution of fast and slow wind flows (and that can only generate
an orthogonal vorticity component).

The MHD model setup relies upon a number of simplifications to the full
physical problem. It uses a polytropic description of the plasma
thermodynamics, meaning that the heating and cooling mechanisms are not
modelled in detail, but that the main dynamical and geometrical features of
the solar wind are retained. This approach furthermore leads to a solar wind
with speed variations which are much weaker than those found on the real solar
atmosphere (smaller contrast between typical fast and solar wind speeds and
broader transitions). We settled on the limiting case in which the rotating
solar corona and solar wind are perfectly axisymetric (with CH/streamer
boundaries perfectly parallel to the direction of rotation). Rotation-induced
interchange reconnection is completely inhibited in this configuration, as is
the development of shear instabilities. The formation of full vortical flows
in the $\left(\theta,\phi\right)$ plane and the injection of helicity into the
wind flow are inhibited. This choice of problem symmetry has, however, a
number of advantages in respect to the full 3D equivalent, namely that it
allows running many more variations of parameters (different phases of the
cycle, different solar surface rotation profiles), and that the runs can be
easily made at a higher spatial resolution. As a consequence, the simulations
develop sharper CH/streamer boundaries than their full 3D counterparts, which
help making the rotation shears more apparent. These boundaries should,
notwithstanding, be much sharper in the real solar corona. For these reasons
altogether, the amplitudes of solar wind speed and rotational shearing layers
should be higher in the real Sun than those that the MHD simulations are
capable of producing. We also used an idealised solar dynamo model to
constrain the large-scale magnetic field topology at each moment of the solar
cycle, meaning that our MHD simulations do not aim at modelling a specific
event, but rather at letting us understand the dynamics of the regions of
interest (the model produces a full set of typical solar coronal structures –
streamers, pseudo-streamers and coronal holes – placed at different latitudes
according to solar activity).

As we have shown with the help of the IRAP’s connectivity tool
(http://connect-tool.irap.omp.eu/), the solar wind streams that reached Parker
Solar Probe during its first few encounters most often traversed the vicinity
of the boundaries between a large equatorial streamer and the adjacent coronal
holes. The dynamics at play in such regions suggest that they are potential
hosts for the development of shearing instabilities (among others) that are
continuously driven by the large-scale wind shears at these boundaries. Such
processes can potentially introduce alfvénic perturbations into the solar wind
and/or give rise to discrete helical (and pressure-balanced) MHD structures
(depending on the aforementioned instability thresholds, growth rates and
time-scales of ejection onto the wind). Due to the limitations intrinsic to
the global MHD approach expressed above, the simulations do not let us
directly identify the onset of shear instabilities (such as Kelvin-Helmhöltz).
The velocity and magnetic field gradients remain, in practice, much smoother
than those that real solar wind can have, and the global flow remains rather
laminar (without fully developed turbulence). This is issue will be addressed
in future work. Interestingly, however, the solar wind shear – due both to the
transition from slow to fast wind and to the rotation rate gradients – extends
well beyond the height of the highest streamers in our simulations, meaning
that the driving mechanism could be effective over a rather large range of
heights. One significant feature revealed by our simulations is the formation
of a very long column with a strong field-aligned vorticity signature atop of
a pseudo-streamer (bottom right panel in Fig. 6). Coronal rotation forces a
strong shear across the pseudo-streamer axis in the plane perpendicular both
to the wind flow and magnetic field direction that acts on the solar wind over
large distances. This provides, perhaps, a more efficient mechanism to inject
helicity (and/or perturbations in the transverse direction, such as those that
characterize switchbacks) than the milder (and effective over shorter
distances) shearing present at the boundaries of large equatorial streamers.
During its first solar encounter, Parker Solar Probe switched connectivity
temporarily from an equatorial streamer boundary region to a low latitude
pseudo-streamer that formed near the small equatorial coronal hole visible in
the top left panel of Fig. 8. During this period, PSP magnetic field
measurements show a transition from a period of strong amplitude and very
frequent switchbacks to another with lower amplitude and more spaced events,
hence supporting the hypothesis above. Our results therefore give support to a
hybrid view on the origin of magnetic switchbacks, by providing both a
generation mechanism acting in the low corona, and a way to sustain (if not
amplify) at least part of them on their way through the high corona and
heliosphere.

A more detailed investigation of these hypothetical mechanisms is yet to be
undertaken, and will require substantial modeling efforts. We believe,
however, that the scenario we describe in this manuscript can shed light on
how to bring different characteristics of SBs (generation, propagation) come
together harmoniously, and also provides new insights on the mechanisms
related to plasma transport between closed and open field regions of the solar
corona. We look forward to Solar Orbiter (Müller et al., 2020) and to its
combined in-situ and remote sensing campaigns, that will be made from
increasingly higher latitudes and at different phases of the solar cycle,
hence providing a unique opportunity to detect wind flows that are formed and
accelerated in a larger range of coronal contexts. Parker Solar Probe will
keep reducing its perihelium distance, and will certainly provide a closer
look into the effects of solar wind shear and rotation on increasingly more
pristine wind flows.

###### Acknowledgements.

We acknowledge support from the French space agency (Centre National des
Études Spatiales; CNES; https://cnes.fr/fr) that funds activity of the space
weather team in Toulouse (Solar-Terrestrial Observations and Modelling
Service; STORMS; http://storms-service.irap.omp.eu/) that developed the
Connectivity Tool. The work of L. G., A. P. R., and N. P. was funded by the
ERC SLOW_SOURCE project (SLOW_SOURCE – DLV-819189). R. F. P. and A. S. B.
acknowledge support by the ERC synergy grant Whole Sun #810218. The numerical
simulations were performed using HPC resources from CALMIP (Grant 2020-P1504).

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