Annales Geophysicae (2005) 23: 643–650SRef-ID: 1432-0576/ag/2005-23-643© European Geosciences Union 2005

AnnalesGeophysicae

Reforming perpendicular shocks in the presence of pickup protons:initial ion acceleration

R. E. Lee1, S. C. Chapman1,2, and R. O. Dendy1,3

1Space and Astrophysics Group, Department of Physics, University of Warwick, Coventry CV4 7AL, UK2Radcliffe Institute for Advanced Study, Harvard University, USA3UKAEA Culham Division, Culham Science Centre, Abingdon, Oxfordshire, OX14 3DB, UK

Received: 03 August 2004 – Revised: 3 December 2004 – Accepted: 6 December 2004 – Published: 28 February 2005

Abstract. Acceleration processes associated with the helio-spheric termination shock may provide a source of anoma-lous cosmic rays (ACRs). Recent kinetic simulations ofsupercritical, quasi-perpendicular shocks have yielded timevarying shock solutions that cyclically reform on the spatio-temporal scales of the incoming protons. Whether a shocksolution is stationary or reforming depends upon the plasmaparameters which, for the termination shock, are ill definedbut believed to be within the time-dependent regime. Herewe present results from high phase space resolution particle-in-cell simulations for a three-component plasma (solar windprotons, electrons and pickup protons) appropriate for thetermination shock. We find reforming shock solutions whichgenerate suprathermal populations for both proton compo-nents, with the pickup ions reaching energies of order twentytimes the solar wind inflow energy. This suprathermal “in-jection” population is required as a seed population for sub-sequent acceleration at the shock which can in turn generateACRs.

Key words. Interplanetary physics (Cosmic rays, He-liopause and solar wind termination) – Space plasma physics(Shock waves)

1 Introduction

With the approach of the Voyager spacecraft towards the he-liopause and in particular the termination shock (Burlagaet al., 2003; Gurnett et al., 2003), together with possiblecrossings thereof (Krimigis et al., 2003; McDonald et al.,2003), the problem of the acceleration of anomalous cos-mic rays (ACRs) (Fisk et al., 1974; Pesses et al., 1981) inthe outer heliosphere is at the forefront of current research(see, for example,Kucharek and Scholer, 1995; Zank et al.,1996; Rice et al., 2000; Zank et al., 2001; Lever et al., 2001).This provides an opportunity for comparing theoretical re-sults with observations.

Correspondence to:R. E. Lee(leer@astro.warwick.ac.uk)

Kinetic simulations are a useful tool to investigate, self-consistently, local acceleration processes at the shock.Particle-in-cell (PIC) simulations of quasi-perpendicularshocks (see, for example,Biskamp and Welter, 1972;Lembege and Dawson, 1987; Lembege and Savoini, 1992;Shimada and Hoshino, 2000; Schmitz et al., 2002a; Hadaet al., 2003; Lee et al., 2004a), have found that, for a cer-tain range of parameters, the shock solutions are not static,rather reforming on the gyro scales of the incoming bulk pro-tons. Hybrid simulations, on the other hand, typically gen-erate static solutions (see, for example,Burgess et al., 1989;Kucharek and Scholer, 1995; Lipatov and Zank, 1999), al-though reforming solutions are recovered for a range of val-ues of resistivity (Quest, 1985; Hellinger et al., 2002). Thetime dependent electromagnetic fields at reforming shockshave been found to accelerate a fraction of the bulk pro-ton population to suprathermal energies (Lee et al., 2004a),and this is a possible cosmic ray injection mechanism at su-pernova remnants. Parameters relevant to the heliospherictermination shock, although ill defined, are consistent withthose expected to yield reforming solutions in a quasi-perpendicular geometry. The use of a quasi-perpendiculargeometry is appropriate if we make the assumption thatParker’s solar wind model is still valid in the outer helio-sphere. FollowingPesses et al.(1981) one concludes that theshock is close to perpendicular (the angle between magneticfield and shock normal>80◦) at heliographic latitudes lessthan 75◦. Pioneer 11 measurements show the existence oftwo broad distributions of shock propagation angle, centredon 90◦ and 270◦, with half widths of 25◦ (Smith, 1993).

Importantly, the heliospheric termination shock is gener-ated in the presence of pickup ions originating from the tran-sheliopause interstellar medium. Two questions then arise,which we address here: first, whether the reforming shocksolutions found previously persist with the self consistent in-clusion of a pickup ion population; and second, to what ex-tent are both solar wind and pickup protons accelerated in theresulting time dependent electromagnetic fields.

644 R. E. Lee et al.: Reforming shocks with pickup ions’

2 Simulation

We use a 1.5-D (1x, 3v) relativistic electromagnetic PICcode to simulate the structure and evolution of a supercrit-ical, collisionless, perpendicular magnetosonic shock. Simu-lations of reforming shocks in quasi-perpendicular and pure-perpendicular reforming shocks have been performed, forexample, byScholer et al.(2003), where it is shown thatthe additional wavemodes supported by oblique geometrydo not strongly affect the overall structure and dynamics ofthe reforming shock. In a PIC simulation, the distributionfunctions of all particle species are represented by computa-tional super-particles, the phase space locations of which areevolved via the Lorentz force law, whilst the electromagneticfields are defined on a spatial grid and evolved via the fullMaxwell equations (see, for example,Birdsall and Langdon,1991). The simulation setup is that described inLee et al.(2004a), with the addition of pickup protons, the results ofwhich are presented for the first time here. We find reform-ing shock solutions in this 1.5-D geometry, where scalar andall three components of vector quantities are functions of onespatial coordinate and time. Reforming shock solutions havealso been found in higher dimensions (Lembege and Savoini,1992).

To accurately reproduce the effects of pickup protons,we simulate three species within our code: electrons, so-lar wind protons, and pickup protons. Particles are injectedon the left-hand side of the simulation box with a driftspeedvinj . The simulation is conducted in the downstreamrest frame, so ifv1 is the upstream andv2 the downstreamflow speed derived from the Rankine-Hugoniot relations,vinjcan be defined from the Alfvenic Mach number,MA by:vinj=MAvA−v2. For solar wind protons and electrons, eachindividual particle has its initial speed modified by an addi-tional Maxwellian random velocity, chosen from a thermaldistribution characterised by avthm for each species. Pickupprotons are born with similar values of perpendicular veloc-ity and arbitrary gyrophase, centred on the drift speed. Thepickup protons are modelled by a population with velocitiesin the form of a spherical shell, centred onvinj , radiusv1. Onthe particle injection boundary, the magnetic field (Bz,1) isconstant and the electric field (Ey,1) is calculated self con-sistently. We use the piston method to generate the shock,and shock-following algorithms, to obtain a sufficiently longrun time, as described inSchmitz et al.(2002a).

Here results are presented from simulations of a perpen-dicular shock propagating into a collisionless plasma, at anAlfv enic Mach number (MA) of 8. The upstream ratio ofplasma thermal pressure to magnetic field pressure for thesolar wind thermal protons is taken asβi=0.2, and for theelectronsβe=0.5. The pickup proton effective temperature ischaracterised byv1 due to their initial shell distribution (afterKucharek and Scholer, 1995; Zank et al., 1996; Ellison et al.,1999). To enable ion and electron time scales to be capturedwithin the same simulation, with reasonable computationaloverheads, it is necessary to reduce the simulation mass ra-tio for ions and electrons toMR=mi/me=20, and the ratio

of electron plasma frequency to electron cyclotron frequencyωpe/ωce=20 (in common withShimada and Hoshino, 2000;Schmitz et al., 2002a,b; Lee et al., 2004a). Simulations athigher mass ratios have been conducted by, for example,Lembege and Savoini(2002) atMR=400, andScholer et al.(2003) at MR=1840; however, to achieve reasonable run-times and simulation domains, a ratio ofωpe/ωce=[2,

√8]

was used, for example, in the latter. In our simulations alower mass ratio is chosen to allow for sufficient resolutionof the ion phase space to study the ion dynamics.Scholeret al.(2003) demonstrate that reformation is a lowβ process,and is not an artifact of unrealistic mass ratios, and that thedifferences that occur between simulations atMR=20 andMR=1840 are in the nature of the instabilities in the footregion. Previous work suggests (see,Lee et al.2004a, Leeet al.2004b) that the reflected ion dynamics are insensitiveto these instabilities, and importantly we resolve the samelength scales for the foot region (∼λci) and shock ramp (ofthe order ofc/ωpe), thus we expect our conclusions to holdatMR=1840.

3 Results

The key phenomenology found in our PIC simulations arisesfrom the non-time-stationary nature of the reforming shock.This is shown in Figs.1 and 2; these plot magnetic fieldstrength and

∫Exdx as functions ofx and time for two sim-

ulations, on the left, for solar wind protons only and on theright, with the addition of a pickup proton population at 10%of the density of the background. As with all figures in thispaper, they are presented in a frame where the downstreamplasma is at rest, using data collected from a time segment ofthe simulation when the shock has formed and is propagat-ing independently of the boundary conditions. Units are nor-malised to upstream parameters for the solar wind protons,thusλci is the upstream solar wind proton cyclotron radius,ωci is the upstream solar wind proton cyclotron frequency,andE inj,SW=miv

2inj/2 is the solar wind proton injection en-

ergy. The shock can be seen to move upstream, from theright to the left of the simulation box, over time, with peaksin magnetic field (the shock ramp) recurring, and the poten-tial well expanding and collapsing, on the time scale of thelocal proton cyclotron period. Over the course of each cycle,a stationary shock ramp forms at the turnaround point of pro-tons in the foot region. Then a new foot region extends intothe upstream region as protons reflect from this new shockfront. A new ramp (magnetic field peak) then starts to format the upstream edge of this foot region, and another cyclebegins. The last of the original foot region protons gyrateinto the downstream plasma as the new shock starts to reflecta new foot region population. The shock thus propagates up-stream (left) in a stepwise fashion. Comparison of a shocksimulation containing no pickup ions (panel A in Figs.1 and2, and see also,Lembege and Savoini, 1992; Shimada andHoshino, 2000; Schmitz et al., 2002a; Lee et al., 2004a) toa shock simulation containing 10 % pickup protons (panel B

R. E. Lee et al.: Reforming shocks with pickup ions’ 645

Fig. 1. Evolution of perpendicular magnetic field strength, overtime (vertical axis), and space (horizontal axis), for a simulation inthe absence of pickup ions (panel A) and a simulation containing10% pickup ions (panel B).

Fig. 2. Evolution ofφ=∫Exdx, over time (vertical axis), and space

(horizontal axis). In the absence of pickup protons (A), and with10% pickup protons (B).

in Figs.1 and2), shows that the reformation spatio-temporalscales are similar.

Figure3 shows a snapshot of the spatial distributions ofx-velocity and kinetic energy for both proton species, togetherwith

∫Exdx, and perpendicular magnetic field, at the begin-

ning of a reformation cycle (t=34.5×2πω−1ci ). The tempo-

ral evolution of both proton species in phase space over areformation cycle is shown in Fig.4, together with the cor-responding scalar

∫Exdx and perpendicular magnetic field

in Fig. 5. From this figure we see that the addition of 10%pickup protons does, however, lead to an extended foot re-gion, observable by the slow decline into the shock’s poten-tial structure starting∼6λci upstream, and in the downstreamacts to suppress magnetic field fluctuations over a distance of∼5λci downstream of the shock front.

In Fig. 3 protons from both species flow in from the left-hand, injection boundary. Solar wind protons continue untilthey reach the shock front (currently situated at∼75λci). Atthis stage of the reformation cycle, the shock front is almoststationary (see Fig.1), so that a fraction of the solar wind pro-tons specularly reflect, in the downstream rest frame, fromthe ramp in the magnetic field and its accompanying poten-tial, with vx→−vx (panel 5). The reflected solar wind pro-

66 68 70 72 74 76 78 80 82 84

−2

0

2

v x,S

W/v

inj

x / λci

0

2

4

6

ε SW

/εin

j,SW

−5

0

5

v x,P

I/vin

j

0

10

20

ε PI/ε

inj,S

W

−1.5−1

−0.50

φ/ε in

j,SW

02468

Bz/B

z,1

Fig. 3. Spatial cross section of the simulation in the shock regionat t=34.5×2πω−1

ci. The top panel (panel 1) shows perpendicular

magnetic field strength,Bz, normalised to the upstream value,Bz,1.Panel 2 showsφ=

∫Exdx normalised to the solar wind proton in-

jection energyEinj,SW. Panels 3 and 5 show the energies of thepickup protons and the solar wind protons, respectively, again nor-malised toEinj,SW. Panels 4 and 6 show the pickup and solar windprotonx-velocities, normalised to the injection velocityvinj .

tons move upstream to form a relatively energetic popula-tion, as shown in panel 4, gaining energies dispersed from1−6×Einj,SW (as seen inLee et al., 2004a) as they cross theshock front and move into the downstream region.

Pickup protons also flow in until they reach the shockfront, at which stage they, too, are reflected (panel 3). Thepresence of pickup protons leads to an extended foot re-gion with two components: the reforming foot region asso-ciated with the solar wind protons close to the shock, plusa weak foot region extending up to∼6λci upstream. Therelative spatial scales of these foot region populations maybe understood in terms of specular reflection at the shockramp. If the protons simply reflect specularly, that is, re-verse their component of velocity normal to the shock front(vn) as they encounter the ramp, then the length-scale foreach population isLfoot=|vn|/ωci . This is governed byvn=vinj +thm sin(ωci t+φ): for thermal protons this impliesvn'vinj becausevthm,SW�vinj ; whereas for the pickup pro-tonsvthm,PI'vinj , so that 0<vn<2vinj . Thus, after specularreflection, the pickup protons create an extended foot regionby moving further back into the upstream region, with a cor-responding larger excursion iny.

646 R. E. Lee et al.: Reforming shocks with pickup ions’

Fig. 4. Phase space (vx/vinj vs. x/λci ) for solar wind protons(lower panel of each pair) and pickup protons (upper panels) inthe shock region. Four pairs of panels at equal time intervals of0.4×2πω−1

ci, starting at timet=33.3×2πω−1

ci(lowest two pan-

els). The maximum upstream position for the reflected ions in eachspecies are indicated for each snapshot.

The evolution of the phase space of both proton speciesover the course of a reformation cycle is shown in Fig.4; thisis accompanied by Fig.5 which shows the magnetic fieldand

∫Exdx at the same times. Visible in

∫Exdx in Fig. 5

is the weak potential feature, associated with the pickup pro-tons, that extends∼6λci upstream of the potential featureassociated with the solar wind protons. This extended poten-tial changes little over a reformation cycle, particularly whencompared to the changes visible in the potential associatedwith reflecting solar wind protons. The maximum upstreampositions for reflected pickup protons and solar wind protonsare indicated by arrows on Fig.4 and on the

∫Exdx panels

in Fig.5, to highlight this point. The presence of an extendedfoot region is also visible to a lesser degree in the magneticfield profiles; these also clearly show the nontimestationarynature of the shock ramp. When it is possible to do so (panelsA, C and D) we observe a shock ramp thickness of the orderof a few electron inertial lengths (in common withLembegeand Savoini, 1992; Scholer et al., 2003). This value of theshock thickness can also be seen in Fig.6 which shows mag-netic field strength and

∫Exdx profiles for a simulation in

the absence of pickup protons, at four times over the courseof a reformation cycle comparable to those shown in Fig.5.Comparing Figs.5 and6 shows

∫Exdx and magnetic field

structure associated with the bulk solar wind protons, that arequalitatively similar to that arising in the presence of pickupprotons.

68 70 72 74 76 78 80 82 84 86 88

−1.5−1

−0.50

x / λci

φ/ε in

j,SW

02468

Bz/B

z,1

−1.5−1

−0.50

φ/ε in

j,SW

02468

Bz/B

z,1

−1.5−1

−0.50

φ/ε in

j,SW

02468

Bz/B

z,1

−1.5−1

−0.50

φ/ε in

j,SW

02468

Bz/B

z,1

A

B

C

D

Fig. 5. Four pairs of panels, corresponding to the spatio-temporalcoordinates of the panels in Fig.4, showing perpendicular magneticfield strength (upper panel of each pair), andφ=

∫Exdx (lower pan-

els) in the shock region. As in Fig.4, the maximum upstream po-sitions of reflected protons for each species are indicated, with thefurthest upstream arrow relating to the pickup ion species.

From the magnetic field profiles it can be seen that the fieldis Bz/Bz,1∼3, on average, in the downstream region. Takingthe average time between minima in the shock’s foot regionvalue of

∫Exdx, the reformation time is∼1.5×2πω−1

ci in thesimulations with both 0% and 10% pickup protons, as can beseen in Fig.2. In terms of the average field just downstreamof the overshoot, the reformation time is then∼0.5 local iongyroperiods.

The differences in temporal evolution for the two speciescan be seen when the counts of reflected protons are com-pared. We count the number of protons upstream of the min-ima in

∫Exdx with, in the case of solar wind protons, ve-

locities directed away from the shock front and in the caseof the pickup protons, those protons with velocities outsideof the injected shell in velocity space. The time variation ofthese reflected populations is shown in Fig.7. In contrast tothe solar wind protons, the pickup proton reflected ion count(and consequently foot region structure) is stable over a ref-ormation cycle, with slight changes coinciding with times ofchanging position for the minima in

∫Exdx as the reforming

shock advances. The solar wind proton reflection count, onthe other hand, varies by a factor of∼2.5 over the course ofa reformation cycle.

To understand the energy gain processes, it is useful toconsider the evolution of

∫F·vdt , and its components, cal-

culated along proton trajectories. Specifically, in Fig.8 weexamine a solar wind proton that reaches∼6Einj,SW, and a

R. E. Lee et al.: Reforming shocks with pickup ions’ 647

52 54 56 58 60 62 64 66

−1.5−1

−0.50

x / λci

φ /ε

inj

02468

Bz /B

z,1

−1.5−1

−0.50

φ /ε

inj

02468

Bz/ B

z,1

−1.5−1

−0.50

φ /ε

inj

02468

Bz/ B

z,1

−1.5−1

−0.50

φ /ε

inj

02468

Bz/ B

z,1

Fig. 6. Four pairs of panels, showing perpendicular magnetic fieldstrength (upper panel of each pair), andφ=

∫Exdx (lower panels)

in the shock region for a simulation in the absence of pickup ions.The panels are separated by equal time intervals of 0.4×2πω−1

ci,

starting at timet=33.3×2πω−1ci

(lowest two panels).

pickup proton that journeys∼8λci upstream after reflection,before returning and reaching high energy on crossing theshock front. To relate these changes to the shock dynam-ics, the proton trajectories are superimposed upon the spatio-temporally evolving

∫Exdx in the bottom panel of Fig.8.

The corresponding local∫Exdx experienced by these two

protons is plotted in the middle panel of Fig.8. Detailed anal-ysis of the energetics of the solar wind protons (Lee et al.,2004a) suggests that the electromagnetic fields relevant toacceleration to suprathermal energies are on ion scales. Asdiscussed inLee et al.(2004a,b), for example, the fluid equa-tions treated appropriately lead to

Ex,i ' −1

enµ0

∂(B2z /2 + B2

y/2)

∂x− vi,yBz; (1)

defining the bulk potential in terms of integralExidx, thenremoves features, such as electron phase space holes, butmaintains ion scale features, such as the foot region po-tential structure. The rate of change of energyeE·v inte-grated over the trajectories of energetic ions with the elec-tric field from Eq. (1) is then found to track that calculatedusing the full electric field from the PIC simulation. Previ-ous PIC simulations for a range of values ofmi/me (see, forexample,Lembege and Savoini, 1992; Scholer et al., 2003)show a variety of kinetic instabilities in the foot region, andin our results, features, possibly related to the Buneman in-stability, are visible in the foot region of the solar wind ionphase space in Figs.3 and4. These features occur on elec-

30 30.5 31 31.5 32 32.5 33 33.5 34 34.5 35 35.50

500

1000

1500

2000

2500

3000

t / 2πωci−1

Cou

nt

Fig. 7. Reflection rates over a series of reformation cycles. Solarwind protons are in black, and pickup protons in blue. Counts areof particles upstream of the minima in

∫Exdx associated with the

shock front, with velocities, in the case of solar wind ions directedin the direction of shock propagation, or for pickup ions, outsideof the initial injected spherical distribution. Pickup ion counts arenormalised to solar wind counts.

Fig. 8. Top panel (panel 1) shows the energy of a pickup proton, cal-culated from

∫F·vdt and its components (x−red,y−blue,z−green,

total−black), panel 2 the energy of a solar wind proton. Panel 3shows

∫Exdx at the position of both protons (solar wind−black,

pickup−purple), panel 4 they-positions, calculated fromy=∫vydt ,

and panel 5 thex-positions overlayed onto the shock potential onion scales (plotted as

∫Ex,idx).

tron spatio-temporal scales, and here we simply note that,as shown in Lee et al. (2004b), the ion dynamics will, fromthe above argument, be insensitive to structures on electronkinetic scales.

The proton trajectories in Fig.8 display both similaritiesand differences during their shock interactions. Both par-ticles encounter the shock when the magnetic field is at itsstrongest (Fig.1), and the potential at its narrowest (panel5). They are reflected off the shock front, moving upstreaminto the foot region gaining energy (panels 1 and 2) in thelocal time dependent electric field. Their gyration and thepropagation of the shock then brings them back towards theshock front, where their energies are too high for a secondreflection, and they enter the downstream region (panel 5).

The x-components of∫

F·vdt for both protons are simi-lar (red, panels 1 and 2 of Fig.8). Initially, as the protonsmove towards the shock, there is anx-energy loss of roughly

648 R. E. Lee et al.: Reforming shocks with pickup ions’

0 1 2 3 4 5 6 7 8 9 10−10

−8

−6

−4

−2

0

εSW

/ εinj,SW

f(ε S

W)

Fig. 9. Solar wind proton energy distribution functions, kinetic en-ergy normalised toEinj,SW vs. normalised frequency on a log10scale. Results are shown both in the absence of pickup protons(black) and with 10% pickup protons (blue). For comparison, theupstream distribution functions are also shown, in the absence ofpickup protons in red and with 10% in magenta.

0 5 10 15 20 25 30−10

−8

−6

−4

−2

0

εPI

/ εinj,SW

f(ε P

I)

Fig. 10.Pickup proton energy distribution functions, kinetic energynormalised toEinj,SW vs. normalised frequency on a log10 scale.Colours are as in Fig.9.

equal magnitude (∼Einj,SW). After reflection this is followedby an x-energy gain, also of roughly equal magnitude, asthe protons move out into the foot region. The magnitudesare similar because the protons experience similar potentialstructures. However, there is a slight disparity because thepickup proton interacts with the shock front and transferssome energy from itsx-component to itsy-component (att∼30.3×2πω−1

ci ), whereas this process is absent from thesolar wind proton trajectory which remains within the footregion (t=30.3 to 32.8×2πω−1

ci in panel 5).Examining they-components of energy (blue, panels 1

and 2 of Fig.8), the origin of the disparity in final energiesbecomes clear. The pickup proton moves into the shock ata higher velocity, and is therefore reflected at that higher ve-locity back into the upstream region (compare panels 3 and5 of Fig. 3). Consequently, the distance travelled by pickupprotons in the foot region, upstream of the shock, is greaterthan that for the solar wind protons. It follows that the driftin they-position is greater (panel 4), so that the gain in they-component of energy, throughEy motional drift, is greater.Examining

∫Exdx at the particle positions (panel 3), we see

that the overall potential structures around the times ofy-energy gain are similar for both particles. Energisation oc-curs during a phase when the local potential is approximatelyconstant: again, as the pickup proton travels further in thisstate, and as they-energy gain is roughly linear, it gainsmore energy. As the particle passes through the foot regionfor a second time, the energisation finishes and the particle

crosses the shock front. Consequently, solar wind protonsare seen at energies up to and beyond∼6Einj,SW and pickupprotons at up to∼20Einj,SW. The self-consistent inclusionof pickup ions leads to additional downstream magnetic fieldstructures, that can lead to further acceleration of the pickupprotons, as indicated by the scattering aroundx'83λci in thepickup proton energies in Fig.3.

To illustrate changes in energisation mechanisms that oc-cur due to the presence of pickup ions, the normalized dis-tribution functions of the proton energies,f (E), for eachspecies are plotted on semi-log axes verses energy, normal-ized to the solar wind proton injection energy, in Figs.9and 10. The distribution functions shown here were col-lected over ten upstream proton gyroperiods from a regionstarting atx'12λci downstream of the shock front, and ex-tending for x'5λci . These distances are sufficiently fardownstream to be in a region of strongly fluctuating fieldsand over spatio-temporal scales sufficiently large as to aver-age over the small-scale variations present within the down-stream populations. For comparison the distribution func-tions in the upstream region are also shown. These space andtime averaged distributions capture the extended tail off (E)

which cannot be seen in the snapshots of phase space (Figs. 3and 4).

Figure 9 shows the solar wind proton distribution func-tions. Comparing the distribution functions of the simula-tions containing no pickup protons to one containing 10%pickup protons shows only minor differences. The peakin the downstream distribution functions is at zero, as theplots refer to the downstream rest frame. The distributionfunctions decay exponentially, with changes in exponent at∼2Einj,SW, the energy gained by protons that specularly re-flect form the stationary shock front (see,Lee et al., 2004a),and at∼5Einj,SW, close to the maximum energy observed forprotons as they leave the shock front and enter the down-stream region. Energisation can also be seen to be weaker inthe presence of pickup protons.

Examining the pickup proton downstream distributionfunction (Fig.10, blue), we see a peak at around 2Einj,SW,the maximum energy of the inflowing pickup protons. Thedistribution function varies slowly on this log axis up to∼17Einj,SW, this value is the maximum energy observedfor pickup protons leaving the shock front and entering thedownstream (Fig.3, panel 3). At higher energies the distri-bution decays exponentially, which is suggestive of acceler-ation in the downstream region.

4 Conclusions

We have performed PIC simulations for a quasi-perpendicular shock in a parameter range where theshock solutions are reforming and which is appropriatefor the heliospheric termination shock. We include a shelldistribution of 10% pickup protons, centred in velocity spaceonvinj , with a radiusv1'25×vthm,SW.

R. E. Lee et al.: Reforming shocks with pickup ions’ 649

1. The addition of 10% pickup protons does not modifythe solar wind proton phase space and dynamics ofthe reforming shock found previously (Lembege andSavoini, 1992; Shimada and Hoshino, 2000; Schmitzet al., 2002a; Lee et al., 2004a).

2. The dynamics of the energised solar wind proton popu-lation proceeds as in simulations where pickup protonsare absent. Typical energies reach∼6Einj,SW and be-yond, however, energisation is weaker in the presenceof pickup protons.

3. A subset of the pickup protons reflect off the shock toform a weak foot region which stands upstream of theshock and is quasi-stationary.

4. The reflected pickup protons gainE∼20×Einj,SW due tothis dynamics. This acts as a suprathermal populationwhich could then act as a seed population for subse-quent acceleration at the shock, which can in turn gen-erate ACRs.

5. There is some evidence of further acceleration of pickupprotons downstream of the shock ramp.

The energisation seen in our simulations takes place ina broad volume spanning the shock, its extended upstreamfoot, and downstream. Both the degree of energisation, andthe extent of the region in which it occurs, are substantial.This implies that some of these effects may be observable,depending upon instrumental availability and resolution, andmay assist the interpretation of heliopause data.

Finally, our simulations, whilst generating a suprather-mal population of protons, do not directly generate ACRs.The reforming shock solutions, under the restrictions ofthese simulations, provide an “injection” or seed populationfor other mechanisms, such as diffusive Fermi acceleration(Bell, 1978; Malkov and Drury, 2001). Importantly, the re-forming shock solutions have an effective reflection proba-bility for the protons which differs from that of a static so-lution (Lee et al., 2004b). Whether or not a solar wind pro-ton is reflected depends critically upon when it encountersthe shock ramp, rather than where it is located in the phasespace (Burgess et al., 1989, compare). This may have impli-cations for processes such as shock surfing (as conjecturedby Scholer et al., 2003) and multiple reflection (as found forstatic shock solutions byZank et al., 1996; Lipatov and Zank,1999). In this contextZank et al.(1996) have noted that theshock ramp thickness is a critical parameter. In commonwith previous simulations (see, for example,Shimada andHoshino, 2000; Schmitz et al., 2002a) and with the assump-tion of (Zank et al., 1996), the shock ramp width in the sim-ulation presented here is∼2c/ωpe. However, the extensionof these PIC simulations to higher dimensions, whilst retain-ing reforming shock solutions (Lembege and Savoini, 1992),will allow wavemodes withk vectors oblique to the magneticfield rather than strictly perpendicular, as in our 1.5-D sim-ulations. The occurrence of multiple reflection thus remainsan open question.

Acknowledgements.R. E. Lee acknowledges a CASE ResearchStudentship from the UK Particle Physics and Astronomy ResearchCouncil in association with UKAEA, and a Warwick PostgraduateResearch Fellowship from the University of Warwick. This workwas also supported in part by the UK Engineering and Physical Sci-ences Research Council. S. C. Chapman acknowledges a RadcliffeFoundation Fellowship.

Topical Editor R. Forsyth thanks two referees for their help inevaluating this paper.

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