Switchbacks Explained: Super-Parker Fields -- the Other Side of the Sub-Parker Spiral
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Switchbacks Explained: Super-Parker Fields – the Other Side of the Sub-Parker Spiral
N. A. Schwadron
1, 2 and D. J. McComas University of New Hampshire, Durham, NH, 03824 Department of Astrophysical Sciences, Princeton University, Princeton, NJ, 08544 (Accepted December 17, 2020; December 31, 2020)
Submitted to ApJABSTRACTWe provide a simple geometric explanation for the source of switchbacks and associ-ated large and one-sided transverse flows in the solar wind observed by Parker SolarProbe. The more radial, Sub-Parker Spiral structure of the heliospheric magnetic fieldobserved previously by Ulysses, ACE, and STEREO is created within rarefaction re-gions where footpoint motion from the source of fast into slow wind at the Sun createsa magnetic field line connection across solar wind speed shear. Conversely, when foot-points move from the source of slow wind into faster wind, a Super-Parker Spiral fieldstructure is formed: below the Alfv´en critical point, one-sided transverse field-alignedflows develop; above the Alfv´en critical point, the field structure contracts betweenadjacent solar wind flows, and the radial field component decreases in magnitude withdistance from the Sun, eventually reversing into a switchback. The Sub-Parker andSuper-Parker Spirals behave functionally as opposites. Observations from Parker SolarProbe confirm the paucity of switchbacks within rarefaction regions and immediatelyoutside these rarefaction regions, we observe numerous switchbacks in the magneticfield that are directly associated with abrupt transients in solar wind speed. The ra-dial component of the magnetic field, the speed gradients, radial Alfv´en speed, and theratio of the sound speed to the radial Alfv´en speed all conform to predictions basedon the Sub-Parker and Super-Parker Spirals within rarefaction regions and solar windspeed enhancements (spikes or jets), respectively. Critically, the predictions associatedwith the Super- Parker Spiral naturally explain the observations of switchbacks beingassociated with unexpectedly large and one-sided tangential flows.
Keywords:
Solar Magnetic Field, Solar Wind, Heliosphere INTRODUCTIONThe solar wind rapidly accelerates in the corona at ∼ s and then becomes super-Alfv´enic at ∼ s (Katsikas et al. 2010; Goelzer et al. 2014). It is this latter transition where the solar windram pressure becomes dominant, overcoming both the magnetic and thermal pressure. The processesthat transfer energy and dissipate this energy to heat the corona and power the solar wind remain a r X i v : . [ a s t r o - ph . S R ] F e b critical questions in heliophysics and astrophysics, and are at the heart of the scientific motivationsfor Parker Solar Probe (PSP) (McComas et al. 2007; Fox et al. 2016).On PSP, the solar wind is observed by the Solar Wind Electrons Alphas and Protons Investigation(SWEAP) (Kasper et al. 2016) and the magnetic field by the Electromagnetic Fields Investigation(FIELDS) (Bale et al. 2016). The Integrated Science Investigation of the Sun (IS (cid:12)
IS ) instrumentsuite (McComas et al. 2016) provides comprehensive measurements of energetic particles over therange 0.02 – 200 MeV/nucleon.PSP observes thousands of intervals (duration from seconds to tens of minutes) where the speedof the solar wind flow suddenly jumps and includes a large, one-sided transverse flow, while simulta-neously the magnetic field orientation rotates through large angles, before returning to roughly theprior solar wind conditions. The observed switchbacks (radial magnetic field reversals) are associatedwith the change in magnetic field direction and velocity spikes associated with the sharp increase insolar wind speed (Bale et al. 2019; Kasper et al. 2019; de Wit et al. 2020; Horbury et al. 2020; Mozeret al. 2020; Rouillard et al. 2020; Tenerani et al. 2020).Magnetic switchbacks have been studied extensively in fast wind from coronal holes (e.g., Kahleret al. 1996) at 1 au, and beyond 1 au with Ulysses (e.g., Balogh et al. 1999; Yamauchi et al. 2004b;Neugebauer & Goldstein 2013). Observations of switchbacks have also been made inside 1 au withHelios (Borovsky 2016; Horbury et al. 2018) prior to the observations by PSP. Strong magnetic-fielddeviations from the Parker Spiral are observed where there are local increases in the radial solar windspeed (Michel 1967), and associated with one-sided or pulsed Alfv´enic fluctuations (Gosling et al.2009, 2011). The one-sided nature of switchbacks is especially clear in PSP observations. Kasperet al. (2019) states that “Transients, including the Alfv´enic jets, are one-sided, in that if the fieldrotates more than ∼ ◦ , then B T is always positive and V pT always exceeds 33 km s − .” (Here, B T is the tangential magnetic field component, and V pT is the proton tangential component.)The transverse flows observed by PSP far exceed those in the Weber–Davis model (Weber & Davis1967) where the lower corona is taken to rotate rigidly at the mean rotational period of the Sun. Tothis point Kasper et al. (2019) state: ”The large rotational velocities measured .. exceed the valuein the axisymmetric Weber-Davis model, in which V pT ( R A ) < . (cid:12) R A , by more than an order ofmagnitude.” Here, Ω (cid:12) refers to the solar rotation rate and R A refers to the Alfv´en radius. For anAlfv´en point of 15 R s , the V pT ( R A ) < − . Thus, the one-sided transverse flows are a keyobservable from PSP that any switchback theory must also explain.Table 1 summarizes these key observations from PSP related to switchbacks and tangential flows.This paper provides a simple and natural geometric explanation for all of these apparently disparateobservations that unifies the interpretation of switchbacks and the transverse flows observed by PSP.There are already a variety of models and conjectures to explain various (although not all) aspectsof switchbacks. One example relates to non-linear shear driven turbulence (Ruffolo et al. 2020). Inthe presence of large speed gradients within the solar wind, there are a number of important effectsthat should be considered. As already indicated, the situation where faster wind outruns slower windleads to the formation of rarefaction regions. However, if speed gradients exist across magnetic fluxtubes, the effects of non-linear shear driven turbulence can result in switchbacks in the magneticfield (Ruffolo et al. 2020). Prior remote observations (DeForest et al. 2016) show a transition fromstriated solar coronal structures to more isotropic “flocculated” fluctuations in the transition just Table 1.
PSP observations of switchbacks and transverse flowsObs. ReferenceSwitchbacked magnetic field Kasper et al. (2019); Bale et al. (2019)Transient jets or pulsations Kasper et al. (2019)One-sided tangential flows Kasper et al. (2019)Large transverse, co-rotational flows Kasper et al. (2019)Alfv´enic correlation δ v to δ B Bale et al. (2019)Anti-correlated plasma density n e and | B | Bale et al. (2019) outside the Alfv´en critical point. This transition in the geometry of solar wind structures is poweredby the relative velocities of adjacent coronal magnetic flux tubes.Another conjecture relates to the effects of interchange reconnection (ICX) on the magnetic struc-ture of the solar wind. Fisk & Kasper (2020) argue that the large transverse flows observed byPSP are a result of transverse flows in the corona and are part of a closed global circulation pat-tern of magnetic flux open to the solar wind. The circulation pattern at the Sun is sustained bythe combined effects of differential motion and interchange reconnection (Fisk & Schwadron 2001).Zank et al. (2020) develop an evolution equation for the development of switchbacks resulting frominterchange reconnection between coronal loops and the open magnetic field. Results from the modelinclude complex aggregated groups of switchbacks and an example event from the model resemblesPSP observations.The concept of interchange reconnection was developed by Crooker et al. (2002) to explain howmagnetic flux injected by coronal mass ejections can be reduced without disconnecting magneticfields entirely. Coronal mass ejections (CMEs) originate in closed magnetic field structures and addmagnetic flux to the heliosphere as they move away from the Sun. Observations of ejecta in thesolar wind show signatures including counter-streaming suprathermal electrons that show outflowingelectron heat flux moving in both directions along closed field lines (e.g., Gosling et al. 1987). Theseobservations confirm the addition of magnetic flux from ejecta, and show that without changing themagnetic topology of CME ejecta, the total flux in the heliosphere would continue to increase, whichleads to a “magnetic field magnitude catastrophe” (Gosling 1975; McComas 1995). Disconnectionwas pictured as magnetic reconnection between magnetic field lines to create U-shaped structuresthat are disconnected entirely from the Sun and are then convected through the heliosphere by thesolar wind (McComas et al. 1991). While disconnection is one possible solution to the magneticfield catastrophe, it was noted that signatures of heat flux dropouts (McComas et al. 1989) shouldbe associated with disconnection. However, the number of heat flux dropouts observed were foundto be at least a factor of 4 too small to account for the loss of magnetic flux from that added bycoronal mass ejections (McComas et al. 1992). Crooker et al. (2002) suggested that the primary fluxbalancing mechanism is instead from interchange reconnection associated with magnetic reconnectionbetween closed fields from coronal mass ejecta and open magnetic fields in the surrounding solar wind.Interchange reconnection was considered as way for open magnetic fieldlines to reconnect with loopsin the corona, and thereby enable large-scale redistribution of open magnetic fields beyond coronalhole boundaries (Fisk et al. 1999; Fisk & Schwadron 2001).The conjecture that footpoint motion is driven by differential motion in coronal holes and inter-change reconnection beyond coronal holes was made previously by Fisk et al. (1999) and Fisk &Schwadron (2001). The basis of this conjecture is rooted in the difference between coronal holes andthe surrounding source regions of solar wind, which remains an area of active research. Potentialfield models that approximate the corona as having zero current below the source surface radius ( ∼ • The photosphere rotates differentially (e.g., Snodgrass 1983); • Coronal holes tend to rotate rigidly with the Sun, at approximately the equatorial rotation rate(e.g., Bird & Edenhofer 1990).In this picture, coronal holes represent the boundary where there is a transition between the openmagnetic flux concentrated within the coronal hole and the more distributed open magnetic flux thatmoves through differential rotation and interchange reconnection beyond the coronal hole.This picture of the Sun’s open magnetic field has remained difficult to verify observationally. Oneobservational signature of footpoint motion was found in rarefaction regions where the magnetic fieldline is stretched between faster solar wind that streams out ahead of slower wind (Murphy et al. 2002;Schwadron 2002), leading to the formation of the Sub-Parker Spiral (Schwadron & McComas 2005;Schwadron et al. 2005). Note that rarefaction regions are formed where faster solar wind outrunsslower solar wind, and these structures are characterized by an almost monotonic decrease in thespeed of solar wind. Rarefaction regions often map back to regions with small longitudinal extentson the Sun, termed “dwells” Schwenn (1990). Because Stream interfaces typically co-rotate with theSun, co-rotating rarefaction regions are formed from the trailing edge of coronal holes (Smith et al.2000).Without footpoint motion, magnetic field lines are not connected between faster and slower wind,and conform to a Parker Spirals associated with their various solar wind speeds. In contrast, thestraightening of the magnetic field in the rarefaction region due to footpoint motion at the Suncreates a magnetic field structure with a larger radial component than the Parker Spiral magneticfield. The deviations in field direction are extremely prominent and commonly observed by Ulyssesin co-rotating rarefaction regions (Murphy et al. 2002; Schwadron & McComas 2005; Schwadronet al. 2005). The magnetic structure in rarefaction regions is referred to as the
Sub-Parker Spiral (Schwadron & McComas 2005; Schwadron et al. 2005) as it is less tightly wound than would be ex-pected for the observed solar wind speed. Another interpretation (Gosling & Skoug 2002) associatedradial magnetic fields with abrupt temporal changes in the solar wind: the “radially directed kink inthe magnetic field connects the different spirals associated with the faster and slower flows immedi-ately preceding and following the temporal flow speed discontinuity.” This description is appropriateprior to or without the development of a rarefaction region.In this paper, we show that the Sub-Parker Spiral depends on the direction of footpoint motionrelative to the gradient between fast and slow solar wind. If the direction of footpoint motion isreversed with respect to the solar wind speed gradient, a different and distinct magnetic field structureis produced in which the field line connection between fast and slow wind contracts and ultimatelyreverses the magnetic field, producing a switchback. We begin by conceptually describing the Super-Parker Spiral and its obvious extension to switchbacks as Super-Parker Spirals in §
2. In §
3, we useobservations from SWEAP and FIELDS to test for the presence of the Sub-Parker Spiral, and theSuper-Parker field structures. We report observations from several rarefaction regions and in nearbyintervals of switchbacked fields and compare them to a simple quantitative model for such magneticstructures (see Appendix). Finally, in § FROM SUB-PARKER TO SUPER-PARKER SPIRALSThe concept of the Sub-Parker Spiral is illustrated in Figure 1 (top panel). Near the Sun, footpointmotion from the source of coronal-hole associated fast wind into slow wind creates a magnetic fieldlineconnection across the interface between fast and slow solar wind. The fast solar wind draws themagnetic field out more quickly than the slow solar wind, therefore forming a rarefaction region. Themagnetic field, stretched between faster solar wind flow and slower solar wind becomes increasinglyradial with distance from the Sun. The basic prediction in this case is the association betweenrarefaction and the Sub-Parker Spiral magnetic field, which has a stronger radial component thanthe Parker Spiral. A recent study by Schwadron et al. (2020) shows that the Sub-Parker Spiralprovides relatively short fieldline connections from the PSP spacecraft to the compressions and shockssurrounding co-rotating interaction regions in the inner heliosphere (at ∼ ∼
100 keVto > MeV observed by IS (cid:12)
IS up to ∼ S l o w e r W i n d F a s t e r W i n d B F i e l d Sub-Parker Spiral
PSP A l f v en C r i t i c a l P o i n t t t t t F o o t p o i n t M o t i o n F a s t - t o - S l o w x F a s t e r W i n d Slower Wind F o o t p o i n t M o t i o n S l o w - t o - F a s t t T - F l o w B F i e l d Super-Parker Spiral P S P A l f v en C r i t i c a l P o i n t t t t SDO AIA 2018-11-02 x Switchbacks
Figure 1.
The Sub-Parker Spiral (top panel) and the Super-Parker Spiral (bottom panel) result fromfootpoint motion between source regions of fast and slow solar wind. In the case of the Sub-Parker Spiral,footpoint motion from the source of fast to slow wind creates a fieldline connection that gets straightened asfast wind drags out magnetic fieldlines more quickly than the slow wind. The Sub-Parker Spiral is associatedwith magnetic fieldlines with larger radial components than the Parker Spiral. When footpoint motion isreversed (bottom panel), and footpoints move from the source of slow into fast wind, then the wind shearkinks the magnetic field. In this case, the faster wind moves along the magnetic field below the Alfv´encritical point, forming compressions and tangential flows and develops into a switchback above the Alfv´encritical point. tend to rotate rigidly implies that differential motions in the photosphere drive open magnetic fieldfootpoints into coronal holes on their leading edge (leading with respect to rigid rotation), and out ofcoronal holes on their trailing edge. Thus, this sense of footpoint motion consistently connects slowerwind from outside the coronal hole with faster wind from within the coronal hole in the oppositesense on the leading and trailing edges of a coronal hole. PSP OBSERVATIONS OF RAREFACTION REGIONS AND SWITCH-BACKSWe re-examine some PSP observations from Orbit 1 to test the prediction that switchbacks occurfar less often in rarefaction regions. Observations from November 20 and November 21, 2018 areshown in Figure 2 and Figure 3, respectively. Within these observational periods, we identify tworarefaction regions as relatively steady, multi-hour periods of decreasing radial solar wind speed. R B T B N Figure 2.
Observations from PSP: FIELDS data (panels 1-3), and SWEAP data (panels 4,5) on November20, 2018. In panels 6, and 7 we form the radial Alfv´en speed and the ratio of the sound speed to the radialAlfv´en speed from the SWEAP and FIELDS observations. Radial distance from the Sun is shown in thebottom panel. Green shaded regions show intervals where switchbacks are observed. The yellow shadedregion shows a rarefaction region.
Within rarefactions it is important to separate the underlying trend of a reduction in radial windspeed from small speed fluctuations ( <
50 km s − ) occurring over short ( <
10 min) periods. In asimilar vein, we identify 11 switchback intervals as periods larger than ∼
10 min where the radial fieldcomponent vanishes or switches sign. Note that switchback intervals often include many subintervalswhere the radial field may switch sign or vanish intermittently.The trends within rarefactions agree with the key features associated with the Sub-Parker Spiral: R B T B N Figure 3.
Observations from PSP: FIELDS data (panels 1-3), and SWEAP data (panels 4,5) on November21, 2018. The format is identical to Figure 3. • The prediction that switchbacks should preferentially occur outside of rarefaction regions isclearly supported by the observations in Figures 2 and 3. The only weak possible exceptionis in the first rarefaction region on November 20, 2018 06:45 where we observe a reduction inradial field strength and the radial component of the magnetic field. However, even for thisevent, we observe a slight increase in the radial component of the solar wind speed, suggestingthat there is a Super-Parker Structure associated with the speed jump embedded within therarefaction region. • Since footpoints move from the source of fast wind to slow wind in the direction opposite ofco-rotation, we predict a tangential flow component that is negative. However, interchangereconnection and potentially changes along the flow history may lead to intermittent increasesin the tangential flow component. Because the rarefaction region expands between faster andslower flow, any variations in the tangential flow will be accentuated. We therefore predicteda mix of positive and negative tangential flows, which is observed in both rarefaction regions. • We observe an increase in the radial magnetic field in both rarefaction regions. The secondrarefaction region is more developed than the first, due to the larger and longer duration dropin speed through the rarefaction, and possibly the slightly larger distance from the Sun. In thissecond rarefaction, the large increase in the radial magnetic field relative to the other two fieldcomponents is particularly evident. This strongly supports the example calculation detailedin Appendix A that the magnetic field within the Sub-Parker Spiral tends to make the fieldstructure more radial than what would be observed in a Parker Spiral. This observation alsodirectly supports the concept that the magnetic field is connected between faster and slowersolar wind flow, contrary to the assumption of no footpoint motion associated with the ParkerSpiral magnetic fields. • The increase in radial magnetic field strength within the Sub-Parker Spiral should increase theradial Alfv´en speed magnitude and reduce the ratio of the sound speed to the radial Alfv´enspeed. Both of these effects are observed in each rarefaction region.The trends (rows 3-13 in Table 2) observed where there are increases or jumps in solar wind speedcounter those within rarefactions, and agree with the Super-Parker Spiral predictions in AppendixA: • Switchbacks occur in regions where there are abrupt increases in solar wind speed, as opposedto the reduction in solar wind speed in rarefaction regions. • Tangential flows should be positive, in the direction of co-rotation, as discussed in the previoussection. • The decrease in the radial magnetic field is evident in each of the regions identified. • There is a decrease in the radial magnetic field strength observed within each of the switchbackfield intervals. As shown in the Appendix, this trend supports the concept that the magneticfield contracts where the faster wind overtakes adjacent flow of slower solar wind. • The decrease in radial magnetic field strength within the Super-Parker Spiral decreases theradial Alfv´en speed magnitude and increases the ratio of the sound speed to the radial Alfv´enspeed, which is consistently observed in ten of the eleven switchback regions observed.PSP observations of magnetic and plasma structures show a number of common features as sum-marized in Table 2. Rows RF1 and RF2 apply to rarefaction regions, and rows SB1-SB11 applyto intervals with switchbacks. The Sub-Parker Spiral should show enhanced radial magnetic fieldstrength, a dropping radial solar wind speed, variable tangential flow, enhanced radial Alfv´en speedmagnitude, and reductions in the ratio of the sound speed to the radial Alfv´en speed. In contrast, theSuper-Parker spiral should show the opposite trends including reduced radial magnetic field strength,decreased or reversed radial magnetic field component, an abrupt increase in radial solar wind speed, The radial Alfv´en speed is defined, v Ar = B r / √ πρ where B r is the radial magnetic field, and ρ is the mass density. Table 2.
PSP observations within rarefactions and switchbacksEvent Fig. | B r | B r V r V t | v Ar | c s /v Ar Sub-PS a Super-PS b RF-1 c ↑ ↑ (cid:38) ↑↓ ↑ ↓ c ↑ ↑ (cid:38) ↓↑ ↑ ↓ d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑ ↓ ? 0/6 5/6SB-3 d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑ ↓ ↑ d ↓ ↓ ↑ ↑↓ ↓ ↑ a Sub-Parker Spiral - Sub-PS b Super-Parker Spiral – Super-PS c Rarefaction – RF d Switchbacks – SB positive values for the tangential flow (flows in the direction of co-rotation), an abrupt decrease inthe radial Alfv´en speed magnitude, and abrupt increases in the ratio of the sound speed to the radialAlfv´en speed. These predictions provide specific observational signatures, the presence of which aretested using PSP observations. The observational signatures are related analytically and are all im-portant for identifying the presence of the Sub-Parker Spiral or the Super-Parker Spiral. Therefore,we use these observational signatures to “score” of each of the 13 intervals against all six criteria.The last two columns of Table 2 provide this scoring. For both of the rarefactions studied, each ofthese 6 trends are observed, and the consistency with the Sub-Parker Spiral is six-for-six (column 8in Table 2). For the switchback intervals, we find perfect agreement between predicted and observedbehaviors for all but two of the intervals and even for those two switchbacks, the score was 5 out of6. The only two exceptions include one case (switchback 2) resulting from missing or inconclusivedata, and one case (switchback 11) where the tangential flows are not uniformly positive. This lattercase however appears to have a more variable radial magnetic field and variable properties in general,suggesting that the interval includes an array of switchbacks embedded on smaller scales. CONCLUSIONSThis study examines the development of magnetic structures in the solar wind observed by ParkerSolar Probe. The Sub-Parker Spiral structure of the heliospheric magnetic field is created withinrarefaction regions where footpoint motion at the Sun creates a magnetic field line connection acrossthe gradient between fast and slow solar wind. As fast solar wind moves outward more quickly than1slower solar wind, the magnetic structure across the rarefaction region becomes more radial than theParker Spiral. We have examined rarefaction regions observed by Parker Solar Probe and find theythat are consistent with the Sub-Parker Structure.The direction of footpoint motion at the Sun between the source of fast and slow wind is critical indefining the magnetic structure in the heliosphere. For situations in which magnetic field footpointsmove from the source of fast wind into slow wind, the magnetic structure in the heliosphere isconsistent with the Sub-Parker Spiral due to the expansion within the rarefaction region and themagnetic field line connection across it. In contrast, when footpoint motion is reversed from thesource of slow wind into faster wind, the magnetic structure contracts between adjacent solar windstreams and forms into a Super-Parker Spiral as the solar wind moves out into the heliosphere.The radial component of the magnetic field decreases in magnitude with distance from the Sun andeventually reverses into a switchback.The Sub-Parker Spiral and the Super-Parker Spiral behave functionally as opposites. The ob-servations from Parker Solar Probe confirm the paucity of switchbacks within rarefaction regions.Immediately outside these rarefaction regions, we observe numerous intervals of switchbacks in themagnetic field that are directly associated with abrupt transients in solar wind speed. In contrast tothe smooth monotonic speed transition within rarefaction regions, the switchbacks occur in “bursts”.The clustering and plenitude of these bursts occurring in close proximity are consistent with PSPbeing magnetically connected to the leading edge of the coronal hole. The observations confirm thefeatures of the Sub-Parker Spiral and the Super-Parker Spiral: the magnetic field strength, the radialcomponent of the magnetic field, the speed gradients, tangential flows, Alfv´en speed, and plasmabeta all conform to predictions based on the Sub-Parker and Super-Parker Spiral within rarefactionregions and solar wind speed enhancements (spikes or jets), respectively.Table 3 describes key observations identified in the Introduction (Table 1) and how the Super-ParkerSpiral accounts for these observations. In the Super Parker Spiral, the presence of footpoint motionat the Sun from the slow to fast wind source provides a configuration that leads to switchbacks as thefaster wind overtakes adjacent slower wind flows. The source of faster flow can come from coronalholes, transients associated with interchange reconnection, loop sources, plumes, spicules or maco-scipules. These sources of solar wind variability have been noted as potential causes of transient jetsor speed variations in the solar wind.The one-sided, co-rotation-directed tangential flows observed by PSP has been one of the mostimportant and baffling pieces of the puzzle. As shown in Figure 1, differential motion determinesthe orientation of tangential magnetic field variations, and the field-aligned flow below the Alfv´encritical point. Tangential flow is thus oriented in the direction of co-rotation, naturally explaining theobserved one-sided tangential flows. The field-aligned flow below the Alfv´en critical point also causesthe development of compressions, and explains the anti-correlation between density and magneticfield strength.The Alfv´enic correlation between velocity and field variations results from the development ofAlfv´enic structures in the solar wind. Beyond the Alfv´en critical point, large-scale variations in theflow inevitably develop Alfv´enic characteristics. Moreover, the ejection of magnetic field variationsclose to the Sun are consistent with Alfv´enic structures (Schwadron & McComas 2003), as are theexhausts from magnetic reconnection. Therefore, whether the source involves Alfv´en waves or inter-2
Table 3.
PSP observations of switchbacks and transverse flows relatedthe Super-Parker Spiral magnetic field (mechanisms in bold indicate newaspects introduced in this paper).Observation MechanismSwitchbacked
Super Parker Spiral, magnetic field fast flow overtakes adjacent slower flowabove Alfv´en critical point
Transient jets Coronal holes, Interchange Reconnection a ,or pulsations loop sources b , plumes c , spicules d One-sided
Directional field-aligned flow tangential flows below Alfv´en critical point
Transverse,
Differential motion drives footpoints co-rotational flows counter to co-rotation
Alfv´enic correlation Alfv´enic wave development e , δ v to δ B Poynting flux from Alfv´en wave energy source f Anti-correlated
Flow compression n e and | B R | below Alfv´en critical point a Fisk & Schwadron (2001); Fisk & Kasper (2020); Zank et al. (2020) b Schwadron & McComas (2003); Fisk & Kasper (2020); Zank et al.(2020) c Poletto (2015) d Yamauchi et al. (2004a) e Gosling et al. (2009); Kasper et al. (2019) f Schwadron & McComas (2003) change reconnection exhausts, an outcome is the development of Alfv´enic variations with correlationbetween velocity and field deviations.Thus, this paper reveals the origin for switchbacked magnetic field structures and one-sided tangen-tial flows from the formation of the Super-Parker Spiral, the counterpart of the Sub-Parker Spiral.These magnetic structures are produced fundamentally through footpoint motion, respectively, intoand out of coronal holes at the Sun caused by differential motion and interchange reconnection acrossregions with a strong gradient in solar wind speed. The existence of these field structures representsa significant departure from the standard Parker Spiral, and naturally explains fundamental rela-tionships between their solar wind source at the Sun and the magnetic and flow structures out in theheliosphere.We are deeply indebted to everyone that helped make the Parker Solar Probe (PSP) missionpossible. We thank the reviewer for their helpful comments and suggestions. We thank Dr. StuartBale for spotting an error in our initial plotting of the magnetic field. This work was supported as a3part of the PSP mission under contract NNN06AA01C. Parker Solar Probe was designed, built, andis now operated by the Johns Hopkins Applied Physics Laboratory as part of NASA’s Living with aStar (LWS) program (contract NNN06AA01C). Support from the LWS management and technicalteam has played a critical role in the success of the Parker Solar Probe mission.REFERENCES
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P., Nakanotani, M., Zhao, L.-L.,Adhikari, L., & Kasper, J. 2020, TheAstrophysical Journal, 903, 1 A. SUB-PARKER AND SUPER-PARKER SPIRALSIn this appendix, we develop an analytical derivation that provides solutions for the Sub-Parkerand the Super-Parker Spiral. Schwadron et al. (2020) follow the ballistic propagation of plasmaparcels from the Sun within the solar wind to determine the structure of the magnetic field in theinner heliosphere. Figure 4 shows the configuration near the Sun in the co-rotating reference framealong a boundary surface (at radius R B ) where footpoint motion moves magnetic footpoints betweenregions of faster wind (with speed V + δV /
2) and regions of slower wind (with speed V − δV / ω φ = −| ω φ | in the opposite directionof the Sun’s rigid rotation (the rigid rotation rate is Ω (cid:12) ). Footpoints also move in co-latitude at rate ω θ = | ω θ | .We take the interface between fast and the rarefaction region tilted by angle Ψ with respect to theazimuthal direction. On the inner boundary surface, at radius R B , the unit vector along the streaminterface is defined ˆ e I = sin Ψˆ e θ + cos Ψˆ e φ . (A1)On the inner boundary the unit vector normal to the stream interface is defined,ˆ e ⊥ | r = R B = − cos Ψˆ e θ + sin Ψˆ e φ . (A2)Therefore the footpoint rotation rate normal to the stream interface is ω ⊥ = − ω θ cos Ψ + ω φ sin θ sin Ψ . (A3)The footpoint rotation rate times the velocity gradient is α = R B (cid:126)ω B · ∇ V | r = R B = ω θ ∂V∂θ + ω φ ∂V∂φ (A4)= R B ω ⊥ ∂V∂s ⊥ (A5)where the velocity gradient normal to the interface is R B ∂V∂s ⊥ = (cid:18) ω θ ∂V∂θ + ω φ ∂V∂φ (cid:19) ( ω ⊥ ) − . (A6)In the application to the rarefaction region considered here ω φ is a negative quantity since longitudinalfootpoint motion opposes solar rotation, and ω θ is positive. This implies that ω ⊥ < ω ⊥ = − ( ω θ cos Ψ + | ω φ | sin θ sin Ψ) . (A7)The velocity gradient in co-latitude is a negative quantity, ∂V /∂θ <
0, and the velocity gradient inthe azimuthal direction is a positive quantity, ∂V /∂φ >
0. Therefore, the velocity gradient normalto the interface is a positive quantity, R B ∂V∂s ⊥ = (cid:18) ω θ (cid:12)(cid:12)(cid:12)(cid:12) ∂V∂θ (cid:12)(cid:12)(cid:12)(cid:12) + | ω φ | ∂V∂φ (cid:19) | ω ⊥ | − > . (A8)6Given these properties of the velocity and footpoint rotation rates, Schwadron et al. (2020) expressedthe magnetic field in the rarefaction region as follows B = A ( r ) (cid:26)(cid:18) | α | ( r − R B ) V (cid:19) ˆ e r − (cid:126)ω B rV − Ω (cid:12) r sin θV ˆ e φ (cid:27) (A9)where (cid:126)ω B = ω θ ˆ e θ + ω φ sin θ ˆ e φ , (A10) A ( r ) = B fB (cid:18) R B r (cid:19) (cid:18) | α || ω ⊥ | ( r − R B ) V Ω (cid:12) sin θ sin Ψ (cid:19) − (A11)and B fB normalizes the magnetic field in the ambient fast solar wind. The corresponding magneticfield in the ambient fast solar wind is: B f = B fB (cid:18) R B r (cid:19) (cid:18) ˆ e r − (cid:126)ω B rV − Ω (cid:12) r sin θV ˆ e φ (cid:19) . (A12)A specific realization of the Sub-Parker Spiral with Ψ = 0 reveals the essence of the magneticstructure. In this case, the vector perpendicular to the stream interface is ˆ e ⊥ = − ˆ e θ and the Sub-Parker magnetic field is given by B a = B fB (cid:18) R B r (cid:19) (cid:40)(cid:32) ω θ (cid:12)(cid:12)(cid:12)(cid:12) ∂V∂θ (cid:12)(cid:12)(cid:12)(cid:12) r = R B ( r − R B ) V (cid:33) ˆ e r − (cid:126)ω B rV − Ω (cid:12) r sin θV ˆ e φ (cid:41) (A13)In this expression, for simplicity we have taken the solar wind speed as a function of latitude, withhigher speed wind at higher latitudes and slower solar wind at lower latitudes. The expression inequation (A13) can be generalized for any solar wind speed gradient, and any value of ω θ : B a = B fB (cid:18) R B r (cid:19) (cid:40)(cid:32) − ω θ ∂V∂θ (cid:12)(cid:12)(cid:12)(cid:12) r = R B ( r − R B ) V (cid:33) ˆ e r − (cid:126)ω B rV − Ω (cid:12) r sin θV ˆ e φ (cid:41) . (A14)In the case that ω θ ∂V /∂θ | r = R B <
0, the expression conforms to the Sub-Parker Spiral solution.However, for ω θ ∂V /∂θ | r = R B >
0, we arrive at a solution where the radial component of the magneticfield decreases with distance, and then reverses sign, leading to the formation of a switchbackedmagnetic field. This is the solution for Super-Parker Spiral. With faster wind at higher latitudes,and slower wind at lower latitudes, the gradient ∂V /∂θ | r = R B <
0. As a result, the sign of ω θ determines whether the solution conforms to Sub-Parker Spiral ( ω θ >
0) or a Super-Parker Spiral( ω θ < ω θ ∂V /∂θ | r = R B < ω θ ∂V /∂θ | r = R B > Faster WindSlowerWindV + δ V /2 V- δ V /2 φ + δφ /2 φ - δφ /2 φ ω φ (r a , φ a ) (r b , φ b ) R B M a g ne t i c F i e l d parcel (b) parcel (a) Solar Equator Projection
Figure 4.
Footpoint motion across gradients in radial solar wind speed from faster into slower create theconditions for the sub-Parker spiral. Black-curves show the streamlines associated with parcel (a) and parcel(b) in the co-rotating reference frame. Footpoint motion provides a magnetic connection between parcel (a)and (b), which implies that the magnetic field is directed along the displacement between these plasmaparcels. This projection is on the Sun’s equatorial plane.
It is interesting to note that equation (A14), which admits both Sub-Parker and Super-Parkersolutions, requires a configuration where the solar wind speed is a function of latitude, as consistentwith a latitudinal shear in the solar wind. In contrast, if speed variations exist such that faster windtrails slower wind in longitude, these structures must lead to the formation of compression regions,which complicate, or disallow steady-state solutions for the magnetic field structure. The treatmentof latitudinal shear in the solar wind is similar and allows straightforward insight into the structuralevolution of the magnetic field.In the case of the Super-Parker Spiral, the switchbacked magnetic field appears at radial distanceswhere 1 − | ω θ | (cid:12)(cid:12)(cid:12)(cid:12) ∂V∂θ (cid:12)(cid:12)(cid:12)(cid:12) r = R B ( r − R B ) V < , (A15)or equivalently where r > R B + V | ω θ || ∂V /∂θ | r = R B . (A16)8 Sub-Parker Magnetic Field
Faster WindSlowerWind ω θ R B p a r c e l ( b ) p a r c e l ( a ) M a g ne t i c F i e l d parcel (b) p a r c e l ( a ) parcel (b) p a r c e l ( a ) T i m e Faster WindSlowerWind ω θ R B p a r c e l ( b ) Super-Parker Magnetic Field p a r c e l ( a ) M a g ne t i c F i e l d p a r c e l ( b ) p a r c e l ( a ) p a r c e l ( b ) p a r c e l ( a ) Meridional Projection
Figure 5.
Comparison between the effects of footpoint motion from faster into slower solar wind inthe case of the Sub-Parker magnetic field (left panel) and from slower into faster solar wind in the caseof the Super-Parker magnetic field (right panel). Black-curves show the streamlines associated with parcel(a) released earlier and parcel (b) released later after footpoints have moved in latitude in the directionindicated. In the case (left panel) that footpoints move from the source of faster solar wind to slowerwind, magnetic field lines are stretched in the radial direction forming the Sub-Parker Spiral. Note that ameridional projection is used in this figure, as opposed to the equatorial projection in Figure 4. Reversingthe sense of the footpoint motion (right panel) leads to contraction of the magnetic field by the solar windspeed gradient, and ultimately leads to reversal of the field polarity associated with switchbacks.
As a specific example, if we consider an azimuthal motion of footpoints in the direction of solarrotation at 17% of the solar rotation rate, a radial distance of 0.3 au, a boundary surface at R B = 25solar radii, a speed change of 100 km s − , and an average wind speed of 300 km s − , then a switchbackis created if the speed gradient exists over a displacement of < . ◦ .In Figure 6, we show the radial component of the magnetic field and its strength as a function ofradial distance based on equation (A14). For simplicity, we consider a specific instance where there9is a speed change of 50 km s − across a region of 0.1 ◦ and an average wind speed of 400 km s − .Note that a 0.1 ◦ structure co-rotates past the a stationary observer in ∼
10 minutes. The three casesshown are for the Sub-Parker Spiral (blue curve), the Parker Spiral (black curve) and the Super-Parker Spiral (green). The only factor differentiating between these instances is the rate of footpointmotion. In the case of the Parker Spiral, the footpoints are fixed and the rotation rate of footpointsis zero. For the Sub-Parker spiral the footpoints rotate from the faster wind at higher latitudes intothe slower wind at lower latitudes and the rate of footpoint motion is 10% of the solar rotation rate.For the Super-Parker Spiral, the footpoint motion is reversed, with footpoints moving from slowerwind at lower latitudes into faster wind at higher latitudes, again at 10% of the solar rotation rate. Inthis case, the Super-Parker Spiral magnetic field reverses its radial component forming a switchbackat 0.22 au.The key features associated with the three field configurations are as follows: • the Parker Spiral magnetic field has a radial component that drops as ∼ /r close to the Sun. • the Sub-Parker Spiral magnetic field has a radial component that also drop as ∼ /r closeto the Sun, but the magnitude of the radial component is significantly larger than that of theParker Spiral. This increase in magnetic field strength is counter-intuitive and results from theamplification of the radial component of the magnetic field as the field is stretched radially inbetween the faster and slower solar wind flow. • the Sub-Parker Spiral magnetic field has a radial component that also drops as ∼ /r close tothe Sun, but the radial component is significantly larger than that of the Parker Spiral. Thisincrease in radial magnetic field strength is counter-intuitive and results from the amplificationof the radial component of the magnetic field as the field is stretched radially in between thefaster and slower solar wind flow. • the Super-Parker Spiral initially has the weakest radial component of the three magneticconfigurations. The radial component of the field is reduced faster than the Parker Spiralmagnetic field, and ultimately reverses to form a switchback, in this case at ∼ .
22 au. Beyondthe switchback, the radial component of the field continues to drop with distance, and themagnitude of the radial magnetic field ultimately overtakes that of the Parker Spiral magneticfield. This interesting increase in the radial field magnitude is also the result of the stretchingof the reversed magnetic field between the faster and slower solar wind flow.It is important to note that the Sub-Parker Spiral magnetic field can persist within the rarefactionregion many au from the Sun, growing to fill an increasing volume of the inner heliosphere. Incontrast, the Super-Parker Spiral is associated with compression between faster and slower solarwind flow. In the case considered, where the gradient in wind speed is across latitude, the fasterand slower flows can continue to stretch the field. However, this idealization is unlikely to applyover broad regions, and eventually a compression region will form that slows the faster wind andspeeds up the slower wind. This compression region will subsume the Super-Parker Spiral within theinherently turbulent flow within compression regions.Another important feature associated with Sub-Parker Spiral and the Super-Parker Spiral is thespecific alignment of flow deviations with the direction of the magnetic field. Generally, the non-radialmagnetic field components will be directed opposite from the direction of footpoint motion. In the0 | B r | ( nT ) B r ( nT ) Sub-Parker SpiralParker SpiralSuper-Parker Spiral
Figure 6.