Solar Energetic Particle transport near a Heliospheric Current Sheet

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by  Markus Battarbee

Draft version January 17, 2017
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SOLAR ENERGETIC PARTICLE TRANSPORT NEAR A HELIOSPHERIC CURRENT SHEET
Markus Battarbee,1 Silvia Dalla,1 and Mike S. Marsh2
7 1Jeremiah Horrocks Institute, University of Central Lancashire, PR1 2HE, UK
1 2Met Office, Exeter, EX1 3PB, UK
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(Accepted to The Astrophysical Journal on January 15, 2017)

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ABSTRACT
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6 SolarEnergeticParticles(SEPs),amajorcomponentofspaceweather,propagatethroughtheinterplanetarymedium
1 strongly guided by the Interplanetary Magnetic Field (IMF). In this work, we analyse the implications a flat Helio-
spheric Current Sheet (HCS) has on proton propagation from SEP release sites to the Earth. We simulate proton
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h propagation by integrating fully 3-D trajectories near an analytically defined flat current sheet, collecting comprehen-
p sive statistics into histograms, fluence maps and virtual observer time profiles within an energy range of 1–800 MeV.
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We show that protons experience significant current sheet drift to distant longitudes, causing time profiles to exhibit
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c multiple components, which are a potential source of confusing interpretation of observations. We find that variation
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of current sheet thickness within a realistic parameter range has little effect on particle propagation. We show that
p
s IMFconfigurationstronglyaffectsdecelerationofprotons. Weshowthatinourmodel,thepresenceofaflatequatorial
. HCS in the inner heliosphere limits the crossing of protons into the opposite hemisphere.
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s Keywords: Sun: magneticfields–Sun: activity–Sun: particleemission–Sun: heliosphere–methods:
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numerical
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Correspondingauthor: MarkusBattarbee
[email protected]
2 Battarbee et al.
1. INTRODUCTION andGuo&Florinski2014). TheroleoftheHCSinSEP
propagation has previously been briefly investigated in
Asignificantcomponentofspaceweatheristhefluxof
Kubo et al. (2009).
SolarEnergeticParticles(SEPs),acceleratedduringen-
In this paper, we present a first analysis of how the
ergyreleaseeventssuchasflaresandCoronalMassEjec-
presence of the HCS affects the propagation of SEPs
tions (CMEs) at the Sun. These high-energy charged
from the Sun to the Earth. We consider a flat current
particles can, after propagating to the Earth, disrupt
sheet and assess effects of current sheet thickness and
satellite communications and impact astronaut health
different dipole configurations on SEP propagation for
and safety (Turner 2000). Charged particles propagat-
protons of different energies. We also present SEP time
ing through interplanetary space are guided and de-
profiles at virtual observers, providing a basis of com-
flected by the solar wind’s magnetic field and its spatial
parison with real observations.
and temporal variations. Modern efforts in modeling
spaceweathereffectsincludeperformingnumericalsim- 2. HELIOSPHERIC CURRENT SHEET MODEL
ulations to solve particle fluences at the Earth based on
In this work, we model the HCS as a flat plane sep-
parent active region and observer locations (see, e.g.,
arating two hemispheres of opposite polarity, with each
Chollet et al. 2010 and Marsh et al. 2015). The most
hemisphere based on a simple analytical magnetic field
common approach is to use a transport equation (see,
model. We model purely radial outflow of solar wind
e.g., Roelof 1969, Aran et al. 2005, and Luhmann et al.
plasma, which, combined with solar rotation and flux
2007),whereparticlesareeffectivelyboundtotheInter-
freeze-in, results in a non-radial magnetic field. The
planetary Magnetic Field (IMF) lines, described as the
IMF is described through spherical heliocentric coordi-
Parker spiral (Parker 1958).
nates as a scaled Parker spiral magnetic field B
Recent research (Marsh et al. 2013, Dalla et al. 2013,
Dalla et al. 2015) has shown that particle drifts, which B=S(θ)BParker, (1)
arenotmodeledbyaclassicaltransportequation,playa
where the Parker field is defined as
significant role in SEP propagation to the Earth. They
r2
have been shown to be significant for protons and es- B =B 0 (2)
r,Parker 0r2
pecially for heavier elements (Dalla et al. 2017). Other
B =0 (3)
significant factors include field-line meandering (Laiti- θ,Parker
nen et al. 2016) and cross-field diffusion (Zhang et al. B r2Ω sinθ
B =− 0 0 (cid:12) . (4)
2003; He et al. 2011). One significant characteristic of φ,Parker u r
sw
the IMF which has not been previously modeled in the
Here B is the field strength at 1r , normalized to
0 0
context of SEP propagation is the Heliospheric Current
provide a field strength of B(1au) = 3.85nT, Ω =
(cid:12)
Sheet (HCS), providing the boundary between the two 2.87 × 10−6rads−1 is the average solar rotation rate,
hemispheres of the Solar dipole field. The presence of u =500kms−1istheradialsolarwindspeedandS(θ)
sw
a current sheet changes motion of charged particles due
is a scaling function providing the change of polarity in
imposing two distinct regions of drifts and the break-
a gradual fashion and describing current sheet thick-
down of guiding centre motion at the sheet (Speiser
ness. DuetoS beingonlyafunctionofcolatitudeθ,the
1965).
analytical field remains divergence-free. This simplified
The HCS is a vast area of space where the magnetic
HCSmodel,wherethecurrentsheetiscompletelyflat,is
fields associated with the northern and the southern
thussymmetricinrespecttotheheliocentriccoordinate
hemispheres of the solar magnetic field transition be-
φ. It is an approximation which is strictly valid only
tween outward and inward-directed polarities. Due to
withintheinnerheliosphereandduringsolarminimum.
the varying and complicated distribution of mean mag-
Modeling of a non-planar current sheet is postponed to
netic flux direction on the solar surface, and the tilt of
further studies.
thesolarmagneticaxiswithrespecttotherotationaxis,
As the field magnitude, and thus the HCS profile, de-
the HCS consists of a complex 3D structure, especially
pendssolelyonθ,thusvaryingalongadirectionperpen-
at greater heliocentric distances. The HCS has been
dicular to the solar wind flow, there is no compression
the topic of much research, but mainly from the point
of the current sheet and thus no driven reconnection.
of view of very energetic particles called galactic cos-
Therefore, the current sheet modeled in this work does
mic rays (GCRs), propagating inwards from the outer
not contain additional electric fields beyond the regular
boundary of the heliosphere (references include, but are
motional electric field
not limited to, Jokipii & Levy 1977, Burger et al. 1985, u
E=− sw ×B, (5)
K´ota&Jokipii2001,Peietal.2012,Straussetal.2012,
c
Solar Energetic Particle transport near a Heliospheric Current Sheet 3
where c is the speed of light. This electric field causes the coordinate system. Below, we summarize the non-
particlestoexperienceE×Bdrift,analogouswithcoro- relativistic forms of the main drifts, the electric field,
tationoffieldlines. InthecaseofawavyHCS(see,e.g., ∇B,andcurvaturedrifts,forthesimplecaseofaunipo-
Straussetal.2012,Peietal.2012,andBurger2012),es- lar IMF (S(θ)≡1), as
pecially with greater heliocentric distance, an assumed u r
v = sw eˆ (8)
radial solar wind flow will no longer be wholly in the E (r2+a2)1/2 φ(cid:48)
current sheet plane, requiring more detailed analysis of µc rcotθ µc r2+2a2
possible reconnection. v∇ = q r2+a2eˆφ(cid:48) − q (r2+a2)3/2eˆθ(cid:48) (9)
Observations estimate the HCS thickness to be in the
mc rcotθ mc r2+2a2
region of between 5000 and 40000 km at 1 au (see, e.g., v =− v2 eˆ − v2 eˆ , (10)
c qB (cid:107)r2+a2 φ(cid:48) qB (cid:107)(r2+a2)3/2 θ(cid:48)
Eastwood et al. 2002 and Winterhalter et al. 1994). We
examine effects of a gradual transition between hemi- where a is a function of colatitude a = u (Ω sinθ)−1
sw (cid:12)
spheres, and the effects of current sheet thickness. Al- andµistheparticlemagneticmomentµ=mv2(2B)−1.
⊥
though energetic protons may have Larmor radii much Here m and q are the particle mass and charge, and v
(cid:107)
larger than the listed current sheet thicknesses, effects and v are the components of velocity parallel and per-
⊥
such as beamed injection and adiabatic focusing may pendiculartothemagneticfield,respectively. See(Dalla
cause the perpendicular velocity component of particles etal.2013)forthemoregeneralrelativisticexpressions.
to be quite small, resulting in smaller than expected Theanalyticalformsshowthatfornear-equatoriallat-
Larmor radii, thus warranting this approach. Thus, we itudes, thetermalignedwitheˆ dominatesbothcurva-
θ(cid:48)
definetheHCSthicknessshapefunctionS tobeafunc- ture and gradient drifts. For both these drifts, when
tion of colatitude through latitude δ =90◦−θ, as considering the two polarity configurations of the So-
(cid:18) (cid:19) lar dipolar field, we find that for A+, both hemispheres
2δ
S(θ)=A −1+2S(1 + ) (6) cause drift of positively charged particles towards the
2 l
HCS
equator, and for A−, away from it, the patterns well
where A is a configuration parameter with values +1
known from GCR studies. Thus, for the A+ configura-
or −1, l is the thickness of the HCS, and S is the
HCS tion, the equator is a stable position, and for the A−
Smootherstep function (Ebert 2003) which maps the
configuration, a labile position.
parameter range [0,1] to the values [0,1] as S(x) =
Inclusion of the HCS, for example defined through a
6x5−15x4+10x3,resultinginasmoothtransitionwith
shape function S(θ), will cause additional drifts due to
nil first and second-order derivatives at the endpoints.
change of magnetic field as a function of θ. The first
ClosertotheSun,thisparametrizationresultsinsmaller
drift,validforbothsmoothandstep-modecurrentsheet
current sheet thicknesses. The parameter A defines the
profiles, is the current sheet drift, described commonly
polarity of the dipolar field according to cosmic ray
as Speiser motion (Speiser 1965). With B approaching
physics standard notation, i.e. a configuration of A+
zero, the guiding centre approximation of particle mo-
(A=+1)hasanoutwards-pointingfieldinthenorthern
tion breaks down. Particles oscillate between the two
hemisphere,andaconfigurationofA−(A=−1)hasan
magnetic field polarities by performing partial gyromo-
inwards-pointing field in the northern hemisphere, with
tionineachside,thencrossingthesheet,andperforming
the direction of the field in the southern hemisphere re-
gyromotion of opposite chirality on the other side. For
versed. Weadditionallyassessthevalidityofimplement-
particles with positive charge, this motion is in a west-
ing a HCS with zero thickness, using a shape function
ern direction for A+ and an eastern direction for A−.
S , which implements the Heaviside step function H as
H For a step-mode field transition and an isotropic distru-
S (θ)=A(−1+2H(δ)). (7) bution, this was found to lead to an average velocity of
H
(cid:104)v (cid:105)=0.463v (Burger et al. 1985).
S
Protons propagating within the fields given by equa-
If the gyroradius of particles is smaller than the char-
tions (1)–(7) will experience drifts due to the electric
acteristic length scale describing the rate of change for
field, and the gradients and curvature of the magnetic
BduetotheshapefunctionS(θ),aseconddriftisfound
field. A full analytical treatise of particle drifts in a
atthecurrentsheet,takingtheformofclassicalgradient
Parker spiral, far from the HCS, can be found in Dalla
drift, and defined as
etal.(2013),whereabetter-suitedfield-alignedframeof
cmv2
reference(eˆl,eˆφ(cid:48),eˆθ(cid:48))isintroduced. Withinthissystem, v = ⊥B×(∇B).
eˆ isdirectedalongtheParkerspiralmagneticfieldline, g 2q B3
l
outwards from the Sun. eˆ is antiparallel to the stan- If this drift is present, then ∇B would be aligned with
θ(cid:48)
dard spherical coordinate vector eˆ , and eˆ completes θ(cid:48), leading to the gradient drift being aligned with φ(cid:48).
θ φ(cid:48)
4 Battarbee et al.
Thedirectionofthisgradientdriftwouldbeoppositeto nesses, by varying the parameter l in equation (6).
HCS
that of current sheet drift (or Speiser motion). EachcurrentsheetthicknesswassimulatedforbothA+
and A− configurations, as described in section 2. The
current sheet was simulated with thicknesses of 0 km,
3. SIMULATIONS
5000 km, and 40000 km at 1 au. The first case was
In our simulations, we investigate SEP trajectories
in fact modeled as a Heaviside step function using the
in the fixed frame (spacecraft frame) in the presence
shape function (7). A plot of S(θ) at 1 au for vari-
of a flat HCS using a numerical test particle model
ous current sheet thicknesses is shown in Figure 1. The
(Dalla & Browning 2005; Kelly et al. 2012) with mod-
shownthicknessesof5000 kmand40000 kmat1 aucor-
ifications suited to heliospheric problems introduced in
respond with angular extents of 0.0019◦ and 0.015◦, re-
Marsh et al. (2013). Instead of using the focused trans-
spectively.
port equation (see, e.g., Roelof 1969), we solve the full
In order to simulate the infinitesimally thin current
three-dimensional differential equations of motion for
sheet, henceforth referred to as the Heaviside case, we
each particle. In our model, drifts are not introduced
could not use the regular Bulirsch-Stoer algorithm as it
intotherelevantequationsanalytically,butinsteadarise
could not automatically optimise particle propagation
naturally from the Lorentz and electric force due to the
over the step function. Instead, we used an adaptive-
fields given by equations 1–7 acting on particles during
step leapfrog Boris-push method (Boris 1970), which is
each step of their motion.
asolvercommonlyusedinParticle-In-Cell(PIC)codes.
We simulate the propagation of energetic protons, in-
jected instantaneously at time t=0 from a heliocentric
distance of 2R . Protons are injected from a region
(cid:12)
withangularextent6◦×6◦,centeredattheheliographic
equator. For select studies, the injection latitude was
variedinordertoassesslatitudinaldrifts. Particleshave
initialpitch-anglespointinginarandomdirectionwithin
a hemisphere pointing outwards from the Sun along the
Parker spiral. The relativistic differential equations of
particle motion and acceleration are solved using a self-
optimizing Bulirsch-Stoer method (Press 1996). Parti-
clesarepropagatedintheprescribedmagneticandelec-
tric fields, where the motional electric field is solved us-
ing a solar wind speed of u = 500kms−1. In order
sw
to model the effects of turbulence and wave-particle ef-
fects, particles experience large-angle scattering in the
solarwindframe,withPoisson-distributedscatteringin-
tervals, and a constant rigidity-independent mean free
path of 1 au, in agreement with an assumed low level of
scattering.
We inject N =105 particles and trace their propaga-
Figure 1. ShapefunctionS(θ)asseenataheliocentricdis-
tion within the heliosphere for 100 hours. Snapshots of
tanceof1 au. Thedashed,solid,anddottedlinescorrespond
particleprofilesareprovidedevery60minutes. Acollec- with HCS thicknesses of 0 km, 5000 km, and 40000 km, re-
tion sphere is placed at a heliocentric distance of 1 au, spectively. The shape function is displayed with both lin-
over which crossings are tracked, allowing the genera- ear (top) and logarithmic (bottom) distance from the helio-
tion of fluence maps, histograms, and virtual observer graphic equator, where the logarithmic plot shows only the
positive half of the function.
timeprofiles. Fluencemapsweregeneratedwithtilesof
angular extent 1◦ ×1◦ over the full length of the sim-
ulation, whereas time profile generation used 6◦ × 6◦ For each run, protons are injected as either mo-
windows and 30 minute time binning. noenergetic populations with initial energies of 1 MeV,
We chose eight different magnetic field configurations 10 MeV, 40 MeV, 100 MeV, 400 MeV, or 800 MeV, or
for use in our simulations. As reference cases, we as a power-law between 10 MeV and 400 MeV with a
simulated particle propagation in both inwards- and spectral index of γ =−1.1.
outwards-pointing unipolar fields (S(θ) = ±1). For he-
4. RESULTS
liospheric current sheets we used three different thick-
Solar Energetic Particle transport near a Heliospheric Current Sheet 5
Our first step was to perform qualitative assessment well as one in longitude, moving protons away from the
of apparent HCS drifts as a function of sheet thickness. well-connected field lines (see also Marsh et al. 2013).
In Figure 2 we show comparisons between all eight sim- We now refer to Dalla et al. (2013) as a theoretical
ulated IMF configurations. We plot the distribution of basis of drift analysis. The strongest drifts in longitude
protons injected at 100 MeV after 1 hour of propaga- (gradientandcurvature)arefoundtobeproportionalto
tion flattened to the x−y plane (the equatorial plane a function g(θ), which approaches zero at the equator.
of the Sun, with the x-axis pointing in the direction of Thisexplainsprotonsdisplayingsignificantlongitudinal
0◦ longitude). Injection was centered at (0◦,0◦). The driftonlyafterhavingdriftedtohigherlatitudes. Inthis
unipolar cases (leftmost column) show that within 1 hr, fieldconfiguration,gradientdrift(∝v2)pushesprotons
⊥
little drift has taken place. The three A+ panels (top to the west whereas curvature drift (∝ v2) causes drift
(cid:107)
row)showthatthepresenceofacurrentsheetgenerates towards the east. Both gradient and curvature drifts
significantcurrentsheetdrifttotheright(west),andthe areinthesamelatitudinaldirection,whichforthisfield
three A− panels (bottom row) show current sheet drift configuration is to higher colatitudes. The polarisation
to the left (east). Gradient drift associated with the drift is of smaller magnitude, and thus, ignored in this
variation of B over the thickness of the current sheet is work.
found to be negligible. In the presence of the heliospheric current sheet, the
We also performed a check to verify that the protons latitudinal drifts in each hemisphere play a significant
whichappeartohavedriftedareindeeddriftingprotons, role to how protons propagate (see, e.g., Jokipii & Levy
not a projection effect due to the x−y plot. Protons 1977). For protons injected at and near the HCS, as in
experiencing current sheet drift were confirmed to be our simulations, we find the dynamics presented to dif-
located in the vicinity of the HCS. Plots performed for fer significantly from the unipolar case. In Figure 4, we
other proton energies show comparable results, with in- plot fluence maps of 1 au crossings of protons, injected
crease in proton energy resulting in greater deviation at 100 MeV, for all eight simulated IMF configurations,
from the well-connected field lines. At later stages of inthesameformatasinthelowerpanelofFigure3. The
thesimulation,upto100 hrs,thedistributionofprotons latitudinal drifts in the A+ configuration are found to
in the inner heliosphere remains characteristically com- efficientlytrapprotonsclosetothecurrentsheet, where
parable with the 1 hr case, although corotation causes theyexperiencecurrentsheetdrift. FortheA−configu-
an westward transition of all protons, and the general ration, curvature and gradient drifts push protons away
propagation of protons outwards from the Sun causes from the HCS, but Speiser motion nevertheless allows
the proton counts close to the Sun to decrease. some protons to propagate along the current sheet, un-
In order to assess the magnitude of proton drifts, we tiltheyareejectedanddriftawayfromit. Asourmodel
gatheredallprotoncrossingsacrossthe1 ausphere,sav- does not include an intrinsic electric field at the sheet,
ing the time of crossing, the longitude and the latitude ejection happens due to particle scattering.
of each proton. In Figure 3 we show a map of 100 MeV Protons experiencing current sheet drift are visible at
proton crossing counts, for a unipolar inwards-pointing western heliolongitudes for the A+ configuration (left
magnetic field, relative to the injection coordinates, us- column) and at eastern heliolongitudes for the A− con-
ing 1◦×1◦ binning, adding up all counts over the 100 h figuration(rightcolumn). OfparticularnotefortheA−
duration of the simulation (top panel). We also show configuration is how, if looking closely at the cells clos-
a comparative picture where we have removed the ef- est to the current sheet at the best-connected field line,
fects of corotation (bottom panel). Corotation, also de- intensities are smaller than just above or below it. In
scribed as the E×B drift, is caused by the field lines Figure 5, we plot histograms of 100 MeV proton lati-
alongwhichtheparticlespropagatebeingfrozenintothe tudes at the time they cross the 1 au sphere boundary,
radially outflowing solar wind plasma, resulting in the foreightdifferentIMFconfigurations. ForanA+config-
intersectionpointsat1 aubeingrotatedwestwards. We uration, protons are preferentially located at the centre
also added a longitudinal offset to the bottom panel, so of the current sheet, whereas for the A− configuration,
coordinates are shown in relation to the best-connected two peaks further out are seen. Thus, the depletion at
field line. Henceforth, we will utilise these corrections. the sheet is shown to be real, not caused by current
The proton distributions in Figure 3 show that the sheet drift spreading a constant amount of protons over
effect of corotation is significant, which is unsurprising a wider range of longitudes.
considering the 100 hr extent of the simulation. The In Figure 6, we plot histograms of 100 MeV proton
strongestfluenceisfoundatthewell-connectedfieldline. longitudesatthetimetheycrossthe1 auspherebound-
Adrift inlatitude (upwardsfor thispolarity) isseen, as ary, for eight different IMF configurations. The current
6 Battarbee et al.
Figure 2. Projectionofprotonsinjectedat100 MeV,after1 hrofsimulation,ontothex−y planeforeightdifferentmagnetic
field configurations. Top row, from left: outwards-pointing unipolar field, followed by A+ configurations with current sheet
thicknessparameterscorrespondingwith1 authicknessesof0 km(Heavisidestep),5000 km,and40000 km. Bottomrow,from
left: inwards-pointingunipolarfield,followedbyA−configurationswithcurrentsheetthicknessparameterscorrespondingwith
1 au thicknesses of 0 km (Heaviside step), 5000 km, and 40000 km. A distance of 1 au is displayed with a dashed circle. The
protonspreadsshowthatcurrentsheetdrift(toprow: totheright,bottomrow: totheleft)isnoticeableforallcurrentsheets.
sheet drift is seen to have a significant effect, allowing due to average Speiser motion being linked with energy
protonstowrapatleast180degreesaroundtheSun. An (Burger et al. 1985). The A− configuration displays
A+ configuration is seen to have slightly stronger cur- a much stronger energy dependence for current sheet
rent sheet drift, which is in agreement with the equator drift,asthereachofparticlegyromotionplaysacritical
being a stable position in A+, and a labile position in role in sampling of the magnetic field reversal, due to
A−. lateral drifts transporting protons away from the HCS.
In Figure 7, we display histograms of proton longi- ThemainpeakforA−,however,doesspreadout,aslon-
tudes at the time they cross the 1 au sphere bound- gitudinal drifts outside the current sheet can also cause
ary, for energies of 10 MeV, 40 MeV, 100 MeV, and protons to spread westward.
400 MeV. The left column shows results for an IMF The maps and histograms presented in Figures 3
with an A+ configuration, the right column for one through 7 do not explicitly display the time profiles
with an A− configuration, with HCS thickness set to of proton crossings at 1 au. In order to allow com-
5000 km at 1 au. Both the maximum amount drifted parisons with real-world observations, we simulated
and the count of protons at each drifting distance are virtual observers at 1 au, by collecting proton counts
found to increase with energy. This is as expected, as over neighbouring regions of 6◦×6◦ extent on the sur-
fasterprotonsareabletosweepacrossthecurrentsheet face of the 1 au sphere. For this analysis, we per-
from a wider region due to a larger gyroradius, and also formedsimulationsusingaprotoninjectiondistribution
Solar Energetic Particle transport near a Heliospheric Current Sheet 7
of the source population. We also note that separation
15
between the observer and the well-connected field line
5 increases the onset time difference between different en-
ergies.
5
60 30 0 30 With the A− IMF configuration, shown in Figure 9,
weseeacaseverysimilartotheunipolarone,withrapid
15
orprolongedrisephasesofintensity,dependingonlongi-
5 tude. For this case, however, intensities extend to both
positive and negative heliolatitudes. Again, the process
5
60 30 0 30
of latitudinal proton drifts causes apparent hardening
ofobservedprotonspectranorthandsouthoftheinjec-
tion region. We also note that a relatively small abrupt
100 101 102 103
Particle Counts componentisseenattheequator,ateasternlongitudes,
due to protons experiencing current sheet drift. Due to
Figure 3. Map of protons, injected at 100 MeV, crossing
the combined effect of current sheet drift and latitudi-
the1 ausphereusing1◦×1◦ binningforaunipolarinward-
nal drifts, high energy protons are much less abundant
pointing magnetic field. Fluence colours and contours are
at western longitudes than for the unipolar case.
on a logarithmic scale, with two contours per decade. Top
panel: Proton crossing coordinates (in degrees) relative to With the A+ IMF configuration, shown in Figure 10,
injection site, showing how the E×B drift and latitudinal we find that the gradient and curvature drifts prevent
drifts both work concurrently. Bottom panel: Proton cross- anysignificantprotonfluxfromextendingtopositiveor
ing coordinates relative to the best-connected fieldline with negative heliolatitudes. Protons at lower energies dis-
the effects of E×B drift removed.
play the same longitudinal characteristic of more pro-
longed event rise with increasing longitude. Of particu-
given by a power-law with γ = −1.1, extending from
lar interest is the abrupt current sheet drift associated
10 MeV to 400 MeV. We inject N = 106 particles
component at early phases of the simulation, as can be
in order to attain better statistics. For gathering of
seen by comparing Figure 10 with Figure 8. This ad-
time profiles, we introduced energy channels spanning
ditional component is found at western observers, caus-
the extents of 10.0−40.0 MeV, 60.0−100.0 MeV, and
ingthetimeprofilestoexhibittwodistinctcomponents.
200.0−400.0 MeV. In Figures 8, 9, and 10, we display
Thus,asingleinjectioneventcould,withasuitableIMF
time profiles for an outwards-pointing unipolar field, an
configuration, be observed as two particle events.
A− IMF configuration, and an A+ IMF configuration,
Comparisons of different HCS thickness parameters
respectively.
did not result in noticeable variation in the characteris-
The unipolar field depicted in Figure 8 shows how a
tics of proton time profiles. The additional plots have
single injection event can cause different kinds of obser-
thus been omitted.
vations, depending on virtual observer location, similar
Solar active regions are usually associated with
to the findings of Marsh et al. (2015). At the best-
sunspots above or below the solar equator. The re-
connected field line, the proton flux increases abruptly
sults presented in Figures 8 to 10 are applicable if the
and then decays exponentially. At eastern longitudes,
acceleration region, for example a coronal shock front,
flux is nearly non-existant. With increasing longitu-
spans all the way to the equator. If the injection lo-
dinal separation to the west, the onset is delayed and
cation is above the HCS, it will take time for particles
the shape of the profile becomes more gradual. As flux
to reach it and feel its effects. In Figure 11, we display
at western longitudes is influenced by corotation, high
virtual time profiles for an observer at the heliographic
energy protons are less numerous, due to propagating equator, when the injection region of 6◦ ×6◦ was cen-
rapidly out of the inner heliosphere. At negative he-
tered at a latitude of +6 degrees. The top row shows
liolatitudes, where in our setup all counts are due to
profilesforanunipolaroutwards-pointingIMF,andthe
drifting effects, we also find an abrupt rise in flux at
bottomrowforanIMFwithanA+HCSconfiguration.
connected longitudes and slower rises at western longi-
For high energy protons, the HCS facilitates arrival at
tudes. However, due to latitudinal drifts being energy-
theobserverearlierthanforanunipolarfield. However,
dependent, these time profiles emphasise high energy
astheHCSspreadsprotonsacrossawiderangeoflongi-
protons. Thus, if the proton flux of a solar event at an
tudes, the achieved peak flux is lower than without the
observer is strongly influenced by latitudinal drifts, the
HCS. At low energies the difference between the two
observed spectrum can appear much harder than that
cases is insignificant, possibly due to protons drifting
8 Battarbee et al.
20 20
unipolar out 100 MeV unipolar in 100 MeV
10 10
0 0
10 10
20 20
45 15 15 45 15 15
20 20
A+ Heaviside 100 MeV A- Heaviside 100 MeV
10 10
0 0
10 10
20 20
45 15 15 45 15 15
20 20
A+ 5000 km 100 MeV A- 5000 km 100 MeV
10 10
0 0
10 10
20 20
45 15 15 45 15 15
20 20
A+ 40000 km 100 MeV A- 40000 km 100 MeV
10 10
0 0
10 10
20 20
45 15 15 45 15 15
Figure 4. Map of crossings of protons, injected at 100 MeV, across the 1 au sphere, over a time of 100 hr, relative to best-
connected fieldline at injection time, with the effects of corotation removed. Fluence colours and contours are on a logarithmic
scale, with two contours per decade. Top row: Unipolar field, pointing outwards (left) and inwards (right). Second to fourth
rows: HCSthicknessscaledto0 km,5000 km,and40 000 kmat1 au,respectively,withwithanA+(leftcolumn)orA−(right
colum) field configuration.
Solar Energetic Particle transport near a Heliospheric Current Sheet 9
104 104
unipolar out 100 MeV unipolar in 100 MeV
103 103
unts102 unts102
o o
C C
101 101
100 100
20 10 0 10 20 10 0 10
Latitude Latitude
104 104
A+ Heaviside 100 MeV A- Heaviside 100 MeV
103 103
unts102 unts102
o o
C C
101 101
100 100
20 10 0 10 20 10 0 10
Latitude Latitude
104 104
A+ 5000 km 100 MeV A- 5000 km 100 MeV
103 103
unts102 unts102
o o
C C
101 101
100 100
20 10 0 10 20 10 0 10
Latitude Latitude
104 104
A+ 40000 km 100 MeV A- 40000 km 100 MeV
103 103
unts102 unts102
o o
C C
101 101
100 100
20 10 0 10 20 10 0 10
Latitude Latitude
Figure 5. Histograms depicting counts of protons, injected at 100 MeV, crossing the 1 au sphere, as a function of latitude,
relative to injection at the equator. Layout as in Figure 4.
close to the equator but not quite reaching the current as over 50% of their initial energy. However, for an A+
sheet. IMF configuration, protons are confined to the vicinity
In Figure 12 we show fluence maps for the same sim- of the HCS, and deceleration due to drifts is suppressed
ulations as seen in Figures 8 through 10. Drifts extend to as little as <25%.
protons for significant distances in latitude and longi- The crossing of SEPs from one IMF polarity to an-
tude. Therelativespread,comparedwith4,isnotdras- other, across sector boundaries caused by a wavy HCS,
tically different, as the number of injected particles for is a complex question which we can not fully analyse
the power-law runs was increased tenfold. One should within the scope of this work. A first step, however,
note that contours are spaced two per decade. is to assess the efficiency of particle drifts and scatter-
As described in Dalla et al. (2015), SEPs experience ing in transporting SEPs across a flat HCS. In order
deceleration during propagation through interplanetary to analyze this, we injected protons of six different en-
space due to adiabatic deceleration and drift effects. In ergies (1 MeV, 10 MeV, 40 MeV, 100 MeV, 400 MeV,
the work presented in this manuscript, protons have and 800 MeV) from a 6◦×6◦ angular injection window,
been injected into the simulation at the described en- centered at +3◦ within an A+ configuration. In Figure
ergies, with deceleration happening by the time they 14, we plot these results with the effects of corotation
reach1 au. Thus,protonswhicharedetectedat1 auas, removed, for three different current sheet thicknesses.
e.g., 100 MeV protons, will have likely been injected at At small energies, only the current sheet drift spreads
higher energies, and will thus have experienced greater particles outside the well-connected region, but at en-
driftsduetothevelocitydependenciesinvolved. InFig- ergies above 40 MeV, some drifts in both latitude and
ure13,weplothistogramsof1 aucrossingenergies,over longitude are visible. However, proton energies need to
the durationof the simulation, for protons injected at exceed 100 MeV in order to be ejected from the current
energies of 10 MeV, 40 MeV, 100 MeV, and 400 MeV, sheet to the southern hemisphere. For comparison, the
for three different magnetic field configurations. For an Larmor radius of 400 MeV protons at 1 au, assuming,
unipolar outwards-pointing IMF, and for an IMF with for example, a pitch-angle of α ≈ 5◦, is of the order of
anA−configuration,protonscandeceleratebyasmuch 40000 km.
10 Battarbee et al.
105 105
104 unipolar out 100 MeV 104 unipolar in 100 MeV
nts103 nts103
Cou102 Cou102
101 101
100 100
180 150 120 90 60 30 0 30 60 90 120 150 180 150 120 90 60 30 0 30 60 90 120 150
Longitude Longitude
105 105
104 A+ Heaviside 100 MeV 104 A- Heaviside 100 MeV
nts103 nts103
Cou102 Cou102
101 101
100 100
180 150 120 90 60 30 0 30 60 90 120 150 180 150 120 90 60 30 0 30 60 90 120 150
Longitude Longitude
105 105
104 A+ 5000 km 100 MeV 104 A- 5000 km 100 MeV
nts103 nts103
Cou102 Cou102
101 101
100 100
180 150 120 90 60 30 0 30 60 90 120 150 180 150 120 90 60 30 0 30 60 90 120 150
Longitude Longitude
105 105
104 A+ 40000 km 100 MeV 104 A- 40000 km 100 MeV
nts103 nts103
Cou102 Cou102
101 101
100 100
180 150 120 90 60 30 0 30 60 90 120 150 180 150 120 90 60 30 0 30 60 90 120 150
Longitude Longitude
Figure 6. Histograms depicting counts of protons, injected at 100 MeV, crossing the 1 au sphere, as a function of longitude,
relative to the best-connected field line. The effect of corotation has been removed. Layout as in Figure 4.
In Figure 15, we plot the same crossings as in Figure rent sheet are significant, allowing high-energy protons
14, but for an IMF with a A− configuration. Again, at to drift over 180 degrees in longitude. We show that
low energies, the current sheet drift is the primary way both A+ and A− configurations of the IMF allow for
particlesspreadoutsidethewell-connectedregion. How- significant current sheet drift which helps protons reach
ever, as general drift directions are away from the cur- regions far from the injection longitude.
rentsheet,anyparticleswhicharetransportedalongthe We assessed the effects of current sheet thickness on
current sheet and then scatter away from it can easily proton propagation, and found there to be negligible
propagatefurtherawayfromit. Thus,atenergiesaslow difference between simulations using realistic parame-
as40 MeV,protonsareseentoscatterintothesouthern ters or a step function. Gradient drifts due to sheet
hemisphere. We note, however, that if the injection re- thickness profiles are found to be non-existant. Thus,
gionofprotonsdoesnotcoincidewiththecurrentsheet, we conclude that using a step-mode current sheet is an
protonswithinanA−configurationareunlikelytoreach adequate tool in numerical simulations.
thecurrentsheet,andthus,unlikelytoscatteracrossit. Weplacedvirtualobserversatadistanceof1 aufrom
Thus,weconcludethataninjectioneventconstrained the Sun and generated time profiles mimicking space
to one hemisphere can, due to lateral drifts and the he- observations. The IMF configuration was confirmed to
liospheric current sheet, remain undetectable in the op- significantly affect the qualitative shape of time profiles
posite hemisphere. atdifferentobserverlocations. Foraninjectionlocation
centeredattheHCS,theA+configurationconfinedpro-
5. CONCLUSIONS tons to the vicinity of the HCS, whereas the A− config-
We simulated the propagation of solar energetic pro- urationcausedobserversatbothnorthernandsouthern
tons with energies ranging from 1 to 800 MeV within latitudes to observe particle fluxes. Latitudes separated
multipledifferentIMFconditions,assessingtheeffectsa from the injection region exhibited harder power-laws
flatheliosphericcurrentsheetlocatedattheheliographic in particle flux compared with latitudes with injection,
equator has on proton drifts and propagation. We show due to energy dependence of drifts. The current sheet
that, in the presence of a flat HCS, drifts along the cur- driftofprotonswasdetectableforanobserverattheso-