A Self-consistent Simulation of Proton Acceleration and Transport Near a High-speed Solar Wind Stream
Nicolas Wijsen, Evangelia Samara, ?ngels Aran, David Lario, Jens Pomoell, Stefaan Poedts
DDraft version February 23, 2021
Typeset using L A TEX twocolumn style in AASTeX63
A self-consistent simulation of proton acceleration and transport near a high-speed solar wind stream
Nicolas Wijsen , Evangelia Samara ,
2, 1 `Angels Aran , David Lario , Jens Pomoell , andStefaan Poedts
1, 6 Centre for mathematical Plasma-Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium Royal Observatory of Belgium, Brussels, Belgium Department of Quantum Physics and Astrophysics, Institute of Cosmos Sciences (ICCUB), Universitat de Barcelona (UB-IEEC), Spain NASA, Goddard Space Flight Center, Heliophysics Science Division, USA Department of Physics, University of Helsinki, Helsinki, Finland Institute of Physics, University of Maria Curie-Sk(cid:32)lodowska, Lublin, Poland (Received January 13, 2021; Revised February 1, 2021; Accepted February 1, 2021)
Submitted to ApJ lett.ABSTRACTSolar wind stream interaction regions (SIRs) are often characterised by energetic ion enhancements.The mechanisms accelerating these particles, as well as the locations where the acceleration occurs,remain debated. Here, we report the findings of a simulation of a SIR event observed by Parker SolarProbe at ∼ ∼ Keywords:
Solar wind (1534) — Corotating streams (314) — Interplanetary particle acceleration (826) INTRODUCTIONThe interaction of a high-speed solar wind stream(HSS) with slower solar wind ahead results in the for-mation of a stream interaction region (SIR; Gosling &Pizzo 1999). SIRs observed at 1 au are usually boundedby forward and reverse compression waves (FCW andRCW). The FCW accelerates and compresses the slowsolar wind propagating in front of the HSS, whereas theRCW decelerates and compresses the fast solar windof the HSS. At larger heliocentric distances, such pres-
Corresponding author: Nicolas [email protected] sure waves commonly steepen into forward and reverseshocks (FS and RS) (Gosling & Pizzo 1999).It was originally suggested that recurrent energeticion intensity enhancements seen at 1 au in associationwith HSSs were produced by these outer forward–reverseshock pairs (e.g., Barnes & Simpson 1976; Fisk & Lee1980). Intensity radial gradients (Van Hollebeke et al.1978) and flow directions (Marshall & Stone 1978) of theSIR-associated energetic ions were consistent with theorigin of the particles at several au (Mason & Sander-son 1999). However, Giacalone et al. (2002) realisedthat compression waves can already induce a first-orderFermi acceleration process, without needing to steepeninto shock waves. This happens when the spatial extentof the compression is significantly smaller than the meanfree path of the particles across the wave, such that the a r X i v : . [ phy s i c s . s p ace - ph ] F e b Wijsen et al. particles experience the compression wave much like ashock (Giacalone et al. 2002). In addition, a second-order Fermi acceleration process may also contribute tothe local ion acceleration in SIRs (e.g., Richardson 1985;Schwadron et al. 1996, 2020).An important question remains with regard to thedominant source of the SIR-associated energetic ions de-tected at 1 au, namely whether they originate from dis-tant shock waves or from close-by compression waves.Observations from Helios (Porsche 1975), Parker SolarProbe (PSP; Fox et al. 2016), and Solar Orbiter (SolO;M¨uller et al. 2013) have shown that SIR energetic ionsare detected well within the Earth’s orbit (e.g., VanHollebeke et al. 1978; McComas et al. 2019; Desai et al.2020; Cohen et al. 2020; Allen et al. 2020a,b,c; Joyceet al. 2020; Schwadron et al. 2020). Hence, the study andmodelling of SIR particle events at different heliocentricdistances is essential in order to address the accelerationand transport of energetic particles.In this work, we study a SIR event observed (Allenet al. 2021) in September 2019 by both PSP (lo-cated at ∼ .
56 au) and the Solar Terrestrial Rela-tions Observatory-Ahead (STEREO-A) spacecraft (lo-cated at ∼ .
95 au). The close radial alignment betweenSTEREO-A and PSP at that time makes this event es-pecially interesting because the evolution of the coronalhole responsible for the HSS observed by both space-craft is minimal (Allen et al. 2021). To understand thisSIR event in depth, a realistic scale-bridging model ofthe solar wind and energetic particle acceleration andtransport is necessary. In this work, this is achieved byusing the combination of two novel models, namely, theEUropean Heliospheric FORcasting Information Asset(EUHFORIA; Pomoell & Poedts 2018) and the PAr-ticle Radiation Asset Directed at Interplanetary SpaceExploration (PARADISE; Wijsen et al. 2019a; Wijsen2020). EUHFORIA is a data-driven magnetohydrody-namic (MHD) model of the solar wind, whereas PAR-ADISE simulates energetic particle distributions in thesolar wind provided by EUHFORIA, using a quasi-linear(Jokipii 1966) approach to capture the interaction be-tween solar wind turbulence and energetic particles. Weillustrate how EUHFORIA+PARADISE captures, in aself-consistent manner: (1) the complex HSS structure,(2) the formation of the SIR, (3) the acceleration ofprotons at both the FCW and the RCW, and (4) thepropagation of the energetic protons into the inner he-liosphere. We show how the spatial distribution of SIRparticles varies as the compression waves tend to steepenat large helioradii. Whereas low-energy particles ( (cid:46) . RESULTS2.1.
The Solar Wind
Figure 1 shows the solar wind bulk speed, proton num-ber density, and magnetic field magnitude observed byPSP (left) and STEREO-A (right). The gaps in thePSP data correspond to time periods when the instru-ments on board the spacecraft were powered off. Theresults of the data-driven EUHFORIA solar wind sim-ulation are shown in red. The precise As can be seen,the EUHFORIA simulation succeeds at reproducing theobserved speed profile at STEREO-A, with the simu-lation capturing the amplitude of the HSS as well asthe decreasing speed profile in the rarefaction region be-hind the HSS. Reproducing the speed profile of the solarwind is essential when studying energetic particle pop-ulations. This is because the frozen-in condition of thesolar wind magnetic field implies that the large-scalemagnetic field connectivity is largely dominated by thesolar wind plasma flow.The arrival of the HSS at PSP occurred in the mid-dle of a data gap (Fig. 1a). The EUHFORIA model(red solid line) predicts a solar wind speed increasemore gradual than what the available PSP data sug-gests. This simulated gradual increase translates intoa small magnetic field increase at the SIR in compar-ison with the observations (cf. Fig. 1e). STEREO-A data suggests the presence of a developing RS on22 Sep 01:35 UT (Allen et al. 2021) as shown by theabrupt decrease in magnetic field magnitude. Allen et al.(2021) found that PSP solar wind data aligns well withSTEREO-A data when the latter is shifted 1.77 daysin time to account for corotation. The dashed line inFig. 1a shows the simulated speed profile observed bySTEREO-A, shifted 1.77 days earlier in time. For sucha more abrupt speed increase, the maximum speed atPSP would be obtained 18h earlier than in the currentsimulation. The gradual speed increase at PSP in thesimulation results partly from an overestimation of thesolar wind speed just ahead of the HSS (see Fig. 1a).The existence of a strong speed gradient in the solarwind at 0.5 au suggests a very abrupt transition betweenthe slow and fast solar wind source regions.Figure 2 displays the three-dimensional structure ofthe HSS. Figure 2a shows the solar wind speed V sw atthe constant latitude ϑ = 2 . ◦ (using the HeliocentricEarth Equatorial (HEEQ) coordinate system), which isapproximately the latitude of PSP and STEREO-A dur-ing the SIR passage. The symbols indicate the locationof STEREO-A (black dots) and PSP (grey triangles) self-consistent simulation of a SIR event Figure 1.
Comparison between the proton solar wind speed (a–b), density (c–d), and magnetic field magnitude (e–f) observed(blue dots) by PSP (left) and STEREO-A (right) with the EUHFORIA simulation results (red solid lines). The dashed linein panel (a) shows the simulated HSS onset observed at STEREO-A shifted earlier in time by 1.77 days (Allen et al. 2021) toaccount for corotation. at different times. The black lines are interplanetarymagnetic field (IMF) lines crossing the spacecraft andprojected on the ϑ = 2 . ◦ plane. The apparent crossingof these IMF lines inside the SIR is solely a projectioneffect. The non-zero latitudinal component of the IMFarises mainly due to the deflection of the magnetic fieldat the FCW and the RCW (Wijsen et al. 2019b).Figure 2c shows V sw at a constant helioradius r =1 . ∼ . ∇ · V sw , which is a measure of the localplasma compression. Most of the solar wind is in anexpanding state, characterised by ∇ · V sw >
0. Theseare the regions where particles undergo adiabatic cool-ing while propagating (Ruffolo 1995). In contrast, theFCW and the RCW bounding the SIR are regions where ∇ · V sw <
0. Both waves are clearly visible in Fig. 2bas two spiral-shaped blue regions, with larger negativevalues of ∇ · V sw at the RCW. Figure 2d illustratesthe complex latitudinal structure of the SIR, showingthat the solar wind contains several additional com-pressed plasma regions that are all characterised by a Wijsen et al.
Figure 2.
Simulated solar wind speed (panels a and c) and the divergence of the solar wind velocity (panels b and d) at HEEQlatitude ϑ = 2 . ◦ (upper row) and at the radial distance r = 1 . clear two-component structure, that is, a FCW followedby a (stronger) RCW. Each of these compression regionsforms a potential source of accelerated particles. How-ever, most of these regions are located well above or be-low the ecliptic. Figure 2d shows that, after passing theRCW, the magnetic connection of both STEREO-A andPSP skim the northern edge of another RCW, associatedwith the southern edge of the HSS. As discussed below,this second RCW may be responsible for the small par-ticle intensity enhancements seen at STEREO-A on 23Sep (see Fig. 3).2.2. Energetic Particles at STEREO-A
Figure 3a shows the 10 minute averaged ion intensitiesobserved by the Solar Electron and Proton Telescope(SEPT; M¨uller-Mellin et al. 2008) on board STEREO-A in four different energy channels, after subtractingthe corresponding pre-event intensity values (averaged from Sep 18 00:00 UT to Sep 19 12:00 UT). Intensityenhancements at energies below ∼
200 keV were observedalready on Sep 20 09:00 UT, whereas (cid:38)
500 keV protonintensities did not increase until mid 21 Sep. Low-energy( (cid:46)
500 keV) proton intensities continued to be elevateduntil 23 Sep 12:00 UT.To simulate these particle populations, a seed popu-lation of 40 keV protons is injected into both the RCWand the FCW bounding the SIR. The PARADISE modelpropagates these protons through the EUHFORIA solarwind, taking into account the effect of small-scale so-lar wind turbulence on the energetic particles. The lat-ter is done using results from quasi-linear theory, whichtranslates the scattering processes of particles by mag-netic fluctuations into an anisotropic pitch-angle diffu-sion process in the reference frame co-moving with thesolar wind (Jokipii 1966). This diffusion process drivesthe particle acceleration in the solar wind regions where self-consistent simulation of a SIR event Figure 3.
Observed (left) and simulated (right) omnidirectional ion intensities at STEREO-A. The vertical lines indicate theonset time of the SIR event (Sep 20 09:00 UT), the SI (Sep 21 09:30 UT), the developing RS (22 Sep 01:35 UT), and the stoptime of the SIR event (23 Sep 12:00 UT). ∇ · V sw <
0, as it models the interaction of particleswith converging scattering centres (e.g., le Roux & Webb2012; Zank 2014). In addition to pitch-angle diffusion, aweak spatial diffusion process perpendicular to the IMFis considered. The precise set-up of the PARADISE sim-ulation is described in Appendix B.Figure 3b shows the simulated intensity profiles, un-der the assumption that the protons propagate withconstant mean free paths along and across the IMFof λ (cid:107) = 0 . λ ⊥ = 10 − au, respectively. Theassumption that λ ⊥ /λ (cid:107) ∼ − implies that the ener-getic particles are predominantly propagating along theIMF lines. Figure 3b shows that the PARADISE sim-ulation successfully reproduces several features seen inFigure 3a, specifically: (1) the onset and ending timesof the SIR event; (2) the double-peaked structure of theintensity-time profiles, with the second peak showing thehighest intensities; (3) the soft energy spectrum of thefirst peak compared to the second peak, with the firstpeak showing very few protons above ∼
500 keV; (4) thesudden strong increase in the particle intensities of thetwo lowest energy channels around 21 Sep 10:00 UT be-fore reaching a second peak; (5) the gradual increase inthe two high-energy channels, hence producing an over-all energy spectrum that hardens with the passing ofthe SIR; (6) the fact that intensity increases are beingobserved at MeV energies; and (7) the energy spectrumobtained from integrating the intensity over the entireevent that has a power-law dependence of E − . ± . in the data and E − . ± . in the simulation. Here, E denotes the particle energy and the power-law fit wasperformed for the energy range 84.1 – 1985.3 keV.The double-peaked profile is a result of the particlesbeing accelerated at either the FCW or the RCW. TheFCW is weaker than the RCW, and hence less efficientin accelerating particles to high energies. Because ofthe ∇· V sw configuration (Figures 2b), particle accelera- tion occurs from ∼ ∼ (cid:46)
90 keV) are already acceleratedwithin 1 au. In addition, since our solar wind simula-tion underestimates the speed gradient at the locationof PSP, it is very likely that, in reality, more particleswere already accelerated within 0.8 au.The observed particle intensities show a sudden spikeon 22 Sep 01:35 UT, which coincides with the passageof the developing RS. However, in the EUHFORIA sim-ulation, no shock wave is present as the RCW is stillsteepening beyond 0.95 au. Only at radial distances be-yond 1 au does the RCW steepens enough to efficientlyaccelerate particles to MeV energies. This explains why,in the simulation, the MeV energy channel peaks too latecompared to the data. In contrast, the peak intensitiesof the two lowest energy channels in Fig. 3b coincideswith the passage of the simulated FSW and the RSW,indicating that these particles are accelerated locally.In the simulation, the widths of the compression re-gions remain larger than the widths of the observed tran-sitions due to the size of the EUHFORIA computationalgrid. However, the diffusive length scale of the particlesacross the compression waves is, in the simulation, sub-stantially larger than the width of the compression wavesitself. As a result, particles accelerate similarly to a dif-fusive shock acceleration process (Giacalone et al. 2002;Wijsen et al. 2019a). The good agreement between thesimulated and observed energy spectra, confirms thatshock waves are not required for producing energeticparticle enhancements near SIRs.The intensity dip observed around Sep 21 09:30 UT co-incides with the stream interface (SI) of the SIR, that is,the region inside the SIR where the compressed fast so-
Wijsen et al. lar wind meets the compressed slow solar wind (Goslinget al. 1978). The SI separates the two energetic par-ticle populations that are accelerated at the FCW andthe RCW. The IMF lines directly adjacent to the SIare located inside the developing SIR from small radialdistances onward, and as a result, they never cross theFCW or the RCW where particles may accelerate. In asimulation with zero cross-field diffusion and no parti-cle drifts, the suprathermal particle intensity would thusdrop to zero at the SI. In contrast, if the cross-field dif-fusion would be more efficient, the two intensity peakswould merge into a single enveloping peak (Richardson1985; Wijsen et al. 2019c). Since this is not the case inthe observed intensity profiles, strong cross-field diffu-sion near the SI is inconsistent with the observations.The observed intensities show a sudden discontinuityon 22 Sep 7:00 UT, which coincides with a rotation inthe magnetic field vector (not shown here). This small-scale structure is not resolved by EUHFORIA and hencethe same is true for PARADISE.Finally, Fig. 3a shows that the particle intensities inthe two lowest-energy channels had an additional smallthird increase on 23 Sep 01:00 UT. In the simulation,this date corresponds to the time when the magneticconnection of STEREO-A is skimming the northernedge of the second RCW shown in Fig. 2.2.3.
Energetic Particles at PSP
Figure 4 shows the simulated particle intensities atPSP (red line) together with the count rate of energeticions measured by the Time-of-Flight (ToF) system ofthe EPI-Lo instrument (McComas et al. 2016) on boardPSP (blue dots). We use hourly averages accumulatedover all the apertures of EPI-Lo and consider the energyranges provided in the NASA Space Physics Data Facil-ity . EPI-Lo data was only available for those periodsplotted in Figure 4, suggesting that intensity enhance-ments were present between 19 Sep 14:00 UT and 21Sep 23:30 UT.In our simulation, the omnidirectional intensities atPSP peak around 21 Sep 00:00 UT, which does not agreewith the decreasing trend of the count rates observed atthat time. However, we recall that, in the solar windsimulation, the transition from the slow to fast solarwind is inferred to be too smooth at PSP. In particular,the time difference between the peaks of the dashed andsolid lines in Fig.1a is ∼ cdaweb.gsfc.nasa.gov the event onset and the decay phase. This further con-firms that the solar wind speed gradient at PSP mostlikely was nearly as steep as the one observed at STA.Such a steep speed gradient indicates that particle accel-eration may already have been occurring inside 0.5 au.However, the shifted curve in Fig. 4 shows a first peakon Sep 18 associated with the FCW, suggesting thatthis enhancement was likely below the background levelof the instrument, and indicating that the accelerationprocess at the FCW was not efficient at ∼ . Three-dimensional Proton Distributions
Figure 5 shows the omnidirectional particle intensityon spherical shells located at 0.5 au (top row) and 1.5 au(bottom row) for the energy channels 84.1 - 92.7 keV(left) and 496.4 - 554.8 keV (right). These figures showhow the spatial dependence of the particle population isstrongly modulated by the shape of the HSS. Whereas ∼ ∼
80 keV particles extend also to the slower so-lar wind. At low energies, the maximum intensities at1.5 au are centred around the RCW and FCW, whereasat 0.5 au the particles are more dispersed and two in-tensity peaks are attained within the slow solar windin front of the FCW and in the fast solar wind streambehind the RCW. Such a spatial dependence is oftenobserved in SIR events (see Richardson 2018, and refer-ences therein) and is a consequence of nonlocal particleacceleration.Figure 5 also illustrates how the magnetic connec-tion of PSP (red triangles) only crosses the southwardedge of the intensity peak associated with the FCW inthe simulation. If the spacecraft had been located atslightly higher latitude, it would have observed signif-icantly higher particle intensities at the FCW, accord-ing to the simulation. This might further explain whyPSP did not see any particle enhancement associatedwith the FCW (Figure 4). The strong latitudinal de-pendence of the particle distribution follows from theintricate structure of the fast solar wind stream (Fig-ure 2c). This illustrates that two spacecraft located atslightly different latitudes can see very different particlepatterns. This finding is in agreement with the resultsof Jian et al. (2019), who analysed 151 pairs of SIRs seenby STEREO A and B and showed that, even within 5 ◦ of latitude, the solar wind properties of a single SIR canbe strongly variable. SUMMARY AND CONCLUSIONIn this work, we have presented the simulation resultsof the SIR event observed by both PSP and STEREO-Ain September 2019. By coupling the energetic particle self-consistent simulation of a SIR event Figure 4.
Count rates of 30-585 keV ions measured by the Time-of-Flight (ToF) system of the EPI-Lo instrument on PSP (bluedots) and the simulated 50-554 keV proton omnidirectional intensity (red solid line), along with the same simulated intensitybut shifted 18h in time (red dashed line)
Figure 5.
Rainbow-colour maps showing the solar wind speed at r = 0 . r = 1 . Wijsen et al. model PARADISE to the MHD model EUHFORIA, wewere able to model a solar wind configuration and ener-getic particle enhancements that are in good agreementwith the observations of STEREO-A and PSP. In oursimulations, the energetic protons are accelerated self-consistently at the compression waves bounding the SIR,assuming a seed population of 40 keV protons. Sucha seed population may originate from the solar windsuprathermal tail, especially near compression or shockwaves where the solar wind gets heated adiabatically,thus producing more suprathermal protons (Prinslooet al. 2019). Our simulations are consistent with the hy-pothesis that energetic particle enhancements near SIRsare mainly produced by the diffusive acceleration of so-lar wind suprathermal tail particles at the compressionwaves bounding SIRs. Low-energy particles ( (cid:46)
500 keV)can already accelerate at small radial distances, beforethe compression waves have steepened sufficiently to ac-celerate MeV particles. Therefore, particles observed ata given heliospheric location may come from different re-gions, with the higher-energy particles being acceleratedat larger helioradii. The stronger RCW (Fig. 2b) leadsto a more efficient acceleration of high-energy particlesnear the trailing edge of the SIR. For these reasons, thesimulated energy spectrum hardens toward the end ofthe SIR event, in agreement with the observations.Earlier models predicted that the energy spectrum as-sociated with SIR events would exhibit a turnover at lowenergies ( < . self-consistent simulation of a SIR event (cid:12) IS/EPI-Lo and magneticfield and plasma data available at https://cdaweb.gsfc.nasa.gov. Parker Solar Probe was designed, built, andis now operated by the Johns Hopkins Applied PhysicsLaboratory as part of NASA’s Living with a Star (LWS)program (contract NNN06AA01C). Support from theLWS management and technical team has played a crit-ical role in the success of the Parker Solar Probe mission.APPENDIX A. EUHFORIA SETUPThe solar wind is modelled by using EUHFORIA, a physics-based coronal and heliospheric model designed for spaceweather research and prediction purposes (Pomoell & Poedts 2018). The coronal module of EUHFORIA uses as inputa synoptic magnetogram from the Global Oscillation Network Group (GONG) and applies the semi-empirical Wang–Sheeley–Arge model (WSA; Arge et al. 2004) to provide the solar wind plasma and magnetic quantities at 0 . r, ∆ ϑ, ∆ ϕ ) = (1 . R (cid:12) , ◦ , ◦ ),where r , ϑ , and ϕ denote the radial, latitudinal, and longitudinal coordinates, respectively.In this study, the GONG magnetogram of 2019-09-18 06:14UT is used as input to the coronal module and thestandard setup (Pomoell & Poedts 2018) of EUHFORIA is used, except for the following three changes:1. A constant value of 30 km/s is added to the resulting WSA speed profile, and in addition, the speed profile iscapped to be in the range [340 , ◦ instead of 10 ◦ , to account for rotation of the coronal magneticfield.3. The magnetic field polarity is assumed to be everywhere positive at 0.1 au.The first change is necessary to avoid a systematic underestimation of the slow and fast solar wind speed in thevicinity of the SIR. The second change is performed to ensure that the simulated arrival time of the HSS at STEREO-A corresponds with the observed arrival time. The third change is done to avoid having a heliospheric current sheet(HCS) in the simulation, which is required by PARADISE as explained in the next paragraph. B. PARADISE SETUPThe PARADISE (Wijsen 2020) model is used to calculate the temporal and spatial evolution of energetic particle pop-ulations that propagate through EUHFORIA solar wind configurations. This is achieved by solving the 5-dimensionalfocused transport equation (FTE; e.g., Roelof 1969; Skilling 1971; le Roux & Webb 2012), which takes into accountthe effects of solar wind turbulence through a set of phase-space diffusion processes. The FTE utilised in PARADISEis expressed in guiding centre (GC) coordinates, and contains therefore the effects of GC drifts and cross-field diffusion(e.g., Zhang 2006; Wijsen 2020; Strauss et al. 2020). The GC drifts have, however, a negligible effect on the transportof the low-energetic protons studied in this work (Wijsen et al. 2020). The FTE accounts for particles’ pitch-angleand momentum changes in compressional, shear, and accelerating solar wind flows (e.g., le Roux et al. 2007; Zank2014). In particular, it takes into account the acceleration of particles in high-amplitude compression and shock waves(Giacalone et al. 2002; Wijsen et al. 2019a).The FTE assumes that the solar wind varies on temporal and length scales that are larger than the energeticparticles’ gyroperiods and gryoradii, respectively. These assumptions may not be valid near the HCS (Burger et al.1985; Wijsen 2020), which is why we do not include a HCS in our solar wind simulations.0
Wijsen et al.
Figure 6.
Simulated 94-1985 keV proton peak intensities as a function of heliocentric radial distance for different parallelmean free paths. The intensities have been normalised to their value at 1 au. The black dots and the grey squares correspondto SIR events observed by Helios (Van Hollebeke et al. 1978) and SolO (Allen et al. 2020c), respectively.
In this study, 40 keV protons are injected in solar wind regions where the divergence of the solar wind velocity isnegative, i.e., ∇ · V sw <
0. As illustrated in Fig. 2, these regions include the RCW and the FCW bounding the SIR.In addition, we scale the injected particle distribution so that f inj ( t, x ) ∝ |∇ · V sw | (Prinsloo et al. 2019). This isdone because a negative ∇ · V sw gives a measure of local compression and therefore of adiabatic heating of the solarwind. In addition, regions where ∇ · V sw < . πr sin( θ )∆ ϑ ∆ ϕ ∆ r ∆ E ∆ µ , where E denotes the energy coordinate, µ the cosine of the pitch-angle, and ( r, ϑ, ϕ ) are spherical coordinates. We choose∆ r = 0 .
02 au, ∆ ϑ = ∆ ϕ = 0 . ◦ and ∆ µ = 0 .
1. The energy E is computed in the HEEQ reference frame uponsampling the pseudo-particles.The FTE includes the effect of turbulence on the energetic particle transport through a set of diffusion processes inphase space. In this work, we include a cross-field diffusion process and a pitch-angle diffusion process. A constantperpendicular mean free path λ ⊥ = 10 − au is assumed to describe the cross-field diffusion coefficient as κ ⊥ = vλ ⊥ .For the pitch-angle diffusion, we prescribe an anisotropic diffusion coefficient based on Quasi-Linear theory (Jokipii1966), as implemented in PARADISE (Wijsen et al. 2019a). The magnitude of the pitch-angle diffusion is fixed byassuming a constant parallel mean free path λ (cid:107) everywhere in the heliosphere.Figure 6 shows the radial variation of the 84 - 1985 keV proton peak intensities for different parallel mean freepaths. The intensities are calculated as a six-hour average around the peak intensity seen by virtual spacecraft thatare radially aligned and located at 2 . ◦ latitude (HEEQ). The intensities have been normalised to the value attainedat 1 au. The dots give the 0.96 - 2.2 MeV proton peak intensities determined by Van Hollebeke et al. (1978) of SIR self-consistent simulation of a SIR event He intensities determined by Allen et al. (2020c) of SIR events observed by SolO,relative to the intensities of the corresponding SIR events observed by the Advanced Composition Explorer (ACE;Stone et al. 1998) at 1 au. Despite the differences in ion species and energy ranges, a good match is obtained betweenthe simulations and the observations. Data of the Sep 2019 SIR event discussed in this work are not included in thefigure, since PSP did not observe the peak intensity. A mean free path of 1.0 au produces high intensities at smallradial distances, since for such a large mean free path, the particles are only weakly coupled to the solar wind plasma.As a result, magnetic focusing is the dominant process in reversing the propagation direction of sunward streamingparticles. In contrast, for the mean free path of 0.09 au, particles are more tightly coupled to the solar wind plasma,and consequently they are efficiently advected with the solar wind toward larger radial distances, resulting in particleintensities that decrease strongly with decreasing radial distance. Most observations included in Fig. 6 fit a parallelmean free path in the range 0 .