11institutetext: Predictive Science Inc., San Diego, California, USA 11email:
<EMAIL_ADDRESS>22institutetext: Physics Department, University of
California, Berkeley, CA 94720-7300, USA 33institutetext: Space Sciences
Laboratory, University of California, Berkeley, CA 94720-7450, USA
44institutetext: Smithsonian Astrophysical Observatory, Cambridge, MA 02138,
USA

# Using Parker Solar Probe observations during the first four perihelia to
constrain global magnetohydrodynamic models

Pete Riley 11 Roberto Lionello 11 Ronald M. Caplan 11 Cooper Downs 11 Jon A.
Linker 11 Samuel T. Badman 2233 Michael L. Stevens 44

(Received 30 October 2020; accepted N/A)

###### Abstract

Context. Parker Solar Probe (PSP) is providing an unprecedented view of the
Sun’s corona as it progressively dips closer into the solar atmosphere with
each solar encounter. Each set of observations provides a unique opportunity
to test and constrain global models of the solar corona and inner heliosphere
and, in turn, use the model results to provide a global context for
interpreting such observations.

Aims. In this study, we develop a set of global magnetohydrodynamic (MHD)
model solutions of varying degrees of sophistication for PSP’s first four
encounters and compare the results with in situ measurements from PSP,
Stereo-A, and Earth-based spacecraft, with the objective of assessing which
models perform better or worse. We also seek to understand whether the so-
called ‘open flux problem’, which all global models suffer from, resolves
itself at closer distances to the Sun.

Methods. The global structure of the corona and inner heliosphere is
calculated using three different MHD models. The first model (“polytropic”),
replaced the energy equation as a simple polytropic relationship to compute
coronal solutions and relied on an ad hoc method for estimating the boundary
conditions necessary to drive the heliospheric model. The second model
(“thermodynamic”) included a more sophisticated treatment of the energy
equation to derive the coronal solution, yet it also relied on a semi-
empirical approach to specify the boundary conditions of the heliospheric
model. The third model (“WTD”) further refines the transport of energy through
the corona, by implementing the so-called wave-turbulence-driven
approximation. With this model, the heliospheric model was run directly with
output from the coronal solutions. All models were primarily driven by the
observed photospheric magnetic field using data from Solar Dynamics
Observatory’s Helioseismic and Magnetic Imager (HMI) instrument.

Results. Overall, we find that there are substantial differences between the
model results, both in terms of the large-scale structure of the inner
heliosphere during these time periods, as well as in the inferred timeseries
at various spacecraft. The “thermodynamic” model, which represents the “middle
ground”, in terms of model complexity, appears to reproduce the observations
most closely for all four encounters. Our results also contradict an earlier
study that had hinted that the open flux problem may disappear nearer the Sun.
Instead, our results suggest that this “missing” solar flux is still missing
even at $26.9R_{S}$, and thus it cannot be explained by interplanetary
processes. Finally, the model results were also used to provide a global
context for interpreting the localized in situ measurements.

Conclusions. Earlier studies suggested that the more empirically-based
polytropic solutions provided the best matches with observations. The results
presented here, however, suggest that the thermodynamic approach is now
superior. We discuss possible reasons for why this may be the case, but,
ultimately, more thorough comparisons and analyses are required. Nevertheless,
it is reassuring that a more sophisticated model appears to be able to
reproduce observations since it provides a more fundamental glimpse into the
physical processes driving the structure we observe.

###### Key Words.:

Sun: corona – Sun: heliosphere – Sun: magnetic fields – (Sun:) solar wind –
Sun: evolution

##  1 Introduction

NASA’s Parker Solar Probe (PSP) spacecraft launched on 12 August 2018 and
reached its first of 24 perihelia (P1) on 5 November 2018. Since then it has
successfully completed six perihelion encounters (as of 27 September 2020),
with ever decreasing distances of closest approach. PSP’s primary scientific
goals are to: (1) better understand what heats the solar corona and
accelerates the solar wind; (2) determine the underlying structure and
dynamics of the coronal magnetic field; and (3) better identify the mechanisms
that accelerate and transport energetic particles in the corona (Fox et al.,
2016).

PSP carries four instrument packages. Of these, two are particularly relevant
for studying the large-scale magnetic and plasma properties of the solar
corona and inner heliosphere. FIELDS (Electro- magnetic Fields Investigation)
consists of two flux-gate magnetometers, a search-coil magnetometer, and five
plasma voltage sensors (Bale et al., 2016). It measures electric and magnetic
fields, as well as radio waves, Poynting flux, plasma density, and electron
temperature. Solar Wind Electrons Alphas and Protons (SWEAP) is composed of
three instruments: two electrostatic analyzers and one Faraday cup, from which
estimates of velocity, density, and temperature of electrons, protons, and
alpha particles can be made (Kasper et al., 2016).

Global models of the solar corona and inner heliosphere can provide crucial
support for interplanetary missions (e.g., Riley et al., 2001; Török et al.,
2018). Not only do they provide a global picture of the properties and
structure of the heliosphere, but they allow observers to connect the
otherwise disparate observations to one another. In addition to computing
basic magnetic and plasma variables, models can also be used to reconstruct
convolved measurements, such as extreme ultraviolet (EUV) and white light
images (Mikic et al., 2018).

A number of numerical models have been developed and applied to help in the
interpretation of PSP observations. Potential Field Source Surface (PFSS)
models represent the simplest approach, where: (1) the corona is assumed to be
current free; (2) time-dependent effects are ignored (or treated as a series
of quasi-equilibria); and (3) a source surface outer boundary condition is
imposed, where the field lines are forced to become radial (typically set to
be $2.5R_{S}$ (e.g., Badman et al., 2020a)). In spite of their inherent
simplicity, PFSS models have been shown to compare well with MHD solutions, at
least under some conditions (Riley et al., 2006). Badman et al. (2020a)
demonstrated the utility of applying PFSS models to interpret the magnetic
connectivity between PSP and the solar surface, and, in particular, in being
able to predict the location of these foot-points. Of course, the PFSS
approach is limited to inquiries concerning the magnetic structure, and cannot
directly address the plasma properties within the corona or further out.

van der Holst et al. (2019) applied the Alfvén Wave Solar atmosphere Model
(AWSoM) to predict the in situ measurements that would be returned from PSP
during P1. In this model, outwardly propagating low-frequency Alfvén waves,
which are partially reflected, provide both the coronal heating and
acceleration. Although no comparisons were made with PSP in that study, a
visual comparison with OMNI data at 1 AU, suggests that the model had captured
the large-scale stream structure of the solar wind. A qualitative comparison
with the predictions and subsequent observations at PSP supports the general
state of the solar wind (slow, dense) at perihelion, although the modest
structure that was apparent was not reproduced by the model. Moreover, the
model underestimated the strength of the magnetic field. In this case, they
relied on ADAPT-GONG maps, which include a more modest 1.85 correction factor
to amplify the photospheric magnetic fields. This need to boost either the
photospheric magnetic field boundary conditions, or the resulting model fields
at 1 AU, is a consistent requirement for all global MHD models, and, depending
on the input magnetogram, can range from 1.5 to 3.0.

We also developed a 3D wave-turbulence-driven (WTD) MHD prediction for the
state of the corona and inner heliosphere for P1. The model was driven by
photospheric magnetic field observations several weeks prior to the encounter
(Riley et al., 2019a), thus, it was a true prediction. This served as both as
a test of our model’s predictive capabilities as well as what we hoped would
be an aid in mission planning. For example, it would inform ground-based
observers where they should target their smaller-scale campaign observations,
based on our predictions of the foot-point locations during perihelion. We
inferred that, in the days prior to first encounter, PSP would be immersed in
wind emanating from a well-established, positive-polarity northern polar
coronal hole. During the encounter, however, field lines from the spacecraft
would map to a negative-polarity equatorial coronal hole, within which it
would remain for the entire encounter, before becoming magnetically connected
to a positive-polarity equatorial coronal hole. In that study, we also
compared MHD and PFSS predictions, noting that while there are an overall
agreement in the forecasts, there were some notable differences. The
equatorial coronal holes, for example, predicted by the PFSS solutions were
substantially smaller than those inferred from the MHD model results.

Most recently, Réville et al. (2020) modified the astrophysical Pluto
magnetohydrodynamic (MHD) code (Mignone et al., 2007) by accounting for Alfvén
wave transport and dissipation. The wave-turbulence approach was similar to
the previously described studies (Riley et al., 2019a; van der Holst et al.,
2019) but simpler in that the Alfvén reflection process was not explicitly
included. Nevertheless, they found that, at least for P1, the large-scale
amplitude of the plasma parameters were well reproduced by the model. Although
the modeled values of the magnetic field strength could be interpreted as
being an underestimate, they argued that, in fact, if the amplitude of the
waves used to heat the corona and accelerate the solar wind are added post hoc
to the radial and total magnetic fields, this would bring the model results
into agreement with the observations. While intriguing, this requires more
substantiation through comparisons with subsequent perihelia, at different
spacecraft locations, and with simultaneous comparison with other
observational metrics, such as EUV and white-light observations,

Finally, bridging the gap between the fully MHD and PFSS approaches, Kim et
al. (2020) presented model results for each of the first three PSP perihelia
using a model composed of a coronal PFSS model, connected to a heliospheric
MHD model. They used the Wang-Sheelely-Arge (WSA) methodology for prescribing
the solar wind speed at the inner boundary of the heliospheric model (Arge et
al., 2003), which gives largely similar results to the Distance from the
Coronal Hole (DCHB) method that we have developed (Riley et al., 2001, 2015).
The most significant disagreements arise in the vicinity of pseudo-streamers,
where the WSA model predicts slow solar wind, in contradiction to
observations.

In this study, we compare MHD results from three distinct approaches with in
situ measurements made by PSP, Stereo-A, and Earth-based spacecraft (ACE and
Wind) in an effort to identify any systematic differences between the model
results. Based on this, we then use the best model results to infer the global
structure of the heliosphere during each of the first four perihelia
encounters. Additionally, we highlight how sensitive the timeseries
comparisons are to the precise trajectory of the spacecraft through the model
solution, suggesting that even when differences exist, the overall global
structure may still be reasonably accurate. Finally, given PSP’s ever smaller
point of closest approach we interpret the comparisons in terms of whether an
“open flux problem” remains.

Figure 1: Comparison of HMI magnetograms for each for the four Perihelia
intervals (P1, P2, P3, and P4), which occurred during CR 2210, 2215, 2221, and
2226. In the left column are the original HMI synoptic maps, with the polar
regions filled in. In the middle column are the processed magnetograms used to
drive the polytropic and thermodynamic models, and in the right column are the
synoptic maps used to drive the WTD model. Blue/red corresponds to
negative/positive polarities, respectively, and the maps have been saturated
at $\pm 10$ G; thus, some of the ARs such as during CR2215, contain field
strengths larger than this maximum.

##  2 Methods

###  2.1 Data

Data used to drive the model were obtained from the Helioseismic and Magnetic
Imager (HMI) onboard the Solar Dynamics Observatory (SDO) spacecraft (Scherrer
et al., 2012), and, specifically, from jsoc.stanford.edu/ajax/exportdata.html.
These data provide an estimate for the radial component of the field on a
uniform grid size of $3600\times 1440$ in longitude and sin(latitude),
respectively.

Figure 1 summarizes each synoptic map for CR 2200, 2215, 2221, and 2226, which
contained each of the perihelia P1, P2, P3, and P4, respectively. We note
several points. First, to produce MHD solutions, the raw magnetograms (left-
most column) must be processed to: (1) remove small-scale features (e.g.,
parasitic polarities) that are below the resolution of the model; and (2)
provide a reasonable estimate for the polar fields, which are poorly, if at
all observed. As can be seen from the middle and right-most columns, the two
processing pipelines we currently use generate similar but not identical
processed maps. The middle column relies on our simpler, but mature technique
that is used to develop our online, standard solutions (e.g.,
http://www.predsci.com/mhdweb/home.php). Flux is balanced and preserved,
meridional variations are made smoother, but the underlying structure visible
in the original map remains. The right-most column relies on a more recent
approach to developing input magnetograms for the MAS code. It attempts to
maintain more structure , particularly at higher latitudes. Perhaps most
importantly, for the maps for the polytropic and thermodynamic runs, the poles
are filled in by extrapolation of the mid-latitude fields, while for the WTD
runs we use the pole-filled data provided by the HMI instrument team. For our
purposes, however, the main point is that they serve as different, but
potentially equally-valid drivers of the MAS code.

In this study, we use in situ measurements from ACE and Wind, in the form of
the merged OMNI dataset as well as from Stereo-A, and, of course PSP. From
each spacecraft we compare solar wind bulk speed, proton number density, and
radial magnetic field. Together, these three quantities characterize the basic
dynamical properties of the solar wind. All data were obtained from NASA’s
Space Physics Data Facility (SPDF) web services API (e.g., Candey et al.,
2019), and were retrieved at 1-hour resolution.

Figure 2:  Radial speed profiles in the solar equatorial plane for each model
(polytropic, thermodynamic, and WTD) for each perihelion encounter 1-4.
Superimposed are the corotating trajectories of the spacecraft during each
encounter, expanded by 10 days on either side of the nominal encounter dates.
The approximate location of the spacecraft at the time of PSP’s point of
perihelion pass is indicated by the location of each spacecraft label.

###  2.2 Models

In this study, we use PSI’s MAS (Magnetohydrodynamic Algorithm outside a
Sphere) code, which solves the usual set of resistive MHD equations in
spherical coordinates on a nonuniform mesh. The details of the model have been
described elsewhere (e.g., (Mikić & Linker, 1994; Riley et al., 2001; Lionello
et al., 2001; Riley et al., 2012c; Mikic et al., 2018; Mikić et al., 2018;
Caplan et al., 2019)). Here, we restrict our description to several relevant
points. First, the model is driven by the observed photospheric magnetic
field. We use HMI observations from the SDO spacecraft to construct a boundary
condition for the radial magnetic field at $1R_{S}$ as a function of latitude
and longitude. Second, the model is run in two stages: First the region from
$1-30R_{S}$ is modeled, followed by the region from $30R_{S}$ to 1 AU, being
driven directly by the results of the coronal calculation. Computationally,
this approach is much more efficient. Third, for these solutions, we use a
single map to cover an entire solar rotation, thus, although the model is
time-dependent, it is run forward in time until a dynamic steady-state is
achieved. This is a reasonable approximation when structure at the Sun is not
appreciably changing from one rotation to the next. However, during intervals
with significant active region activity, such as E2 and E4 (Figure 1), it may
lead to inaccuracies. For E2, in particular, the active region was responsible
for observed solar impulsive events (Pulupa et al., 2020). Fourth, MAS relies
on a variety of approximations to reconstruct (or predict) the large-scale
structure and properties of the solar corona and inner heliosphere. In order
of increasing complexity (and historical development), we refer to them as the
‘polytropic’, ’thermodynamic’ and ’WTD’ models.

The polytropic approximation solves the usual set of MHD equations in
spherical coordinates with the energy equation being approximated by a simple
adiabatic approximation (that is, we set all energy source terms to zero).
This requires us to choose a polytropic index, $\gamma=1.05$ to reflect the
near-isothermal nature of the corona. In the solar wind, it is set to 1.5.
These simplifications result in a fast, robust code that reproduces the
structure of the magnetic field reasonably well, but fails to generate
solutions with sufficient variation in solar wind speeds or densities. To
address this, we use an empirically-based approach, DCHB, to specify the solar
wind speed at the inner boundary of the heliospheric code (Riley et al.,
2001). Although semi-empirical, it generally produces results that match 1 AU
observations as good as, or better than those computed using the WSA
approximation (Riley et al., 2015).

The thermodynamic approximation replaces the polytropic assumption with an
empirically-based treatment of energy transport processes (radiation losses,
heat flux, and coronal heating) in the corona (Lionello et al., 2001, 2009).
In this case, $\gamma$ now returns to a more defensible value of
$\frac{5}{3}$. Development of this model focused on improving the density and
temperature structure in the solar corona through comparisons with EUV and
X-ray images from a variety of spacecraft. Relatively little direct comparison
was performed with in situ measurements, Thus, we also implement the DCHB
approximation to derive the heliospheric boundary conditions from the
thermodynamic solution. In this sense, the model is not strictly – or fully –
thermodynamic, and should strictly be labeled the “semi-empirical
thermodynamic” model. When we have produced complete end-to-end coronal-
heliospheric thermodynamic solutions, we have found that they cannot reproduce
the observed structure in the solar wind at 1 AU, even though they correctly
match the observed amplitude of the quantities.

Finally, the Wave-Turbulence-Driven (WTD) model is a generalization of the
thermodynamic approach by self-consistently heating the corona and using the
WKB approximation for wave pressures, providing the necessary acceleration of
the solar wind (Mikić et al., 2018). The physical motivation for this heating
model is that outward and reflecting Alfvén waves interact with one another,
resulting in their dissipation, and heating of the corona (Zank et al., 1996;
Verdini & Velli, 2007). And, while the WTD model is significantly more
physics-based than, say, the polytropic model, it should be noted that it does
require a careful choice of two free parameters. We have found that this
approach can account for both the acceleration of solar wind along open field
lines, as well as the heating of plasma entrained within closed-field regions
(Lionello et al., 2014; Downs et al., 2016); however, again, relatively little
exploration of its abilities to reproduce in situ measurements has thus far
been performed.

In summary then, our analysis relies on three models: (1) the polytropic
model; (2) the (semi-empirical) thermodynamic model; and (3) the WTD model.
Although it is conceptually convenient to think of them as distinct
approaches, in reality, they represent snapshots of a continually evolving
model; a model that has been developed over $\sim 25$ years and one that has
been used to interpret a disparate set of observations at different stages
during this evolution. In particular, the polytropic model has been most
extensively compared against in situ measurements (e.g., Riley et al., 2001,
2012b, 2015) while the WTD model has been almost exclusively compared against
remote solar observations, and, in particular, white-light and EUV images
(e.g., Mikic et al., 2018; Linker et al., 2019).

##  3 Results

Figure 3:  Comparison of model results with PSP in situ measurements during
encounter 1. From top to bottom: Speed, radial magnetic field, and number
density ($V_{r}$, $B_{r}$, and $N_{p}$, respectively) are compared with WTD
(red), polytropic (green), and thermodynamic (blue) solutions. Additionally,
for P1, the original prediction made by Riley et al. (2019a) is also shown
(magenta). Values of $B_{r}$, with the exception of the prediction, were
multiplied by a factor of three. In the bottom panel, heliocentric distance
(cyan) and Carrington longitude (magenta) are shown as a function of time.
Figure 4:  Same as Figure 3, but for model and data P1 comparisons at the
location of Earth (with in situ measurements supplied by NASA’s SPDF OMNI
dataset.  Figure 5:  Same as Figure 3, but for PSP data/model comparisons
during P2.  Figure 6:  Same as Figure 3, but for model and data comparisons at
the location of Stereo-A and for P2.  Figure 7:  Same as Figure 3, but for
model and data comparisons at the location of PSP and for P3.  Figure 8:  Same
as Figure 3, but for model and data comparisons at the location of Earth and
for P3.

Before beginning a detailed comparison of the timeseries measurements with the
model results, it is instructive to explore the large-scale structural
differences predicted by each of the models. Figure 2 summarizes the radial
speed of the solar wind from $1R_{S}$ to 1 AU for each of the models and for
each of the first four PSP encounters (from October 2018 through March 2020).
These are displayed in Carrington coordinates with $\phi=0$ along the positive
x-axis. Superimposed on each panel is the corotating trajectory of the
spacecraft during each encounter, expanded by 10 days on either side of the
nominal encounter dates, to encompass the possibility of additional data
captured by the spacecraft in some encounters. The approximate location of the
spacecraft at the time of PSP’s point of perihelion pass is indicated by the
location of each spacecraft label. Thus, we can infer that Earth-based
observing platforms were able to see the solar origin of the plasma measured
at PSP for P4, and, to a lesser extent P2. However, for P1 and P3, the
relative position of the spacecraft meant that Earth could not, in particular,
provide a direct estimate of the photospheric magnetic field at the time of
perihelion. From a modeling perspective then, the boundary conditions used to
drive the models for P4 and P2 were intrinsically more accurate than for P1
and P3. We also note how the angular separation of Earth and Stereo-A
decreases from P1 to P4. During this interval, Earth was in fact leading
Stereo-A; however, this is in fact a consequence of Stereo-A outpacing Earth
and lapping it. In progressively later PSP encounters, Stereo-A will again
catch (August 2023) and advance past the longitudinal position of Earth. In
summary, each encounter presents a modestly different configuration of the
three spacecraft. Finally, note that, unlike an inertial projection of
spacecraft trajectories, in the corotating frame, the spacecraft travel in a
clockwise direction, thus, for example, when interpreting the thermodynamic
solution for P1 (top middle panel), PSP measures slow solar wind initially,
and, after traversing two modest streams, is immersed in slow solar wind at
perihelion (where it hovers at approximately the same Carrington longitude
while performing a ‘loop’), then exits the perihelion as it encounters a
significant high-speed stream.

These maps emphasize the differences in the structure of the solar wind
predicted by each model for each encounter. In general, the polytropic model
produces stronger high-speed streams in the equator than either of the other
two models. Additionally, the WTD model tends to produce broader (in
longitude) streams with less contrast between the slower and faster wind
streams. Comparing the thermodynamic model to the polytropic model, we note
that the structure - as a function of longitude - is quite similar. Based on
this, it is likely that the band of solar wind variability for the polytropic
model is narrower than the thermodynamic model.

To assess the quality of the model results, we compare data from PSP, Earth-
based spacecraft (ACE and Wind, through the OMNI dataset), and Stereo-A for
each of the first four perihelia passes (labeled P1 through P4). Beginning
with P1, Figure 3 compares modeled and observed radial velocity, radial
magnetic field, and number density for the interval from day-of-year (DOY) 294
through 325. This includes the nominal 12 days that bracket the date of
closest approach (from DOY 304 to 315) but adds an additional 10 days on
either side to define an interval that is marginally longer than a solar
rotation period. Perihelion occurs near the center of the panel (DOY 310, or
06 November 2020), coinciding with the central portion of the interval, and
associated with a positive increase in longitude with respect to time (bottom
panel). During the interval from approximately DOY 303 to 316, PSP remained at
roughly the same Carrington longitude ($\sim 330^{\circ}$), which, at the time
was an unprecedented position for an interplanetary spacecraft to hold.
Focusing first on the radial velocity, we note that during this first
encounter, PSP was immersed in slow ($<500$ km s-1 solar wind, only rising
substantially beyond this on DOY 319. These variations are well captured by
the thermodynamic model results; however, the other models fail to reproduce
these basic variations. The WTD model, while reasonably predicting an average
and unchanging speed of $\sim 350$ km s-1 does not capture any of the
structure within the slow stream, nor does it predict the rise in speed later
in the interval. The polytropic solution qualitatively matches the variations
from slower to faster wind, but the amplitude is far too large, predicting
speeds approaching 700 km s-1 at the peak of the small stream (DOY 313).
Finally, the actual prediction made by Riley et al. (2019a) is grossly
inaccurate (magenta line), with a forecast of constant, 600 km s-1 solar wind
throughout most of the interval.

Figure 9:  Same as Figure 3, but for model and data comparisons at the
location of Stereo-A and for P3.  Figure 10:  Same as Figure 3, but for model
and data comparisons at the location of PSP and for P4.

Similar inferences can be made for the comparisons with the radial magnetic
field, $B_{r}$. It should be emphasized that the values for the polytropic,
thermodynamic, and WTD models have been multiplied by a factor of 3, which
reflects the well-known problem with global models that, while they are able
to capture variations in the field strength, they significantly underestimate
their amplitude (Riley et al., 2012b; Linker et al., 2017; Riley et al.,
2019b). This factor is also the same as used by other global modelers relying
on HMI data to drive their simulations (e.g., van der Holst et al., 2013).
Intriguingly, and a point we will return to later, the solution used to make
the prediction (magenta) was not multiplied by any corrective factor. From
these comparisons, we infer that the prediction, polytropic, and thermodynamic
solutions appear to have captured the large-scale variations in the magnetic
field, and, in particular, the immersion into a negative polarity field for
the entire interval, only switching to a positive polarity around DOY 318.
Although not shown here, the origin of this wind was an equatorial coronal
hole (Riley et al., 2019a; Badman et al., 2020a). Both the polytropic and
thermodynamic solutions suggest some structure in the field around DOY 308
that is not reflected in the observations.

Finally, we consider density. The third panel of Figure 3 compares the
model/prediction results with the observed density during the first encounter.
Both the prediction and the WTD model fail to capture the substantial increase
in density as PSP swept into 35.7 $R_{S}$. In fact, given that all four model
results were approximately the same on DOY 303 and again on DOY 314, this
further suggests that the WTD and prediction results were actually regions of
lower intrinsic density (a fact confirmed by plots of scaled density, not
shown here but presented for the prediction by Riley et al. (2019a)). Of the
two models that capture both the peak at perihelion and the subsequent high-
density region following it (DOY 315-319), the thermodynamic results (blue)
most closely match the observations, although the maximum value is
overestimated.

The PSP P1 results can be contrasted with Stereo-A and near-Earth spacecraft
measurements made at 1 AU. In Figure 4 we compare the same three parameters at
the location of Earth, using data from the OMNI dataset (composed of
measurements from both ACE and Wind). During its passage from the location of
PSP to Earth, the streams have evolved, steepening where fast wind is
overtaking slower wind ahead, and expanding where slower wind is trailing
faster wind. The resulting “sawtooth” profile in radial velocity, covering
slightly more than a Carrington rotation, is typical of ambient solar wind
conditions in the absence of transient phenomena. In this specific case, there
are two high-speed streams beginning on DOY 309 and 314. This profile is
captured well by both the polytropic (green) and the thermodynamic (blue)
models. Of these, the thermodynamic results are again a closer match with
observations, mimicking the size and structure of the streams better. Note, in
particular, that the first stream has a sharp rise, followed by an immediate
decay, while the second one exhibits a flat top before decaying, indicating
that the spacecraft was immersed more fully in an equatorial coronal hole
(likely the result of an equatorward extension to the polar coronal hole - see
later). Again, the WTD results – both from the prediction and the
retrospective run – do not match the observed structure of the solar wind
speed.

Considering next the radial magnetic field, the observations suggest that the
spacecraft skimmed along the heliospheric current sheet (HCS) for most of the
rotation, dipping into a positive polarity region around DOY 307 and into a
negative polarity region on DOY 314. Only the WTD results miss these sector
crossing. We note again, however, that the predicted values have not been
corrected by the factor of three that was applied to the WTD, polytropic, and
Thermodynamic solutions.

Comparison of the plasma density with the measurements suggests that the broad
variations have been captured by both the polytropic and thermodynamic models.
However, both models appear to overestimate the size of the compression
regions, driven by the fast streams overtaking slower streams. This is
particularly true for the two major streams on DOY 309 and 314. This is likely
due to the fact that the speed gradients and overall amplitude predicted by
the models were substantially higher than those observed. Additionally, the
polytropic solutions systematically predict an earlier arrival of the high-
speed streams than was observed.

Comparison of Figures 3 and 4 appears to show that there is more large-scale
structure at 1 AU than at $\sim 40R_{S}$. However, it is important to note
that whereas the same temporal interval at Earth translates into more than
360∘, at PSP, it is approximately half of this (180∘), owing to the
spacecraft’s acceleration into perihelion and deceleration out of it.

Next we consider P2. Figure 5 summarizes the same measurements and model
results (with the exception of the prediction results). The solar wind
measured by PSP was very quiescent: slow, dense, and somewhat variable plasma
emanating from a negative polarity region on the Sun. During the entire
interval, and until DOY 105 the field remained negative, thus, it represents a
unique opportunity to investigate the properties of solar wind from one
specific region, essentially two weeks of measurements came from a Carrington
longitude range of less than 20∘ or less, centered near 0∘. In terms of solar
wind speed, the thermodynamic and WTD models best match the observed low
speed. But, while the actual values are overestimated, only the thermodynamic
model appears to have captured the variations, that is, the stream structure.
The polytropic model is patently wrong. From the perspective of the radial
magnetic field though, both the thermodynamic and polytropic solutions capture
the global features of the field in terms of amplitude and direction. The
thermodynamic model erroneously suggests a small polarity reversal starting on
DOY 85, but both results are vastly better than the WTD results. Finally,
comparing density profiles, once again, the thermodynamic model reproduces the
large-scale variations observed in the plasma data, although it somewhat
overestimates them.

Thus far, we have limited ourselves to a qualitative comparison between the
model results and the observations. Quantitative estimates, such as mean
absolute error (MAE) or correlation coefficient (CC) have their place,
particularly for space weather applications, but can result in inferences that
do not match our subjective interpretation (e.g., Owens et al., 2005; Riley et
al., 2015, 2013), particularly when scientific understanding is the goal. For
example, a model that produces a constant velocity at say, 400 km s-1, would
produce a lower MAE than one that reproduced the structure of the solar wind
streams, but not their exact phasing in time. Clearly the latter model is of
more scientific value (Riley et al., 2017). To better illustrate this, we
computed CCs for each of the model results against their observed values for
P2 at PSP (Figure 5). The results were as follows: (1)
$CC_{vr}$(WTD,Poly,Thermo): 0.027, -0.197, 0.016; (2)
$CC_{Br}$(WTD,Poly,Thermo): -0.672, 0.886, 0.8923; and (3)
$CC_{Np}$(WTD,Poly,Thermo): 0.625, 0.456, 0.804. Thus, the quantitative
estimates suggest that, in terms of speed, the WTD performs slightly better
than the thermodynamic solution. However, practically speaking, based on these
results, none of the models shows any significant correlation with the
observations. The CCs for the radial field match our qualitative
interpretation, with both the polytropic and thermodynamic models
outperforming the WTD results. Finally, although our subjective inference that
the thermodynamic model substantially outperformed the other two, in terms of
density, this is not captured by the CC values, where the CC for the WTD
solution (0.625) does not accurately reflect its lack of agreement in terms of
structure, as compared to the thermodynamic solution (CC=0.804). Thus, bearing
these points in mind, for the remainder of the study, we continue to provide
qualitative descriptions of the model comparisons.

Contrasting the comparison of PSP data/model results with those at Stereo-A
(Figure 6) leads to similar statements as for the P1 comparison. Here, it is
even more apparent that Stereo-A was skimming along the HCS for the entire
interval (as indicated by the radial field values ‘hugging’ a value of zero).
When spacecraft are so situated, this makes comparisons with models even more
precarious, as small shifts in latitude can lead to substantial changes in the
values of the plasma and magnetic field values. The relatively poor comparison
across all variables and models reinforces this point. At least qualitatively,
one could argue that the thermodynamic solution appears to more closely match
the observations; however, this is a weak inference at best. Finally, it is
worth noting that the amplitude of the compression regions, seen as the peaks
in the number density profiles are largest for the thermodynamic model. This
may appear paradoxical because the differences in speed between the slow and
fast streams are larger for the polytropic results than the thermodynamic
results. However, the reason is that the base density in the slow solar wind
of the thermodynamic model is substantially higher. Thus, the compression of
an already denser medium by a relatively smaller high-speed stream leads to a
larger compression region (as measured by peak density) than the compression
of a more tenuous region, even if the fast wind compressing it is
substantially faster. This is reinforced by comparing with Figure 4, which
shows similar effects. In general, the thermodynamic compression regions are
significantly larger than are observed, suggesting that the model could be
improved by setting the base number density for the slow wind, which is a
model parameter, to a lower value.

Turning next to P3, in Figure 7 we compare radial speed, radial magnetic
field, and number density once again with the three model results.
Unfortunately, during this perihelion some of the plasma measurements were
unavailable. Nevertheless, what was recovered, again, demonstrates that the
thermodynamic solutions most closely match the data, although again, the base
number density is too high. Note, in particular, the generally low speed prior
to the perihelion, and the large high-speed stream on DOY 63, which the
thermodynamic model captures well both in terms of phase and amplitude. This,
comparison, in turn, suggests a potentially interesting application for the
models: The thermodynamic solution may be a reasonable proxy for the
unobserved data during the actual perihelion portion of the mission. This is
also supported by the reasonable match between the model results and the
observations of the radial component of the magnetic field shown in the second
panel, as well as comparisons with data at 1 AU, which we turn to next.

Figure 11:  Same as Figure 3, but for model and data comparisons at the
location of Earth and for P4.

Perihelion 3 at Earth looked quite different (Figure 8). Although there was a
single large stream of $\sim 700$ km s-1 this occurred some 20 days prior to
the arrival of the stream at PSP. Although it is tempting to associate this
stream with the single high-speed stream observed at PSP, inspection of Figure
2 shows that, at least in the equatorial plane, there were two possible
candidates for the stream. To assess this more carefully, we note that at the
time of PSP’s perihelion, Earth was located at approximately $232^{\circ}$
Carrington longitude, which coincided with the arrival of the second and more
significant high-speed stream. Additionally, referring back to Figure 7, we
can infer that the observed high-speed stream at PSP was probably the second
of the two, based on the thermodynamic profile showing two peaks, with the
second one being more pronounced and aligning well with the measurements.
Unfortunately, we cannot be certain because of the data gap. Nevertheless,
based on the global perspective provided by the model, it is reasonable to
infer that the high-speed stream observed at PSP on DOY 263 was the same one
observed at Earth on DOY 243, one solar rotation earlier.

More generally, we can make several comments about the comparisons between
models and observations. First, although the single high-speed stream is
produced by both the polytropic and thermodynamic models, little of the
structure between this and the next stream on DOY 268 is captured. Similarly,
only some of the structure during the first half of the interval matches. On
the other hand, the two-sector pattern shown in the radial magnetic field is
reproduced, with the spacecraft remaining predominantly on the positive side
of the HCS. All models, to varying degrees, capture this sector structure,
with the thermodynamic model performing best. Finally, the modeled densities
during this interval are systematically either too high (polytropic and
thermodynamic) or too low (WTD), which is related to the fact that the modeled
speeds are systematically too low or high, respectively.

The comparison between model results for P3 at Stereo-A is considerably better
(Figure 9). There were five/six high-speed streams of various strengths during
this two-rotation interval, almost all of which were captured by the
thermodynamic model. This is reinforced by the density measurements, for which
the compression regions (peaks in density) match with the observations, albeit
in most cases being too strong. The polytropic and WTD solutions did not
reproduce this structure with any degree of fidelity, and, as with the Earth
comparison, tended to over-estimate or under-estimate the densities.
Unfortunately, magnetic field data was only available for a portion of this
interval. Nevertheless, the thermodynamic solution, in particular, matches
most of the large-scale variations, and, in particular, the transition from a
toward to an away sector on DOY 237. Based on the close match between the
thermodynamic solution and the plasma measurements at Stereo-A, it would not
be unreasonable to infer the unobserved large-scale magnetic structure from
the thermodynamic model results, and, in particular, the transition from
negative to positive polarity occurring rapidly and strongly on DOY 264. A
final point that is worth reinforcing is that, again, while the properties of
the plasma speeds are remarkably well captured by the thermodynamic model, the
modeled density peaks in the compressions are significantly larger than were
observed. This, again, we believe is a result of setting the boundary value
for the slow solar wind to be larger than was observed, providing more
material for the high-speed streams to compress.

The final PSP encounter analyzed here, P4, occurred during the first two
months of 2020. Figure 10 contrasts the model results with the observations.
In general, we remark that the overall features observed in the observations
are matched by the thermodynamic solutions. These include: (1) generally slow
(300-400 km s-1) solar wind; (2) peak densities, likely exceeding 800 cm-3;
and (3) peak radial fields reaching almost -150 nT. If the model results are
an accurate predictor of what PSP would have observed later, we might predict
the appearance of a high-speed stream (700 km s-1) on DOY 51. However, given
the limited data and discrepancies, this would be a tentative conclusion at
best. These discrepancies include: (1) the earlier rise in number density in
both the thermodynamic and polytropic solutions than was observed; (2) the
failure to capture the switch from negative to positive magnetic polarity on
DOY 32; and (3) an apparent discontinuous change in the speed and density of
the polytropic and thermodynamic solutions close to perihelion. This last
disagreement reveals an interesting limitation of the models, only appearing
because of the distance of closest approach during P4. For our standard web-
based model solutions, the boundary separating the coronal and heliospheric
models was set to $30R_{S}$. For the previous encounters, this did not impact
the simulated spacecraft fly-throughs since they always occurred within the
domain of the heliospheric model. For P4, however, PSP’s closest approach was
26.9 $R_{S}$, which is inside this boundary. Since both the polytropic and
thermodynamic models are not seamless at this boundary, artifacts are
introduced when the simulated spacecraft is flown through the merged solution.
This is particularly noticeable for the polytropic solution, for which the
coronal model cannot adequately calculate the speeds and densities of the
plasma. And, while the thermodynamic solution is better, because we did not
drive the heliospheric model directly with the coronal solutions,
discontinuities are inevitably introduced. Accepting these limitations,
however, we remark that, again, the thermodynamic model appears to estimate
the densities and speeds best. For the magnetic field, such discontinuities
are not as significant, since the field at the outer boundary of the coronal
solution is used directly to drive the inner boundary of the heliospheric
model.

Figure 12:  Comparison of MHD modeled solar wind speeds at the location of the
three spacecraft (PSP, Earth, and Stereo-A (STA)) with the interval defined by
each PSP perihelia pass (1-4). In each panel, speeds from 200 to 800 km s-1
are shown in the rainbow spectrum from blue to red, using the same transitions
as in Figure 2’s colour bar. The orbit of each spacecraft is shown by the
white curve and the location of the HCS is marked by the red contour
($B_{r}=0$).

As a final comparison, in Figure 11 we compare in situ measurements with model
results at the location of Earth. This is more representative of model
comparisons in the ecliptic plane, particularly when there are not any strong
equatorial sources of high-speed wind and/or the spacecraft is skirting close
to the HCS. In such cases, an argument can be made that some of the stream
structure is captured, but that the phasing of the streams in time is not
precise. This, however, is a subjective interpretation, requiring support from
other data sources. One such dataset is the magnetic field, which, in this
case, matches reasonably well: most of the interval is associated with
negative polarity field. An excursion into positive polarity field on DOY 42
is well matched by the thermodynamic and polytropic solutions, as well as the
initial descent from positive to negative polarity at the start of the
interval; However, another potentially significant excursion into positive
polarity on DOY 25 is completely missed by all models. Comparisons with the
number density highlight the mismatch in phasing of the modest compression
regions and the overestimates of the base densities for both the thermodynamic
and (to a lesser extent) polytropic solutions and the underestimate by the WTD
model. Overall, this comparison does not provide compelling support that any
of the models have reproduced the large scale structure of the heliosphere for
this interval. On the other hand, we recognize that the spacecraft’s proximity
to the HCS suggests that the model results would necessarily introduce large
uncertainties.

In summary, this detailed comparison between several models and observations
at PSP, Stereo-A, and Earth for each of the first four perihelia passes,
provides a means to evaluate the model results in terms of our confidence in
their ability to reproduce the global structure of the heliosphere. The
results suggest that, in general, the ’thermodynamic’ approach reproduces the
observations most closely, and thus, we advocate that people use this model
from PSI’s website. Additionally, it supports the view that the underestimate
of the magnetic field by the models is a feature that persists at least to
within 26.9 $R_{S}$, and, thus, cannot be resolved by interplanetary
processes. Finally, these comparisons suggest that the model results are most
accurate for P1 and P3.

Based on these comparisons, we can now investigate the global structure of the
inner heliosphere predicted by these models. Given the particularly good
comparisons for P1 and P3, we infer the accuracy of these results is highest,
but that, likely, the large-scale picture drawn from the models for P2 and P4
is also reasonable, although, subject to the caveats noted earlier about the
presence and likely short-time-scale evolution of the active regions during
these intervals. Also, since the thermodynamic solutions consistently
performed better than the other two model approaches we limit our analyses to
these results.

Figure 12 summarizes the radial velocity profiles from the thermodynamic model
for each of the four perihelia at each of the spacecraft. Note that the maps
for Earth and STA are virtually the same, since they are both close to 1 AU.
The latitudinal position of the spacecraft, however, can be quite different,
leading to the striking differences in the in situ comparisons. Since PSP was
also moving in heliocentric distance as well as longitude and latitude, this
slice is taken from the distance of closest approach; thus, it is most
representative of the point of perihelion, which is indicated by the loop in
the white trajectory curve. Perhaps the most visually significant difference
between the maps is the slower speeds seen at PSP, reflecting the fact that
the solar wind is continuing to expand between PSP and 1 AU. Several other
features are worth noting. First, between PSP and Earth (or STA) there is a
general increase in the complexity of the structure. Whereas at PSP, there is
a simpler two-state picture of slow and fast wind, at Earth (or STA), the fast
wind within the latitudinal bands of the HCS has become more isolated, almost
forming islands (or beams in the radial sense) of fast wind. Second, the
structure of the overall band of solar wind variability (Gosling et al.,
1995), that is to say, the region within $\pm 30^{\circ}$ latitude has
increased: Whereas at PSP it was relatively flat, by 1 AU it arcs
substantially more in latitude. Third, the latitudinal gradient in solar wind
speed at PSP has disappeared and wind beyond this band of solar wind
variability is approximately 750 km s-1. Fourth, the underlying reason for the
differences in the timeseries profiles seen in Figures 3 \- 11 between the
three spacecraft can be, at least in part, interpreted as differences in the
latitudinal position of the spacecraft at each of these times. In fact,
generally speaking, Earth and PSP/STA were always separated the most in
latitude, with the former generally in the southern hemisphere and the latter
in the northern hemisphere. Moreover, this suggests that PSP and STA should
have much more similar profiles, at least based on the relatively similar
latitudinal position. As an example, comparison of PSP and Earth for P3
reveals that PSP was immersed in slow solar wind for most of the time, and
particularly at perihelion, whereas Earth partially intercepted a high speed
stream, but was generally embedded within solar wind of more variable speed.

Finally, comparison between PSP and either Earth or STA maps, particularly for
the first and third perihelia, highlights the evolution of stream structure in
moving outward from the location of PSP (a few tens of solar radii) and Earth
(215 $R_{S}$). Note in particular, how the HCS (red trace) becomes distorted,
with regions at highest latitude getting pulled to earlier longitudes in some
locations and stretched to later longitudes at other locations. This is due to
the dynamical effects of the surrounding stream structure. Fast solar wind at
earlier longitudes (smaller heliocentric distances) attempts to overtake
slower wind at later longitudes (farther heliocentric distances), decelerating
the slower wind within which the HCS is embedded. Similarly, when fast solar
wind outruns slower solar wind behind, the current sheet is stretched out in
longitude (as it is embedded within the expansion wave (rarefaction region)
that is created through this process).

The HCS is effectively a surface separating regions of opposite magnetic
polarity, and, although not directly observable, it is the largest coherent
structure within the heliosphere. It is intimately related to the large-scale
dynamical flow of the solar wind, and although passive, it responds to the
dynamics of interplanetary streams and thus provides an indirect measure of
stream evolution. At least during relatively quiescent times (and within, say,
20 AU), corotating interaction regions (CIRs) are organized about the HCS
(Pizzo & Gosling, 1994). The large-scale structure of the HCS is summarized in
Figure 13 for each of the four perihelia passes. In general, we note the
relatively flat orientation of the HCS for all four perihelia passes. These
pictures can be contrasted with the shape of the HCS during more active
conditions, which can span effectively the entire spherical domain (Riley et
al., 2002). More specifically, however, we note that for P1 and P3, there was
a vertical, and hence sharp crossing of the HCS from negative polarity
(southern hemisphere) to positive polarity (northern hemisphere), whereas for
perihelia P2 and P4, any crossings of the HCS were at more inclined angles.
Additionally, the HCS is notably flatter during the interval surrounding P4.
These are consistent with the inferences drawn from Figure 12, and explain why
the model comparisons with observations were generally better for P1 and P3,
for which there were well-defined crossing of the HCS and immersion into
distinct sources of solar wind on either side.

##  4 Conclusions and discussion

In this study we have modeled the global structure of the inner heliosphere
for each of the first four PSP perihelia. We found that our semi-empirical
thermodynamic model consistently produced better matches with observations
than either a less sophisticated polytropic approach, or a more advanced wave-
turbulence-driven model. Our results provided a global perspective from which
to interpret the localized in situ measurements made during P1 through P4 and
connect observations made at PSP with those at 1 AU from Earth-based
spacecraft as well as Stereo-A. We did not find any evidence for the
resolution of the open flux problem at least within the heliocentric distance
reached by PSP during P4 ($\sim 26.9R_{S}$).

While the thermodynamic model results were, in some cases, quite remarkable,
the agreement between model output and observations may be improved upon in a
number of ways. First, and foremost, and as described previously (e.g., Riley
et al., 2012b; Linker et al., 2017; Riley et al., 2019b, a), the model results
are extremely sensitive to the boundary conditions, and, in particular, the
observed photospheric magnetic field. The limitations of these measurements
include: (1) No observations beyond the view afforded by Earth-based
spacecraft, which includes limited inferences on the evolving field on the
backside of the Sun as well as little-to-no reliable information from polar
latitudes; (2) No consensus on the actual “ground truth” values of the
photospheric field (Riley et al., 2014); and (3) Limited availability or
accuracy of nonradial magnetic fields at the base of the model.

Second, and related to this first point, is the lack of availability of
accurate time-dependent boundary conditions.Currently, flux-transport models,
such as ADAPT (Arge et al., 2010), are the likely the best quasi-time-
dependent solution to this problem; however, they are driven by Earth-based
observations of the photosphere. Only with the availability of non-Earth views
of the Sun, or the accurate assimilation far-side data from helioseismic
techniques (e.g., Liewer et al., 2017), can we begin to build truly time-
dependent, synchronic maps of the Sun. With the successful launch of Solar
Orbiter and the availability of observations away from the Sun-Earth line from
the polarimetric and helioseismic imager (PHI) (Solanki et al., 2014), we can
begin to assemble boundary conditions that mitigate and quantitatively assess
the impact of this limitation. Additionally, PFSS solutions, particularly for
PSP encounters is reinforcing the idea that even daily updated quasi-
synchronic maps, such as produced by ADAPT (using either GONG or HMI
observations) can produce significantly better matches in terms of the
observed polarity of the magnetic field measured by FIELDS (Badman et al.,
2020a). Thus, a natural but challenging next step would be to assess how
driving the MHD solutions with a sequence of ADAPT maps affects the quality
and accuracy of the solutions.

Third, the free parameters set in each of the three models have not been
rigorously tested in sensitivity studies. Over the years, we have explored
heuristically how different values might impact specific comparisons with data
(e.g., white-light, EUV/x-ray, in situ); however, no systematic study has been
performed. Moreover, these values were, in some cases, set during periods of
solar activity that was substantially different to the current state of the
corona. For example, the free parameters used to specify the mapping of solar
wind speed along field lines in the corona, to drive the heliospheric boundary
conditions of the polytropic and thermodynamic models were essentially fixed
based on specific (but detailed) analysis of the time period surrounding the
solar minimum of 1996 (Riley et al., 2001). Although they remain reasonable,
based on the comparisons presented here, it is likely that the change in the
overall state of the corona during that past quarter of a century might
require a re-examination of these parameters.

Fourth, and finally, model comparisons with observations may be improved by
incorporating more datasets into the assessment of the model results, as well
as defining and using more quantitative metrics for accessing accuracy.
Currently, we use an ad hoc approach of comparing subsets of the available
data, depending on the specific datasets we are hoping to interpret with the
model results. For example, in eclipse predictions (Mikic et al., 2018), the
focus is to produce the best match with white-light observations of the actual
eclipse. This is, however, at the expense of matching in situ measurements. We
have suggested that an approach that incorporates metrics for all available
metrics, within the framework of a Pareto frontier (Camporeale, 2020), may
optimize model solutions. Of course, it could be that the resulting solutions
are neither the best ones at reproducing coronal observations nor in situ
measurements.

It is noteworthy that, overall, the thermodynamic model appears to be
outperforming both the polytropic and WTD approaches. The polytropic model
represents the most empirically-based technique, while the WTD model is the
most physics-based. For many years, the polytropic model performed best in
comparisons with in situ measurements over a wide range of intervals covering
the space era (e.g., Riley et al. (2001, 2012a, 2012b). The motivation for the
thermodynamic model was, at least in part, to address the limitation that the
polytropic approximation resulted in poor comparisons with white-light
observations. It is not yet clear whether the improved in situ results of the
thermodynamic solutions are due to the overall improvement of parameters in
the model controlling the heating of the corona, the new lower-level of solar
activity that the Sun has entered, or some other phenomena. Either way, it is
encouraging that a more sophisticated model is now capable of providing more
accurate solutions. Of course this requires further, more systematic
comparisons with observations over a broader range of the solar cycle.

Figure 13: Comparison of the shape of the HCS for each of the first four
perihelia passes (P1, P2, P3, and P4). The HCS is approximated by the
isosurface of $B_{r}=0$ from the solar surface to 1 AU. The thermodynamic
solutions were used to estimate the location of the HCS for each perihelia,
and the slight discontinuity visible at $30R_{S}$ is due to the merging of the
coronal and heliospheric solutions.

The primary distinction between the polytropic and thermodynamic solutions
originates in the structure of the magnetic field in the corona, since both
heliospheric solutions are driven by this. Thus, the fact that the
thermodynamic solutions are providing closer matches with in situ measurements
reinforces previous coronal comparisons with white light observations,
particularly during eclipses, which also demonstrated the superiority of the
thermodynamic results. A further refinement to the thermodynamic approach
could be made, bridging the gap with the WTD model, by directly driving the
heliospheric model with the output from the thermodynamic solution. However,
previous explorations of this have resulted in worse comparisons with 1 AU
measurements (Riley et al., 2012b).

The investigation undertaken here was an exploration of several different
modes of operation of several models, to understand, at least qualitatively,
the capabilities of the different approaches. It should not be interpreted as
a systematic parameter phase-space of sensitivity study, which would require a
more careful analysis, changing only one variable at a time and assessing the
impact of that change on the model results and their similarities and
differences with observations. For example, in comparing the
polytropic/thermodynamic solutions with the WTD approach, the biggest
distinguishing factor might be that the maps were processed using different
approximations and had little to do with the underlying physics in the model.
We found that the updated pipeline for the WTD solutions resulted in better
comparisons with white-light observations during the last two total eclipses
in 2017 (Mikic et al., 2018) and 2019 (Linker et al., 2019), and we had
anticipated that this would translate into better comparisons with in situ
measurements. That it didn’t, now requires another suite of runs where we use
the same input maps for all modeling approaches.

Intriguingly, our study showed that, in spite of our initial prediction for
PSP’s first perihelion, there remains a significant deficit in the value of
the radial (and hence total) magnetic field predicted by the models. Our
prediction, which relied on older photospheric magnetic field data, likely
estimated the value of $B_{r}$ poorly, at least in part due to the significant
evolution of the active region located at PSP’s subsolar point as it reached
closest approach. Unfortunately, this conspired in just the right way so that
when the measurements were returned from the spacecraft, the agreement
appeared to be remarkable. Although this hinted at the possibility that the
so-called open flux problem might arise in processes in the solar wind (beyond
several tens of solar radii), it is clear from the results presented here
(requiring a factor of 3 correction to bring the model results into agreement
with the observations) that the problem exists throughout the heliosphere, at
least to within $26.9R_{S}$. Badman et al. (2020b) came to a similar
conclusion through the analysis of PFSS model solutions for encounters 1-5. As
such, the open flux problem must be resolved by processes or uncertainties
occurring closer to the Sun. As outlined by Linker et al. (2017) and Riley et
al. (2019b), several possibilities could, in principle, resolve this mismatch,
and it is likely that a combination of factors play a role. One of the
promising ideas is that an unobserved concentration of magnetic flux lies at
the solar poles and provides the “missing” open flux to the heliosphere (Riley
et al., 2019b). This may, at least indirectly, be supported by the results
here. In particular, the peak values of the polar fields used to drive the WTD
model solution for P1 (Figure 3 (red)) were only 4.7 G, whereas the peak
values in the prediction (Figure 3 (magenta)), while more spatially localized,
reached up to 65 G. Of course other ideas have also been proposed, including
that of Réville et al. (2020), who argued that if the amplitude of the waves
used to heat the corona and accelerate the solar wind are added to the radial
and total magnetic field kinematically, this would bring the model results
into agreement with the measurements. Ultimately, the extent to which polar
flux may resolve this “missing flux” problem will be addressed by Solar
Orbiter. During its extended mission, it will reach $34^{\circ}$ heliolatitude
and such flux, if it exists, should be clearly visible by the Polarimetric and
Helioseismic Imager (PHI) onboard.

Finally, we used the model results to explore the global structure of the
inner heliosphere during each of the first four perihelia. This allowed us to
provide an explanation for the better matches between models and observations
for P1 and P3, due to the sharp fold in the HCS and, hence, more separation of
the spacecraft from the HCS for large parts of these intervals. On the other
hand, during P2 and P4, the spacecraft were often skimming through, or
adjacent to the current sheet, making model predictions very sensitive to the
precise location of the HCS. While current solar wind conditions remain
extremely quiet, as we follow the ascending phase of the solar activity cycle,
such global pictures of the properties and structure in the heliosphere will
become ever more useful for interpreting more complex in situ structure.

This study suggests several possible avenues for future investigations. First,
and as noted above, a natural extension to the modeling pipeline would be to
assess a fully thermodynamic approach, where all the boundary values from the
coronal solution are used to drive the heliospheric model. Based on earlier
comparisons, we anticipate that this would, at least initially, yield worse
comparisons with in situ measurements; however, this remains to be determined.
Second, a systematic sequence of parametric studies, again targeting a smaller
number of intervals, such as those studied here, where model parameters and
boundary conditions are systematically optimized to produce the best matches
with observations. As noted previously (e.g., Riley et al. (2012b, 2019b)),
however, boundary conditions, and, in particular, the radial component of the
magnetic field, likely play a – if not the – crucial role in the quality of
the solution. With the availability of PSP measurements and, most recently,
Solar Orbiter data, together with Stereo-A and Wind/ACE, we can now assess the
fidelity of the solution at multiple points in longitude, and, eventually
latitude, potentially allowing us to estimate what the intrinsic limitations
may be in our ability to reproduce the measurements. Finally, and related to
this, systematic studies aimed at estimating the contribution of polar fields
to in situ measurements, as well as improved models for these values (e.g.,
ADAPT) may further reduce the uncertainty in model predictions, although,
ultimately, the most substantial gains will come from direct measurements of
the photospheric field both from polar regions as well as at different
heliocentric longitudes, that is, simultaneous $4\pi$-steradian coverage of
the Sun’s magnetic field.

###### Acknowledgements.

The authors gratefully acknowledge support from NASA (80NSSC18K0100,
NNX16AG86G, 80NSSC18K1129, 80NSSC18K0101, 80NSSC20K1285, 80NSSC18K1201, and
NNN06AA01C), NOAA (NA18NWS4680081), and the U.S. Air Force (FA9550-15-C-0001).

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