Particle acceleration and wave excitation in quasi-parallel high-Mach-number collisionless shocks: Particle-in-cell simulation
aa r X i v : . [ a s t r o - ph . H E ] F e b Draft version July 9, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
PARTICLE ACCELERATION AND WAVE EXCITATION IN QUASI-PARALLEL HIGH-MACH-NUMBERCOLLISIONLESS SHOCKS: PARTICLE-IN-CELL SIMULATION
Tsunehiko N. Kato Department of Physical Science, Graduate School of Science, Hiroshima University and1-3-1, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
Draft version July 9, 2018
ABSTRACTWe herein investigate shock formation and particle acceleration processes for both protons andelectrons in a quasi-parallel high-Mach-number collisionless shock through a long-term, large-scaleparticle-in-cell simulation. We show that both protons and electrons are accelerated in the shock andthat these accelerated particles generate large-amplitude Alfv´enic waves in the upstream region ofthe shock. After the upstream waves have grown sufficiently, the local structure of the collisionlessshock becomes substantially similar to that of a quasi-perpendicular shock due to the large transversemagnetic field of the waves. A fraction of protons are accelerated in the shock with a power-law-likeenergy distribution. The rate of proton injection to the acceleration process is approximately constant,and in the injection process, the phase-trapping mechanism for the protons by the upstream waves canplay an important role. The dominant acceleration process is a Fermi-like process through repeatedshock crossings of the protons. This process is a ‘fast’ process in the sense that the time required formost of the accelerated protons to complete one cycle of the acceleration process is much shorter thanthe diffusion time. A fraction of the electrons is also accelerated by the same mechanism, and havea power-law-like energy distribution. However, the injection does not enter a steady state during thesimulation, which may be related to the intermittent activity of the upstream waves. Upstream of theshock, a fraction of the electrons is pre-accelerated before reaching the shock, which may contributeto steady electron injection at a later time.
Subject headings: acceleration of particles – cosmic rays – ISM: supernova remnants – methods: nu-merical – plasmas – shock waves INTRODUCTIONCollisionless shocks, which are driven by various vio-lent phenomena throughout the universe, are believed tobe sites of particle acceleration. In particular, cosmicrays with energies below the knee energy (approximately10 . eV) are considered to be accelerated by shocks insupernova remnants (SNRs) in our galaxy. A number ofX-ray observations have revealed synchrotron X-rays ra-diated from high-energy electrons around the shocks inseveral young SNRs, and these synchrotron X-rays areregarded as evidence of electron acceleration around theshocks to energies of approximately 10 eV (Koyama etal. 1995; Long et al. 2003; Bamba et al. 2003). Recentobservations have also revealed gamma rays associatedwith the decay of neutral pions ( π mesons), which oc-curs as a result of proton-proton collisions (Ackermannet al. 2013). These observations are regarded as directevidence of high-energy protons in the vicinity of theshocks.One of the most plausible candidates for the accelera-tion process that acts in shocks is first-order Fermi ac-celeration (Drury 1983; Blandford & Eichler 1987). Inparticular, first-order Fermi acceleration can explain thepower-law energy spectrum for the accelerated particlesand the power-law index expected from the observationsof cosmic rays. However, the acceleration process that is [email protected] Current address: Center for Computational Astrophysics,National Astronomical Observatory of Japan, 2-21-1 Osawa, Mi-taka, Tokyo 181-8588, Japan actually operating in the SNR shocks has not yet beendetermined, especially with respect to electron accelera-tion, and further observational and theoretical investiga-tions are needed.Generally, it is considered that the acceleration effi-ciency is strongly dependent on the shock velocity or theshock Mach number, and the process would be more ef-ficient for larger velocity or higher Mach number shocks.For example, young SNRs have large shock velocities(approximately 1,000 to 10,000 km s − ), and the cor-responding Mach number is very large (approximately100). Young SNRs are believed to be efficient acceler-ators of particles. In addition, the orientation of thebackground magnetic field upstream of the shock canalso have a significant influence on the efficiency of theparticle acceleration process. For example, the bipolarmorphology of the emission region of non-thermal syn-chrotron X-rays in the supernova remnant SN1006 isconsidered to be related to the orientation of the mag-netic field in the interstellar medium around the remnant,which is upstream of the shock. Recent observations ofradio polarization by Reynoso et al. (2013) suggest thatthe electron acceleration is efficient when the direction ofthe ambient magnetic field is approximately parallel tothe shock normal. Hence, the particle acceleration pro-cess can be more efficient in quasi-parallel shocks thanin quasi-perpendicular shocks. Here, shocks in which theupstream magnetic field lies along the shock normal arereferred to as parallel shocks, and shocks in which the up-stream field lies perpendicular to the shock normal arereferred to as perpendicular shocks. Tsunehiko N. KatoIn the heliosphere, collisionless shocks (e.g., Earth’sbow shock, interplanetary shocks associated with coro-nal mass ejections, solar wind termination shocks, etc.)are also formed. These shocks have been investigatedthrough a number of direct in situ observations by space-craft: For example, the Earth’s bow shock (Burgess et al.2012) and those associated with solar energetic particlesand energetic storm particle events (Lee et al. 2012). Inparticular, these observations indeed showed that quasi-parallel shocks can be efficient ion accelerators and theirefficiency is generally dependent on the shock strength(Reames 2000). The wave generation in the upstreamregion of shocks by reflected energetic diffuse ions arealso observed in Earth’s bow shock (Hoppe et al. 1981;Burgess et al. 2005). Although the Alfv´en Mach numbersof these shocks are generally smaller than those of SNRshocks, recently, a quasi-parallel collisionless shock atvery high Mach number of ∼
100 have been observed inSaturn’s bow shock by Cassini spacecraft (Masters et al.2013). This observation also showed an efficient electronacceleration there, suggesting that quasi-parallel shocksat very high Mach numbers can be efficient electron ac-celerators, too.The acceleration processes in shocks have also been in-vestigated through numerical simulations. Recent large-scale hybrid simulations, in which protons are treatedas discrete particles whereas electrons are approximatedas a massless fluid, have shown that protons are accel-erated efficiently in quasi-parallel shocks with a power-law-like energy distribution (Giacalone 2004; Sugiyama2011; Gargat´e & Spitkovsky 2012; Caprioli & Spitkovsky2014). The acceleration process observed in these studiesis often a ‘fast’ process in the sense that the acceleratedprotons cross the shock front back and forth repeatedlywithin a much shorter timescale than that of the diffusivemotion. Another prominent feature of these particle ac-celerating shocks is the wave excitation in the upstreamregion of the shock by the accelerated protons, whichwould be a similar processes to those observed in quasi-parallel Earth’s bow shocks by in situ observations. Theamplitude of these waves can be even larger than thestrength of the background magnetic field. These large-amplitude upstream waves can strongly influence boththe particle acceleration process and the shock structure.Hybrid simulations, however, cannot deal with the ki-netic dynamics of electrons, especially non-thermal elec-tron acceleration. In the present paper, we investigatethe particle acceleration process for both protons andelectrons in high-Mach-number quasi-parallel shocks, aswell as the shock formation process and structure, tak-ing into account the electron dynamics self-consistentlythrough particle-in-cell (PIC) simulations in which bothprotons and electrons are treated as discrete particles.For protons to be accelerated to sufficiently high ener-gies, long-term (typically hundreds of proton gyro-time)and therefore large-scale simulations are required. In or-der for the calculation time to be as long as possible, wecarry out a large-scale one-dimensional simulation. InSection 2, we describe the simulation model. The pri-mary results of the simulation are shown in Section 3.The obtained results are discussed in Section 4, and con-clusions are presented in Section 5. METHOD In order to investigate the particle acceleration processin quasi-parallel collisionless shocks in an electron-protonplasma, we carry out a large-scale numerical simulation.The simulation code is a one-dimensional electromag-netic particle-in-cell code with one spatial dimension andthree velocity dimensions (1D3V) that was developedbased on a standard method described by Birdsall &Langdon (1991). The basic equations are the Maxwell’sequations and the (relativistic) equation of motion of par-ticles. In the following, we take the x -axis as the one-dimensional direction.In the simulation, a collisionless shock is driven accord-ing to the “injection method”. There are two conductingrigid walls at both ends of the simulation box. Thesewalls reflect both incident particles and electromagneticwaves specularly. Initially, the plasma is moving in the+ x -direction at a bulk velocity V . Both electrons andprotons are loaded uniformly in the region between thetwo walls with an average velocity of V plus the ther-mal velocity, where the temperatures of the electronsand protons are initially set to be equal. As the plasmais magnetized, the plasma convects the ordered back-ground magnetic field B . Since the electric field shouldvanish in the plasma rest frame, the motional electricfield E = − V × B appears in the simulation frame,in which the plasma is moving. In the early stage ofthe simulation, the particles that collide with the wallon the + x side are reflected specularly and then interactwith the incoming plasma. This interaction causes someinstabilities and eventually leads to the formation of acollisionless shock. The frame of the simulation is therest frame of the shock downstream plasma (hereinafter,the downstream rest frame). Thus, in the simulation, weobserve the propagation of the collisionless shock in the − x -direction in the downstream rest frame.In the following, we take ω − as the unit of timeand λ e = cω − as the unit of length, where ω pe =(4 πn e0 e /m e ) / is the electron plasma frequency definedfor the far upstream plasma number density n e0 withelectron mass m e and magnitude of the electron charge e = | e | . The units of the electric and magnetic fields aregiven by E ∗ = B ∗ = c (4 πn e0 m e ) / . RESULTSWe carry out a large-scale one-dimensional PIC simu-lation under the following conditions. We use a reducedproton mass of m p = 30 m e for which the proton inertiallength is given by λ p = 5 . λ e . The number of spatialgrids is N x = 2 . × , and there are N PPC = 160 super-particles per cell per species. The physical dimension ofthe simulation box is L x = 1 . × λ e = 2 . × λ p .The size of a cell is thus ∆ x = 0 . λ e . The time step istaken to be ∆ t = 0 . ω − .The initial bulk velocity of the plasma is given by V =0 . c in the + x -direction. The corresponding Lorentzfactor is given by Γ = 1 .
08, and thus the relativisticeffect is not significant for the shock formation. Theordered background magnetic field is set on the x - y plane,and, in order to study a quasi-parallel shock, the anglebetween it and the x -axis (i.e., the shock normal) is takento be Θ = 30 ◦ in the downstream rest frame. Hence,the motional electric field lies in the − z -direction. Thestrength of the background magnetic field B = | B | isarticle Acceleration in Quasi-parallel Shocks 3 Figure 1.
Time evolution of the energy density of the transversemagnetic field B t = ( B y + B z ) / . The horizontal and vertical axesindicate the x -coordinate and the time, respectively. The value oflog ( B t /B ) is shown in grayscale (color). A collisionless shockis formed on the right-hand side and propagates to the left. Large-amplitude ( B t ≫ B ) waves are evident upstream (left-hand side)of the shock. set so that B /B ∗ = ω ce /ω pe = 8 . × − , where ω ce = eB /m e c is the electron cyclotron frequency. Theproton gyro frequency is given by ω cp = 2 . × − ω pe .The Alfv´en velocity in the upstream region is thus givenby v A = 1 . × − c . In terms of the magnetization, σ ≡ B / πn e0 ( m e + m p ) V , which corresponds to σ =1 . × − , i.e., it is very weakly magnetized. The plasmabeta parameters defined for electrons and protons areboth set to be 0 .
5, so that the total plasma beta becomesunity ( β = β e + β p = 1) and the initial temperatures ofelectrons and protons are determined accordingly ( T e = T p = 1 . × − m e c ).3.1. Structure of Collisionless Shock
Figure 1 presents the evolution of the magnetic energydensity of the transverse components, B t = ( B y + B z ) / ,normalized by that of the background field B / π . Acollisionless shock is observed to form in the vicinityof the wall located at x = 1 . × λ e and then prop-agate to the left at an approximately constant speed.The shock speed is obtained as approximately − . c in the downstream rest frame (namely, the simulationframe) and approximately − . c in the upstream restframe. The corresponding Alfv´en Mach-number is givenby M A ∼ M A ∼ B t ≫ B ) waves in theupstream region (left-hand side) of the shock. Thesewaves appear around ω pe t ∼ . × , and the re-gion in which they exist, which is sometimes referredto as the foreshock region, extends upstream with time.This excitation of the waves in the upstream region canbe attributed to the appearance of the energetic par-ticles (Bell 1978). Indeed, excitation is commonly ob-served in numerical simulations in which a fraction ofthe particles is accelerated in the shock efficiently un-der the quasi-parallel condition, for example, in non-relativistic hybrid simulations (Giacalone 2004; Capri-oli & Spitkovsky 2014) and relativistic PIC simulations(Sironi & Spitkovsky 2011).Figure 2(a) shows the profiles of the magnetic fieldcomponents B y and B z around the shock at ω pe t =1 × , where the shock front is located at x ∼ . × Figure 2. (a) Magnetic fields B y (black (red)) and B z (gray(blue)) around the shock at ω pe t = 1 × normalized by thebackground field B . (b) Close-up of the region enclosed by thedotted box in (a). (c) Angle of the local magnetic field to the shocknormal ( x -axis). (d) x − u x phase-space plot of the protons, where u x denotes the x -component of the particle 4-velocity. λ e . Large-amplitude waves occur upstream of theshock, and the magnetic fields of the waves are muchlarger than the background field B in the vicinity ofthe shock. Figure 2(b) shows a close-up of the regionindicated by the dotted box in Fig. 2(a). The trans-verse magnetic fields are dominant around the shock.As shown in Fig. 2(c), the local magnetic field and theshock normal (i.e., the x -axis) are approximately per-pendicular. Since the shock structure is much smallerthan the typical wavelength of the upstream waves, theshock experiences an almost uniform perpendicular mag-netic field. Therefore, the local shock structure itself be-comes essentially that of the quasi-perpendicular shock.Indeed, Fig. 2(d) shows that some of the incoming up-stream protons are reflected at the shock front, which is awell-known characteristic of quasi-perpendicular shocks.Due to this strong perpendicular magnetic field, the lo-cal Alfv´en Mach number becomes M A ∼ .
6. Thus, theshock itself is locally not a high-Mach-number shock.3.2.
Wave Excitation in the Upstream Region
Figure 3(a) shows the profiles of the magnetic fieldaround the shock, as in Fig. 2(a). Figure 3 (b) showsa close-up of the region 1 . × < x/λ e < . × indicated by the dotted box in Fig. 3(a) together withthe electric fields. The generated waves are monochro-matic rather than turbulent, as is clear from the fig-ure. The wavelengths of these waves are typically λ ∼ λ e − λ e , which is comparable to or somewhat largerthan the gyro-radius of the protons reflected at the shock(defined for the background field strength B ). Thewavelength satisfies the condition λ ≫ πv A /ω cp ∼ λ e ,where ω cp = eB /m p c is the cyclotron frequency of theupstream protons, and so are regarded as Alfv´enic elec-tromagnetic waves. Although the waves actually propa-gate obliquely with respect to the background field, thestructures of the electric and magnetic fields of the wavesare essentially the same as the structure of right-hand Tsunehiko N. Kato ! " $ % & $ ’ ( ) % & $ " $ + & $ ’ ( $ , & $ "-.*/" * "-. !- ( & ( "-0* * "-0.*"-0. Figure 3. (a) Magnetic fields around the shock at ω pe t = 1 × ,as in Fig. 2(a). (b) Close-up of the region enclosed by the dottedbox in (a). The normalized electric fields E y /B and E z /B arealso shown by the dashed curves. (c) Number density of protons n p normalized by the number density far upstream n e0 . circularly polarized waves and are naturally explained tobe generated by the resonant mode instability (Winske& Leroy 1984). The number density of the incoming up-stream plasma is also modulated because of the existenceof the waves, as shown in Fig. 3(c). This compressivebehavior can be explained as a non-linear effect or a fea-ture of obliquely propagating unstable modes (Gary etal. 1981).Figure 4 shows the features of the upstream waves ob-served in the upstream rest frame. The fluctuation in theproton number density δn , the fluctuation in the mag-netic pressure δP B , and the product of these fluctuationsare presented in Figs. 4(a), 4(b), and 4(c), respectively.In Fig. 4(a), the shock front is visible as the slightly in-clined horizontal discontinuity at nearly ω pe t ∼ × and the region below this shock front is the upstreamregion of the shock. Note that when the amplitude ofthe waves is small ( ω pe t . . × in this figure), thewaves propagate in the upstream ( − x ) direction, whichis consistent with the explanation, which indicates thatthe waves are generated by the resonant mode instability.In addition, there is a positive correlation between thedensity fluctuation and the magnetic fluctuation. Thiscan also be explained as a feature of the oblique mode(Gary et al. 1981). In the nonlinear regime, the wavesare almost at rest in the upstream rest frame and growin wavelength and amplitude.3.3. Particle Acceleration
Figure 5(a) presents the energy spectra of the pro-tons and the electrons in the downstream region of theshock at ω pe t = 2 × together with the fitted ther-mal Maxwellian distributions. Here, E kin = ( γ − mc denotes the kinetic energy of particles of mass m andLorentz factor γ measured in the downstream rest frame.Note that the bulk kinetic energy of the protons and thatof the electrons of the incoming plasma in the upstreamregion are given as approximately 2 . m e c and approxi-mately 7 . × − m e c , respectively. In this figure, high-energy and non-thermal populations with power-law-likedistributions exist not only in protons (for E kin & m e c ) Figure 4.
Evolution of upstream waves observed in the upstreamrest frame. (a) Fluctuation in the proton number density δn p .(b) Fluctuation in the magnetic pressure δP B . (c) The product δn p × δP B , which shows the correlation of the density fluctuationand the magnetic fluctuation. In all panels, positive values areshown in white (red), and negative values are shown in black (blue).The horizontal coordinate is x − V t , where V is the upstream flowvelocity measured in the simulation frame. ! " ! "$ ! " ! ! ! ! $ ! % &’( )* + , $ ! " ! ! "- ! ". ! "/ ! "0 ! "1 ! "2 ! " ! "$ ! " ! ! ! ’ % & ’ ( ! " ! "$ ! " ! ! ! ! $ ! % &’( )* + , $
465 475
Figure 5. (a) Energy spectra of protons (black (blue) solid curve)and electrons (gray (red) solid curve) in the downstream regionof the shock obtained from the simulation at ω pe t = 2 × .Thermal Maxwellian distributions for protons with temperature T p = 0 . m e c and for electrons with temperature T e = 0 . m e c are also shown by the black (blue) dashed curve and the gray (red)dashed curve, respectively. (b) Development over time of the en-ergy spectra. The spectra for ω pe t = 5 × (dotted curves),1 . × (dashed curves), and 2 . × (solid curves) are shown. but also in electrons (for E kin & m e c ). The power-lawindices obtained from these portions of the distributionsare approximately 2 . . E kin ∼ m e c ) is smaller than that ofthe protons ( E kin ∼ , m e c ).The number densities and temperatures of theMaxwellian distributions for fitting the thermal com-ponents in Fig. 5(a) are given by n p , th = 3 . n e0 and T p = 0 . m e c for protons and n e , th = 3 . n e0 and T e = 0 . m e c for electrons, while the total number den-sities (including the non-thermal components) are givenby n p ∼ n e ∼ . n e0 . Here, since a fraction of the ther-mal electrons can be relativistic, we adopted a relativisticMaxwellian (or J¨uttner-Synge) distribution (c.f. Landauarticle Acceleration in Quasi-parallel Shocks 5 ! " ! " ! " !"" $ % & ’ () * ) + , + ! " ! " !" " " " " ’ -$. */ Figure 6.
Cumulative energy spectra of the protons (black(blue)) and the electrons (gray (red)) in the downstream regionof the shock at ω pe t = 2 × normalized by the mean total en-ergy density. & Lifshitz 1980). The temperature ratio T p /T e ∼ . T eq = 0 . m e c .Therefore, both temperatures are lower than the equilib-rium temperature. This is attributed to the fact that afraction of energy is transferred to the non-thermal parti-cles through the particle acceleration process. Note thatthe fitting by the Maxwellian distribution is not satisfac-tory for the electrons. This can be improved by addinga high-temperature thermal component with a tempera-ture of approximately 0 . m e c to the first Maxwelliancomponent. As shown later, a fraction of the incomingelectrons are accelerated in the upstream region beforearriving at the shock front and these ‘pre-accelerated’electrons would result in a high-temperature component.Figure 5(b) shows the development over time of theenergy spectra in the shock downstream region. For pro-tons, the high-energy power-law-like portion extends overtime, which is an expected behavior for the Fermi-like ac-celeration process with a constant injection of seed par-ticles. However, this is not the case for electrons. Theamount of non-thermal electrons decreases with time. Aswill be shown later, this is because the injection of theelectrons to the acceleration process is not constant butrather occurs only at a particular time interval.Figure 6 shows the cumulative energy distributions de-fined by F i ( E ) ≡ Z ∞ E N i ( E ′ ) E ′ dE ′ (1)in the downstream region at ω pe t = 2 × normal-ized by the mean total energy density, where i = e , p,and N i ( E ) is the energy spectrum for species i shownin Fig. 5. The mean total energy density is given by F tot = F e (0) + F p (0). The total energy ratio of pro-tons to electrons is approximately 2 .
5. The non-thermalprotons and the non-thermal electrons contain approxi-mately 10% and approximately 0 . Acceleration of Protons
Figure 7.
Acceleration histories of the highest-energy protons.The x − t trajectories are shown in (a). The E kin − t histories areshown in (b), in which the kinetic energies E kin are measured inthe downstream rest frame. The dotted line denotes the case oflinear acceleration at a rate of dE kin /dt ∼ . × − ( m e c ω pe ). Figure 7 presents the acceleration histories of the sixprotons that are accelerated to the highest energies in thesimulation. The trajectories of these protons on the x − t plane are shown in Fig. 7(a) and on the E kin − t planein Fig. 7(b), where E kin denotes the kinetic energy mea-sured in the downstream rest frame. All of these protonsare accelerated upon repeatedly crossing back and forthacross the shock to energies up to E kin ∼ , m e c .While the acceleration process is essentially a stochasticprocess, these protons are, on average, accelerated lin-early with time. The average acceleration rate is roughlygiven by dE kin /dt ∼ . × − ( m e c ω pe ), which is indi-cated by the dotted line in Fig. 7(b).Figure 8 shows a representative acceleration history ofthe protons. In Figs. 8(a) and 8(b), the energy histo-ries of the proton are presented on the x s − E kin plane,where the kinetic energy E kin is measured in the up-stream rest frame in Fig. 8(a) and in the downstreamrest frame in Fig. 8(b), respectively. Here, x s is the co-ordinate in which the shock front is fixed at the origin x s = 0 and given by x s ( t ) = x ( t ) − x sh ( t ), where x ( t )is the particle position and x sh ( t ) is the shock position,both measured in the downstream rest frame. As shownin Fig. 7, the proton is accelerated by repeatedly cross-ing the shock. Figures 8(c) and 8(d) show the variationof the corresponding particle kinetic energies over time.Figures 8(a) through 8(d) show that the energy of theparticle is almost conserved while it remains in one of theregions if the energy is measured in the plasma rest framein that region. Therefore, the acceleration process is nota resonant process but is essentially Fermi accelerationin the sense that particles are accelerated via repeatedelastic scatterings off the scattering centers at differentvelocities. The proton’s trajectory on the x s − z planeis shown in Fig. 8(e). The trajectory is approximately asimple gyro-orbit and is unlike that of the diffusive mo-tion. The non-diffusive features of the acceleration pro-cess are also observed in recent simulations (Sugiyama2011; Sironi & Spitkovsky 2011). Furthermore, there isno apparent average drift in the z -direction, namely thedirection of the motional electric field, suggesting thatthe acceleration process is not the shock drift accelera-tion.Figure 9 shows the history of the local magnetic fieldstrength at the particle position normalized by the up-stream background field, | B | /B , for the same proton, Tsunehiko N. Kato !"""" ’ ( ) ( l * %""" " %"""+ , ()(l * &-"+!" . !-.!-""-."-" w /* ( ) ( * & ( ) ( * & %""" " %"""+ , ()(l * :;<:=< :8<:9<:*< Figure 8.
Representative acceleration history of protons. (a) and(b) show the acceleration history on the x s − E kin plane, where x s is the coordinate for which the shock front is fixed at the origin.The kinetic energy E kin is measured in the upstream and down-stream rest frames. (c) and (d) show those on the t − E kin plane.The shaded regions denote the time intervals in which the protonremains in the downstream region and the non-shaded regions de-note the time intervals in which proton remains in the upstreamregion. The trajectory on the x s − z plane is shown in (e). !" $ % $ & % ! ’!("! "’ )* + Figure 9.
Local strength of magnetic field at the particle positionnormalized by the background field B as a function of time forthe same proton as in Fig. 8. The shaded regions indicate that theproton is in the downstream region. as shown in Fig. 8. After the large-amplitude waves arewell developed in the upstream region ( ω pe t & × ),the strength of the local magnetic field is at least dou-bled in both the upstream and downstream regions. Ac-cordingly, the typical gyration time τ B = 2 πω − c , where ω c = e | B | /γm p c is the local proton cyclotron frequency,becomes approximately half in each region, shorteningthe acceleration timescale.Figure 10(a) presents the energy gain factors per cycleof the shock crossing of the protons as a function of theenergies before the shock cycle for a sample of 1,000 pro-tons that are accelerated to energies higher than 300 m e c at ω pe t = 2 × . The factors are distributed around ap-proximately 1 to 2, and for high energies they convergeto approximately 1.4. The particles for which the ener-gies more than double during a half cycle are acceleratedwithin that region rather than upon shock crossing, al-though these acceleration processes are not dominant.There also exists a small fraction of particles that loseenergy. Figure 10(b) shows the residence time of the par-ticles in the upstream region and that in the downstreamregion for one cycle normalized by the average gyrationtime τ B . Here, the average gyration time is taken overeach half cycle of the shock crossing. Most of the protonsreturn to the shock front within times on the order of theaverage gyration time τ B .For one cycle of the shock crossing of the particles, !" & ’ ( ’ $ ) * + * , - . / ( t $%%%2%%3%%!%% . * &8)&9) Figure 10. (a) Energy gain factor per cycle of shock crossing, E /E , where E and E are the kinetic energies of the particlebefore and after the cycle, respectively, as a function of the kineticenergy before the crossing, E kin = E , for a sample of 1,000 protonsthat are accelerated to energies higher than 300 m e c at the end ofthe simulation ω pe t = 2 × . (b) Residence times in the upstreamregion (black (red) dots) and the downstream region (gray (green)crosses) for one cycle. The residence times in the upstream anddownstream regions derived from the Bohm diffusion model arealso shown by the dashed and dotted lines, respectively. (See thetext for details.) the residence time until return to the shock front in eachregion for diffusive motion is approximately given by (cf.Kato & Takahara 2003) t res = 43 c | V | τ , (2)where τ denotes the mean free time of the particle, and V is the flow speed of the scattering centers measuredin the shock rest frame. If we adopt the Bohm diffusionmodel, the mean free time of particles is given by τ = τ B ,where τ B is the gyration time of the particle. For thismodel, the residence times are given by t res , u ∼ . τ B for the upstream region and t res , d ∼ τ B for the down-stream region. These values are indicated by the dashedand dotted lines in Fig. 10(b). Most protons return tothe shock within much shorter times than those obtainedfrom the diffusion model, while a small fraction of pro-tons return to the shock with times that are comparableto or even longer than the diffusive time for low energies( E kin < E kin ∼ . m e c ) without pre-acceleration. Theseprotons will result in a constant injection of seed particlesfor the acceleration mechanism. On the other hand, it isalso seen within some time intervals (3 × < ω pe t < × , ω pe t ∼ . × , ω pe t ∼ . × ) that theprotons are pre-accelerated before the first shock cross-ing and gain energies up to approximately 20 m e c . Thiswould reflect the intermittent activity of the waves inthe upstream region, which is also visible in Fig. 1. InFig. 11(b), the times and the positions of the protons atwhich they attain energies of E kin /m e c = 5 , ,
20, andarticle Acceleration in Quasi-parallel Shocks 7
Figure 11.
Injection properties for the same protons as in Fig. 10.(a) Energies of the protons at the first crossing of the shock as afunction of the first crossing time. (b) Time and position when theprotons attain the energies of E kin /m e c = 5 , ,
20, and 50 forthe first time.
50 for the first time are shown. We again confirm thatthe accelerated protons are almost constantly injectedinto the acceleration process at the shock front withoutpre-acceleration.Figure 12 shows the phase angles of the same sample ofprotons as in Fig. 10 when they cross the shock front fromthe upstream side to the downstream side as a functionof the energy at the crossing. Here, the phase angle θ isdefined as the angle between the transverse component ofthe wave magnetic field, B t = (0 , B y , B z ), at the particleposition and that of the 4-velocity of the protons, u t =(0 , u y , u z ), so that ˆ B t · ˆ u t = cos θ and n · ˆ u t = sin θ, (3)where ˆ B t = B t / | B t | , ˆ u t = u t / | u t | , and n = ˆ x × ˆ B t .This figure shows that, for the first shock crossings, i.e.,the injection, there is a concentration of the phase an-gles around θ ∼ .
5. This would show the phase trappingof the injected protons by the upstream large-amplitudewaves (Sugiyama 1999; Sugiyama et al. 2001), suggest-ing that the phase trapping can play an important role inthe injection process for protons. On the other hand, forlater crossings or larger energies the phase angles are dis-tributed almost uniformly, indicating that at that stage,these protons are no longer trapped by the waves andthe acceleration process becomes almost independent ofthe phase angle.3.3.2.
Acceleration of Electrons
The acceleration process of the highest-energy elec-trons is shown in Figs. 13 and 14 as in Figs. 7 and8. These electrons are accelerated by essentially thesame process as that of the protons. However, thereare some differences from the proton case. The accel-eration process works somewhat intermittently for theelectrons. Indeed, five of the six electrons shown inFig. 13 are injected into the acceleration process around ω pe t ∼ × . The linear acceleration rate is given by dE kin /dt ∼ . × − ( m e c ω pe ), which is slightly larger !" ’ ( ) * + , ) - . / + ,, q ,,, ) % $ %& $ %&& $ %&&&8 ;< + = $ Figure 12.
The angles between the transverse component ofthe wave magnetic field and that of the 4-velocity of the accel-erated protons at the shock crossings form the upstream side tothe downstream side. The sample of the protons is the same asthat in Fig. 10. Those for the first shock crossing are indicated bythe black dots, and the others are indicted by the gray dots as afunction of their kinetic energy at the crossing.
Figure 13.
Acceleration histories of the highest-energy electronsas in Fig. 7. As a guide, the linear acceleration rate dE kin /dt ∼ . × − ( m e c ω pe ) is shown by the dotted line in (b). !""" $ % & % l ’ !""" ) %&%l ’ *+"(," ! -+".+"/+" w % & % ’ : < % & % ’ : ) %&%l ’ =>?=@? =:?=
Representative acceleration history of electrons, asin Fig. 8. than the case of the protons shown in Fig. 7.Figure 15 shows the injection properties for the 321electrons that are finally accelerated to energies higher Tsunehiko N. Kato
Figure 15.
Injection properties for the 321 electrons that areaccelerated to energies higher than 60 m e c at the end of the simu-lation ω pe t = 2 × , as in Fig. 11. (b) Times and positions whenthe electrons attain the energies of E kin /m e c = 1 , , , ,
20, and50 for the first time. !" $ % & $ ’ ( $ ) & $ !"*+,! + !"**!"*-!"*.!"*!!"* / !" Figure 16.
Evolution of the upstream waves around ω pe t =6 × . The components B y (in black (red)) and B z (in gray(blue)) are shown from top to bottom for ω pe t = 5 . × , × , . × , and 7 × . than 60 m e c , as in Fig. 11. In contrast to the case inwhich the protons are injected into the acceleration pro-cess at an approximately constant rate, almost all of theelectrons are injected around ω pe t ∼ × , with someexceptions. The energies at the first shock crossing aremuch higher than the upstream bulk kinetic energy of theelectrons (approximately 0 . m e c ), indicating a pre-acceleration in the upstream region. The time of theefficient injection coincides with the end of the time in-terval observed in the proton injection in which the pre-acceleration of protons is efficient (see Fig. 11(a)). Dur-ing this time interval, the amplitudes of the upstreamwaves become very low (see Fig. 16). On the other hand,just before that time, the wave amplitude grows to veryhigh level. The injection efficiency of both protons andelectrons would be related with these wave activities.The injection properties for the 1,000 sample electronsthat achieve slightly lower energies 50 < E kin /m e c < Figure 17.
Same as Fig. 15, but for the 1,000 electrons acceler-ated to energies within the range of between 50 m e c and 60 m e c at ω pe t = 2 × . are pre-accelerated in a part of the upstream region andthe width of this region is extended with time, as shownin Fig. 17(b). This region coincides with the foreshockregion, where large-amplitude waves exist (see Fig. 1).With this extension of the pre-acceleration region, thetypical energies at the first shock crossing are also in-creasing, as shown in Fig. 17(a) for ω pe t > × .In particular, for ω pe t > . × , approximately halfof the injected electrons have gained energy in the up-stream region to higher than 10 m e c before the electronsreach the shock front. Although these pre-acceleratedelectrons have not contributed to the highest-energy elec-trons shown in Fig. 15 by the end of the simulation, theymay contribute at a later time.Figure 18 shows the energy spectra of the electronscalculated for three regions upstream of the shock at ω pe t = 1 × . The bulk upstream electrons are heatedby advection as they approach the shock. Their energyspectra in each region are well fitted by Maxwellian dis-tributions. Moreover, a fraction of electrons are accel-erated to high energies in the foreshock region, wherethe amplitude of the upstream waves becomes large, asshown in Fig. 17. This component can also be fitted bya Maxwellian although its temperature (approximately0 . m e c ) is much higher than that of the bulk electrons.As already mentioned, these pre-accelerated electronswould result in the high-temperature Maxwellian compo-nent in the downstream region. These pre-accelerationprocesses may later contribute to the steady electron in-jection for the Fermi-like acceleration process. DISCUSSIONIn the present simulation of a quasi-parallel shock, weobserved ion (proton) acceleration and the associatedwave generation around the shock. Such processes arealso observed in quasi-parallel shocks in the heliosphereby in situ observations with spacecraft. In particular, theion acceleration is observed in, for example, the Earth’sbow shock (Burgess et al. 2012) and the shocks associ-ated with the solar energetic particle events (Lee et al.2012). For the latter, the ion energy spectrum shows aarticle Acceleration in Quasi-parallel Shocks 9 ! " ! "$ ! "% ! "& ! " ! ! ! ! & ! % ’ ( ) * + ( , - ! " ! "$ ! "% ! "& ! " ! ! ! * +(, ./ & Figure 18.
Electron energy spectra in three upstream regions at ω pe t = 1 × , where the shock is located at x ∼ . × λ e andthe energies are measured in the upstream rest frame. Electronenergy spectra obtained for the regions of 9 × < x/λ e < . × (dotted curve), 1 . × < x/λ e < . × (dashed curve),and 1 . × < x/λ e < . × (solid curve) are shown inblack. The curves in gray show the fitted relativistic Maxwellianswith temperatures of T/m e c = 0 . . . . power-law shape and considered to be accelerated by thediffusive first-order Fermi acceleration mechanism. Onthe other hand, in the present simulation, as shown inSection 3.3.1, the dominant acceleration mechanism arenot diffusive but a process that has a shorter accelerationtimescale, which is similar to the “scatter-free” accelera-tion mechanism (Sugiyama et al. 2001; Sugiyama 2011).Regardless of the difference in the dominant accelerationmechanism, the resulting energy spectrum still becomes apower-law shape. The reason would be that it still satis-fies the fundamental requirements for the Fermi-type ac-celeration mechanism, i.e., the repeated shock crossings,the elastic scattering in the scattering center rest frame,and an approximately constant escape probability fromthe acceleration cycle, although the spatial motion of theparticles is not diffusive. The large-amplitude upstreamwaves observed in the simulation have a similar structureto the ULF waves observed in the quasi-parallel Earth’sbow shock by in situ observation (Hoppe et al. 1981); inthe present simulation, the reflected/accelerated protonswould play a similar role to the energetic diffuse ionsin Earth’s bow shock for generating these waves. Thesomewhat regular perpendicular magnetic structure dueto the generated waves in the upstream region observedin the simulation may be caused by the 1D dimension-ality of the simulation. In fact, recent multi-dimensionalhybrid simulations of high-Mach-number quasi-parallelshocks show more turbulent magnetic structures aroundthe shock transition region (Caprioli & Spitkovsky 2014).To see whether the 1D dimensionality affects the resultssignificantly, some multi-dimensional simulations are de-sired.Amano & Hoshino (2010) proposed an electron injec-tion model with a critical Alfv´en Mach number abovewhich the electron injection occurs. From observations in the heliosphere, it seems that this criterion is satisfied inseveral shocks in which electron acceleration is observed(e.g., Oka et al. 2006; Masters et al. 2013). It would beworthwhile to see whether this electron injection mech-anism operated in the present simulation. The electronacceleration occurred only around ω pe t ∼ × in thesimulation. In this time interval, the amplitude of the up-stream waves is temporally reduced as shown in Fig. 16and the local inclination angle of the magnetic field atjust upstream of the shock is given by Θ ∼ ◦ − ◦ . Thecritical Mach number is thus given by M inj A = 0 . − . M A = 28.Hence, the condition for injection M A ≫ M inj A is satisfiedand, from the criterion argument, the electron injectionis expected to occur. In their injection model, it is as-sumed that a fraction of the incoming electrons are re-flected at the shock front to form the electron beam andthen generate the whistler waves in the upstream region,and finally these waves scatter the electrons leading tothe diffusive acceleration. Thus, we seek such electronbeams reflected at the shock front in the phase-spaceplots at ω pe t = 4 × , 5 × , and 6 × . However,such beams couldn’t be found. In addition, the trajecto-ries of the accelerated electrons show no clear indicationof the interaction with the whistler waves. Therefore,it seems that the injection mechanism of the electronsobserved in the simulation is different from those consid-ered in Amano & Hoshino (2010). On the other hand, inthe other times, their injection mechanism does not op-erate, too, although the criterion is still satisfied, wherethe local inclination angle of the magnetic field is typi-cally given by Θ ∼ ◦ and the critical Mach number be-comes M inj A <
1. Regarding this point, it should be notedthat in the present simulation the superluminal conditioncan be realized because the shock velocity is relativelylarge ( ∼ . c ) and also the local inclination angle ofthe magnetic field becomes quasi-perpendicular after theupstream waves grow substantially; the condition for theshock to be subluminal is Θ < ◦ for V s = 0 . c and thisis difficult to be satisfied after the upstream waves growto large amplitude. In such cases, the de Hoffmann-Tellerframe does not exist and so the incoming electrons can-not be reflected at the shock front. Thus, to see whetherthe electron reflection at the shock front occurs and theinjection process proposed by Amano & Hoshino (2010)operates, simulations with smaller shock velocities wouldbe needed. This should be investigated in the futurestudies.In Section 3.1, the local shock structure becomes sub-stantially quasi-perpendicular after the amplitude of theupstream waves grow to be sufficiently large. This wouldbe a common feature of the quasi-parallel shocks in whichthe particle acceleration is efficient and the acceleratedparticles excite large-amplitude waves in the upstreamregion. In such local quasi-perpendicular conditions,as observed in several simulations of the perpendicularshocks (Amano & Hoshino 2009; Kato & Takabe 2010;Sironi & Spitkovsky 2011; Riquelme & Spitkovsky 2011;Matsumoto et al. 2013), the electron heating and/or ac-celeration in the foot region of the shock structure canalso occur (depending on the shock parameters and thedimensionality of the simulation) in addition to the elec-0 Tsunehiko N. Katotron acceleration/heating in the upstream wave regionfound in Section 3.3.2. Regarding this shock structure,while we observe that the shock is simply formed in theenvironment of the local quasi-perpendicular magneticfield, the shock formation due to the nonlinear steepen-ing of the upstream waves themselves was also reportedfor relativistic parallel shocks (Sironi & Spitkovsky 2011).This would indicate that the shock structure can dependon the shock speed. Which shock structure is realizedwould be determined by whether the typical gyro-radiusof the protons reflected at the shock front is larger thanthe typical wavelength of the upstream waves.Note that, for PIC simulations, in particular thosedealing with the particle acceleration process, as in thepresent work, the number of super-particles used in thesimulations can be important because high-energy par-ticles in the simulations generally suffer from the energyloss process due to the stopping power of the plasma(Kato 2013). The energy loss rate is inversely propor-tional to the number of super-particles in the electronskin depth volume, N e . Therefore, if N e is too small,the energy loss process becomes significant making theacceleration process inefficient. For one-dimensional sim-ulations, the energy loss rate for relativistic particles isgiven by dE kin dt ∼ − N e ( m e c ω pe ) . (4)From Figs 7 and 13, if we take the representativevalue of the acceleration rate for the present case to be dE kin /dt ∼ × − ( m e c ω pe ), the value of N e for whichthe energy loss rate is equal to the acceleration rate, isgiven by N e ∼
50. Figure 19 shows the energy spectra ofprotons and electrons in the shock downstream region at ω pe t = 2 × , as in Fig. 5(a), for four simulations thatare identical except for the value of N e . The accelerationefficiency is indeed dependent on N e for both protonsand electrons. In particular, in the case of N e = 50 (inwhich the number of super-particles per cell is N PPC = 3for ∆ x = 0 . λ e ) and N e = 133 ( N PPC = 8), the ac-celeration process becomes significantly inefficient and isalmost completely nonfunctional. Even in the case of N e = 667 ( N PPC = 40), the acceleration efficiency is stillaffected by the energy loss. Thus, for these simulations,the number of super-particles used should be chosen tobe sufficiently large, so that the energy loss is negligible.In the present paper, we observed that electron in-jection occurs only within a particular time interval( ω pe t ∼ × ) and does not settle at a constant rate,at least during the calculation time of the simulation.For longer timescales, the electron injection would beable to occur constantly or repeatedly. Instead, the elec-tron injection observed in the simulation may be a conse-quence of the initial condition and may not occur again.In order to resolve this issue, further long-term simu-lations are required. In addition, in two or three di-mensions, some electromagnetic instabilities, such as theion-Weibel instability, can play important roles, even fornon-relativistic shocks (Kato & Takabe 2008; Niemiec etal. 2012), and these effects can influence the particle ac-celeration process and the shock structure. Thus, multi-dimensional and long-term PIC simulations are also de-sired. ! " ! ! " ! "$ ! "% ! "& ! "’ ! "( ! ") ! "* ! " ! ! ! + , - . / , ! ") ! "* ! " ! ! ! ! * ! ) . /,0 * Figure 19.
Energy spectra in the downstream region of the shockat ω pe t = 2 × for protons (in black (blue)) and electrons (ingray (red)), as in Fig. 5, for four simulations that are identicalexcept for the number of super-particles within the electron skindepth, N e : N e = 50 (dotted curves), N e = 133 (dashed curves), N e = 667 (dot-dashed curves), and N e = 2 ,