Using Parker Solar Probe observations during the first four perihelia to constrain global magnetohydrodynamic models
Pete Riley, Roberto Lionello, Ronald M. Caplan, Cooper Downs, Jon A. Linker, Samuel T. Badman, Michael L. Stevens
AAstronomy & Astrophysics manuscript no. riley-PSP-obs-model-comp-2020-v5 © ESO 2021February 11, 2021
Using Parker Solar Probe observations during the first fourperihelia to constrain global magnetohydrodynamic models
Pete Riley , Roberto Lionello , Ronald M. Caplan , Cooper Downs , Jon A. Linker , Samuel T. Badman , andMichael L. Stevens Predictive Science Inc., San Diego, California, USA e-mail: [email protected] Physics Department, University of California, Berkeley, CA 94720-7300, USA Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA Smithsonian Astrophysical Observatory, Cambridge, MA 02138, USAReceived 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 solaratmosphere with each solar encounter. Each set of observations provides a unique opportunity to test and constrain global models ofthe 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 sophisticationfor 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 fluxproblem’, which all global models su ff er from, resolves itself at closer distances to the Sun. Methods.
The global structure of the corona and inner heliosphere is calculated using three di ff erent MHD models. The first model(“polytropic”), replaced the energy equation as a simple polytropic relationship to compute coronal solutions and relied on an adhoc 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-empiricalapproach to specify the boundary conditions of the heliospheric model. The third model (“WTD”) further refines the transport ofenergy through the corona, by implementing the so-called wave-turbulence-driven approximation. With this model, the heliosphericmodel was run directly with output from the coronal solutions. All models were primarily driven by the observed photosphericmagnetic field using data from Solar Dynamics Observatory’s Helioseismic and Magnetic Imager (HMI) instrument. Results.
Overall, we find that there are substantial di ff erences between the model results, both in terms of the large-scale structureof 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 forall four encounters. Our results also contradict an earlier study that had hinted that the open flux problem may disappear nearer theSun. Instead, our results suggest that this “missing” solar flux is still missing even at 26 . R S , and thus it cannot be explained byinterplanetary processes. Finally, the model results were also used to provide a global context for interpreting the localized in situmeasurements. Conclusions.
Earlier studies suggested that the more empirically-based polytropic solutions provided the best matches with observa-tions. The results presented here, however, suggest that the thermodynamic approach is now superior. We discuss possible reasons forwhy this may be the case, but, ultimately, more thorough comparisons and analyses are required. Nevertheless, it is reassuring that amore sophisticated model appears to be able to reproduce observations since it provides a more fundamental glimpse into the physicalprocesses 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 Au-gust 2018 and reached its first of 24 perihelia (P1) on 5 Novem-ber 2018. Since then it has successfully completed six perihelionencounters (as of 27 September 2020), with ever decreasing dis-tances of closest approach. PSP’s primary scientific goals are to:(1) better understand what heats the solar corona and acceleratesthe solar wind; (2) determine the underlying structure and dy-namics of the coronal magnetic field; and (3) better identify themechanisms that accelerate and transport energetic particles inthe corona (Fox et al. 2016).PSP carries four instrument packages. Of these, two areparticularly relevant for studying the large-scale magnetic and plasma properties of the solar corona and inner heliosphere.FIELDS (Electro- magnetic Fields Investigation) consists of twoflux-gate magnetometers, a search-coil magnetometer, and fiveplasma voltage sensors (Bale et al. 2016). It measures elec-tric and magnetic fields, as well as radio waves, Poynting flux,plasma density, and electron temperature. Solar Wind ElectronsAlphas and Protons (SWEAP) is composed of three instruments:two electrostatic analyzers and one Faraday cup, from which es-timates of velocity, density, and temperature of electrons, pro-tons, and alpha particles can be made (Kasper et al. 2016).Global models of the solar corona and inner heliosphere canprovide crucial support for interplanetary missions (e.g., Rileyet al. 2001; Török et al. 2018). Not only do they provide a globalpicture of the properties and structure of the heliosphere, but
Article number, page 1 of 17 a r X i v : . [ phy s i c s . s p ace - ph ] F e b & A proofs: manuscript no. riley-PSP-obs-model-comp-2020-v5 they allow observers to connect the otherwise disparate obser-vations to one another. In addition to computing basic magneticand plasma variables, models can also be used to reconstructconvolved measurements, such as extreme ultraviolet (EUV) andwhite light images (Mikic et al. 2018).A number of numerical models have been developed and ap-plied to help in the interpretation of PSP observations. Poten-tial Field Source Surface (PFSS) models represent the simplestapproach, where: (1) the corona is assumed to be current free;(2) time-dependent e ff ects are ignored (or treated as a series ofquasi-equilibria); and (3) a source surface outer boundary condi-tion is imposed, where the field lines are forced to become radial(typically set to be 2 . R S (e.g., Badman et al. 2020a)). In spiteof their inherent simplicity, PFSS models have been shown tocompare well with MHD solutions, at least under some condi-tions (Riley et al. 2006). Badman et al. (2020a) demonstratedthe utility of applying PFSS models to interpret the magneticconnectivity between PSP and the solar surface, and, in partic-ular, in being able to predict the location of these foot-points.Of course, the PFSS approach is limited to inquiries concerningthe magnetic structure, and cannot directly address the plasmaproperties within the corona or further out.van der Holst et al. (2019) applied the Alfvén Wave Solaratmosphere Model (AWSoM) to predict the in situ measure-ments that would be returned from PSP during P1. In this model,outwardly propagating low-frequency Alfvén waves, which arepartially reflected, provide both the coronal heating and accel-eration. Although no comparisons were made with PSP in thatstudy, a visual comparison with OMNI data at 1 AU, suggeststhat the model had captured the large-scale stream structure ofthe solar wind. A qualitative comparison with the predictionsand subsequent observations at PSP supports the general state ofthe solar wind (slow, dense) at perihelion, although the modeststructure that was apparent was not reproduced by the model.Moreover, the model underestimated the strength of the mag-netic field. In this case, they relied on ADAPT-GONG maps,which include a more modest 1.85 correction factor to amplifythe photospheric magnetic fields. This need to boost either thephotospheric magnetic field boundary conditions, or the result-ing model fields at 1 AU, is a consistent requirement for allglobal 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 heliospherefor P1. The model was driven by photospheric magnetic fieldobservations several weeks prior to the encounter (Riley et al.2019a), thus, it was a true prediction. This served as both as a testof our model’s predictive capabilities as well as what we hopedwould be an aid in mission planning. For example, it wouldinform ground-based observers where they should target theirsmaller-scale campaign observations, based on our predictionsof the foot-point locations during perihelion. We inferred that,in the days prior to first encounter, PSP would be immersed inwind emanating from a well-established, positive-polarity north-ern polar coronal hole. During the encounter, however, field linesfrom the spacecraft would map to a negative-polarity equatorialcoronal hole, within which it would remain for the entire en-counter, before becoming magnetically connected to a positive-polarity equatorial coronal hole. In that study, we also comparedMHD and PFSS predictions, noting that while there are an over-all agreement in the forecasts, there were some notable di ff er-ences. The equatorial coronal holes, for example, predicted bythe PFSS solutions were substantially smaller than those inferredfrom the MHD model results. Most recently, Réville et al. (2020) modified the astrophys-ical 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 de-scribed studies (Riley et al. 2019a; van der Holst et al. 2019) butsimpler in that the Alfvén reflection process was not explicitlyincluded. Nevertheless, they found that, at least for P1, the large-scale amplitude of the plasma parameters were well reproducedby the model. Although the modeled values of the magnetic fieldstrength could be interpreted as being an underestimate, they ar-gued that, in fact, if the amplitude of the waves used to heat thecorona and accelerate the solar wind are added post hoc to the ra-dial and total magnetic fields, this would bring the model resultsinto agreement with the observations. While intriguing, this re-quires more substantiation through comparisons with subsequentperihelia, at di ff erent spacecraft locations, and with simultaneouscomparison with other observational metrics, such as EUV andwhite-light observations,Finally, bridging the gap between the fully MHD and PFSSapproaches, Kim et al. (2020) presented model results for each ofthe first three PSP perihelia using a model composed of a coro-nal PFSS model, connected to a heliospheric MHD model. Theyused the Wang-Sheelely-Arge (WSA) methodology for prescrib-ing the solar wind speed at the inner boundary of the heliosphericmodel (Arge et al. 2003), which gives largely similar results tothe Distance from the Coronal Hole (DCHB) method that wehave developed (Riley et al. 2001, 2015). The most significantdisagreements arise in the vicinity of pseudo-streamers, wherethe WSA model predicts slow solar wind, in contradiction to ob-servations.In this study, we compare MHD results from three distinctapproaches with in situ measurements made by PSP, Stereo-A,and Earth-based spacecraft (ACE and Wind) in an e ff ort to iden-tify any systematic di ff erences between the model results. Basedon this, we then use the best model results to infer the globalstructure of the heliosphere during each of the first four peri-helia encounters. Additionally, we highlight how sensitive thetimeseries comparisons are to the precise trajectory of the space-craft through the model solution, suggesting that even when dif-ferences exist, the overall global structure may still be reason-ably accurate. Finally, given PSP’s ever smaller point of closestapproach we interpret the comparisons in terms of whether an“open flux problem” remains.
2. Methods
Data used to drive the model were obtained from the Helioseis-mic and Magnetic Imager (HMI) onboard the Solar DynamicsObservatory (SDO) spacecraft (Scherrer et al. 2012), and, specif-ically, from jsoc.stanford.edu / ajax / exportdata.html. These dataprovide an estimate for the radial component of the field on auniform grid size of 3600 × Article number, page 2 of 17iley et al.: Comparing PSP observations with MHD models
Fig. 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 processedmagnetograms used to drive the polytropic and thermodynamic models, and in the right column are the synoptic maps used to drive the WTDmodel. Blue / red corresponds to negative / positive polarities, respectively, and the maps have been saturated at ±
10 G; thus, some of the ARs suchas during CR2215, contain field strengths larger than this maximum. rently use generate similar but not identical processed maps.The middle column relies on our simpler, but mature tech-nique that is used to develop our online, standard solutions (e.g., ). Flux is bal-anced 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 de-veloping input magnetograms for the MAS code. It attempts tomaintain more structure , particularly at higher latitudes. Perhapsmost importantly, for the maps for the polytropic and thermody-namic runs, the poles are filled in by extrapolation of the mid-latitude fields, while for the WTD runs we use the pole-filleddata provided by the HMI instrument team. For our purposes,however, the main point is that they serve as di ff erent, but poten-tially equally-valid drivers of the MAS code.In this study, we use in situ measurements from ACE andWind, in the form of the merged OMNI dataset as well as from Stereo-A, and, of course PSP. From each spacecraft we comparesolar wind bulk speed, proton number density, and radial mag-netic field. Together, these three quantities characterize the basicdynamical properties of the solar wind. All data were obtainedfrom NASA’s Space Physics Data Facility (SPDF) web servicesAPI (e.g., Candey et al. 2019), and were retrieved at 1-hour res-olution. In this study, we use PSI’s MAS (Magnetohydrodynamic Algo-rithm outside a Sphere) code, which solves the usual set of re-sistive MHD equations in spherical coordinates on a nonuniformmesh. The details of the model have been described elsewhere(e.g., (Miki´c & Linker 1994; Riley et al. 2001; Lionello et al.2001; Riley et al. 2012c; Mikic et al. 2018; Miki´c et al. 2018;Caplan et al. 2019)). Here, we restrict our description to several
Article number, page 3 of 17 & A proofs: manuscript no. riley-PSP-obs-model-comp-2020-v5
Fig. 2.
Radial speed profiles in the solar equatorial plane for each model (polytropic, thermodynamic, and WTD) for each perihelion encounter1-4. Superimposed are the corotating trajectories of the spacecraft during each encounter, expanded by 10 days on either side of the nominalencounter dates. The approximate location of the spacecraft at the time of PSP’s point of perihelion pass is indicated by the location of eachspacecraft label.Article number, page 4 of 17iley et al.: Comparing PSP observations with MHD models relevant points. First, the model is driven by the observed photo-spheric magnetic field. We use HMI observations from the SDOspacecraft to construct a boundary condition for the radial mag-netic field at 1 R S as a function of latitude and longitude. Second,the model is run in two stages: First the region from 1 − R S is modeled, followed by the region from 30 R S to 1 AU, beingdriven directly by the results of the coronal calculation. Compu-tationally, this approach is much more e ffi cient. Third, for thesesolutions, we use a single map to cover an entire solar rotation,thus, although the model is time-dependent, it is run forward intime until a dynamic steady-state is achieved. This is a reason-able approximation when structure at the Sun is not appreciablychanging from one rotation to the next. However, during inter-vals 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 impulsiveevents (Pulupa et al. 2020). Fourth, MAS relies on a variety ofapproximations to reconstruct (or predict) the large-scale struc-ture and properties of the solar corona and inner heliosphere. Inorder of increasing complexity (and historical development), werefer to them as the ‘polytropic’, ’thermodynamic’ and ’WTD’models.The polytropic approximation solves the usual set of MHDequations in spherical coordinates with the energy equation be-ing approximated by a simple adiabatic approximation (that is,we set all energy source terms to zero). This requires us tochoose a polytropic index, γ = .
05 to reflect the near-isothermalnature of the corona. In the solar wind, it is set to 1.5. These sim-plifications result in a fast, robust code that reproduces the struc-ture of the magnetic field reasonably well, but fails to generatesolutions with su ffi cient variation in solar wind speeds or den-sities. To address this, we use an empirically-based approach,DCHB, to specify the solar wind speed at the inner boundaryof the heliospheric code (Riley et al. 2001). Although semi-empirical, it generally produces results that match 1 AU obser-vations as good as, or better than those computed using the WSAapproximation (Riley et al. 2015).The thermodynamic approximation replaces the polytropicassumption with an empirically-based treatment of energy trans-port processes (radiation losses, heat flux, and coronal heat-ing) in the corona (Lionello et al. 2001; Lionello et al. 2009).In this case, γ now returns to a more defensible value of .Development of this model focused on improving the densityand temperature structure in the solar corona through compar-isons with EUV and X-ray images from a variety of space-craft. Relatively little direct comparison was performed within situ measurements, Thus, we also implement the DCHB ap-proximation to derive the heliospheric boundary conditions fromthe thermodynamic solution. In this sense, the model is notstrictly – or fully – thermodynamic, and should strictly be la-beled the “semi-empirical thermodynamic” model. When wehave produced complete end-to-end coronal-heliospheric ther-modynamic solutions, we have found that they cannot reproducethe observed structure in the solar wind at 1 AU, even thoughthey correctly match the observed amplitude of the quantities.Finally, the Wave-Turbulence-Driven (WTD) model is a gen-eralization of the thermodynamic approach by self-consistentlyheating the corona and using the WKB approximation for wavepressures, providing the necessary acceleration of the solar wind(Miki´c et al. 2018). The physical motivation for this heatingmodel is that outward and reflecting Alfvén waves interact withone another, resulting in their dissipation, and heating of thecorona (Zank et al. 1996; Verdini & Velli 2007). And, while theWTD model is significantly more physics-based than, say, the polytropic model, it should be noted that it does require a care-ful choice of two free parameters. We have found that this ap-proach can account for both the acceleration of solar wind alongopen field lines, as well as the heating of plasma entrained withinclosed-field regions (Lionello et al. 2014; Downs et al. 2016);however, again, relatively little exploration of its abilities to re-produce 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) thermodynamicmodel; and (3) the WTD model. Although it is conceptually con-venient to think of them as distinct approaches, in reality, theyrepresent snapshots of a continually evolving model; a modelthat has been developed over ∼
25 years and one that has beenused to interpret a disparate set of observations at di ff erent stagesduring this evolution. In particular, the polytropic model hasbeen most extensively compared against in situ measurements(e.g., Riley et al. 2001; Riley et al. 2012b; Riley et al. 2015)while the WTD model has been almost exclusively comparedagainst remote solar observations, and, in particular, white-lightand EUV images (e.g., Mikic et al. 2018; Linker et al. 2019).
3. Results
Before beginning a detailed comparison of the timeseries mea-surements with the model results, it is instructive to explore thelarge-scale structural di ff erences predicted by each of the mod-els. Figure 2 summarizes the radial speed of the solar wind from1 R S to 1 AU for each of the models and for each of the firstfour PSP encounters (from October 2018 through March 2020).These are displayed in Carrington coordinates with φ = ff er-ent configuration of the three spacecraft. Finally, note that, unlikean inertial projection of spacecraft trajectories, in the corotatingframe, the spacecraft travel in a clockwise direction, thus, forexample, 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 solarwind at perihelion (where it hovers at approximately the sameCarrington longitude while performing a ‘loop’), then exits theperihelion as it encounters a significant high-speed stream.These maps emphasize the di ff erences in the structure of thesolar wind predicted by each model for each encounter. In gen-eral, the polytropic model produces stronger high-speed streamsin the equator than either of the other two models. Additionally, Article number, page 5 of 17 & A proofs: manuscript no. riley-PSP-obs-model-comp-2020-v5
Fig. 3.
Comparison of model results with PSP in situ measurements during encounter 1. From top to bottom: Speed, radial magnetic field, andnumber 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, weremultiplied by a factor of three. In the bottom panel, heliocentric distance (cyan) and Carrington longitude (magenta) are shown as a function oftime. the WTD model tends to produce broader (in longitude) streamswith 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 quitesimilar. Based on this, it is likely that the band of solar windvariability for the polytropic model is narrower than the thermo-dynamic model.To assess the quality of the model results, we compare datafrom PSP, Earth-based spacecraft (ACE and Wind, through theOMNI dataset), and Stereo-A for each of the first four perihe-lia passes (labeled P1 through P4). Beginning with P1, Figure 3compares modeled and observed radial velocity, radial mag-netic field, and number density for the interval from day-of-year(DOY) 294 through 325. This includes the nominal 12 days thatbracket the date of closest approach (from DOY 304 to 315) butadds an additional 10 days on either side to define an intervalthat is marginally longer than a solar rotation period. Perihelionoccurs near the center of the panel (DOY 310, or 06 Novem-ber 2020), coinciding with the central portion of the interval,and associated with a positive increase in longitude with respectto time (bottom panel). During the interval from approximatelyDOY 303 to 316, PSP remained at roughly the same Carring-ton longitude ( ∼ ◦ ), which, at the time was an unprecedentedposition 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 − solar wind, only ris-ing substantially beyond this on DOY 319. These variations arewell captured by the thermodynamic model results; however, theother models fail to reproduce these basic variations. The WTDmodel, while reasonably predicting an average and unchangingspeed of ∼
350 km s − does not capture any of the structurewithin the slow stream, nor does it predict the rise in speed laterin the interval. The polytropic solution qualitatively matches thevariations from slower to faster wind, but the amplitude is far toolarge, predicting speeds approaching 700 km s − at the peak ofthe small stream (DOY 313). Finally, the actual prediction madeby Riley et al. (2019a) is grossly inaccurate (magenta line), witha forecast of constant, 600 km s − solar wind throughout most ofthe interval.Similar inferences can be made for the comparisons with theradial magnetic field, B r . It should be emphasized that the val-ues for the polytropic, thermodynamic, and WTD models havebeen multiplied by a factor of 3, which reflects the well-knownproblem with global models that, while they are able to capturevariations in the field strength, they significantly underestimatetheir amplitude (Riley et al. 2012b; Linker et al. 2017; Riley et al.2019b). This factor is also the same as used by other global mod-elers relying on HMI data to drive their simulations (e.g., van der Article number, page 6 of 17iley et al.: Comparing PSP observations with MHD models
Fig. 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 SPDFOMNI dataset.
Holst et al. 2013). Intriguingly, and a point we will return tolater, the solution used to make the prediction (magenta) wasnot multiplied by any corrective factor. From these comparisons,we infer that the prediction, polytropic, and thermodynamic so-lutions appear to have captured the large-scale variations in themagnetic field, and, in particular, the immersion into a negativepolarity field for the entire interval, only switching to a positivepolarity around DOY 318. Although not shown here, the originof this wind was an equatorial coronal hole (Riley et al. 2019a;Badman et al. 2020a). Both the polytropic and thermodynamicsolutions suggest some structure in the field around DOY 308that is not reflected in the observations.Finally, we consider density. The third panel of Figure 3compares the model / prediction results with the observed den-sity during the first encounter. Both the prediction and the WTDmodel fail to capture the substantial increase in density as PSPswept into 35.7 R S . In fact, given that all four model resultswere approximately the same on DOY 303 and again on DOY314, this further suggests that the WTD and prediction resultswere actually regions of lower intrinsic density (a fact confirmedby plots of scaled density, not shown here but presented for theprediction by Riley et al. (2019a)). Of the two models that cap-ture both the peak at perihelion and the subsequent high-densityregion following it (DOY 315-319), the thermodynamic results(blue) most closely match the observations, although the maxi-mum 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 wecompare the same three parameters at the location of Earth, us-ing data from the OMNI dataset (composed of measurementsfrom both ACE and Wind). During its passage from the loca-tion of PSP to Earth, the streams have evolved, steepening wherefast wind is overtaking slower wind ahead, and expanding whereslower wind is trailing faster wind. The resulting “sawtooth” pro-file in radial velocity, covering slightly more than a Carringtonrotation, is typical of ambient solar wind conditions in the ab-sence of transient phenomena. In this specific case, there are twohigh-speed streams beginning on DOY 309 and 314. This profileis captured well by both the polytropic (green) and the thermo-dynamic (blue) models. Of these, the thermodynamic results areagain a closer match with observations, mimicking the size andstructure of the streams better. Note, in particular, that the firststream has a sharp rise, followed by an immediate decay, whilethe second one exhibits a flat top before decaying, indicating thatthe spacecraft was immersed more fully in an equatorial coronalhole (likely the result of an equatorward extension to the polarcoronal hole - see later). Again, the WTD results – both from theprediction and the retrospective run – do not match the observedstructure of the solar wind speed.Considering next the radial magnetic field, the observationssuggest that the spacecraft skimmed along the heliospheric cur-rent sheet (HCS) for most of the rotation, dipping into a posi-tive polarity region around DOY 307 and into a negative polarity Article number, page 7 of 17 & A proofs: manuscript no. riley-PSP-obs-model-comp-2020-v5
Fig. 5.
Same as Figure 3, but for PSP data / model comparisons during P2. region on DOY 314. Only the WTD results miss these sectorcrossing. We note again, however, that the predicted values havenot been corrected by the factor of three that was applied to theWTD, polytropic, and Thermodynamic solutions.Comparison of the plasma density with the measurementssuggests that the broad variations have been captured by boththe polytropic and thermodynamic models. However, both mod-els appear to overestimate the size of the compression regions,driven by the fast streams overtaking slower streams. This is par-ticularly true for the two major streams on DOY 309 and 314.This is likely due to the fact that the speed gradients and over-all amplitude predicted by the models were substantially higherthan those observed. Additionally, the polytropic solutions sys-tematically predict an earlier arrival of the high-speed streamsthan was observed.Comparison of Figures 3 and 4 appears to show that thereis more large-scale structure at 1 AU than at ∼ R S . However,it is important to note that whereas the same temporal interval atEarth translates into more than 360 ◦ , at PSP, it is approximatelyhalf of this (180 ◦ ), owing to the spacecraft’s acceleration intoperihelion and deceleration out of it.Next we consider P2. Figure 5 summarizes the same mea-surements and model results (with the exception of the predic-tion results). The solar wind measured by PSP was very qui-escent: slow, dense, and somewhat variable plasma emanatingfrom a negative polarity region on the Sun. During the entireinterval, 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 weeksof measurements came from a Carrington longitude range ofless than 20 ◦ or less, centered near 0 ◦ . In terms of solar windspeed, the thermodynamic and WTD models best match the ob-served low speed. But, while the actual values are overestimated,only the thermodynamic model appears to have captured thevariations, that is, the stream structure. The polytropic modelis patently wrong. From the perspective of the radial magneticfield though, both the thermodynamic and polytropic solutionscapture the global features of the field in terms of amplitudeand direction. The thermodynamic model erroneously suggestsa small polarity reversal starting on DOY 85, but both results arevastly better than the WTD results. Finally, comparing densityprofiles, once again, the thermodynamic model reproduces thelarge-scale variations observed in the plasma data, although itsomewhat overestimates them.Thus far, we have limited ourselves to a qualitative compar-ison between the model results and the observations. Quantita-tive estimates, such as mean absolute error (MAE) or correlationcoe ffi cient (CC) have their place, particularly for space weatherapplications, but can result in inferences that do not match oursubjective interpretation (e.g., Owens et al. 2005; Riley et al.2015; Riley et al. 2013), particularly when scientific understand-ing is the goal. For example, a model that produces a constantvelocity at say, 400 km s − , would produce a lower MAE thanone that reproduced the structure of the solar wind streams,but not their exact phasing in time. Clearly the latter model isof more scientific value (Riley et al. 2017). To better illustrate Article number, page 8 of 17iley et al.: Comparing PSP observations with MHD models
Fig. 6.
Same as Figure 3, but for model and data comparisons at the location of Stereo-A and for P2. this, we computed CCs for each of the model results againsttheir observed values for P2 at PSP (Figure 5). The results wereas 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 quan-titative estimates suggest that, in terms of speed, the WTD per-forms slightly better than the thermodynamic solution. However,practically speaking, based on these results, none of the mod-els shows any significant correlation with the observations. TheCCs for the radial field match our qualitative interpretation, withboth the polytropic and thermodynamic models outperformingthe WTD results. Finally, although our subjective inference thatthe thermodynamic model substantially outperformed the othertwo, in terms of density, this is not captured by the CC values,where the CC for the WTD solution (0.625) does not accuratelyreflect its lack of agreement in terms of structure, as comparedto the thermodynamic solution (CC = / model results withthose at Stereo-A (Figure 6) leads to similar statements as for theP1 comparison. Here, it is even more apparent that Stereo-A wasskimming along the HCS for the entire interval (as indicated bythe radial field values ‘hugging’ a value of zero). When space-craft are so situated, this makes comparisons with models evenmore precarious, as small shifts in latitude can lead to substantialchanges 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 thatthe thermodynamic solution appears to more closely match theobservations; 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 thethermodynamic model. This may appear paradoxical because thedi ff erences in speed between the slow and fast streams are largerfor the polytropic results than the thermodynamic results. How-ever, the reason is that the base density in the slow solar windof the thermodynamic model is substantially higher. Thus, thecompression of an already denser medium by a relatively smallerhigh-speed stream leads to a larger compression region (as mea-sured by peak density) than the compression of a more tenuousregion, even if the fast wind compressing it is substantially faster.This is reinforced by comparing with Figure 4, which showssimilar e ff ects. In general, the thermodynamic compression re-gions are significantly larger than are observed, suggesting thatthe model could be improved by setting the base number densityfor the slow wind, which is a model parameter, to a lower value.Turning next to P3, in Figure 7 we compare radial speed, ra-dial magnetic field, and number density once again with the threemodel results. Unfortunately, during this perihelion some of theplasma measurements were unavailable. Nevertheless, what wasrecovered, again, demonstrates that the thermodynamic solutionsmost closely match the data, although again, the base numberdensity is too high. Note, in particular, the generally low speedprior to the perihelion, and the large high-speed stream on DOY63, which the thermodynamic model captures well both in terms Article number, page 9 of 17 & A proofs: manuscript no. riley-PSP-obs-model-comp-2020-v5
Fig. 7.
Same as Figure 3, but for model and data comparisons at the location of PSP and for P3. of phase and amplitude. This, comparison, in turn, suggests apotentially interesting application for the models: The thermo-dynamic solution may be a reasonable proxy for the unobserveddata during the actual perihelion portion of the mission. This isalso supported by the reasonable match between the model re-sults and the observations of the radial component of the mag-netic field shown in the second panel, as well as comparisonswith data at 1 AU, which we turn to next.Perihelion 3 at Earth looked quite di ff erent (Figure 8). Al-though there was a single large stream of ∼
700 km s − this oc-curred some 20 days prior to the arrival of the stream at PSP.Although it is tempting to associate this stream with the sin-gle high-speed stream observed at PSP, inspection of Figure 2shows that, at least in the equatorial plane, there were two possi-ble candidates for the stream. To assess this more carefully, wenote that at the time of PSP’s perihelion, Earth was located atapproximately 232 ◦ Carrington longitude, which coincided withthe arrival of the second and more significant high-speed stream.Additionally, referring back to Figure 7, we can infer that the ob-served high-speed stream at PSP was probably the second of thetwo, based on the thermodynamic profile showing two peaks,with the second one being more pronounced and aligning wellwith the measurements. Unfortunately, we cannot be certain be-cause of the data gap. Nevertheless, based on the global perspec-tive provided by the model, it is reasonable to infer that the high-speed stream observed at PSP on DOY 263 was the same oneobserved at Earth on DOY 243, one solar rotation earlier. More generally, we can make several comments about thecomparisons between models and observations. First, althoughthe single high-speed stream is produced by both the polytropicand thermodynamic models, little of the structure between thisand the next stream on DOY 268 is captured. Similarly, onlysome of the structure during the first half of the interval matches.On the other hand, the two-sector pattern shown in the radialmagnetic field is reproduced, with the spacecraft remaining pre-dominantly on the positive side of the HCS. All models, tovarying degrees, capture this sector structure, with the thermo-dynamic model performing best. Finally, the modeled densitiesduring this interval are systematically either too high (polytropicand thermodynamic) or too low (WTD), which is related to thefact that the modeled speeds are systematically too low or high,respectively.The comparison between model results for P3 at Stereo-Ais considerably better (Figure 9). There were five / six high-speedstreams of various strengths during this two-rotation interval, al-most all of which were captured by the thermodynamic model.This is reinforced by the density measurements, for which thecompression regions (peaks in density) match with the obser-vations, albeit in most cases being too strong. The polytropicand WTD solutions did not reproduce this structure with anydegree of fidelity, and, as with the Earth comparison, tendedto over-estimate or under-estimate the densities. Unfortunately,magnetic field data was only available for a portion of this in-terval. Nevertheless, the thermodynamic solution, in particular,matches most of the large-scale variations, and, in particular, the Article number, page 10 of 17iley et al.: Comparing PSP observations with MHD models
Fig. 8.
Same as Figure 3, but for model and data comparisons at the location of Earth and for P3. transition from a toward to an away sector on DOY 237. Basedon the close match between the thermodynamic solution and theplasma measurements at Stereo-A, it would not be unreasonableto infer the unobserved large-scale magnetic structure from thethermodynamic model results, and, in particular, the transitionfrom negative to positive polarity occurring rapidly and stronglyon DOY 264. A final point that is worth reinforcing is that,again, while the properties of the plasma speeds are remarkablywell captured by the thermodynamic model, the modeled den-sity peaks in the compressions are significantly larger than wereobserved. This, again, we believe is a result of setting the bound-ary 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 duringthe first two months of 2020. Figure 10 contrasts the model re-sults with the observations. In general, we remark that the overallfeatures observed in the observations are matched by the ther-modynamic solutions. These include: (1) generally slow (300-400 km s − ) solar wind; (2) peak densities, likely exceeding 800cm − ; and (3) peak radial fields reaching almost -150 nT. If themodel results are an accurate predictor of what PSP would haveobserved later, we might predict the appearance of a high-speedstream (700 km s − ) on DOY 51. However, given the limiteddata and discrepancies, this would be a tentative conclusion atbest. These discrepancies include: (1) the earlier rise in numberdensity in both the thermodynamic and polytropic solutions thanwas observed; (2) the failure to capture the switch from negativeto positive magnetic polarity on DOY 32; and (3) an apparent discontinuous change in the speed and density of the polytropicand thermodynamic solutions close to perihelion. This last dis-agreement reveals an interesting limitation of the models, onlyappearing because of the distance of closest approach during P4.For our standard web-based model solutions, the boundary sep-arating the coronal and heliospheric models was set to 30 R S . Forthe previous encounters, this did not impact the simulated space-craft fly-throughs since they always occurred within the domainof the heliospheric model. For P4, however, PSP’s closest ap-proach was 26.9 R S , which is inside this boundary. Since both thepolytropic and thermodynamic models are not seamless at thisboundary, artifacts are introduced when the simulated spacecraftis flown through the merged solution. This is particularly notice-able for the polytropic solution, for which the coronal model can-not adequately calculate the speeds and densities of the plasma.And, while the thermodynamic solution is better, because we didnot drive the heliospheric model directly with the coronal solu-tions, discontinuities are inevitably introduced. Accepting theselimitations, however, we remark that, again, the thermodynamicmodel appears to estimate the densities and speeds best. For themagnetic field, such discontinuities are not as significant, sincethe field at the outer boundary of the coronal solution is useddirectly to drive the inner boundary of the heliospheric model.As a final comparison, in Figure 11 we compare in situ mea-surements with model results at the location of Earth. This ismore representative of model comparisons in the ecliptic plane,particularly when there are not any strong equatorial sources ofhigh-speed wind and / or the spacecraft is skirting close to the Article number, page 11 of 17 & A proofs: manuscript no. riley-PSP-obs-model-comp-2020-v5
Fig. 9.
Same as Figure 3, but for model and data comparisons at the location of Stereo-A and for P3.
HCS. In such cases, an argument can be made that some of thestream structure is captured, but that the phasing of the streams intime is not precise. This, however, is a subjective interpretation,requiring support from other data sources. One such dataset isthe magnetic field, which, in this case, matches reasonably well:most of the interval is associated with negative polarity field. Anexcursion into positive polarity field on DOY 42 is well matchedby the thermodynamic and polytropic solutions, as well as theinitial descent from positive to negative polarity at the start of theinterval; However, another potentially significant excursion intopositive polarity on DOY 25 is completely missed by all models.Comparisons with the number density highlight the mismatchin phasing of the modest compression regions and the overes-timates of the base densities for both the thermodynamic and(to a lesser extent) polytropic solutions and the underestimateby the WTD model. Overall, this comparison does not providecompelling support that any of the models have reproduced thelarge scale structure of the heliosphere for this interval. On theother hand, we recognize that the spacecraft’s proximity to theHCS suggests that the model results would necessarily introducelarge uncertainties.In summary, this detailed comparison between several mod-els and observations at PSP, Stereo-A, and Earth for each ofthe first four perihelia passes, provides a means to evaluate themodel results in terms of our confidence in their ability to repro-duce the global structure of the heliosphere. The results suggestthat, in general, the ’thermodynamic’ approach reproduces theobservations most closely, and thus, we advocate that people use this model from PSI’s website. Additionally, it supports the viewthat the underestimate of the magnetic field by the models is afeature that persists at least to within 26.9 R S , and, thus, cannotbe resolved by interplanetary processes. Finally, these compar-isons suggest that the model results are most accurate for P1 andP3. Based on these comparisons, we can now investigate theglobal structure of the inner heliosphere predicted by these mod-els. Given the particularly good comparisons for P1 and P3, weinfer the accuracy of these results is highest, but that, likely, thelarge-scale picture drawn from the models for P2 and P4 is alsoreasonable, although, subject to the caveats noted earlier aboutthe presence and likely short-time-scale evolution of the activeregions during these intervals. Also, since the thermodynamicsolutions consistently performed better than the other two modelapproaches we limit our analyses to these results.Figure 12 summarizes the radial velocity profiles from thethermodynamic model for each of the four perihelia at each ofthe spacecraft. Note that the maps for Earth and STA are virtu-ally the same, since they are both close to 1 AU. The latitudinalposition of the spacecraft, however, can be quite di ff erent, lead-ing to the striking di ff erences in the in situ comparisons. SincePSP was also moving in heliocentric distance as well as longi-tude and latitude, this slice is taken from the distance of clos-est approach; thus, it is most representative of the point of per-ihelion, which is indicated by the loop in the white trajectorycurve. Perhaps the most visually significant di ff erence betweenthe maps is the slower speeds seen at PSP, reflecting the fact that Article number, page 12 of 17iley et al.: Comparing PSP observations with MHD models
Fig. 10.
Same as Figure 3, but for model and data comparisons at the location of PSP and for P4. the solar wind is continuing to expand between PSP and 1 AU.Several other features are worth noting. First, between PSP andEarth (or STA) there is a general increase in the complexity ofthe structure. Whereas at PSP, there is a simpler two-state pictureof slow and fast wind, at Earth (or STA), the fast wind within thelatitudinal bands of the HCS has become more isolated, almostforming islands (or beams in the radial sense) of fast wind. Sec-ond, the structure of the overall band of solar wind variability(Gosling et al. 1995), that is to say, the region within ± ◦ lati-tude has increased: Whereas at PSP it was relatively flat, by 1 AUit arcs substantially more in latitude. Third, the latitudinal gradi-ent in solar wind speed at PSP has disappeared and wind beyondthis band of solar wind variability is approximately 750 km s − .Fourth, the underlying reason for the di ff erences in the timeseriesprofiles seen in Figures 3 - 11 between the three spacecraft canbe, at least in part, interpreted as di ff erences in the latitudinal po-sition of the spacecraft at each of these times. In fact, generallyspeaking, Earth and PSP / STA were always separated the mostin latitude, with the former generally in the southern hemisphereand the latter in the northern hemisphere. Moreover, this sug-gests that PSP and STA should have much more similar profiles,at least based on the relatively similar latitudinal position. As anexample, comparison of PSP and Earth for P3 reveals that PSPwas immersed in slow solar wind for most of the time, and par-ticularly at perihelion, whereas Earth partially intercepted a highspeed stream, but was generally embedded within solar wind ofmore variable speed. Finally, comparison between PSP and either Earth or STAmaps, particularly for the first and third perihelia, highlights theevolution of stream structure in moving outward from the lo-cation 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 lon-gitudes in some locations and stretched to later longitudes atother locations. This is due to the dynamical e ff ects of the sur-rounding stream structure. Fast solar wind at earlier longitudes(smaller heliocentric distances) attempts to overtake slower windat later longitudes (farther heliocentric distances), deceleratingthe slower wind within which the HCS is embedded. Similarly,when fast solar wind outruns slower solar wind behind, the cur-rent sheet is stretched out in longitude (as it is embedded withinthe expansion wave (rarefaction region) that is created throughthis process).The HCS is e ff ectively a surface separating regions of op-posite magnetic polarity, and, although not directly observable,it is the largest coherent structure within the heliosphere. It isintimately related to the large-scale dynamical flow of the so-lar wind, and although passive, it responds to the dynamics ofinterplanetary streams and thus provides an indirect measure ofstream evolution. At least during relatively quiescent times (andwithin, say, 20 AU), corotating interaction regions (CIRs) are or-ganized about the HCS (Pizzo & Gosling 1994). The large-scalestructure of the HCS is summarized in Figure 13 for each of thefour perihelia passes. In general, we note the relatively flat ori-entation of the HCS for all four perihelia passes. These pictures Article number, page 13 of 17 & A proofs: manuscript no. riley-PSP-obs-model-comp-2020-v5
Fig. 11.
Same as Figure 3, but for model and data comparisons at the location of Earth and for P4. can be contrasted with the shape of the HCS during more ac-tive conditions, which can span e ff ectively the entire sphericaldomain (Riley et al. 2002). More specifically, however, we notethat for P1 and P3, there was a vertical, and hence sharp cross-ing of the HCS from negative polarity (southern hemisphere)to positive polarity (northern hemisphere), whereas for perihe-lia P2 and P4, any crossings of the HCS were at more inclinedangles. Additionally, the HCS is notably flatter during the in-terval surrounding P4. These are consistent with the inferencesdrawn from Figure 12, and explain why the model comparisonswith observations were generally better for P1 and P3, for whichthere were well-defined crossing of the HCS and immersion intodistinct sources of solar wind on either side.
4. Conclusions and discussion
In this study we have modeled the global structure of the innerheliosphere for each of the first four PSP perihelia. We found thatour semi-empirical thermodynamic model consistently producedbetter matches with observations than either a less sophisti-cated polytropic approach, or a more advanced wave-turbulence-driven model. Our results provided a global perspective fromwhich to interpret the localized in situ measurements made dur-ing P1 through P4 and connect observations made at PSP withthose at 1 AU from Earth-based spacecraft as well as Stereo-A.We did not find any evidence for the resolution of the open fluxproblem at least within the heliocentric distance reached by PSPduring P4 ( ∼ . R S ). While the thermodynamic model results were, in some cases,quite remarkable, the agreement between model output and ob-servations may be improved upon in a number of ways. First, andforemost, and as described previously (e.g., Riley et al. 2012b;Linker et al. 2017; Riley et al. 2019b,a), the model results areextremely sensitive to the boundary conditions, and, in partic-ular, the observed photospheric magnetic field. The limitationsof these measurements include: (1) No observations beyond theview a ff orded by Earth-based spacecraft, which includes limitedinferences on the evolving field on the backside of the Sun aswell as little-to-no reliable information from polar latitudes; (2)No consensus on the actual “ground truth” values of the photo-spheric field (Riley et al. 2014); and (3) Limited availability oraccuracy of nonradial magnetic fields at the base of the model.Second, and related to this first point, is the lack of availabil-ity of accurate time-dependent boundary conditions.Currently,flux-transport models, such as ADAPT (Arge et al. 2010), arethe likely the best quasi-time-dependent solution to this prob-lem; however, they are driven by Earth-based observations ofthe photosphere. Only with the availability of non-Earth views ofthe Sun, or the accurate assimilation far-side data from helioseis-mic techniques (e.g., Liewer et al. 2017), can we begin to buildtruly time-dependent, synchronic maps of the Sun. With the suc-cessful launch of Solar Orbiter and the availability of observa-tions away from the Sun-Earth line from the polarimetric andhelioseismic imager (PHI) (Solanki et al. 2014), we can begin toassemble boundary conditions that mitigate and quantitativelyassess the impact of this limitation. Additionally, PFSS solu- Article number, page 14 of 17iley et al.: Comparing PSP observations with MHD models
Fig. 12.
Comparison of MHD modeled solar wind speeds at the location of the three spacecraft (PSP, Earth, and Stereo-A (STA)) with the intervaldefined by each PSP perihelia pass (1-4). In each panel, speeds from 200 to 800 km s − are shown in the rainbow spectrum from blue to red, usingthe 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 bythe red contour ( B r = tions, particularly for PSP encounters is reinforcing the idea thateven daily updated quasi-synchronic maps, such as produced byADAPT (using either GONG or HMI observations) can producesignificantly better matches in terms of the observed polarity ofthe magnetic field measured by FIELDS (Badman et al. 2020a).Thus, a natural but challenging next step would be to assess howdriving the MHD solutions with a sequence of ADAPT mapsa ff ects the quality and accuracy of the solutions.Third, the free parameters set in each of the three modelshave not been rigorously tested in sensitivity studies. Over theyears, we have explored heuristically how di ff erent values mightimpact 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 ofsolar activity that was substantially di ff erent to the current stateof the corona. For example, the free parameters used to specifythe mapping of solar wind speed along field lines in the corona,to drive the heliospheric boundary conditions of the polytropicand thermodynamic models were essentially fixed based on spe-cific (but detailed) analysis of the time period surrounding thesolar minimum of 1996 (Riley et al. 2001). Although they re-main reasonable, based on the comparisons presented here, it islikely that the change in the overall state of the corona duringthat past quarter of a century might require a re-examination ofthese parameters. Fourth, and finally, model comparisons with observationsmay be improved by incorporating more datasets into the as-sessment of the model results, as well as defining and usingmore quantitative metrics for accessing accuracy. Currently, weuse an ad hoc approach of comparing subsets of the availabledata, depending on the specific datasets we are hoping to inter-pret with the model results. For example, in eclipse predictions(Mikic et al. 2018), the focus is to produce the best match withwhite-light observations of the actual eclipse. This is, however,at the expense of matching in situ measurements. We have sug-gested that an approach that incorporates metrics for all availablemetrics, within the framework of a Pareto frontier (Camporeale2020), may optimize model solutions. Of course, it could be thatthe resulting solutions are neither the best ones at reproducingcoronal observations nor in situ measurements.It is noteworthy that, overall, the thermodynamic model ap-pears to be outperforming both the polytropic and WTD ap-proaches. 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 incomparisons with in situ measurements over a wide range of in-tervals covering the space era (e.g., Riley et al. (2001); Rileyet al. (2012a,b). The motivation for the thermodynamic modelwas, at least in part, to address the limitation that the polytropicapproximation resulted in poor comparisons with white-light ob- Article number, page 15 of 17 & A proofs: manuscript no. riley-PSP-obs-model-comp-2020-v5 servations. It is not yet clear whether the improved in situ resultsof the thermodynamic solutions are due to the overall improve-ment of parameters in the model controlling the heating of thecorona, the new lower-level of solar activity that the Sun hasentered, or some other phenomena. Either way, it is encourag-ing that a more sophisticated model is now capable of providingmore accurate solutions. Of course this requires further, moresystematic comparisons with observations over a broader rangeof the solar cycle.The primary distinction between the polytropic and thermo-dynamic solutions originates in the structure of the magneticfield in the corona, since both heliospheric solutions are drivenby this. Thus, the fact that the thermodynamic solutions are pro-viding closer matches with in situ measurements reinforces pre-vious coronal comparisons with white light observations, partic-ularly during eclipses, which also demonstrated the superiorityof the thermodynamic results. A further refinement to the ther-modynamic approach could be made, bridging the gap with theWTD model, by directly driving the heliospheric model with theoutput from the thermodynamic solution. However, previous ex-plorations of this have resulted in worse comparisons with 1 AUmeasurements (Riley et al. 2012b).The investigation undertaken here was an exploration of sev-eral di ff erent modes of operation of several models, to under-stand, at least qualitatively, the capabilities of the di ff erent ap-proaches. It should not be interpreted as a systematic parame-ter phase-space of sensitivity study, which would require a morecareful analysis, changing only one variable at a time and assess-ing the impact of that change on the model results and their sim-ilarities and di ff erences with observations. For example, in com-paring the polytropic / thermodynamic solutions with the WTDapproach, the biggest distinguishing factor might be that themaps were processed using di ff erent approximations and had lit-tle to do with the underlying physics in the model. We foundthat the updated pipeline for the WTD solutions resulted in bet-ter comparisons with white-light observations during the last twototal eclipses in 2017 (Mikic et al. 2018) and 2019 (Linker et al.2019), and we had anticipated that this would translate into bet-ter comparisons with in situ measurements. That it didn’t, nowrequires another suite of runs where we use the same input mapsfor all modeling approaches.Intriguingly, our study showed that, in spite of our initialprediction for PSP’s first perihelion, there remains a significantdeficit in the value of the radial (and hence total) magnetic fieldpredicted by the models. Our prediction, which relied on olderphotospheric magnetic field data, likely estimated the value of B r poorly, at least in part due to the significant evolution of theactive region located at PSP’s subsolar point as it reached closestapproach. Unfortunately, this conspired in just the right way sothat when the measurements were returned from the spacecraft,the agreement appeared to be remarkable. Although this hintedat the possibility that the so-called open flux problem might arisein processes in the solar wind (beyond several tens of solar radii),it is clear from the results presented here (requiring a factor of3 correction to bring the model results into agreement with theobservations) that the problem exists throughout the heliosphere,at least to within 26 . R S . Badman et al. (2020b) came to a simi-lar conclusion through the analysis of PFSS model solutions forencounters 1-5. As such, the open flux problem must be resolvedby processes or uncertainties occurring closer to the Sun. Asoutlined by Linker et al. (2017) and Riley et al. (2019b), sev-eral possibilities could, in principle, resolve this mismatch, andit is likely that a combination of factors play a role. One of thepromising ideas is that an unobserved concentration of magnetic flux lies at the solar poles and provides the “missing” open fluxto the heliosphere (Riley et al. 2019b). This may, at least indi-rectly, be supported by the results here. In particular, the peakvalues of the polar fields used to drive the WTD model solutionfor P1 (Figure 3 (red)) were only 4.7 G, whereas the peak val-ues in the prediction (Figure 3 (magenta)), while more spatiallylocalized, reached up to 65 G. Of course other ideas have alsobeen proposed, including that of Réville et al. (2020), who ar-gued that if the amplitude of the waves used to heat the coronaand accelerate the solar wind are added to the radial and totalmagnetic field kinematically, this would bring the model resultsinto agreement with the measurements. Ultimately, the extent towhich polar flux may resolve this “missing flux” problem willbe addressed by Solar Orbiter. During its extended mission, itwill reach 34 ◦ heliolatitude and such flux, if it exists, shouldbe clearly visible by the Polarimetric and Helioseismic Imager(PHI) onboard.Finally, we used the model results to explore the global struc-ture of the inner heliosphere during each of the first four peri-helia. This allowed us to provide an explanation for the bettermatches between models and observations for P1 and P3, dueto the sharp fold in the HCS and, hence, more separation of thespacecraft from the HCS for large parts of these intervals. On theother hand, during P2 and P4, the spacecraft were often skim-ming through, or adjacent to the current sheet, making modelpredictions very sensitive to the precise location of the HCS.While current solar wind conditions remain extremely quiet, aswe follow the ascending phase of the solar activity cycle, suchglobal pictures of the properties and structure in the heliospherewill become ever more useful for interpreting more complex insitu structure.This study suggests several possible avenues for future in-vestigations. First, and as noted above, a natural extension tothe modeling pipeline would be to assess a fully thermodynamicapproach, where all the boundary values from the coronal so-lution are used to drive the heliospheric model. Based on ear-lier 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 sequenceof parametric studies, again targeting a smaller number of in-tervals, such as those studied here, where model parameters andboundary conditions are systematically optimized to produce thebest matches with observations. As noted previously (e.g., Rileyet al. (2012b); Riley et al. (2019b)), however, boundary condi-tions, and, in particular, the radial component of the magneticfield, likely play a – if not the – crucial role in the quality ofthe solution. With the availability of PSP measurements and,most recently, Solar Orbiter data, together with Stereo-A andWind / ACE, we can now assess the fidelity of the solution at mul-tiple points in longitude, and, eventually latitude, potentially al-lowing us to estimate what the intrinsic limitations may be inour ability to reproduce the measurements. Finally, and relatedto this, systematic studies aimed at estimating the contribution ofpolar fields to in situ measurements, as well as improved modelsfor these values (e.g., ADAPT) may further reduce the uncer-tainty in model predictions, although, ultimately, the most sub-stantial gains will come from direct measurements of the photo-spheric field both from polar regions as well as at di ff erent helio-centric longitudes, that is, simultaneous 4 π -steradian coverage ofthe 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).
Article number, page 16 of 17iley et al.: Comparing PSP observations with MHD models
P1 P2P3 P4
Fig. 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 theisosurface of B r = R S is due to the merging of the coronal and heliospheric solutions. References
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