tablenum SIXtablenum

11institutetext: Institute of Physics, University of Graz, Austria
22institutetext: Columbia Astrophysics Laboratory, Columbia University, New
York, USA 33institutetext: Leibniz Institut for Astrophysics Potsdam, Germany
44institutetext: Department of Physics, University of Helsinki, Finland
55institutetext: Institute of Experimental and Applied Physics, University of
Kiel, Germany 66institutetext: CAS Key Laboratory of Geospace Environment,
University of Science and Technology of China, Hefei, China 77institutetext:
Max Planck Institute for Solar System Research, Göttingen, Germany
88institutetext: Centre for mathematical Plasma Astrophysics (CmPA),
Department of Mathematics, KU Leuven, Belgium 99institutetext: Solar-
Terrestrial Centre of Excellence - SIDC, Royal Observatory of Belgium,
Brussels, Belgium 1010institutetext: Institute of Physics, University of Maria
Curie-Skłodowska, Lublin, Poland 1111institutetext: DTU Space, Copenhagen,
Denmark 1212institutetext: Kanzelhöhe Observatory for Solar and Environmental
Research, University of Graz, Austria 1313institutetext: Hvar Observatory,
Faculty of Geodesy, University of Zagreb, Croatia

# How the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical model

Stefan J. Hofmeister How the area of solar coronal holes affects the
properties of high-speed solar wind streams near Earth - I. An analytical
modelHow the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical modelHow the area of solar
coronal holes affects the properties of high-speed solar wind streams near
Earth - I. An analytical modelHow the area of solar coronal holes affects the
properties of high-speed solar wind streams near Earth - I. An analytical
modelHow the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical modelHow the area of solar
coronal holes affects the properties of high-speed solar wind streams near
Earth - I. An analytical model Eleanna Asvestari How the area of solar coronal
holes affects the properties of high-speed solar wind streams near Earth - I.
An analytical modelHow the area of solar coronal holes affects the properties
of high-speed solar wind streams near Earth - I. An analytical model Jingnan
Guo How the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical modelHow the area of solar
coronal holes affects the properties of high-speed solar wind streams near
Earth - I. An analytical modelHow the area of solar coronal holes affects the
properties of high-speed solar wind streams near Earth - I. An analytical
modelHow the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical model Verena Heidrich-Meisner
How the area of solar coronal holes affects the properties of high-speed solar
wind streams near Earth - I. An analytical modelHow the area of solar coronal
holes affects the properties of high-speed solar wind streams near Earth - I.
An analytical model Stephan G. Heinemann How the area of solar coronal holes
affects the properties of high-speed solar wind streams near Earth - I. An
analytical modelHow the area of solar coronal holes affects the properties of
high-speed solar wind streams near Earth - I. An analytical model Jasmina
Magdalenic How the area of solar coronal holes affects the properties of high-
speed solar wind streams near Earth - I. An analytical modelHow the area of
solar coronal holes affects the properties of high-speed solar wind streams
near Earth - I. An analytical modelHow the area of solar coronal holes affects
the properties of high-speed solar wind streams near Earth - I. An analytical
modelHow the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical model Stefaan Poedts How the
area of solar coronal holes affects the properties of high-speed solar wind
streams near Earth - I. An analytical modelHow the area of solar coronal holes
affects the properties of high-speed solar wind streams near Earth - I. An
analytical modelHow the area of solar coronal holes affects the properties of
high-speed solar wind streams near Earth - I. An analytical modelHow the area
of solar coronal holes affects the properties of high-speed solar wind streams
near Earth - I. An analytical model Evangelia Samara How the area of solar
coronal holes affects the properties of high-speed solar wind streams near
Earth - I. An analytical modelHow the area of solar coronal holes affects the
properties of high-speed solar wind streams near Earth - I. An analytical
modelHow the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical modelHow the area of solar
coronal holes affects the properties of high-speed solar wind streams near
Earth - I. An analytical model Manuela Temmer How the area of solar coronal
holes affects the properties of high-speed solar wind streams near Earth - I.
An analytical modelHow the area of solar coronal holes affects the properties
of high-speed solar wind streams near Earth - I. An analytical model Susanne
Vennerstrom How the area of solar coronal holes affects the properties of
high-speed solar wind streams near Earth - I. An analytical modelHow the area
of solar coronal holes affects the properties of high-speed solar wind streams
near Earth - I. An analytical model Astrid Veronig How the area of solar
coronal holes affects the properties of high-speed solar wind streams near
Earth - I. An analytical modelHow the area of solar coronal holes affects the
properties of high-speed solar wind streams near Earth - I. An analytical
modelHow the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical modelHow the area of solar
coronal holes affects the properties of high-speed solar wind streams near
Earth - I. An analytical model Bojan Vršnak How the area of solar coronal
holes affects the properties of high-speed solar wind streams near Earth - I.
An analytical modelHow the area of solar coronal holes affects the properties
of high-speed solar wind streams near Earth - I. An analytical model Robert
Wimmer-Schweingruber How the area of solar coronal holes affects the
properties of high-speed solar wind streams near Earth - I. An analytical
modelHow the area of solar coronal holes affects the properties of high-speed
solar wind streams near Earth - I. An analytical model

(Received ¡date¿ / Accepted ¡date¿)

Since the 1970s, it is empirically known that the area of solar coronal holes
affects the properties of high-speed solar wind streams (HSSs) at Earth. We
derive a simple analytical model for the propagation of HSSs from the Sun to
Earth, and thereby show how the area of coronal holes and the size of their
boundary regions affect the HSS velocity, temperature, and density near Earth.
We presume that velocity, temperature, and density profiles form across the
HSS cross-section close to the Sun, and that these spatial profiles translate
into corresponding temporal profiles in a given radial direction due to the
solar rotation. These temporal distributions drive the stream interface to the
preceding slow solar wind plasma, and disperse with distance from the Sun. The
HSS properties at $1\text{\,}\mathrm{A}\mathrm{U}$ are then given by all HSS
plasma parcels launched from the Sun that did not impinge into the stream
interface at Earth distance.

We show that the velocity plateau region of HSSs as seen at
$1\text{\,}\mathrm{A}\mathrm{U}$, if apparent, originates from the center
region of the HSS close to the Sun, whereas the velocity tail at
$1\text{\,}\mathrm{A}\mathrm{U}$ originates from the trailing boundary region.
The peak velocity of HSSs at Earth depends on the longitudinal width of the
HSS close to the Sun. The shorter the longitudinal width of a HSS close to the
Sun, the more of its ’fastest’ HSS plasma parcels from the HSS core and
trailing boundary region have impinged into the stream interface to the
preceding slow solar wind, and the smaller is the peak velocity of the HSS at
Earth. As the longitudinal width is statistically correlated to the area of
coronal holes, this explains also the well-known empirical relationship
between coronal hole areas and HSS peak velocities. Further, the temperature
and density of HSS plasma parcels at Earth depend on their radial expansion
from the Sun to Earth. The radial expansion is determined by the velocity
gradient across the HSS boundary region close to the Sun, and gives the
velocity-temperature and density-temperature relationships at Earth their
specific shape. When considering a large number of HSSs, the presumed
correlation between the HSS velocities and temperatures close to the Sun
degrades only slightly up to $1\text{\,}\mathrm{A}\mathrm{U}$, but the
correlation between the velocities and densities is strongly disrupted up to
$1\text{\,}\mathrm{A}\mathrm{U}$ due to the radial expansion. Finally, we show
how the number of particles of the piled-up slow solar wind in the stream
interaction region depends on the velocities and densities of the HSS and
preceding slow solar wind plasma.

###### Key Words.:

Sun - Solar Wind - Solar-Terrestrial Relations

##  1 Introduction

Since the 1970s, it is well-known that high-speed solar wind streams (HSSs),
i.e., the fast, Alfvénic component of the solar wind, originate from solar
coronal holes (Nolte et al., 1976). Solar coronal holes are regions of reduced
density and temperature compared to the ambient solar corona, and feature an
open-to-interplanetary-space magnetic field topology. Along these open
magnetic field lines, solar plasma is accelerated radially away from the
rotating Sun, reaching supersonic speeds forming the large-scale HSSs that
transcend our solar system.

The acceleration process of HSSs, and in general of the solar wind, is one of
the large unsolved problems in solar physics (Cranmer, 2002). There are two
concurring theories for the acceleration. On the one hand, there are
wave/turbulence driven models, which are based on the observation of slow
magneto-acoustic waves traveling along the magnetic field lines upwards to
larger heights (Banerjee et al., 2009), and their turbulent dissipation by
interaction between outgoing waves and incoming waves reflected in the outer
corona (e.g., review by Cranmer, 2009; Hahn et al., 2012; Hahn & Savin, 2013).
On the other hand, there are reconnection/loop-opening models, which are based
on continuous magnetic reconnection processes, e.g., with ephemeral magnetic
regions within coronal holes (Wang, 2020). Although it is not clear yet which
of the two processes provides most of the energy needed for the solar wind
acceleration, we are sure that the main part of the acceleration takes place
at heights $<20\text{\,}\frac{\mathrm{\text{R${}_{\textup{sun}}$}}}{}$, i.e.,
very close to the Sun (Chashei et al., 2005; Wang et al., 2000; Yakovlev &
Yakovlev, 2019).

The direction of propagation of HSSs throughout the heliosphere is set at
similar distances close to the Sun. Since up to the Alfvénic point, which is
probably located at about
$1217\text{\,}\frac{\mathrm{\text{R${}_{\textup{sun}}$}}}{}$ (Pizzo et al.,
1983; Marsch & Richter, 1984), the magnetic energy density dominates both the
thermal and kinetic energy density of HSSs. Thus, HSSs mainly follow the
global magnetic field configuration at these heights. At the Alfvénic point,
the magnetic and kinetic energy density are equal. The magnetic field
configuration gets affected by the outflowing solar wind, and consequently,
magnetic field lines get stretched into the radial direction away from the Sun
(Pneuman & Kopp, 1971).

Starting from the Alfvénic point, the kinetic energy density becomes dominant,
and the solar wind plasma mainly propagates radially away from the rotating
Sun. Since the magnetic field is frozen into the solar wind but also anchored
in the rotating Sun, the heliospheric magnetic field winds up and forms an
Archimedean spiral (Parker, 1958). Different coronal sources of the various
types of solar wind thereby correspond to different spiral sectors in the
Archimedean spiral. Consequently, also the interface between HSS plasma and
preceding ambient slow solar wind plasma forms a spiral. At the back of this
interface, the faster HSS plasma impinges into the stream interface and gets
thermalized, whereas in the front of this interface, preceding slow solar wind
plasma gets piled up, forming a dense, hot stream interaction region (SIR)
around the stream interface.

At Earth distance, HSSs usually appear as a gradual increase of velocity
across the SIR, an occasionally apparent extended velocity plateau, and a slow
decline of velocities back to slow solar wind velocities. Nolte et al. (1977)
studied eight HSSs in 1973 at Earth distance and found that most of the HSS
plasma during its declining phase seen at Earth originated from a boundary
region at the trailing edge of the coronal hole, having a width of $\approx
6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$. This approximate size of the
boundary regions has further been confirmed by a fast-latitude scan of
Ulysses, finding a boundary length of $\approx
6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ for the equator-ward boundary
of a HSS originating from a polar coronal hole (McComas et al., 1998). The
velocities, temperatures, and ion charge states of undisturbed HSS plasma
parcels are generally well correlated to each other, while their densities
show a weak anti-correlation to the velocities (Neugebauer, & Snyder, 1966;
Burlaga & Ogilvie, 1970a, b; Ogilvie, 1985; Xu & Borovsky, 2015).

Besides the above-mentioned correlations, there are two well-known
relationships between the peak velocity of HSSs near Earth and decisive
properties of their source coronal holes: 1) the relationship to the flux tube
expansion factor of coronal holes (Wang, & Sheeley, 1990), and 2) to the area
of coronal holes (Nolte et al., 1976). The relationship to the flux tube
expansion factor is usually interpreted in terms of at which height energy is
deposited to accelerate the solar wind plasma close to the Sun (Wang, &
Sheeley, 1990), and therefore thought to be related to the main acceleration
phase of HSSs. Consequently, it is often employed by solar coronal models to
provide solar wind properties at a spherical shell at a distance of
$0.1\text{\,}\mathrm{A}\mathrm{U}$ from the Sun. These solar wind properties
are then used as inner boundary conditions to model the solar wind in the
heliosphere, e.g., by time-dependent 3D magnetohydrodynamic (MHD) models like
ENLIL (Odstrcil, 2003) and EUHFORIA (Pomoell, & Poedts, 2018).

The relationship of the HSS speed near Earth to the area of coronal holes has
first been found by Nolte et al. (1976), and confirmed by Robbins et al.
(2006); Vršnak et al. (2007); Abramenko et al. (2009); Karachik, & Pevtsov
(2011); Rotter et al. (2012); Hofmeister et al. (2018); Garton et al. (2018);
Heinemann et al. (2020). The corresponding Pearson’s correlation coefficients
range from $0.40.96$, depending on the periods the datasets cover. Robbins et
al. (2006) have indicated, that the slope of this linear relationship could
depend on the latitude of the coronal hole. In contrary, Hofmeister et al.
(2018) have shown that it is rather the latitudinal separation angle between
the center of mass of the coronal hole and the satellite taking the in-situ
measurements, which affects the peak velocity we see in the ecliptic at
$1\text{\,}\mathrm{A}\mathrm{U}$. Garton et al. (2018) have shown that the
peak velocities of HSSs statistically saturate at around
$700\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$ for coronal holes with a
longitudinal width $>70\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and is
also well correlated with the longitudinal widths of the coronal holes.
Hofmeister et al. (2020) showed that the empirical relationship between the
peak velocities of HSSs and the area of their source coronal holes is obtained
in MHD simulations, which neglect the acceleration phase and purely model the
propagation phase of HSSs in the heliosphere. This demonstrates that this
empirical relationship describes the propagation phase of HSSs from the Sun to
Earth.

In this study, we derive a simplified theoretical model for the propagation of
HSSs from the Sun to Earth, and show how the area of coronal holes affects the
solar wind properties near Earth. This model is not self-consistent, but
contains all relevant mechanisms to explain the empirical relationships
between the HSS properties near Earth and the area of their source coronal
holes. The study is divided in five main sections. In Section 2, we introduce
the approximation that disturbances in HSSs only have local effects, such that
disturbances generated at the HSS boundary regions only affect the boundary
regions. This assumption also involves that plasma parcels launched from the
HSS core region into a given radial direction travel at a constant speed
through interplanetary space, before reaching the interface to the preceding
slow solar wind. We refer to this approximation as the assumption of freely
propagating HSSs. In Section 3, we start at the Sun, presume that the cross
section of HSSs close to the Sun is similar to the area of their source
coronal holes, and consists of a core region and a boundary region. We derive
how the properties across the cross section translate into temporal
distributions into a given radial direction due to the solar rotation.
Further, we derive how the plasma parcels evolve with distance from the Sun
under the assumption of freely propagating HSSs. In Section 4, we relax the
assumption of freely propagating HSSs and add the stream interface to our
calculations. We study the kinematics of the stream interface between the HSS
and the preceding slow solar wind, using momentum conservation between the
impinging HSS plasma at its back and piling-up slow solar wind plasma at its
front. In Section 5, we finally postulate that HSSs observed consist of all
HSS plasma parcels launched at the Sun that did not impinge into the stream
interface yet. We derive the evolution of the main properties of HSSs with
distance from the Sun, how their properties at
$1\text{\,}\mathrm{A}\mathrm{U}$ depend on each other, and how they depend on
the cross-section of the HSSs close to the Sun. We also show how the
properties of the stream interface relate to the properties of the HSSs, and
we deliberate the limitations of our model. In Section 6, we discuss our
findings and present our conclusions.

##  2 Starting assumptions

Figure 1: Top: sketch of the formation of the velocity profile across the HSS
cross section close to the Sun, which propagates in a channel within a slow
solar wind environment. Velocities are delineated as arrows. Bottom left:
acoustic, Alfvénic, and magneto-acoustic velocity in a HSS environment defined
by Equations 1 and 3, versus its distance from the Sun. Bottom right:
acceleration of a plasma parcel in a HSS due to gravitation and pressure
gradients, as function of distance from the Sun.

In this section, we discuss the approximation of freely propagating HSS plasma
parcels, which is an important concept needed in this study. This
approximation implies that disturbances in HSSs only have local effects, and
that therefore HSS plasma parcels launched into a given radial direction
travel at a constant speed through interplanetary space as long as they do not
reach the interface to the preceding slow solar wind.

In the following, we assume a HSS that has a circular cross-section at
$0.1\text{\,}\mathrm{A}\mathrm{U}$. Within the HSS, we assume an increased
velocity $v_{f}$, temperature $T_{f}$, and decreased density $n_{f}$ as
compared to the ambient slow solar wind stream properties $v_{s}$, $T_{s}$,
and $n_{s}$. The magnetic flux density $B_{f}$ of the HSS is assumed to be the
same as in the ambient slow solar wind $B_{s}$, as the expansion of the
coronal hole is governed by the magnetic pressure which balances the magnetic
flux densities. In MHD simulations, providing initial values of

$\displaystyle v_{f}=$650\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$,\quad
n_{f}=$150\text{\,}\mathrm{c}\mathrm{m}^{-3}$,\quad
T_{f}=$1.6\text{\,}\mathrm{M}\mathrm{K}$,\quad
B_{f}=$216\text{\,}\mathrm{n}\mathrm{T}$$ (1)

results in HSS velocities of
$730\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$, densities of
$1.5\text{\,}\mathrm{c}\mathrm{m}^{-3}$, temperatures of
$75\text{\,}\mathrm{k}\mathrm{K}$, and magnetic field strength of about
$2\text{\,}\mathrm{n}\mathrm{T}$ near Earth. Initial values of

$\displaystyle v_{s}=$350\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$,\quad
n_{s}=$500\text{\,}\mathrm{c}\mathrm{m}^{-3}$,\quad
T_{s}=$400\text{\,}\mathrm{k}\mathrm{K}$,\quad
B_{s}=$216\text{\,}\mathrm{n}\mathrm{T}$$ (2)

lead to slow solar wind velocities of
$350\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$, densities of
$5\text{\,}\mathrm{c}\mathrm{m}^{-3}$, temperatures of
$20\text{\,}\mathrm{k}\mathrm{K}$, and magnetic field strength of about
$2\text{\,}\mathrm{n}\mathrm{T}$ near Earth. Furthermore, the magnetic flux
density chosen corresponds to a magnetic flux density of
$1\text{\,}\mathrm{G}$ in the low solar corona. We will use these values as
our starting parameters for the HSSs and slow solar wind plasma at
$0.1\text{\,}\mathrm{A}\mathrm{U}$.

Further, throughout the study, we presume 1) that the HSS is a strongly
compressible fluid consisting solely of protons, 2) that HSS plasma parcels
propagate radially away from the rotating Sun with a constant velocity, 3)
that the HSS undergoes adiabatic cooling, and 4) that the magnetic field
follows perfectly Parker’s spiral. Within this section, we also approximate
HSSs to be a steady-state system, which implies conservation of mass flux and
magnetic flux within a solid angle as measured from the Sun; however, we will
lift this restriction in the following sections, allowing the mass flux within
a solid angle to change with distance from the Sun due to compression and
dispersion effects. For a stream propagating freely away from the Sun, i.e.,
experiencing no external forces such as interactions with other streams, these
assumptions result in

$\displaystyle v(r)=v_{f},$ $\displaystyle
B(r)=B_{f}\cdot\frac{r_{0}^{2}}{r^{2}},$ $\displaystyle
n(r)=n_{f}\cdot\frac{r_{0}^{2}}{r^{2}},$ $\displaystyle
B_{r}(r)=B(r)\cdot\cos(\alpha),$ $\displaystyle
T(r)=T_{f}\cdot\left(\frac{r_{0}^{2}}{r^{2}}\right)^{\gamma-1},$
$\displaystyle B_{\lambda}(r)=B(r)\cdot\sin(\alpha),$
$\displaystyle\alpha(r)=\arctan\left(\frac{r\cdot\omega}{v}\right).$ (3)

Here, $r_{0}$ denotes the associated distance from the Sun at which $n_{f}$
and $T_{f}$ are defined, $\omega=14^{\circ}/$\mathrm{d}$$ is the average
rotation rate of the Sun, $\gamma=5/3$ is the adiabatic constant, $\alpha$ is
the angle of the Parker spiral to the radial direction, $B_{r}$ is the
component of the magnetic field strength in the radial direction, and
$B_{\lambda}$ is the component perpendicular to the radial direction in the
solar equatorial plane. In the following, we omit the explicit radial
dependency of these variables whenever possible to improve readability.

Next, we divide the propagation of HSSs from the Sun to Earth into different
regimes. First, close to the Sun, we set a fast, low-density HSS in a slow,
high-density solar wind environment. At the lateral boundaries of the flow,
the HSS will be decelerated due to viscous forces and turbulence close to the
Sun, resulting in a smooth transition of the velocity profile from the slow
solar wind to HSS plasma, and consequently in a velocity profile over the
cross-section of the HSS at $0.1\text{\,}\mathrm{A}\mathrm{U}$ (Fig. 1 top).

The maximum speed with which any disturbances generated at the lateral
boundary can travel from the boundary to the center of the stream is given by
the maximum velocity of fast magneto-sonic waves $v_{ms}$, given by

$v_{ms}(r)=\sqrt{v_{so}^{2}+v_{a}^{2}},\hskip
28.45274ptv_{so}(r)=\sqrt{\frac{\gamma\ k_{B}\ T}{m_{p}}},\hskip
28.45274ptv_{a}(r)=\frac{B}{\sqrt{\mu_{0}\ n\ m_{p}}},$ (4)

where $v_{so}$ is the sonic speed, $v_{a}$ the Alfvén speed, $m_{p}$ the
proton mass, $\mu_{0}$ the vacuum permeability, and $k_{B}$ the Boltzmann
constant. Using the HSS and slow solar wind properties defined above, we find
that close to the Sun at $0.1\text{\,}\mathrm{A}\mathrm{U}$ the magneto-sonic
speed is $415\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$, and strongly
reduces to $93\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$ at
$0.5\text{\,}\mathrm{A}\mathrm{U}$ and
$50\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$ at
$1\text{\,}\mathrm{A}\mathrm{U}$ (Fig. 1 bottom left). Thus, disturbances
generated at the boundaries close to the Sun can travel to and affect large
parts of the HSS close to the Sun. With increasing distance from the Sun, the
wave velocity decreases so that that these disturbances are mostly advected
with the HSS, affecting only the local plasma conditions very close to the HSS
boundary. Therefore, we may assume that the velocity profile of HSSs is formed
mostly close to the Sun, and that with increasing distance from the Sun, the
HSS plasma propagates freely in interplanetary space not seeing boundary
effects.

The radial acceleration of a freely propagating plasma parcel within the HSS
environment can be approximated by the equation of motion, combined with the
radial scaling laws for the plasma density, temperature, and magnetic field
strength (Equ. 3). Assuming that the plasma parcel has the same density as the
HSS, and with $m_{p}$ being the proton mass, $G$ being the gravitational
constant, and $P$ being the thermodynamic pressure, the acceleration of such a
plasma parcel is given by

$\displaystyle\frac{dv}{dt}$ $\displaystyle=\left(-\nabla
P-\nabla\frac{B^{2}}{2\ \mu_{0}}+\frac{(B\cdot\nabla)B}{\mu_{0}}-\frac{G\
m_{p}\ n\ M_{\text{Sun}}}{r^{2}}\right)/(m_{p}\ n)$ $\displaystyle=2\ \gamma\
k_{B}\ m_{p}^{-1}\
T_{f}\frac{r_{0}^{2\gamma-2}}{r^{2\gamma-1}}+\frac{B_{f}^{2}}{\mu_{0}\ m_{p}\
n_{f}}\frac{r_{0}^{2}}{r^{3}}\left(\sin^{2}(\alpha)-r\ \sin\alpha\ \cos\alpha\
\frac{d\alpha}{dr}\right)-\frac{G\ M_{\text{Sun}}}{r^{2}}.$ (5)

In the bottom right panel of Figure 1, we plotted the acceleration of such
plasma parcels versus the distance from the Sun. The acceleration due to the
radial gas pressure gradient, which decreases with $r^{2(\gamma-1)}\approx
r^{2.3}$, clearly dominates the total acceleration, with a minor contribution
of the gravitation, and a negligible contribution due to the magnetic forces.
The total acceleration due to the pressure gradients exceed the deceleration
due to the gravitation at least until $1.5\text{\,}\mathrm{A}\mathrm{U}$, and
consequently the plasma is accelerated at least to this distance from the Sun.
However, since the acceleration decreases with $\approx r^{2.3}$, the
acceleration falls below $0.5\text{\,}\mathrm{m}\mathrm{s}^{-2}$ already at a
distance of $0.2\text{\,}\mathrm{A}\mathrm{U}$. The associated total velocity
increase of a plasma parcel having a velocity of
$650\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$ from
$0.21\text{\,}\mathrm{A}\mathrm{U}$ is only about
$15\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$. Therefore, we neglect the
radial acceleration of plasma parcels with distance as long as the plasma
parcel did not reach the preceding SIR.

##  3 Freely propagating HSS

Figure 2: Top: sketch of the velocity distribution across the cross-section of
the HSS at $0.1\text{\,}\mathrm{A}\mathrm{U}$, having a longitudinal widths of
$22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ and a widths of the
boundary region of $6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$. Bottom
left: velocity (black), temperature (red), and density distribution (blue)
along the cross-section of the HSS at $0.1\text{\,}\mathrm{A}\mathrm{U}$. The
solid lines give the sigmoidal approximations, and the black dashed line the
linear approximation for the velocity profile. Bottom right: dispersed
temporal velocity distribution as seen at $1\text{\,}\mathrm{A}\mathrm{U}$,
and the associated temperature and density distributions. Note, that the
dotted lines represent the mathematical solution for plasma originating from
the leading boundary region at $0.1\text{\,}\mathrm{A}\mathrm{U}$ of the HSS.
These plasma parcels will have impinged into the stream interface at
$1\text{\,}\mathrm{A}\mathrm{U}$ and thus cannot be observed.

In this section, we derive a simplified model for the propagation of freely
propagating HSSs. We assume a circular coronal hole in the low solar corona,
and that the open magnetic field of the coronal hole expands with a constant,
arbitrary expansion factor to $0.1\text{\,}\mathrm{A}\mathrm{U}$, defining the
cross-section of the associated HSS. This cross-section at
$0.1\text{\,}\mathrm{A}\mathrm{U}$ consists of a nearly homogeneous core
region as characterized by its temperature, density, and radial velocity
distribution, and an adjunct boundary region across which the HSS properties
adjust to the properties of the ambient slow solar wind. The rotation of the
Sun translates these spatial distributions of the properties along the
latitudinal slice into temporal distributions at a given radial direction.
From there on, we presume that the HSS propagates freely in interplanetary
space. Under these assumptions, we derive the velocity distribution of HSSs at
a given distance from the Sun by the dispersion of its initial temporal
velocity distribution close to the Sun. Further, we derive the evolution of
the density and temperature of the associated plasma parcels in interplanetary
space by calculating their expansion and presuming adiabatic cooling.

We define the properties of the HSS at $0.1\text{\,}\mathrm{A}\mathrm{U}$ in
the following way, as sketched in Figure 2 top. We denote the circular
diameter of the HSS cross-section, which is also the longitudinal width of the
HSS, as $\lambda_{HSS}$, and the width of the boundary region as
$\lambda_{bd}$. Thereby, $\lambda_{bd}$ is the width over which the HSS
properties change by $85\text{\,}\frac{\mathrm{\char 37\relax}}{}$ from HSS to
slow solar wind properties as measured from the center of the boundary region.
Throughout the study, we keep the width of the boundary region fixed at
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$. We presume that the
properties across the HSS solely depend on the distance to the center of the
boundary region $d$, given by the sigmoid function:

$\begin{pmatrix}v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(d)\\\
T_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(d)\\\
n_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(d)\end{pmatrix}=\begin{pmatrix}v_{s}\\\
T_{s}\\\ n_{s}\end{pmatrix}+\begin{pmatrix}v_{f}-v_{s}\\\ T_{f}-T_{s}\\\
n_{f}-n_{s}\end{pmatrix}\cdot\frac{1}{2}\ \left(1+\tanh{\left(2.5\
\frac{d-\lambda_{bd}/2}{\lambda_{bd}}\right)}\right).$ (6)

The tangens hyperbolicus function was chosen over other sigmoid functions for
calculation reasons.

The associated velocity, temperature, and density distributions along a
longitudinal slice through the center of the HSS is shown in the bottom left
panel of Figure 2. Since, in our formulation, the velocity, density, and
temperature at $0.1\text{\,}\mathrm{A}\mathrm{U}$ depend only on the distance
to the HSS boundary, they are also naturally correlated. This correlation
allows to determine two of these quantities by knowing the third, using the
normalized quantities

$\frac{v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(d)-v_{s}}{v_{f}-v_{s}}=\frac{T_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(d)-T_{s}}{T_{f}-T_{s}}=\frac{n_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(d)-n_{s}}{n_{f}-n_{s}}.$
(7)

This relationship will be used later in this study.

###  3.1 Dispersion of the velocity distribution with distance

|  | $\text{for }0.5\ \lambda_{HSS}-\varphi\geq\lambda_{bd}$ | $\text{for }0.5\ \lambda_{HSS}-\varphi<\lambda_{bd}$
---|---|---|---
$\lambda_{HSS,eff}$ | $=$ | $2\ \sqrt{\left(0.5\ \lambda_{HSS}\right)^{2}-\varphi^{2}}$ | $2\ \sqrt{\left(0.5\ \lambda_{HSS}\right)^{2}-\varphi^{2}}$
$\lambda_{bd,eff}$ | $=$ | $0.5\ \lambda_{HSS,eff}-\sqrt{(0.5\ \lambda_{HSS}-\lambda_{bd})^{2}-\varphi^{2}}$ | $0.5\ \lambda_{HSS,eff}$
$v_{sl,max}$ | $=$ | $v_{f}$ | $v_{s}+\frac{v_{f}-v_{s}}{\lambda_{bd}}\left(0.5\ \lambda_{HSS}-\varphi\right)$
$\left|\frac{dv}{d\lambda}\right|$ | $=$ | $\frac{v_{sl,max}-v_{s}}{\lambda_{bd,eff}}$ | $\frac{v_{sl,max}-v_{s}}{\lambda_{bd,eff}}$
Table 1: Geometrical relations between the effective longitudinal width of a
HSS along a given longitudinal slice$\lambda_{HSS,eff}$, the associated
effective width of the HSS boundary region $\lambda_{\text{bd,eff}}$, the peak
velocity in the slice $v_{sl,max}$, the associated velocity gradient across
the boundary region $\frac{dv}{d\lambda}$, and the overall longitudinal width
of the HSS $\lambda_{HSS}$, the HSS boundary region $\lambda_{bd}$, the
displacement of the slice $\varphi$ to the center of the HSS, the maximum
velocity of the HSS $v_{f}$, and the velocity of the slow solar wind $v_{s}$.
The left solutions correspond to slices which cross the HSS core region, and
the right solution corresponds to slices which lie purely in the
northern/southern HSS boundary region.

Next, we derive the temporal HSS velocity distribution at a given distance
from the Sun for freely propagating HSSs. We start with the spatial velocity
distribution along a longitudinal slice through the HSS close to the Sun (Fig.
2 top and bottom left). Using linear approximations for the velocity increase
across the leading, i.e., western, and trailing, i.e., eastern boundary
region, this is given by

$v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(\lambda)=\begin{cases}v_{s}+\lambda\cdot\frac{dv}{d\lambda}&\text{for
}0\leq\lambda\leq\lambda_{\text{bd,eff}},\\\ v_{sl,max}&\text{for
}\lambda_{\text{bd,eff}}<\lambda<(\lambda_{HSS,eff}-\lambda_{\text{bd,eff}}),\\\
v_{s}+(\lambda_{HSS,eff}-\lambda)\cdot\frac{dv}{d\lambda}&\text{for
}(\lambda_{HSS,eff}-\lambda_{\text{bd,eff}})\leq\lambda\leq\lambda_{HSS,eff}.\end{cases},$
(8)

where $\lambda$ is a location in the slice measured from the leading edge
along the slice, $\lambda_{HSS,eff}$ is the effective longitudinal width of
the slice in the HSS, $v_{sl,max}$ is the peak velocity in the slice,
$\lambda_{\text{bd,eff}}$ is the effective width of the associated boundary
region along the slice, and $\frac{dv}{d\lambda}$ is the velocity gradient
across the effective boundary region. It is important to note that the
effective longitudinal width decreases, the width of the effective boundary
region increases, and the velocity gradient across the effective boundary
region decreases with increasing latitudinal displacement $\varphi$ of the
slice to the center of the HSS. Further, also the peak velocity in the slice
is decreased if either the entire slice is located in the northern/southern
boundary region, or if the HSS is so small that the entire HSS consists solely
of boundary region plasma. The mathematical relations of these parameters to
the latitudinal displacement of the slice, the overall longitudinal width of
the HSS, and the width of the boundary region follow from geometrical
considerations and are given in Table 1.

As the Sun rotates, this longitudinal velocity distribution
$v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(\lambda)$ along a longitudinal slice
in the HSS maps into a temporal velocity distribution
$v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)$ in given, fixed heliospheric
radial direction. Using $\lambda\rightarrow t=\frac{\lambda}{\omega}$, this
results in

$v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)=\left.\begin{cases}v_{s}+t\ \omega\
\frac{dv}{d\lambda}&\text{for }0\leq t\ \omega\leq\lambda_{\text{bd,eff}},\\\
v_{sl,max}&\text{for }\lambda_{\text{bd}}<t\
\omega<(\lambda_{HSS,eff}-\lambda_{\text{bd,eff}}),\\\
v_{s}+(\lambda_{HSS,eff}-t\ \omega)\cdot\frac{dv}{d\lambda}&\text{for
}(\lambda_{HSS,eff}-\lambda_{\text{bd,eff}})\leq t\
\omega\leq\lambda_{HSS,eff}.\end{cases}\right.$ (9)

This temporal velocity distribution disperses with distance from the Sun. As
we presume a freely propagating HSS, the temporal velocity distribution at an
arbitrary distance $r$ from the Sun is related to the temporal velocity
distribution at $0.1\text{\,}\mathrm{A}\mathrm{U}$ by

$v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)=v_{r}\left(t+\frac{r}{v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}}\right).$
(10)

By substituting $t+\frac{r}{v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}}$ with
$\breve{t}$, this leads to the following dispersed velocity distributions of
the leading boundary region, plateau region, and trailing boundary regions of
HSSs at a given distance from the Sun.

  * •

Leading boundary region

$v_{r}(\breve{t})=\begin{cases}0.5\cdot\left(v_{s}+\breve{t}\ \omega\
\frac{dv}{d\lambda}+\sqrt{\left(v_{s}+\breve{t}\ \omega\
\frac{dv}{d\lambda}\right)^{2}-4\ r\ \omega\
\frac{dv}{d\lambda}}\right)&\text{for }r<v_{s}\ v_{sl,max}\ \left(\omega\
\frac{dv}{d\lambda}\right)^{-1}\\\
&\quad\quad\land\,\frac{r}{v_{s}}\leq\breve{t}\leq\left(\frac{r}{v_{sl,max}}+\frac{\lambda_{bd,eff}}{\omega}\right),\\\
0.5\cdot\left(v_{s}+\breve{t}\ \omega\
\frac{dv}{d\lambda}-\sqrt{\left(v_{s}+\breve{t}\ \omega\
\frac{dv}{d\lambda}\right)^{2}-4\ r\ \omega\
\frac{dv}{d\lambda}}\right)&\text{for }r\geq v_{s}\ v_{sl,max}\ \left(\omega\
\frac{dv}{d\lambda}\right)^{-1}\\\
&\quad\quad\land\,\left(\frac{r}{v_{sl,max}}+\frac{\lambda_{bd,eff}}{\omega}\right)\leq\breve{t}\leq\frac{r}{v_{s}}.\end{cases}$
(11)

Note, that the term $v_{s}+\breve{t}\ \omega\ \frac{dv}{d\lambda}$ is
associated with the velocity of plasma parcels originating from the leading
HSS boundary region at $0.1\text{\,}\mathrm{A}\mathrm{U}$, i.e.,
$v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)$. The first line is valid for close
distances from the Sun, as long as $r<v_{s}\ v_{sl,max}\ \left(\omega\
\frac{dv}{d\lambda}\right)^{-1}$, i.e., as long as the HSS plasma stemming
from the edge to the core region did not overtake the HSS plasma originating
from the western edge of the boundary region. The second line is valid for the
complementary case, taking place at distances $r\geq v_{s}\ v_{sl,max}\
\left(\omega\ \frac{dv}{d\lambda}\right)^{-1}$, i.e., when the plasma from the
edge to the core region did overtake the plasma from western boundary edge.
Note, that the second line, which is marked as a black dotted line in Figure 2
bottom left, is purely a mathematical solution. These plasma parcels will have
impinged into the stream interface to the preceding slow solar wind plasma,
breaking down the assumption of freely propagating plasma parcels.

  * •

Plateau region

$v_{r}(\breve{t})=v_{sl,max}\quad\quad\text{for
}\left(\frac{r}{v_{sl,max}}+\frac{\lambda_{bd,eff}}{\omega}\right)<\breve{t}<\left(\frac{r}{v_{sl,max}}+\frac{\lambda_{HSS,eff}-\lambda_{bd,eff}}{\omega_{\text{sun}}}\right).$
(12)

This equation expresses that the peak velocity of the HSS plateau at any
distance to the Sun is given by the maximum velocity in the slice, as long as
one assumes freely propagating HSSs, i.e., neglecting compression effects near
the preceding slow solar wind in the SIRs, and any acceleration effects.

  * •

Trailing boundary region

$\displaystyle v_{r}(\breve{t})$
$\displaystyle=0.5\cdot\left(v_{s}+\left(\lambda_{HSS,eff}-\breve{t}\
\omega\right)\
\frac{dv}{d\lambda}+\sqrt{\left(v_{s}+\left(\lambda_{HSS,eff}-\breve{t}\
\omega\right)\ \frac{dv}{d\lambda}\right)^{2}+4\ r\ \omega\
\frac{dv}{d\lambda}}\right)$ $\displaystyle\quad\quad\quad\text{for
}\left(\frac{r}{v_{sl,max}}+\frac{\lambda_{HSS,eff}-\lambda_{bd,eff}}{\omega}\right)\leq\breve{t}\leq\left(\frac{r}{v_{s}}+\frac{\lambda_{HSS,eff}}{\omega}\right).$
(13)

Note, that the term $v_{s}+\left(\lambda_{HSS,eff}-\breve{t}\ \omega\right)\
\frac{dv}{d\lambda}$ is associated with the velocity of plasma parcels
originating from the trailing HSS boundary region at
$0.1\text{\,}\mathrm{A}\mathrm{U}$, i.e.,
$v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)$.

###  3.2 Density and temperature of freely propagating HSS plasma parcels

Figure 3: Distribution of densities (left) and temperatures (right) of plasma
parcels versus their velocities at $1\text{\,}\mathrm{A}\mathrm{U}$, as
derived by Equation 18 for a HSS having a diameter of
$22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$. Each line represents
plasma parcels originating from a different longitudinal slice through the
HSS, resulting in a different radial expansion. The numbers give the
latitudinal displacement of the slice to the center of the HSS; slices
$>5\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ lie entirely within the
northern boundary region. For comparison, the dashed diagonal line shows the
solution for the case that the radial expansion of plasma parcels would have
been neglected.

Next, we estimate the density and temperature of HSS plasma parcels at a
distance $r$ from the Sun by deriving their expansion of volume with distance
from the Sun,
$\frac{V_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)}{V_{r}(\breve{t})}$, and
presuming adiabatic cooling,

$\displaystyle
n_{r}(\breve{t})=n_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)\cdot\left(\frac{V_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)}{V_{r}(\breve{t})}\right)$
(14) $\displaystyle
T_{r}(\breve{t})=T_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)\cdot\left(\frac{V_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)}{V_{r}(\breve{t})}\right)^{\gamma-1}$
(15)

The density and temperature of plasma parcels at
$0.1\text{\,}\mathrm{A}\mathrm{U}$, $n_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}$
and $T_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}$, can be derived using their
correlation to the velocity at $0.1\text{\,}\mathrm{A}\mathrm{U}$ using
Equation 7. The expansion of the volume of plasma parcels propagating radially
away from the Sun is given by their lateral and radial expansion. As we assume
HSSs propagating radially away from the Sun, the lateral expansion follows an
$r^{2}$ law. The radial expansion within plasma parcels can be determined by
their intrinsic radial velocity gradient. This leads to the expansion of the
volume

$\frac{V_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)}{V_{r}(\breve{t})}=\frac{r_{0}^{2}}{r_{2}}\cdot\left(1+\frac{dv_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)}{dr}\cdot\frac{r-$0.1\text{\,}\mathrm{A}\mathrm{U}$}{v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)}\right)^{-1}.$
(16)

To evaluate this equation, a smooth function for the plasma velocity from the
slow solar wind to HSS plasma is needed. Consequently, we approximate the
velocity distribution across the leading and trailing HSS boundaries at
$0.1\text{\,}\mathrm{A}\mathrm{U}$ again by sigmoidal functions,

$v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(\lambda)=\begin{cases}v_{s}+\frac{v_{sl,max}-v_{s}}{2}\
\left(1+\tanh{\left(2.5\
\frac{\lambda-\lambda_{bd,eff}/2}{\lambda_{bd,eff}}\right)}\right)&\text{for
}\lambda<\lambda_{HSS,eff}/2,\\\ v_{s}+\frac{v_{sl,max}-v_{s}}{2}\
\left(1-\tanh{\left(2.5\
\frac{\lambda-\left(\lambda_{HSS,eff}-\lambda_{bd,eff}/2\right)}{\lambda_{bd,eff}}\right)}\right)&\text{for
}\lambda>\lambda_{HSS,eff}/2,\end{cases}$ (17)

whereby the first line describes the leading HSS boundary region, and the
second line the trailing HSS boundary region. Taking the derivative leads to
the increase of volume dependent on the distance from the Sun, given by:

$\frac{V_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)}{V_{r}(\breve{t})}=\frac{r_{0}^{2}}{r^{2}}\cdot\begin{cases}\left(\left|1-\frac{5\
\omega\
\left(r-$0.1\text{\,}\mathrm{A}\mathrm{U}$\right)\left(v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)-v_{s}\right)\left(v_{sl,max}-v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)\right)}{\lambda_{bd,eff}\
v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)^{2}\
\left(v_{sl,max}-v_{s}\right)}\right|\right)^{-1}&\text{for
}\frac{r}{v_{s}}<\breve{t}<\left(\frac{r}{v_{sl,max}}+\frac{\lambda_{HSS,eff}}{2\
\omega}\right)\\\ \left(1+\frac{5\ \omega\
\left(r-$0.1\text{\,}\mathrm{A}\mathrm{U}$\right)\left(v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)-v_{s}\right)\left(v_{sl,max}-v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)\right)}{\lambda_{bd,eff}\
v_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(t)^{2}\
\left(v_{sl,max}-v_{s}\right)}\right)^{-1}&\text{for
}\left(\frac{r}{v_{sl,max}}+\frac{\lambda_{HSS,eff}}{2\
\omega}\right)<\breve{t}<\left(\frac{r}{v_{s}}+\frac{\lambda_{HSS,eff}}{\omega}\right).\end{cases}.$
(18)

In the bottom right panel of Figure 2, we have derived the temporal density
and temperature behavior for the HSS of the top panel in Figure 2 at Earth
distance. The dotted lines, which correspond to the leading edge of the HSS at
$0.1\text{\,}\mathrm{A}\mathrm{U}$ and which are derived by the first line of
Equation 18, are purely mathematical solutions since these plasma parcels will
have impinged into the stream interface. Following the plateau region of the
HSS, the temperature decreases fast to slow solar wind values, whereas the
density shows a small drop for an extended time, and then increases fast to
slow solar wind values. Note, that this drop in density is an effect of the
radial expansion of plasma parcels with distance from the Sun. It therefore
depends on the chosen sigmoid function we used to model the velocity gradient
across the boundary region and on the presumed boundary width. Using less
steep sigmoid functions or wider boundary regions, respectively, will soften
or even remove this density drop. In solar wind plasma measurements in the
heliosphere, this density drop will only be observable if the HSS has a steep
velocity gradient across its trailing boundary region close to the Sun, and if
the HSS plateau region is very extended so that the reference density level
before the drop is well-defined.

Further, in Figure 3, we show the distribution of densities and temperatures
of HSS plasma parcels at $1\text{\,}\mathrm{A}\mathrm{U}$ versus their
velocities. These have been derived for a HSS having a longitudinal width of
$22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and slices through that
HSS at latitudinal displacements in steps of
$1\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ to the HSS center. For
comparison, the dashed diagonal lines give the temperature and density
distributions for the case of purely lateral expanding plasma parcels, i.e.,
that the radial expansion is neglected. At the endpoint of each line, i.e.,
$v_{r}=v_{sl,max}$, the plasma parcel originates from the center of the slice
where the radial velocity gradient along the slice is
$\approx$0\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$$. Therefore, the
plasma parcels do not expand into the radial direction, and the solutions for
considering and neglecting the radial expansion of plasma parcels with
distance coincide. The same is valid for plasma parcels originating from pure
slow solar wind plasma. For all other points along the lines, the plasma
parcels originate from within the boundary region, where the radial velocity
gradient along the slice is
$\neq$0\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$$. These plasma parcels
expand in the radial direction with distance from the Sun, additionally
reducing their densities and temperatures compared to a purely lateral
expansion.

##  4 Kinematics of the stream interface

Figure 4: Numerical solution of the kinematics of the stream interface. Top
left: temporal velocity profile of the driving HSS plasma. The HSS is presumed
to have a longitudinal width of
$22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and the slices are set at
$0\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$4\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$8\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and
$10\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ to the HSS center. Top
right: resulting velocity of the stream interface, versus its distance from
the Sun. Bottom left: total mass of the stream interface per solid angle,
dependent on its distance from the Sun. Bottom right: numerical arrival time
of the stream interface at $1\text{\,}\mathrm{A}\mathrm{U}$ versus the
analytically approximated arrival time derived by Equation 28.

In this section, we derive the kinematics of the stream interface between the
HSS and the preceding slow solar wind plasma, and estimate its arrival time at
Earth. Due to the very high magnetic Reynolds numbers, the HSS and slow solar
wind plasma are confined within the magnetic field and cannot mix; therefore,
the HSS accumulates at the back of the stream interface, and the preceding
slow solar wind is piled up at its front. By presuming the stream interface to
be a discrete entity, we derive the kinematics of the stream interface by
momentum conservation between the stream interface, the impinging HSS plasma
at its back, and the piling-up slow solar wind plasma at its front.

At a given time $t$ and velocity of the stream interface $v_{SI}$, the number
of accumulating HSS particles per time per area is given by $n_{r}(t)\
(v_{r}(t)-v_{SI})\ dt$, impinging with a velocity $(v_{r}(t)-v_{SI})$.
Analogue, the number area density of the piling-up slow solar wind plasma is
given by $n_{s}\ (v_{SI}-v_{s})\ dt$ impinging with a velocity
$(v_{SI}-v_{s})$. With $N_{SI}$ being the number area density of the stream
interface, the kinematics of the stream interface is therefore determined by

$\displaystyle\frac{dv_{SI}(t)}{dt}=\frac{1}{N_{SI}}\left(n_{r}\
(v_{r}-v_{SI})^{2}-n_{s}\ (v_{s}-v_{SI})^{2}\right),$
$\displaystyle\frac{dN_{SI}(t)}{dt}=n_{s}\ (v_{SI}-v_{s})+n_{r}\
(v_{r}-v_{SI}).$ (19)

When we presume constant properties of the HSS and slow solar wind plasma
impinging into the stream interface per solid angle, the set of differential
equations is solved by

$\displaystyle N_{SI}(t)=\sqrt{t^{2}\ n_{r}\ n_{s}\
\left(v_{r}-v_{s}\right)^{2}+2\ t\ N_{0}\left(n_{r}\ (v_{r}-v_{0})+n_{s}\
(v_{0}-v_{s})\right)+N_{0}^{2}},$ (20) $\displaystyle
v_{SI}(t)=\frac{\frac{d}{dt}N_{SI}(t)+n_{s}\ v_{s}-n_{r}\
v_{r}}{n_{s}-n_{r}},$ (21)

whereby $v_{0}$ and $N_{0}$ are the velocity and number area density of the
stream interface at the time $t_{0}$. When we further assume that the stream
interface starts with a velocity of
$0\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$, that the associated number
area density related to a solid angle at the start is
$0/($\frac{\mathrm{\SIUnitSymbolDegree}}{}$)^{2}$, and that the HSS velocity
is equal to $v_{sl,max}$, the equations simplify. With $n_{sl,min}$ being the
density at the center of the longitudinal slice within the HSS, which can be
derived by Equation 7, the mass increase rate of the stream interface, its
velocity, and its distance traveled at a time $t$ become

$\displaystyle\frac{d}{dt}N_{SI}=\sqrt{n_{sl,min}\ n_{s}}(v_{sl,max}-v_{s}),$
(22) $\displaystyle v_{SI}=\frac{v_{s}\ \left(n_{s}-\sqrt{n_{sl,min}\
n_{s}}\right)+v_{sl,max}\ \left(\sqrt{n_{sl,min}\
n_{s}}-n_{sl,min}\right)}{n_{s}-n_{sl,min}},$ (23) $\displaystyle
r_{SI}(t)=t\cdot v_{SI}.$ (24)

Note, that the velocity of the stream interface is independent on time. This
means that the velocity immediately adjusts so that the accumulated HSS mass
in its back balances the piling-up slow solar wind mass at its front, and it
is constant as long as the impinging HSS and piling-up SSW properties stay
constant.

Next, we split the travel time of the stream interface from the Sun to Earth
into two regions. First, we estimate the time and distance from the Sun at
which the HSS plasma overtakes the preceding slow solar wind plasma, i.e., at
which the SIR is formed. Up to this time, the stream interface roughly
propagates with the velocity of the preceding slow solar wind as no momentum
has yet been transferred from the HSS to the stream interface. Within the
first-order approximation of linearly increasing velocities across the leading
boundary region, all plasma parcels originating from the leading boundary
region impinge into the stream interface at nearly the same time. Evaluating
this condition for plasma parcels originating from the western and eastern
edge of the leading boundary region, i.e.,

$v_{s}\ t_{1}=v_{sl,max}\ (t_{1}-\lambda_{bd,eff}/\omega),$ (25)

one finds for the impingement time $t_{1}$, and for the corresponding distance
$r_{1}$ at which the SIR is formed,

$\displaystyle t_{1}$ $\displaystyle=v_{sl,max}\ \left(\omega\
\frac{dv}{d\lambda}\right)^{-1},$ (26) $\displaystyle r_{1}$
$\displaystyle=v_{s}\ v_{sl,max}\ \left(\omega\
\frac{dv}{d\lambda}\right)^{-1}.$ (27)

From there on, we neglect the following acceleration phase of the stream
interface, presume that it is instantaneously accelerated to its equilibrium
velocity given by Equation 24, and approximate it to further propagate with
this constant equilibrium velocity. The arrival time of the stream interface
$t_{SI,r}$ at a distance $r$ is then given by the time at which the SIR is
formed plus the remaining travel time of the stream interface at its
equilibrium speed up to the distance $r$, i.e.,

$\displaystyle t_{SI,r}=v_{sl,max}\ \left(\omega\
\frac{dv}{d\lambda}\right)^{-1}+\frac{r-v_{s}\ v_{sl,max}\ \left(\omega\
\frac{dv}{d\lambda}\right)^{-1}}{v_{SI}}\quad\quad\text{for }r>r_{1}.$ (28)

In the following, we apply these equations to the HSS plasma defined in
Section 3. We presume that the HSS has longitudinal widths of
$22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ close to the Sun, and
evaluate the HSSs and associated stream interface properties at latitudinal
displacements of $0\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$4\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$8\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and
$10\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ to the HSS center. First,
we derive the numerical solution for the kinematics of the stream interface,
by using its defining set of differential equations combined with the full
time-dependent solution of the impinging HSS plasma. For each time step, we
derive iterative the HSS properties velocity and density at the location of
the stream interface using Equations 11 to 13 and Equation 18, assuming the
preceding slow solar wind to have constant properties defined by Equations 1
and 2, and update the velocity, number density, and location of the stream
interface using Equations 19. In the top right panel of Figure 4, we show the
velocity of the stream interface versus its distance. Starting at the Sun, the
velocity of the stream interface first coincides with the velocity of the
preceding slow solar wind. Next, the velocity increases sharply to the steady-
state solution given by Equations 24, which is related to the formation of the
SIR by the impinging HSS plasma. Finally, the velocity of the stream interface
slowly declines again, which is related to the reduced velocity of HSS plasma
parcels originating from the trailing boundary region of the HSS. The much
slower decrease of the velocity of the stream interface as compared to its
sharp increase is explained by its larger, accumulated area density and
momentum at larger distances, which is shown in the bottom left panel of
Figure 4. While the accumulated particles per area are close to zero near the
Sun, they roughly increase linearly as soon as the SIR is formed. In the
bottom right panel of Figure 4, we compare the estimated arrival times at
$1\text{\,}\mathrm{A}\mathrm{U}$ derived by Equation 28 versus the exact
arrival times derived by solving the differential Equation 19 numerically. The
estimated arrival times match the exact arrival times within an error of
$1\text{\,}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}$, and are therefore good
analytical proxies which are used in the next section.

##  5 Characteristic properties of HSS and the stream interfaces in the inner
heliosphere

In this section, we postulate that the HSS plasma observed by satellites
consist of all HSS plasma parcels launched from the Sun, that did not impinge
into the stream interface at the satellite’s distance. Accordingly, we study
the properties of HSSs by evaluating only freely propagating HSS plasma
parcels, derived in Section 3, that arrive later than the stream interface at
the given distance, derived in Section 4.

We first derive the characteristic properties of HSSs, which we define as the
properties of the fastest HSS plasma parcel, as a function of the distance
from the Sun. In case that not the entire HSS core region has impinged into
the stream interface, the velocity of the fastest parcel is equal to the
velocity of the HSS core region. Else, the velocity is given by the fastest
plasma parcel originating from the trailing boundary region that did not
impinge into the stream interface yet. These cases can be approximated by
freely propagating HSSs (Equ. 12 to 13) evaluated at the arrival time of the
stream interface at the given distance (Equ. 28), which results in:

$\displaystyle v_{p}(r)$ $\displaystyle=\begin{cases}v_{sl,max}&\text{for
}t_{SI}\leq\frac{\lambda_{HSS,eff}-\lambda_{bd,eff}}{\omega}+\frac{r}{v_{sl,max}},\\\
0.5\ \left(v_{s}-v_{sl,max}+\frac{v_{sl,max}\ v_{s}}{v_{SI}}-\frac{r\
\omega}{v_{SI}}\ \frac{dv}{d\lambda}+\lambda_{HSS,eff}\
\frac{dv}{d\lambda}+\vphantom{\quad+\sqrt{\left(v_{s}-v_{sl,max}+\frac{v_{sl,max}\
v_{s}}{v_{SI}}-\frac{r\ \omega}{v_{SI}}\
\frac{dv}{d\lambda}+\lambda_{HSS,eff}\ \frac{dv}{d\lambda}\right)^{2}+4\ r\
\omega\ \frac{dv}{d\lambda}}}\right.\\\
\left.\quad+\sqrt{\left(v_{s}-v_{sl,max}+\frac{v_{sl,max}\
v_{s}}{v_{SI}}-\frac{r\ \omega}{v_{SI}}\
\frac{dv}{d\lambda}+\lambda_{HSS,eff}\ \frac{dv}{d\lambda}\right)^{2}+4\ r\
\omega\ \frac{dv}{d\lambda}}\right)&\text{for
}t_{SI}>\frac{\lambda_{HSS,eff}-\lambda_{bd,eff}}{\omega}+\frac{r}{v_{sl,max}}.\end{cases}$
(29)

Therefore, the peak velocity is dependent on the properties of the slow solar
wind and of the HSS, the velocity of the stream interface, the effective
longitudinal width of the HSS, the velocity gradient across the HSS boundary
close to the Sun, and the distance at which the equation is evaluated. Note,
that the effective longitudinal widths of the HSS and the longitudinal
velocity gradient across the HSS boundary are also implicitly dependent on the
latitudinal position of the longitudinal slice within the HSS (Table 1).

The temperature and density of the plasma parcels associated with the peak
velocity can be determined by first deriving the plasma temperature and
density of these parcels close to the Sun using the correlation between the
velocity, temperature, and density, i.e., Equation 7, and then deriving their
expansion and associated cooling by Equations 15, 14, and 18:

$\displaystyle
n_{@vp}(r)=n_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(v_{p})\cdot\left(\frac{$0.1\text{\,}\mathrm{A}\mathrm{U}$}{r}\right)^{2}\cdot\begin{cases}1&\text{for
}t_{SI}\leq\frac{\lambda_{HSS,eff}-\lambda_{bd,eff}}{\omega}+\frac{r}{v_{sl,max}}\\\
\left(1+\frac{5\ \omega\
\left(r-$0.1\text{\,}\mathrm{A}\mathrm{U}$\right)\left(v_{p}-v_{s}\right)\left(v_{sl,max}-v_{p}\right)}{\lambda_{bd,eff}\
v_{p}^{2}\ \left(v_{sl,max}-v_{s}\right)}\right)^{-1}\quad\quad\quad&\text{for
}t_{SI}>\frac{\lambda_{HSS,eff}-\lambda_{bd,eff}}{\omega}+\frac{r}{v_{sl,max}}\end{cases},$
(30) $\displaystyle
T_{@vp}(r)=T_{$0.1\text{\,}\mathrm{A}\mathrm{U}$}(v_{p})\cdot\left(\frac{$0.1\text{\,}\mathrm{A}\mathrm{U}$}{r}\right)^{2(\gamma-1)}\cdot\begin{cases}1&\text{for
}t_{SI}\leq\frac{\lambda_{HSS,eff}-\lambda_{bd,eff}}{\omega}+\frac{r}{v_{sl,max}},\\\
\left(1+\frac{5\ \omega\
\left(r-$0.1\text{\,}\mathrm{A}\mathrm{U}$\right)\left(v_{p}-v_{s}\right)\left(v_{sl,max}-v_{p}\right)}{\lambda_{bd,eff}\
v_{p}^{2}\ \left(v_{sl,max}-v_{s}\right)}\right)^{1-\gamma}&\text{for
}t_{SI}>\frac{\lambda_{HSS,eff}-\lambda_{bd,eff}}{\omega}+\frac{r}{v_{sl,max}}.\end{cases}$
(31)

In the next sections, we analyze the behavior of theses properties with
distance from the Sun and their interrelations in more detail.

###  5.1 Evolution of the velocity, density and temperature with distance from
the Sun

Figure 5: Schematic cross-section of a HSS at
$0.1\text{\,}\mathrm{A}\mathrm{U}$, having a longitudinal widths of
$22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ (top left). Peak velocity
of the HSS versus its distance from the Sun (top right). Temperature (bottom
left) and density (bottom right) of the associated HSS plasma parcels versus
the distance from the Sun. The HSS properties have been derived at latitudinal
displacements of $0\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ (blue),
$4\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ (orange),
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ (green),
$8\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ (red), and
$10\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ (purple) to the center of
the HSS. The dashed vertical lines mark the distance at which the plasma
parcels start to originate from the trailing boundary region.

First, we present how the velocity, density, and temperature of HSSs,
represented by the plasma parcels having the peak velocity, behave with
distance from the Sun. We again presume a HSS having a longitudinal width of
$22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ close to the Sun, and
evaluate Equations 29, 30 and 31 at latitudinal displacements of
$0\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$4\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$8\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and
$10\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ to the HSS center, as
sketched in the top left panel of Figure 5.

In the top right panel of Figure 5, we show the peak velocity of the HSS
versus its distance from the Sun. In the center of the HSS (blue line), the
peak velocity is constant with
$650\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$ up to a distance of
$0.92\text{\,}\mathrm{A}\mathrm{U}$, and decreases from thereon continuously.
Therefore, up to $0.92\text{\,}\mathrm{A}\mathrm{U}$, all these plasma parcels
originate from the core region of the HSS. Starting from
$0.92\text{\,}\mathrm{A}\mathrm{U}$, the peak velocity is given by the fastest
plasma parcel from the trailing boundary region, that did not impinge into the
stream interface yet. As more ‘fastest’ plasma parcels impinge into the stream
interface, the peak velocity decreases with distance. The rate of decrease
thereby mostly depends on the velocity gradient across the HSS boundary region
close to the Sun, and the velocity with which the stream interface propagates
away from the Sun (Eq. 29). At a latitudinal displacement of
$4\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, the longitudinal slice
still crosses the HSS core region. Accordingly, the maximum velocity in the
longitudinal slice, and thus the peak velocity of the HSS, is still
$650\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$. However, the effective
longitudinal width of the HSS is reduced, which results in an effective
smaller HSS core region along the slice. A smaller core region is equivalent
to less time until all plasma parcels of the core region have impinged into
the stream interface, and thus an earlier onset of the decrease of the peak
velocities with distance. At latitudinal displacements of
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$8\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ and
$10\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, the longitudinal slice
through the HSS crosses only the boundary region of the HSS. This results in a
reduced maximum velocity within the slice, and consequently in smaller peak
velocities already close to the Sun. As the velocity gradient across the
leading HSS boundary region is smaller at larger latitudinal displacements to
the HSS center, the time by which HSS plasma parcels have caught up with the
preceding slow solar wind, and thus the distance at which the SIR forms,
increases (see Equ. 28). Up to that distance, the HSS peak velocity is given
by the maximum velocity within the longitudinal slice. Right afterwards, the
peak velocities start to decrease as there is no HSS plateau region.

In the bottom panels of Figure 5, we show the associated densities and
temperatures of the plasma parcels representing the peak velocities versus the
distance from the Sun, in a double-logarithmic plot. As long as all the plasma
parcels originate either from the HSS core region or from the longitudinal
center of the HSS, i.e., where the velocity gradient along the longitudinal
slice at $0.1\text{\,}\mathrm{A}\mathrm{U}$ is zero, the plasma parcels only
expand laterally. Consequently, the density decreases with $r^{-2}$ and the
temperature with $r^{2(\gamma-1)}$. As soon as the HSS peak velocity gets
associated with plasma parcels originating from the trailing boundary region,
for which the longitudinal velocity gradient across the boundary region is
$\neq$0\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$$, the plasma parcels also
expand radially. Due to the radial expansion, the densities of these plasma
parcels decrease faster than $r^{-2}$, and the temperatures decrease faster
than $r^{2(\gamma-1)}$. Therefore, in general, it should not be assumed that
the densities in HSSs decrease with $r^{-2}$, with the exception of plasma
parcels originating from the HSS core region.

###  5.2 Dependence of the properties of HSSs at
$1\text{\,}\mathrm{A}\mathrm{U}$ on their geometry close to the Sun

Figure 6: Peak velocity of HSSs at $1\text{\,}\mathrm{A}\mathrm{U}$ versus
their longitudinal widths (top left), and versus the areas of their source
coronal holes at the solar surface (top right). Temperatures (bottom left) and
densities (bottom right) of the HSS plasma parcels having the peak velocity at
$1\text{\,}\mathrm{A}\mathrm{U}$, versus the longitudinal widths of the HSSs.
The data were derived for latitudinal displacements of
$0\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$3\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$9\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and
$12\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ to the center of the HSS.

Here, we show how the properties of HSSs at $1\text{\,}\mathrm{A}\mathrm{U}$
depend on each other, and on the geometry of the HSS close to the Sun.
According to Equation 29, the peak velocity of HSSs is dependent on the
longitudinal width of the HSS close to the Sun. The longitudinal width of the
HSS determines the spatial distance between the leading edge of the HSS, which
coincides with the stream interface at this distance, and the trailing
boundary region of the HSS. Consequently, it also affects the time and the
distance from the Sun at which a plasma parcel originating from the trailing
boundary region catches up with the stream interface. The larger the distance
between the leading edge and trailing boundary region, the fewer plasma
parcels from the trailing boundary region have impinged into the stream
interface at a given distance from the Sun, and the higher is the peak
velocity of the HSS at this distance. This explains the dependence of the peak
velocity of HSSs on the longitudinal widths of the HSS. In the top left panel
of Figure 6, we show the peak velocities of HSSs at a distance of
$1\text{\,}\mathrm{A}\mathrm{U}$ versus the longitudinal width of HSSs close
to the Sun, at latitudinal displacements of
$0\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$3\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$9\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and
$12\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ to the HSS center. Dotted
lines are associated with plasma parcels that originate from the HSS core
region. Dashed-dotted lines represent plasma parcels originating from the
trailing boundary region within a slice that crosses the HSS core region. And
solid lines describe plasma parcels that originate from a slice that lies
entirely in the northern or southern boundary region. The peak velocities are
roughly linearly dependent on the longitudinal width of the HSSs and have
similar slopes for all latitudinal displacements. Solely the x-intersections
vary. The x-intersection gives the minimal latitudinal extent of a HSS that is
needed so that the HSS crosses the longitudinal slice, and thus that a HSS
appears in the data.

In the top right panel of Figure 6, we show the HSS peak velocities versus the
areas of their solar source regions, i.e., coronal holes. The areas are
thereby derived by $A_{CH}=\pi\ (\lambda_{HSS}/2)^{2}$ in square degrees, and
subsequently projected to a sphere at $1\text{\,}\mathrm{R}_{sun}$ to derive
their areas in $\mathrm{k}\mathrm{m}^{2}$. This associates the areas of the
HSSs with the areas of coronal holes at the solar surface. This functional
relationship between longitudinal width and area results in that the lines get
slightly skewed, dependent on the latitudinal displacement $\varphi$.

In the bottom panels of Figure 6, we show the dependence of the temperature
and density of the plasma parcels at $1\text{\,}\mathrm{A}\mathrm{U}$ that are
associated with the peak velocity versus the longitudinal width of the HSS.
With increasing longitudinal width, the temperature increases and the density
decreases, and finally saturates. The overshoot in the density decrease is
related to the radial expansion of plasma parcels originating from the
trailing HSS boundary. Presuming a less steep sigmoidal velocity function
across the HSS boundary or a wider HSS boundary region, respectively, reduces
the density overshoot.

Note, that the kinks in the relationships between the longitudinal widths and
areas to the velocities (at velocities of
$550600\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}^{-1}$), in the temperatures (at
$3050\text{\,}\mathrm{k}\mathrm{K}$), and in the densities (at about
$1\text{\,}\mathrm{c}\mathrm{m}^{-3}$) are artifacts due to the approximation
we used for modeling the HSS boundary regions. Different sigmoidal functions
and definitions of the size of the boundary region will lead to slightly
different results for the strength of the kinks. In purely numerical solutions
of our initial equations, the kinks are smoothed out.

###  5.3 Interrelation of HSS properties at $1\text{\,}\mathrm{A}\mathrm{U}$

Figure 7: Temperatures (left column) and densities (right column) versus the
velocities of HSS plasma parcels for three subsets. First row: plasma parcels
having the peak velocities of HSSs. Here, we considered all HSSs having a
longitudinal widths of $040\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, at
latitudinal displacements of the slice of
$0\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$3\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$6\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$,
$9\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, and
$12\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ to the center of the HSS.
Second row: HSS plasma parcels originating from a HSS having a longitudinal
widths of $22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$, for latitudinal
displacements of $011\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ (labeled
on top of the lines). Third row: all HSS plasma parcels, derived from all HSSs
with longitudinal widths of
$040\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$ at latitudinal
displacements of $012\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$.

In this section, we show the interrelation between the velocity, density, and
temperature of HSS plasma parcels at $1\text{\,}\mathrm{A}\mathrm{U}$. We
investigate the interrelation for three cases: 1) the distributions of plasma
parcels that are related to the peak velocity of HSSs, 2) the distribution of
HSS plasma parcels originating from a single longitudinal slice within a HSS,
and 3) the statistical distribution of plasma parcels from large HSS datasets.

In the top panels of Figure 7, we show the temperature and density of the
plasma parcels representing the peak velocities at
$1\text{\,}\mathrm{A}\mathrm{U}$ versus the peak velocities. For most of the
distribution, the temperature of these plasma parcels increases and the
densities decrease with increasing peak velocities. This is a consequence of
their presumed (anti–) correlation close to the Sun, but amended by the radial
expansion of the plasma parcels on their way to
$1\text{\,}\mathrm{A}\mathrm{U}$. Note, that for the longitudinal slice
through the center of the HSS, i.e., at
$\varphi=$0\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$$, we find small
densities even for the small peak velocities. As the overall HSS has to be
very small to provide small peak velocities in the center of HSS, and as
thereby the chance that a satellite takes measurements right in the center of
these HSSs becomes negligible, these low densities can probably be never
observed. Therefore, we expect from observational HSS datasets, that analyze
only the plasma parcels having the peak velocities, a correlation between the
velocity and temperature and an anti-correlation between the velocity and
density.

In the mid panels of Figure 7, we show the temperature and density
distributions of HSS plasma parcels versus their velocity for a HSS having a
longitudinal width of $22\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$. Each
line thereby represents plasma parcels originating from a different
longitudinal slice through this HSS. For each slice, the dependence between
the HSS density and temperature to the velocity has a unique curvature. This
curvature is related to the radial expansion of the plasma parcels on their
way from the Sun to $1\text{\,}\mathrm{A}\mathrm{U}$, and thus to the velocity
gradient across the trailing HSS boundary within that slice. The expansion is
largest for plasma parcels originating from the center of the trailing
boundary region, and less for parcels that arise from closer to western and
eastern edge of the trailing boundary region, i.e., from the transition to the
HSS core region or to the subsequent slow solar wind plasma.

In the bottom panels of Figure 7, we show statistical distributions of the
temperatures and densities versus velocities, by superposing all HSS plasma
parcels originating from all longitudinal slices for all HSSs up to a
longitudinal width of $40\text{\,}\frac{\mathrm{\SIUnitSymbolDegree}}{}$. The
color gives the associated average latitudinal displacement of the origin of
the plasma parcels to the center of the HSS. In this statistical dataset, the
temperature increases with velocity, resulting from the original correlation
of temperature and velocity across the HSS cross section close to the Sun.
Again, the radial expansion of plasma parcels originating from the trailing
boundary region results in additional cooling, which counteracts the
temperature increase. In contrast, the velocity-density distribution is much
less confined. The decrease of density with increasing velocity originates
from the anti-correlation of the density and velocity across the HSS cross
section close to the Sun. Though, the radial expansion of plasma parcels
additionally enhances the density decrease. Consequently, the densities can
decrease very fast from slow solar wind to HSS values, dependent on the
velocity gradient across the trailing boundary region, and thus strongly
dependent on the latitudinal origin of the plasma parcels with respect to the
center of the HSS. This amplification for the density decrease increases the
range at which densities appear at given velocities.

Note, that the curvatures in each of the plots depend on the radial expansion
of plasma parcels from the Sun to Earth, and thus on the widths of the
boundary region and the approximation we have used for the velocity profile
across the boundary region. Larger widths, or less steep velocity gradients,
respectively, result in smaller curvatures in these plots.

###  5.4 Properties of stream interface at $1\text{\,}\mathrm{A}\mathrm{U}$

Figure 8: Velocity of the stream interfaces at
$1\text{\,}\mathrm{A}\mathrm{U}$ versus the longitudinal widths of HSSs (top
left). Accumulated mass within a solid angle at the stream interface at
$1\text{\,}\mathrm{A}\mathrm{U}$ versus the longitudinal widths of the HSSs
(top right). Peak velocities of HSSs versus the velocities of the stream
interfaces at $1\text{\,}\mathrm{A}\mathrm{U}$ (bottom left). Accumulated mass
within a solid angle at the stream interfaces versus their velocities at
$1\text{\,}\mathrm{A}\mathrm{U}$ (bottom right).

Finally, we show how the properties of the stream interface at
$1\text{\,}\mathrm{A}\mathrm{U}$ relate to the properties of HSSs at
$1\text{\,}\mathrm{A}\mathrm{U}$, and on the geometry of the HSSs close to the
Sun. The velocity of the stream interface depends on the momentum transferred
from the HSS to the stream interface, and consequently on the underlying
velocity profile along the longitudinal slice through the HSS. As the velocity
profile depends on the longitudinal width of the HSS and the latitudinal
displacement of the slice to the center of the HSS, the velocity of the stream
interface is also dependent on these quantities, shown in the top left panel
of Figure 8. Furthermore, since both the peak velocity of the HSS and the
velocity of the stream interface depend on the longitudinal width of the HSS,
they are naturally correlated. However, it is important to note that this is a
result of the mutual dependency on the longitudinal width of the HSS, and not
a causal dependency.

The accumulated mass of the stream interface is dependent on the velocity
difference between the stream interface, the preceding slow solar wind, and
the subsequent HSS (Eq. 19). A higher velocity of the stream interface
implicates that less mass from the HSS accumulates, but also that more
preceding slow solar wind plasma gets piled up. As the preceding slow solar
wind plasma has a higher density than the HSS plasma, the slow solar wind term
dominates. Consequently, the total accumulated mass increases with increasing
velocity of the stream interface, which is shown in the bottom right panel of
Figure 8. Since the velocity of the stream interface is again dependent on the
longitudinal width of the HSS, so is the accumulated mass of the stream
interface as shown in the top panels of Figure 8.

###  5.5 Limitations

In this study, a number of simplifications have been made so that the model
was analytically solvable. These simplifications should not strongly affect
the relationships between HSS properties derived throughout the study.
However, some simplifications will affect the absolute values of the
properties near Earth. First, we have implicitly assumed that coronal holes
extend radially into interplanetary space, forming the cross-section of HSSs
at $0.1\text{\,}\mathrm{A}\mathrm{U}$. In reality, each coronal hole has its
unique expansion factor, and thus, the relationship between the area of
coronal holes in the low corona to the area of the associated HSSs at
$0.1\text{\,}\mathrm{A}\mathrm{U}$ is not trivial. Second, we have presumed
that HSSs are propagating radially away from the Sun. HSSs are being bent
towards the ecliptic very close to the Sun. Our calculations do not require
radially propagating HSSs as assumed here for simplicity, but only that the
HSS plasma propagates along a guiding line which does not affect its
kinematics. Bent magnetic field lines provide such guiding lines for plasma,
and thus do not affect our calculations regarding the HSS velocities. Third,
we have neglected a further acceleration of the solar wind in interplanetary
space. Although this acceleration after $0.1\text{\,}\mathrm{A}\mathrm{U}$ is
comparably small, it will still shift all velocities to slightly larger
values. Fourth, we have assumed that the impingement of HSS plasma into the
preceding slow solar wind plasma happens instantaneously, neglecting the
extension of the SIR. This assumption might work well as long as the SIR has a
limited extension compared to the spatial scales of the overall system; but it
will break down at the latest when the backward wave steepens to a shock wave
and decouples from the stream interface in the outer solar system. Fifth, we
have neglected viscosity in the solar wind plasma, which results in a slower
dispersion of the velocity distribution than presumed by our calculations.
Sixth, our model is only one-dimensional for the propagation of plasma parcels
from the Sun to Earth, and therefore neglects a possible redistribution of
plasma along the two-dimensional SIR. This could affect the momentum balance
between the HSS plasma and the preceding slow solar wind plasma at the stream
interface, and thus the velocity of the stream interface and the amount of
piled-up slow solar wind plasma. Seventh, the assumption of radially
propagating HSSs, affect the temperatures we derive at Earth distance. MHD
simulations show that HSSs originating from low and medium-latitude coronal
holes usually are bent towards the ecliptic, whereby their expansion is
inhibited towards higher latitudes. Thus, the plasma expands less into the
latitudinal direction with distance to the Sun, which results in less cooling.
Eighth, we have neglected waves in the plasma, which heat HSSs in
interplanetary space, and could also stabilize HSS plasma, i.e., counteract
velocity gradients. And ninth, adiabatic cooling by expansion, as presumed in
our calculations, requires that work has to be done that transfers the energy
away from the cooled gas. However, it is not clear to us what expansion work
HSSs do during their propagation away from the Sun. This question is
particularly interesting for plasma parcels in the HSS tail, as the preceding
HSS plasma parcels are faster and thus run away, i.e., there should not be
compression work involved. Note, that points seven and nine primarily affect
the temperature; as the temperature is derived independently from the velocity
and density in our calculations, this does not strongly affect the other main
findings of our study. All above points will lead to deviations from the
results presented here, and require more detailed studies. But they are not
expected to change the physical explanation of how the area of solar coronal
holes affects the solar wind properties near Earths.

##  6 Summary, Discussion, and Conclusions

In this study, we presumed that four main effects determine the properties of
HSSs in interplanetary space at a given distance from the Sun. First, a
velocity profile forms across the lateral HSS boundary close to the Sun. Due
to the solar rotation, this spatial velocity profile maps into a temporal
velocity profile into a given radial direction. Second, HSS plasma parcels
propagate freely, i.e., with a constant radial velocity, in interplanetary
space as long as they did not reach the SIR to the preceding slow solar wind.
Third, the leading HSS plasma parcels impinges into the preceding slow solar
wind plasma, thermalizes, and drives the SIR. And fourth, the characteristic
properties of HSSs observed at a given distance from the Sun is given by the
fastest HSS plasma parcels which did not impinge into the SIR yet.

By building an analytical model on these assumptions, we found that:

  * •

The peak velocity of HSSs at Earth distance is dependent on the effective
longitudinal widths of HSSs at $0.1\text{\,}\mathrm{A}\mathrm{U}$, i.e., the
length of a longitudinal slice through the HSS at
$0.1\text{\,}\mathrm{A}\mathrm{U}$ that represents the foot points of the HSS
plasma seen at Earth. The shorter the effective longitudinal widths, the more
plasma from the HSS core region and trailing boundary region has already
impinged into the stream interface at $1\text{\,}\mathrm{A}\mathrm{U}$. The
more ‘fastest’ plasma parcels from the trailing boundary region have impinged
into the stream interface, the smaller is the velocity of the leading plasma
parcel, which did not yet impinge into the stream interface at
$1\text{\,}\mathrm{A}\mathrm{U}$, and thus the smaller is the peak velocity of
the HSS seen at $1\text{\,}\mathrm{A}\mathrm{U}$.

  * •

The lengths of the effective longitudinal width are dependent on the
longitudinal widths of the overall HSS at $0.1\text{\,}\mathrm{A}\mathrm{U}$,
and the latitudinal displacement of the associated slice representing the foot
points to the center of the HSS. When we approximate the longitudinal widths
by its area (by assuming that the HSS has a circular cross-section), and
further presume that the area of HSSs at $0.1\text{\,}\mathrm{A}\mathrm{U}$
statistically depends on the area of their source coronal holes, we can
explain the long-known empirical relationship between the area of solar
coronal holes and the peak velocities of the associated HSSs near Earth.

  * •

The temperature and density of HSS plasma parcels near Earth depend on their
original temperature and density close to the Sun, and thus on their initial
location within the HSS core or boundary region close to the Sun.

  * •

Furthermore, their density and temperature near Earth depend on their
expansion on their way from Sun to Earth. Plasma parcels originating from the
boundary region of HSSs expand not only in the longitudinal and latitudinal
direction, but also in the radial direction on their way from the Sun to
Earth. The radial expansion thereby depends on the strengths of the radial
velocity gradient within the parcels close to the Sun, which is again set by
the initial location of the plasma parcels within the HSS core or boundary
region, and in particular by the widths of the HSS boundary region.

  * •

As we presumed a correlation between the solar wind temperature and velocity
close to the Sun, the radial expansion of the plasma parcels reducing the
temperature counteracts this correlation. However, the velocity and
temperature are still well correlated at Earth distance.

  * •

As we presumed an anti-correlation between the solar wind density and velocity
close to the Sun, the radial expansion enhances the decrease of density with
increasing velocity at larger distances from the Sun. This leads to a fast
drop of density with increasing velocity in the associated scatter plot, and
nearly destroys the initial anti-correlation between the solar wind density
and velocity at Earth distance.

In the following, we will shortly discuss some consequences of our results. 1)
How our calculations help to understand from where within coronal holes the
HSS plasma seen at Earth stems from, 2) how the correlation between HSS
velocity, density, and temperature evolves from the Sun to Earth, and 3) how
our calculations explain the empirical relationship between the HSS peak
velocity at Earth and the area of coronal holes and why this relationship is
similar to the distance to the coronal hole boundary model.

###  6.1 From where within coronal holes does the solar wind plasma stem from
that we see at Earth

Based on our calculations, an extended temporal velocity plateau of HSSs at
Earth indicates that not all plasma stemming from the HSS core close to the
Sun has impinged into the stream interface yet. Consequently, the plasma from
the plateau region at Earth stems from the inner region of coronal holes, away
from the coronal hole boundaries, and the peak velocity of the HSS at Earth is
similar to the peak velocity of the HSS close to the Sun. Suppose an extended
plateau region at Earth is not apparent, i.e., that the temporal velocity
profile is dominated by the tail during which the peak velocity decreases to
slow solar wind velocities. This indicates that all the plasma from the HSS
core region has already impinged into the stream interface. Consequently, the
plasma seen at Earth stems mainly from the trailing boundary region of the
coronal hole. The more plasma from the trailing boundary region has already
impinged into the stream interface at Earth distance, the slower the peak
velocity of the HSS at Earth will be. Thereby, the amount of plasma that has
already impinged into the stream interface depends on the longitudinal
extension of the HSS close to the Sun and the velocity gradient across the
trailing boundary region. The velocity gradient again depends on the
latitudinal distance of the foot points of the solar wind plasma in the
coronal hole from its center. This interpretation goes along with Nolte et al.
(1977), who studied eight HSSs at Earth distance in 1973. By ballistic back-
mapping, they found that all the HSS plasma within the declining velocity tail
at Earth distance stems from the trailing boundary region of the coronal
holes.

Note, that these calculations have two interesting consequences. First, if
almost all of the HSS plasma originating from a small- to medium-sized coronal
hole has already impinged into the stream interface at a given distance from
the Sun, we will see plasma that still has the composition of HSSs, but only
slow speeds. This “slow Alfvénic solar wind” has been described by D’Amicis &
Bruno (2015) and Stansby et al. (2020) as a distinct solar wind class, and
they related their small velocities compared to usual HSSs to differing
magnetic field geometries in the solar corona. We note, that some of these
slow Alfvénic solar wind streams could also be usual HSSs where almost all of
the plasma of the coronal hole has already impinged into the stream interface.
Second, in the very rare case of very small velocity gradients across the
longitudinal HSS boundary region, it can happen that none of the HSS plasma
has yet impinged into the stream interface at Earth distance, i.e., that the
stream interface has not even formed. This case most likely happens if the
plasma stems from very close to the latitudinal edge of coronal hole, where
also the velocity is close to the adjacent slow solar wind velocity. In this
case, one could observe almost undisturbed plasma stemming from leading
boundary regions of coronal holes even at Earth distance.

###  6.2 Evolution of the solar wind speed, density, and temperature with
distance from the Sun

In our calculations, we have presumed a velocity, density, and temperature
gradient across the HSS boundary region close to the Sun, dependent on the
distance to the HSS boundary. Consequently, the pairs of these three
parameters have been assumed to be (anti-) correlated to each other close to
the Sun. Thereupon, we have determined the expansion of the plasma parcels
with distance from the Sun by their longitudinal and latitudinal expansion,
which is related to the spherical geometry, times their radial expansion,
which is related to their radial dispersion due to radial velocity gradients.
Finally, the temperature has been derived under the assumption of adiabatic
cooling and the negligence of wave heating; these limitations on the
temperature are discussed in a subsequent section.

We presumed that the radial plasma velocity across the HSS core region is
roughly constant, and consequently the associated radial velocity gradient is
roughly zero. When we follow an isolated plasma parcel originating from the
HSS core region while it propagates away from the Sun, its volume does not
expand in the radial but only in the longitudinal and latitudinal direction.
Therefore, its density follows a $r^{2}$-law, and its temperature follows a
$r^{2(\gamma-1)}$. In contrast, plasma parcels originating from the boundary
region of HSSs inherit a radial velocity gradient. Therefore, these plasma
parcels naturally expand also in the radial direction while propagating away
from the Sun. The rate of radial expansion depends on the velocity gradient,
and thus on their initial location within the HSS boundary. Therefore, its
density decrease as a function of distance will in general not follow a
$r^{2}$-law, and consequently the temperature also not a $r^{2(\gamma-1)}$
law. Furthermore, note, that the radial expansion of these plasma parcels, in
combination with keeping their mean velocity constant, implies that the mass
flux is in general not conserved. Thus, the functional relationship of the
density and temperature of individual plasma parcels with distance from the
Sun, and whether the mass flux is a conserved quantity, depends on the initial
location of the plasma parcels within the HSS close to the Sun.

Furthermore, when one compares the plasma properties of HSSs at various
distance from the Sun, usually the properties around the peak velocity, or
within the velocity plateau, respectively, are used. With increasing distance
to the Sun, the fastest plasma parcels will have impinged into the stream
interface, and thus the peak velocity will be represented by another plasma
parcel originating from closer to the HSS trailing edge. Consequently, we
should be careful when we compare the properties of HSSs at various distances
from the Sun and interpret the results physically using thermodynamics, as we
possibly compare the properties of distinctly different plasma parcels within
the HSS. Interpreting the evolution of HSS properties with simple
thermodynamics as done by Perrone et al. (2019) might work well for HSSs that
have a clear, extended velocity plateau, i.e., where all plasma parcels
originate from the HSS core region. However, if such an extended velocity
plateau is not apparent, the plasma parcels analyzed at the various distances
will originate from different locations within the trailing boundary region
close to the Sun, and the method will inevitably fail.

Our calculations further explain reasonably well why the velocities of HSS
plasma parcels at Earth distance are still well correlated to their
temperatures but why they are only weakly anti-correlated to their densities
(e.g., Neugebauer, & Snyder, 1966; Burlaga & Ogilvie, 1970a, b; Ogilvie, 1985;
Xu & Borovsky, 2015). The additional decrease of temperature by the radial
expansion of plasma parcels counteracts the original presumed correlation
between the velocities and temperatures close to the Sun. This introduces a
slope into the temperature-velocity distribution at Earth distance, but the
temperature-velocity distribution remains well-defined. In contrast, the
radial expansion further reduces the densities of the plasma parcels and
thereby enhances the decrease of densities with increasing velocities we
presumed close to the Sun. This enhancement results in that the densities can
drop very fast with increasing velocities to the minimum value, in the case
that the associated radial velocity gradients close to the Sun have been
large. This fast drop, superposed with the regular slower decreases of plasma
parcels having smaller velocity gradients, results in the worse-defined
density-velocity distribution shown in Figure 7.

###  6.3 The empirical relationships of the solar wind speed at Earth to the
area of coronal holes, to the distance to the coronal hole boundary, and to
the flux tube expansion factor

The empirical relationship between the areas of solar coronal holes and the
peak velocities of HSSs near Earth, as found by Nolte et al. (1976); Robbins
et al. (2006); Vršnak et al. (2007); Abramenko et al. (2009); Karachik, &
Pevtsov (2011); Rotter et al. (2012); Hofmeister et al. (2018); Garton et al.
(2018); Heinemann et al. (2020), consists of three parts: 1) a roughly linear
relationship between the areas of coronal holes and the peak velocities near
Earths for small- to medium-sized coronal holes, 2) a saturation of the peak
velocities of large coronal holes, and 3) smaller peak velocities for coronal
holes of a given area located at higher latitudes.

In our calculations, we explained the roughly linear relationship between the
areas of solar coronal holes and the peak velocities of the associated HSSs
near Earth as an effect of the reduction of peak velocities by the continuous
absorption of the fastest HSS plasma parcels by the stream interface, combined
with the decreasing starting velocities of HSS plasma parcels across the HSS
boundary region close to the Sun. After all the plasma originating from the
leading boundary region and the HSS core region has impinged into the stream
interface, the fastest plasma parcel originating from the trailing boundary
region, which has not yet impinged into the stream interface, determines the
peak velocity. The amount of plasma that has not yet impinged into the stream
interface depends on the longitudinal widths of the coronal holes, or on their
areas, respectively, and on the displacement of the longitudinal slice to the
center of the coronal hole. Further, the saturation of peak velocities near
Earth for very elongated coronal holes arises from the fact that for very
elongated coronal holes not all plasma parcels originating from the HSS core
region have impinged into the stream interface at Earth distance. Thus, the
peak velocities are given by plasma parcels originating from the HSS core
region, which all have similar velocities. Finally, a larger latitudinal
separation angle between the satellite measuring the HSS properties at Earth
distance to the center of the coronal hole results in that we measure the HSS
properties farther in its latitudinal flanks. Farther in the latitudinal
flanks, the effective longitudinal widths of the HSS are reduced compared to
the overall longitudinal widths of the HSS in its center. A smaller effective
longitudinal width is again related to that more plasma from the trailing
boundary region has already impinged into the stream interface, which results
in smaller peak velocities measured at Earth distance.

Moreover, our results relate the empirical coronal hole area - HSS peak
velocity relationship to the distance to the coronal hole boundary model
(DCHB; Riley et al., 2001) and the Wang-Sheeley-Arge model (WSA; Arge et al.,
2003). The DCHB model, purely depending on the distance to the coronal hole
boundary, and the WSA model, depending on the inverse flux tube expansion
factor and the distance to the coronal hole boundary, are usually used to
derive a HSS velocity, density, and temperature profile above coronal holes at
$0.1\text{\,}\mathrm{A}\mathrm{U}$. Both have in common that the velocities
are large in the center of the HSS, and that they decrease to the ambient slow
solar wind values at the HSS boundary. These are then used as input for MHD
simulations to propagate the HSSs from the Sun to Earth. In this study, we
have shown that the empirical relationship between the area of coronal holes
and the peak velocities near Earth has implicitly incorporated the formation
of the velocity profile above the coronal hole close to the Sun and the
propagation phase from the Sun to Earth. The main difference in these three
models is the presumed physical origin of the velocity profile close to the
Sun. In the DCHB model, the acceleration of HSS plasma is presumed to depend
on the distance to the coronal hole boundary, which is interpreted by
diffusion and/or loop-opening processes at the coronal hole boundary. In the
WSA model, the solar wind acceleration depends on the distance to the coronal
hole boundary and a modeled flux tube expansion factor, whereby flux tubes
near the boundary of coronal holes naturally show a larger expansion. A larger
expansion is again interpreted with slower solar wind velocities. Within this
study on the relationship between the coronal hole areas and HSS peak
velocities at Earth, it does not matter where, or by which physical processes,
the HSS boundary region forms; it is only important that it exists. Despite
the different interpretations involved, all of these models describe a similar
idea: that a velocity profile above the coronal hole forms, which sets the HSS
properties close to the Sun, and that these HSSs are propagated from the Sun
to Earth, taking the interaction with the preceding slow solar wind into
account.
Finally, we like to add that the ongoing Parker Solar Probe and Solar Orbiter
missions (Fox et al., 2016; Müller et al., 2020) will soon allow us to
confront our model with more detailed observations. Parker Solar Probe’s
measurements during its perihelions close to the Sun will give us more insight
into the distribution of the HSS properties across their cross-section near
the Sun, while Solar Orbiter’s unique orbit will allow us to analyze the
radial evolution of HSSs and the associated SIRs in the inner heliosphere.
These measurements will further improve our understanding on how HSSs evolve
in the inner heliosphere, and how correspondingly the geometry of coronal
holes on the Sun affects the properties of HSSs near Earth.

###### Acknowledgements.

We thank Lars Berger for discussions on the expansion of high-speed streams in
interplanetary space and the associated cooling mechanisms. Stefan J.
Hofmeister is grateful for the Marietta-Blau fellowship awarded by the
Austrian mobility service. This fellowship enabled him to visit the research
groups at DTU Space, the University of Kiel, and KU Leuven for a total period
of 10 months, and ultimately to achieve the results presented here.
Furthermore, Stefan J. Hofmeister acknowledges support by the NASA Living with
a Star program grant 80NSSC20K0183 and DFG grant 448336908. The project
EUHFORIA 2.0 has received funding from the European Union’s Horizon 2020
research and innovation program under grant agreement No 870405. These results
were also obtained in the framework of the projects C14/19/089 (C1 project
Internal Funds KU Leuven), G.0D07.19N (FWO-Vlaanderen), SIDC Data Exploitation
(ESA Prodex-12), and Belspo project B2/191/P1/SWiM. Eleanna Asvestari
acknowledges the support of the Academy of Finland, Project TRAMSEP, Academy
of Finland Grant 322455. Evangelia Samara was supported by a PhD grant awarded
by the Royal Observatory of Belgium. Bojan Vršnak acknowledges the support by
the Croatian Science Foundation under the projects IP-01-2018-7549 (MSOC) and
IP-2020-02-9893 (ICOHOSS).

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