Magnetic Cloud and Sheath in the Ground-Level Enhancement Event of 2000 July 14. I. Effects on the Solar Energetic Particles
aa r X i v : . [ phy s i c s . s p ace - ph ] O c t A CC EP T E D D raft version O ctober
16, 2020Typeset using L A TEX manuscript style in AASTeX63
Magnetic Cloud and Sheath in the Ground-Level Enhancement Event of 2000 July 14. I. E ff ects onthe Solar Energetic Particles S.-S. W u and G. Q in ∗ School of Science, Harbin Institute of Technology, Shenzhen, 518055, China; [email protected]
Submitted to ApJABSTRACTGround-level enhancements (GLEs) generally accompany with fast interplanetary coronalmass ejections (ICMEs), the shocks driven by which are the e ff ective source of solar energeticparticles (SEPs). In the GLE event of 2000 July 14, observations show that a very fast andstrong magnetic cloud (MC) is behind the ICME shock and the proton intensity-time profilesobserved at 1 au had a rapid two-step decrease near the sheath and MC. Therefore, we studythe e ff ect of sheath and MC on SEPs accelerated by an ICME shock through numericallysolving the focused transport equation. The shock is regarded as a moving source of SEPswith an assumed particle distribution function. The sheath and MC are set to thick sphericalcaps with enhanced magnetic field, and the turbulence levels in sheath and MC are set to behigher and lower than that of the ambient solar wind, respectively. The simulation resultsof proton intensity-time profiles agree well with the observations in energies ranging from ∼ ∼
100 MeV, and the two-step decrease is reproduced when the sheath and MC arrivedat the Earth. The simulation results show that the sheath-MC structure reduced the protonintensities for about 2 days after shock passing through the Earth. It is found that the sheath
Corresponding author: G. [email protected] W u and Q in contributed most of the decrease while the MC facilitated the formation of the second stepdecrease. The simulation also infers that the coordination of magnetic field and turbulence insheath-MC structure can produce a stronger e ff ect of reducing SEP intensities. Keywords:
Sun: particle emission — Sun: coronal mass ejections (CMEs) — interplanetarymedium — methods: numerical INTRODUCTIONThe solar energetic particle (SEP) events, especially ground-level enhancements (GLEs), are one of thesources of space radiation harmful to the safety of spacecraft and the health of astronauts (e.g., Lanzerotti2017; Mertens et al. 2018; Mertens & Slaba 2019). Therefore, it is important to study the acceleration andpropagation of SEPs both in observation and theory. Over the past several decades research in this field hasmade significant progress.From the observation characteristics SEP events can be divided into two categories: impulsive and gradualones (e.g., Cane et al. 1986; Reames 1999, 2017; Cliver 2009). Impulsive events are believed to be causedby solar flares with low intensity and short duration. On the other hand, gradual events are related to theshocks which are driven by coronal mass ejections (CMEs) and can continuously release particles fromcorona to interplanetary space, so that they usually last longer with higher flux. In addition, each of the twocategories can be further divided into two sub-categories according to recent research (Reames 2020).Based on the classification according to observations, numerical simulations for SEP events with eitherimpulsive sources or continuous sources are carried out (e.g., Dr¨oge 2000; Zhang et al. 2009; Dresing et al.2012; Qin & Wang 2015; Qi et al. 2017; Hu et al. 2018). The modeling work, focusing on the transport ofSEPs in interplanetary space, can be used to deal with a lot of problems such as the e ff ects of adiabatic cool-ing (e.g., Qin et al. 2006), perpendicular di ff usion (e.g., Zhang et al. 2009; Wang et al. 2012; Dresing et al.2012), the reservoir phenomenon (e.g., Zhang et al. 2009; Qin et al. 2013), and the release time of SEPsnear the Sun (e.g., Diaz et al. 2011; Wang & Qin 2015). ∗ Author of correspondence.
Most of GLE events belong to the gradual category, usually accompanying with fast interplanetary coro-nal mass ejections (ICMEs) (Gopalswamy et al. 2012) that are the interplanetary counterpart of CMEs(Luhmann et al. 2020). ICMEs are macro-scale structures, thereby being able to drive many types ofspace weather disturbances, such as geomagnetic storms and Forbush decreases (Fds) in galactic cosmicray (GCR) intensities (Cane 2000; Richardson & Cane 2011; Gopalswamy 2016). The ICMEs can be iden-tified by specific plasma, compositional, and magnetic field signatures. Particularly, if the signatures exhibitstrong and smooth magnetic field, coherent rotation of the magnetic field components, and low proton tem-perature and plasma β values, one can identify a magnetic cloud (MC) embedded in the ICME (Burlaga et al.1981; van Driel-Gesztelyi & Culhane 2009; Richardson & Cane 2010). ICMEs can drive shocks if theirspeed is su ffi ciently faster than the preceding solar wind, and the shock is an e ff ective accelerator of chargedparticles thus producing the gradual SEP event. The region between the shock and ICME’s leading edge iscalled sheath, in which the turbulence level is greater than that in the ambient solar wind due to the fact thatthe magnetic field lines are highly compressed by the ICME and shock.Many research have demonstrated that MCs and turbulent sheath regions can cause Fds (e.g.,Zhang & Burlaga 1988; Cane 1993; Yu et al. 2010; Jordan et al. 2011; Richardson & Cane 2011; Luo et al.2017, 2018) as the result of modulating the intensity of GCRs, so that the other type of energetic particles,SEPs, accelerated by ICME-driven shocks may also be significantly a ff ected by MCs and sheath regions.Therefore, to evaluate the influence of MC and sheath one can better predict SEP intensities. Consequently,based on the prediction of solar activity (e.g., Petrovay 2010; Qin & Wu 2018), the prediction of the trend ofGLE events on solar cycle scale, which is important for preventing major radiation hazards, can be promoted(e.g., Miroshnichenko 2018; Wu & Qin 2018).In this paper, we use the numerical code denoted as Shock Particle Transport Code (SPTC) developedby Wang et al. (2012) based on a stochastic di ff erential equation approach (Zhang 1999; Qin et al. 2006)to study the e ff ects of MC and sheath on SEPs released by an ICME shock, the simulation results will becompared with the observations of GLE59, which occurred on 2000 July 14 and was accompanied witha very fast and strong MC (Lepping et al. 2001). In Section 2, the observational features of GLE59 arepresented, while the simulation model is elaborated in Section 3. We show our simulation results and W u and Q in compare them with observations in Section 4. Conclusions and discussion are presented in Section 5. Notethat, besides this work, we use the same MC and Sheath model to study the Fd occurred following theGLE59, and reproduce the observed Fd successfully (Qin & Wu 2020). OBSERVATIONSThe proton intensity-time profiles of GLE59, the fifth GLE event in solar cycle 23, are exhibited in Fig-ure 1(a) with observations from the Electron, Proton, and Alpha Monitor (EPAM) (Gold et al. 1998) onboard
ACE and Energetic Particle Sensor (EPS) (Onsager et al. 1996) onboard
GOES 8 . Figures 1(b) − Wind spacecraft.In Figure 1(a), there was an X5 . ∼ ∼ / nucleon for ions (e.g., Mazur et al. 2000; Wang et al. 2014; Tan 2017). InGLE59, the decline phase can be observed even in more than a hundred MeV protons, so that it is di ff erentfrom the dropout phenomenon. It is shown that the decline phase has a two-step decrease structure whicha classical two-step Fd has (e.g., Cane 2000) for high energy channels between the MC’s leading edge andthe shock, such as the gray and orange curves. The first step occurred right after the shock arrival, whichindicates that the first step might be caused by the turbulent sheath region. Considering the fact that thesecond step appeared near the arrival time of the MC and the recovery of proton intensities was close to thedeparture time of the MC, we assume that the second step was caused by the MC. In the following, we willreproduce the observed proton intensity-time profiles in simulation considering the e ff ects of sheath andMC that are placed behind an ICME shock.The flare information is from Gopalswamy et al. (2012), and the shock information is from http: // / mag / ace / ACElists / obs list.html SEP TRANSPORT MODELThis section focuses on the simulation model, including the configurations of IMF, shock, MC, and sheath,transport equation, and di ff usion coe ffi cients.3.1. IMF, Shock, MC, and Sheath
The Parker field is adopted as solar wind magnetic field and given by B P = AB P0 (cid:18) r au r (cid:19) e r − ω r sin θ V sw e φ ! , (1)where A = ± B P0 is the radial component of solar wind magnetic field at 1 au, r au is aconstant and equals to 1 au, r , θ , and φ are the solar distance, polar angle, and azimuthal angle of any pointin a non-rotating heliographic coordinate system, respectively, ω is the angular speed of solar rotation, and V sw is solar wind speed.In SPTC, the shock is treated as a spherical cap with uniform speed for releasing SEPs, and the longitudeand latitude of shock nose are set to the same as those of the corresponding solar flare. The distribution func-tion of the source f b ( t , r ) at time t and position r is given by the following equation (Kallenrode & Wibberenz1997; Wang et al. 2012) f b = f δ ( r − v s t ) (cid:18) R in r (cid:19) α p exp " − Ω ( θ, φ ) Ω p p γ p ( Ω ≤ Ω s ) , (2)where f is a constant, v s is shock speed, t is time, R in is inner boundary, α p and Ω p are the attenuationcoe ffi cients of shock strength in radial and angular directions, Ω is the angular width from shock nose to the W u and Q in position r , Ω s is the half angular width of shock, p is the momentum of particles, and γ p is spectral indexthat varies with p . The attenuation coe ffi cients α p and Ω p are functions of the momentum of particles, i.e., α p = α × pp ! η α , (3) Ω p = Ω × pp ! η Ω , (4)where α , Ω , η α , η Ω , and p are constants with p = .
78 MeV / c , and c is the speed of light. Note that, thespectral index γ p is not the energy spectrum index since other parameters, i.e., α p and Ω p in Equation (2)are also functions of energy.In this work, the MC and sheath are set as thick spherical caps behind the shock, with the same direction,velocity, and angular width as those of the shock. On the one hand, Figure 1(b) shows that the magneticfield in sheath-MC structure is greater than that in the ambient solar wind, so that Parker field is not suitablefor representing the magnetic field in this area. On the other hand, it is hard for us to give a self-consistentanalytical magnetic field model to describe the complex three-dimensional magnetic field in sheath-MCstructure. For simplicity, the magnetic field in sheath-MC structure is set to the Parker field B P plus amagnetic field enhancement in radial direction B ejecta = B P + A ∆ B r e r , (5)where ∆ B r can be expressed by the sum of a set of delta-like functions and can be written as ∆ B r = k X i = ∆ B ir , (6) ∆ B ir = B ir r au v s t ! δ n v s t + δ r i − rw i ! , (7) δ n ( x ) = (cid:16) − x (cid:17) n for x ∈ [ − , , , (8)where B ir , δ r i , w i ( i = , , , ..., k ), and n are constants obtained by fitting the observed magnetic fieldwith Equation (5). Figure 2(a) shows the fitting result, and the black solid and red dashed curves arethe observed and fitted magnetic field, respectively. The fitted magnetic field is the sum of the magneticenhancements represented by the colored solid curves and Parker field. Each of the colored solid curves isgiven by Equation (7), and the fitted coe ffi cients are listed in Table 1. Note that, one can find the integralof the divergence of magnetic field enhancement in sheath-MC structure along the radial direction equalsto zero, but ∇ · (cid:16) ∆ B ir e r (cid:17) is not zero in most parts of the sheath-MC structure. Panels(b)-(c) in Figure 2present the comparison between the observed and modeled polar and azimuthal angles of IMF with blacksolid and red dashed lines. It is shown that the polar and azimuthal angles of the simplified magnetic fieldin sheath-MC structure can not fit the observed ones due to the fact that the observed magnetic field insheath-MC is mostly in the azimuthal direction. However, we use this simplified analytical magnetic fieldfor the preliminary study of the transport of SEPs focusing on the general characteristics of the e ff ect of thesheath-MC, for example, the magnetic mirror e ff ect (e.g., Reames et al. 1997; Bieber et al. 2002; Tan et al.2009).Figure 2(d) shows the sectional view of the IMF, shock, sheath, and MC through the ecliptic plane,represented by the black spiral curves, red arc, thick yellow cap, and thick green cap, respectively. Thespiral curves in sheath-MC structure are plotted with dashed lines to indicate that the magnetic field is notParker field in this area. 3.2. Transport Equation
We use the SPTC to model the transport of SEPs based on the previous studies (e.g., Qin et al. 2006;Zhang et al. 2009; Wang et al. 2012). The focused transport equation in three-dimensional space is writtenas (Skilling 1971; Schlicheiser 2002; Qin et al. 2006; Zhang et al. 2009) ∂ f ∂ t + (cid:16) v µ ˆ b + V sw (cid:17) · ∇ f − ∇ · ( κ ⊥ · ∇ f ) − ∂∂µ D µµ ∂ f ∂µ ! − p " − µ (cid:16) ∇ · V sw − ˆ b ˆ b : ∇ V sw (cid:17) + µ ˆ b ˆ b : ∇ V sw ∂ f ∂ p + − µ (cid:20) − vL + µ (cid:16) ∇ · V sw − b ˆ b : ∇ V sw (cid:17)(cid:21) ∂ f ∂µ = , (9)where f ( x , µ, p , t ) is the gyrophase-averaged distribution function and x is the particle position in a non-rotating heliographic coordinate system, v and µ are the speed and pitch-angle cosine of particles, respec-tively, V sw = V sw e r is the solar wind velocity, κ ⊥ and D µµ are the perpendicular and pitch-angle di ff u- W u and Q in sion coe ffi cients of particles, respectively, L = (cid:16) ˆ b · ∇ ln B (cid:17) − is the magnetic focusing length due to thein-homogeneous magnetic field, and ˆ b is the unit vector along the local background magnetic field withstrength B . The equation includes the most of the particle transport mechanisms, i.e., particle streamingalong magnetic field line and solar wind flowing in the IMF (2nd term), perpendicular di ff usion (3rd term),pitch-angle di ff usion (4th term), pitch-angle dependent adiabatic cooling by the expanding solar wind (5thterm), and focusing (6th term). We use a time-backward Markov stochastic process method to solve Equa-tion (9) (Zhang 1999; Qin et al. 2006). 3.3. Di ff usion Coe ffi cient The model of pitch-angle di ff usion coe ffi cient D µµ is set as (Beeck & Wibberenz 1986;Teufel & Schlickeiser 2003) D µµ ( µ ) = δ b slab B ! π ( s − s vl slab R L l slab ! s − (cid:16) µ s − + h (cid:17) (cid:16) − µ (cid:17) , (10)where δ b slab is the slab component of magnetic turbulence, l slab is the correlation length of δ b slab , s = / R L = pc / ( | q | B ) is the Larmorradius with the charge of particle q , and h is a constant for modeling the non-linear e ff ect of pitch-angledi ff usion at µ =
0. The parallel mean free path λ || is given by (Jokipii 1966; Hasselmann & Wibberenz1968; Earl 1974) λ k = v Z + − (cid:16) − µ (cid:17) D µµ d µ. (11)The perpendicular mean free path λ ⊥ is defined from the perpendicular di ff usion coe ffi cient κ ⊥ for conve-nience λ ⊥ ≡ κ ⊥ v , (12)and from nonlinear guiding center theory (Matthaeus et al. 2003) with analytical approximations(Shalchi et al. 2004, 2010) we have λ ⊥ = δ b B ! √ π s − s Γ (cid:16) s + (cid:17) Γ (cid:16) s + (cid:17) l D / λ k / , (13)where δ b is the 2D component of magnetic turbulence, and l is the correlation length of δ b . Inaddition, the perpendicular di ff usion coe ffi cient κ ⊥ can be written as κ ⊥ = κ ⊥ (cid:16) I − ˆ b ˆ b (cid:17) .The turbulence level is given by σ ≡ δ bB = q δ b + δ b B . (14)The ratio of 2D energy to slab energy is found to be 80%:20% (Matthaeus et al. 1990; Bieber et al. 1994)and widely used in the literature (e.g., Zank & Matthaeus 1992, 1993; Hunana & Zank 2010). By using therelation δ b = δ b , the turbulence levels of slab and 2D components in solar wind, sheath, and MC canbe written as δ b slab B ! i = √ σ i ( i = P, S, M) , (15) δ b B ! i = √ σ i ( i = P, S, M) , (16)where P, S, M denote solar wind, sheath, and MC. Because the turbulence level in MC and sheath is lessand greater than that in solar wind, the value of σ M and σ S should set to be lower and higher than that of σ P , respectively. SIMULATIONS AND COMPARISONS WITH OBSERVATIONS4.1.
Parameter Settings
Table 2 lists the main parameters in the simulations in this work. The half angular width of the shock, Ω s ,is set to 45 ◦ . The shock speed is set to 1406 km / s, which is calculated by dividing the Sun-Earth distanceby the shock transit time measured from the flare onset to the 1 au shock arrival. In addition, 450 km / s ischosen as solar wind speed V sw . In order to make the solar wind magnetic field strength B P equal to 5 nT at1 au, the radial strength of solar wind magnetic field at 1 au, B P0 , is set to 3.62 nT. The half thickness of MC, L M , is set to 0.22 au, and the distance d M between the central position of MC and the shock is set to 0.45 au,so that the arrival and departure times of MC agree with the observations obtained by Richardson & Cane(2010). The half thickness of sheath, L S is set to 0.08 au according to the passages of shock and ICME’sleading edge. Furthermore, the angular speed of solar rotation is set to ω = π/ . / day, and the innerand outer boundaries of the simulation are set to R in = .
05 au and R out =
50 au, respectively.The parameters of magnetic turbulence are listed in Table 3. We set l slab = .
025 au, and thus l equals to l slab / . = . u and Q in turbulence levels in solar wind, sheath, and MC are set to 0.3, 1.6, and 0.1, respectively. The Kolmogorovspectral index of the IMF turbulence, s , equals to 5 / ff ect index, h , is set to 0.01.The other shock parameters are obtained by fitting simulated proton intensity-time profiles to observations.Firstly, the attenuation coe ffi cient of shock strength in angular direction, Ω p has relatively low influenceon proton time-intensity profiles, and thus we give certain values manually for attenuation constant Ω and the corresponding power-law index η Ω . Secondly, the attenuation coe ffi cient in radial direction, α p ismore important than other parameters to the shape of proton time-intensity profiles, so that the attenuationconstant α and the corresponding power-law index η α can be derived by fitting the shape of simulatedtime-intensity profiles to that of the observed ones with a certain spectral index γ p . Thirdly, γ p is fitted foreach energy channel with the magnitude of simulated proton time-intensity profile and that of observed onebased on the derived α and η α . Finally, we fine-tune all parameters to obtain the best fitting results. Theattenuation constants α and Ω are equal to 0.5 and 10 ◦ , respectively, and the power-law indices η α and η Ω are equal to 0.86 and 0, respectively. The spectral index γ p for the six energy channels as shown in Figure 1equal to -10.0, -12.2, -15.7, -20.6, -27.6, and -40.2, respectively.Figure 3 shows the values of distribution function at di ff erent heliocentric distances along the shock nosedirection calculated by using the fitted shock parameters. It is shown that the values of distribution functioncan be represented by the power-law shape or the power-law shape with an exponential tail, which is con-sistent with the result of shock acceleration studies (e.g., Ellison & Ramaty 1985; Giacalone & K´ota 2006;Zuo et al. 2011; Kong et al. 2019). 4.2. Results
The simulation results are presented in Figure 4 where the black solid curves are the observations whilethe red solid curves represent the simulation results obtained by incorporating the MC and sheath into theSPTC. The pink vertical line denotes the flare onset, before which the observed proton intensities are usedto obtain the backgrounds that are added to the simulated fluxes. The green vertical line represent theshock arrival while the blue vertical lines show the arrival and departure times of MC. It is shown that the1simulations can fit the observations well, and the two-step decrease is reproduced between the arrivals ofshock and MC’s leading edge.To evaluate the e ff ect of sheath-MC structure on SEPs, the simulation results without MC and sheath arealso presented in Figures 4 by the green dashed curves. It is shown that the shape of the green dashed curvesis the general one exhibited in the literature (e.g., Wang et al. 2012; Qin & Qi 2020) without rapid declinephase after the shock passage of the Earth. The comparison between the green dashed and red solid curvesshows the e ff ect that the sheath-MC structure can reduce SEP intensities when it arrived at the Earth, whichlasted about 2 days. The comparison also indicates that the sheath-MC structure hardly a ff ected the SEPintensities before it reached the Earth.To further explore the respective influences of MC and sheath on SEPs, the simulation results for 11.2MeV protons with only MC or sheath are presented in Figure 5(a) by the blue or orange curves, respectively.All the other lines in Figure 5(a) are the same as that in Figure 4. Figure 5(b) presents the di ff erence betweenthe four curves and the green curve in Figure 5(a). The simulation result with only MC, i.e., the blue curve islower and higher than the green curve before and after the MC arrival, respectively. It is also shown that, thesimulation result with only sheath, i.e., the orange curve is higher and lower than the green curve before andafter the shock arrival, respectively, which is similar to the “di ff usion barrier” e ff ect (e.g., Luo et al. 2017,2018). The simulation result with sheath-MC, i.e., the red curve is almost always lower than the greencurve, and the first decrease is deeper than the second decrease. The event-integrated fluences of the blue,orange, and red curves are 4%, 23%, and 27% less than that of the green curve, respectively. Therefore, thesheath plays an important role in the decrease of SEP intensities, while the MC contributes to the formationof the second step decrease.It is also necessary to investigate the respective impacts of local background magnetic field and turbu-lence level on SEPs. Next, in our study we include the sheath-MC structure. The simulation results withturbulence levels set according to Section 4.1 with the local background magnetic fields the same as thatof ambient solar wind are plotted in Figure 5(c) by the blue curve. The simulation results with only thelocal background magnetic fields di ff erent from that of ambient solar wind is presented by the orange curvein Figure 5(c). The other lines are the same as that in Figure 5(a). Figure 5(d) has the same format as2 W u and Q in Figure 5(b) except that it is originated from Figure 5(c). It is clear that neither the blue curve nor the orangecurve can produce the two-step decrease. We can show that the event-integrated fluences of the blue, or-ange, and red curves are 1%, 16%, and 27% less than that of the green curve, respectively, which indicatesthat the combination of turbulence and the enhancement of local background magnetic field can producestronger e ff ect in the decrease of SEP intensities. CONCLUSIONS AND DISCUSSIONIn this work, we investigate the proton intensity-time profiles observed near the Earth for GLE59. It isshown that, the intensities have a rapid two-step decrease after the ICME shock arriving at the Earth for ∼ ∼
100 MeV protons, which is clearer with higher energy. The two-step decrease is assumed to becaused by the sheath region and MC. To reproduce the phenomenon, a simplified sheath-MC structure isincorporated into the SPTC for simulating the transport of shock accelerated energetic particles.The shock is treated as a moving source of SEPs with uniform speed, which is determined by dividingthe Sun-Earth distance by the shock transit time, and the longitude and latitude of shock nose are set to thesame as that of the corresponding solar flare. For simplicity, the MC and sheath are set as thick sphericalcaps with the direction, speed, and angular width set as the same as that of the shock. The Parker fieldis chosen as the local background magnetic field in solar wind, while the Parker field plus a magneticenhancement represented by Equation (5) is adopted as the local background magnetic field in sheath-MCstructure. Besides, the turbulence levels in MC and sheath are set to lower and higher than that in solarwind, respectively.The simulation results indicate that the observed proton intensity-time profiles of GLE59 can be fittedwell when a sheath-MC structure is placed behind the ICME shock for ∼ ∼
100 MeV protons, and thetwo-step decrease of proton intensities is reproduced when the sheath and MC’s leading edge arrived atthe Earth. Besides, the comparison between the simulation results with and without sheath-MC structureshows that the sheath-MC structure hardly a ff ected the proton intensities before the shock arrived at theEarth, while it reduced the proton intensities after the shock arrival. The reducing e ff ect lasted about 2 days.The first decrease is found to be deeper than the second one. Furthermore, The simulation results withonly sheath or MC infers that the sheath contributed most of the decrease and the MC played an important3role in the formation of the second step decrease. We also investigate that the e ff ect of the combinationof the turbulence and local background magnetic field in sheath-MC structure is greater than the simplysuperimposing of their respective e ff ects on reducing SEP intensities.The simulation results show that the sheath contributed most of the decrease and the first decrease isdeeper than the second one, which can be explained by two reasons. Firstly, the magnetic focusing e ff ectoccurs if magnetic field is in-homogeneous, so that the strong focusing e ff ects in sheath and MC due to therapid change of the local background magnetic fields acting as magnetic mirrors to block the passage ofparticles, thus reducing SEP intensities. The magnetic focusing e ff ect in sheath is greater than that in MCbecause the magnetic field in sheath varies faster than that in MC. Secondly, the turbulence level in sheathis higher than that in MC, resulting in shorter parallel mean free path of energetic particles in sheath thanthat in MC, and consequently the energetic particle intensities will be reduced (e.g., Luo et al. 2017, 2018).It is impossible to measure the time-varying magnetic field of overall space now. For simplicity, the Parkerfield is chosen as the solar wind magnetic field and the Parker field plus a magnetic field enhancement isused as the magnetic field in sheath-MC structure. Furthermore, we use a simplified analytical magneticfield enhancement in sheath-MC model, which is not divergence-free, for the preliminary study of the trans-port of SEPs. The magnetic enhancement is set in radial direction for simplicity since it is di ffi cult for usto provide a self-consistent analytical three-dimensional sheath-MC model, causing the modeled azimuthaland polar angles inconsistent with observations. Since the energetic particles’ di ff usion process dependson the direction of local magnetic field, the inaccuracy of magnetic enhancement direction would makethe simulation result di ff erent from observation. However, we think the magnetic mirror e ff ect from thissimplified model is still successful to reproduce the observational characteristics. It is assumed that we canroughly describe the general characteristics of SEPs flux a ff ected by the sheath and MC with this model.In the future, we may adopt an analytical sheath-MC structure without divergence instead. In addition, toprovide a local background magnetic field, we may use a three-dimensional magnetohydrodynamic simula-tion (e.g., Luo et al. 2013; Pomoell & Poedts 2018; Wijsen et al. 2019) with some divergence-free schemes(Balsara & Kim 2004).4 W u and Q in In this work, the shock is treated as a moving source of energetic particles with a pre-described distribu-tion function Equation (2). According to the fitted values of α and η α in Table 2, the distribution function f b diminishes along the radial direction gradually, and the decay is faster if the energy is higher. It is reasonablebecause high energy particles are believed to be produced near the corona. The research on the accelera-tion of energetic particles (e.g., Kong et al. 2017, 2019; Qin et al. 2018; Kong & Qin 2020) by shocks canprovide a more fidelity source, the including of which in this work may promote the understanding of thee ff ects of sheath and MC.The parallel and perpendicular di ff usion is described by well established models, i.e., Equations (10) − (13). However, the turbulence parameters in these equations are simplified, for example, the radial depen-dent is not considered. With the current progress in turbulence theory by the community (e.g., Zank et al.2018; Zhao et al. 2018; Adhikari et al. 2020), we can better understand the transport of SEPs by usingradial and time dependent turbulence parameters. Besides, The Parker Solar Probe (PSP) can provide near-Sun solar wind and SEP observations (Bale et al. 2019; Kasper et al. 2019; McComas et al. 2019) that canpromote the understanding of the radial evolution of turbulence, ICMEs, shocks, and SEPs, which are im-portant to our studies. Therefore, we will adopt the new achievement about the understanding of solar windturbulence in the future research. ACKNOWLEDGMENTSThis work was supported, in part, under grant NNSFC 41874206. We thank the
ACE
EPAM;
GOES
EPS;
Wind
MFI teams for providing the data used in this paper. The
ACE data are provided by the ACEScience Center and the
GOES data by the NOAA. We appreciate the availability of the
Wind data at theCoordinated Data Analysis Web. The work was carried out at National Supercomputer Center in Tianjin,and the calculations were performed on TianHe-1 (A).REFERENCES
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Day of 2000/07 B (b) Day of 2000/07 −50050 θ (c)
14 15 16 17 18 19 20
Day of 2000/07 ϕ (d) Figure 1.
Observations for GLE59. (a) The proton intensity-time profiles are observed near the Earth for six energychannels ranging from ∼ ∼
100 MeV. The pink and green vertical dashed lines denote the flare onset and thepassages of ICME shocks, respectively. The boundaries of ICME and MC are presented with the red and blue verticalsolid lines, respectively. (b) − (d) are the intensity, polar angle, and azimuthal angle of IMF in GSE angular coordinates,respectively. B (a) ObservationFitting
Day/Hour of 2000/07 −50050 θ (b) Day/Hour of 2000/07 ϕ (c) (d) hockMC heath Figure 2. (a) Fitting (red dashed line) of observed magnetic field (black solid line) in sheath-MC structure withEquation (5), and the red dashed line is the sum of colored solid lines and Parker field at 1 au. The colored solidlines are given by Equation (7), and the coe ffi cients are listed in Table 1. (b)-(c) Comparisons between the polarand azimuthal angles of observed IMF (black solid lines) and those of modeled IMF (red dashed lines), which arecalculated from the fitting result in Figure 2a.(d) A sectional view of the IMF (black spiral curves), shock (red arc), sheath (yellow area), and MC (green area)through the ecliptic plane. u and Q in Energy (MeV) −3 −1 f b Figure 3.
The values of distribution function at shock nose are plotted versus energy for di ff erent heliocentric dis-tances. H / ( c m s r - s e c - M e V ) E = 0.78 MeV
ObservationWith MC-sheathW/O MC-sheath E = 3.00 MeV10 H / ( c m s r - s e c - M e V ) E = 11.2 MeV 10 −1 E = 25.7 MeV14 15 16 17 18 19
Day of 2000/07 −2 H / ( c m s r - s e c - M e V ) E = 56.6 MeV 14 15 16 17 18 19
Day of 2000/07 −3 −1 E = 115 MeV
Figure 4.
Simulation results of GLE59 for six energy channels. The simulation results with and without sheath-MCstructure are presented by the red solid and green dashed lines, respectively, and the black solid lines are the obser-vations. The pink, green, and blue vertical dashed lines denote the flare onset, shock passage, and MC boundaries,respectively. u and Q in H / ( c m s r - s e c - M e V ) (a) With sheath-MCWith only MCWith only sheathW/O sheath-MC −600−400−2000200 (b)
14 15 16 17 18 19
Day of 2000/07 H / ( c m s r - s e c - M e V ) (c) With sheath-MCWith sheath-MC (only turbulence)With sheath-MC (only magnetic)W/O sheath-MC
14 15 16 17 18 19
Day of 2000/07 −500−400−300−200−1000100 (d)
Figure 5.
Simulation results of GLE59 for 11.2 MeV protons. (a) The blue and orange curves show the simulationresults with only MC and sheath, respectively. The other lines are the same as those in Figure 4. (b) The di ff erences ofthe four simulated lines and the green dashed line in Figure 5a. (c) The blue curve presents the simulation result withsheath-MC structure where only turbulence level is di ff erent from that in solar wind, while the orange curve shows thesimulation result with sheath-MC structure where only magnetic field is di ff erent from that in solar wind. The othercurves are the same as those in Figure 5a. (d) The same as Figure 5b but obtained from Figure 5c. Table 1.
Parameters for repre-senting magnetic field enhance-ments in Figure 2(a) with Equa-tions (6) − (8).Color a B r δ r w n Green 10 0.42 0.05 5Pink 10 0.26 0.09 5Orange 4 0.21 0.1 5Blue 24 0.03 0.56 5Red 9 -0.29 0.3 5 a It denotes the color of the col-ored solid curves in Figure 2(a). u and Q in Table 2.
Parameter settings for the simulation.Type Parameter Meaning ValueShock Ω s half angular width 45 ◦ v s speed 1406 km / s α attenuation constant in radial 0.5 η α power-law index of α p Ω attenuation constant in angular 10 ◦ η Ω power-law index of Ω p V sw speed 450 km / s B P0 radial strength of IMF at 1 au 3.62 nT B P | total strength of IMF at 1 au 5 nTMC L M half thickness 0.22 au d M distance between MC center and shock 0.45 auSheath L S half thickness 0.08 auOthers ω angular speed of solar rotation 2 π / / day R in inner boundary of simulation 0.05 au R out outer boundary of simulation 50 au Table 3.
Parameter settings of turbulence for the simula-tion.Parameter Meaning Value σ P turbulence level in solar wind 0.3 σ S turbulence level in sheath 1.6 σ M turbulence level in MC 0.1 l slab slab correlation length 0.025 au l
2D correlation length 0.0096 au s Kolmogorov spectral index 5 / h non-linear e ff ect index 0 ..