Relativistic particle acceleration in developing Alfvén turbulence
aa r X i v : . [ a s t r o - ph . H E ] D ec Not to appear in Nonlearned J., 45.
Relativistic particle acceleration in developing Alfv´en turbulence
S. Matsukiyo and T. Hada
Department of Earth System Science and Technology, Kyushu University,6-1 Kasuga-Koen, Kasuga, 816-8580, Fukuoka, Japan [email protected]
ABSTRACT
A new particle acceleration process in a developing Alfv´en turbulence in thecourse of successive parametric instabilities of a relativistic pair plasma is in-vestigated by utilyzing one-dimensional electromagnetic full particle code. Co-herent wave-particle interactions result in efficient particle acceleration leadingto a power-law like energy distribution function. In the simulation high energyparticles having large relativistic masses are preferentially accelerated as the tur-bulence spectrum evolves in time. Main acceleration mechanism is simultaneousrelativistic resonance between a particle and two different waves. An analyticalexpression of maximum attainable energy in such wave-particle interactions isderived.
Subject headings: particle acceleration, parametric instability, developing turbu-lence, relativistic effect
1. Introduction
Large amplitude Alfv´en waves are ubiquitous in space and astrophysical environments.Such waves are believed to play key roles in the so-called DSA (diffusive shock accelera-tion) process in which charged particles are diffusively accelerated in the course of multiplescattering through turbulent Alfv´en waves upstream and downstream of collisionless shocks.The DSA is widely accepted as one of the most efficient acceleration mechanisms of galac-tic cosmic rays of energy up to ∼ . eV (Krymsky 1977; Axford et al. 1977; Bell 1978;Blandford & Ostriker 1978; Drury 1983; Lagage & Cesarsky 1983; Blandford & Eichler 1987;Jones & Ellison 1991; Malkov & Drury 2001; Duffy & Blundell 2005). In the process it is 2 –implicitly assumed that the Alfv´en turbulence is phase randomized and its spectrum is timestationary. On the other hand, turbulent Alfv´en waves commonly observed in the solar-terrestrial environments are often intermittent, and coherent MHD structures are frequentlysuperposed. Namely, the phase random approximation cannot be assumed (de Wit et al.1999; Hada et al. 2003; Narita et al. 2006) and a spectrum of turbulence may evolve bothin space and time (Bruno & Carbone 2005). This is probably due to the fact that nonlin-ear wave-wave interactions tend to generate coherence among wave phases, and that spatialand temporal scales of relaxation processes in space plasmas are much larger than typicalscales of our solar-terrestrial system. This may hold true with some high energy astro-physical environments. That is, spatial and temporal scales of a relaxation process are notnegligibly small in comparison with scales of a whole acceleration site. Generally speaking,particle acceleration rate through coherent wave-particle interactions is much higher thanthat through incoherent, or fully turbulent, wave-particle interactions (Kuramitsu & Hada2000, 2008). Therefore, acceleration processes through coherent wave-particle interactions ina ’developing’ turbulence, where a turbulence has not been fully developed, should be paidmore serious attention.It is well-known that in various space plasma environments a nonequilibrium ion dis-tribution function generates large amplitude Alfv´en waves through some instabilities andthat those waves nonlinearly evolve to produce coherent magnetic wave forms. One ofthe common interpretations of generation mechanisms of such wave forms accompanyingdensity fluctuations which are frequently observed in the solar wind (Spangler et al. 1997;Spangler & Fuselier 1988) is a parametric instability where nonlinear wave-wave interactionsconvert energy of a parent wave into several daughter waves with different frequencies andwavelengths (Galeev & Oraevskii 1963; Sagdeev & Galeev 1969). Although in the contextof cosmic ray acceleration in turbulent media the above process was taken into accountto estimate steady state distributions of wave intensities (Chin & Wentzel 1972; Wentzel1974; Skilling 1975a,b,c), its developing processes and associated coherent wave-particle in-teractions have never been considered. Recent numerical studies revealed that successiveparametric instabilities result in turbulent wave forms which have not been fully developed(Matsukiyo & Hada 2003; Nariyuki & Hada 2005, 2006). However, only a few past stud-ies paid much attention to particle acceleration processes in such a developing turbulence.Furthermore, only limited spatial and temporal evolutions of wave forms or spectra werediscussed, since a computational resource was limited.In this paper long time evolution of parametric instabilities of a large amplitude Alfv´enwave in a rather large spatial domain is reproduced by utilizing one-dimensional relativis-tic full particle-in-cell (PIC) code. A plasma is assumed to be composed of electrons andpositrons, since electron-positron pairs can be the dominant constituent of some high en- 3 –ergy astrophysical plasmas like in the vicinity of a pulsar, active galactic nuclei (AGN),gamma ray burst (GRB), and so on. In the simulation we observe a particle accelera-tion process which is quite efficient and is due to interactions between coherent Alfv´enwaves and relativistic particles. The process is quite different from some other coherent ac-celeration processes being discussed recently, in which high frequency electrostatic wavesplay essential roles, like electron surfing acceleration induced by a cross field Bunemaninstability (Shimada & Hoshino 2000; McClements et al. 2001; Hoshino & Shimada 2002;Dieckmann et al. 2004, 2005; Amano & Hoshino 2007) and wakefield acceleration (Tajima & Dawson1979; Katsouleas & Dawson 1983; Lyubarsky 2006; Hoshino 2008).Simulation settings and results are represented in section 2. The acceleration processis discussed in detail by using an analytical model and test particle simulation in section 3.Summary and discussions are given in section 4.
2. 1D PIC Simulation
Long time evolution of parametric instabilities of a large amplitude monochromaticAlfv´en wave in a relativistic pair plasma is reproduced by performing a one-dimensional PICsimulation. A parent wave is given only at the beginning of the run as a monochromatic andright-handed circularly polarized Alfv´en wave with amplitude B p /B = 1 and wavenumber k c/ Ω = 6 .
21 (number of wave mode is 512) which gives a frequency ω / Ω = 0 . B is the ambient magnetic field which is along the x − axis, c denotes speed of light,and Ω = eB /m c is nonrelativistic gyrofrequency, respectively. Corresponding velocityperturbations of the pair plasma are given by the relativistic Walen relation (Hada et al.2004). Boundary conditions are periodic for particles and all field components. System size is L = 518 . c/ Ω . Squared ratio of nonrelativistic gyro and plasma frequencies is Ω /ω p = 0 . T /m c = 1 . × − .Fig.1 shows time evolutions of energy densities (upper panel) and Fourier spectra ofwave amplitudes for B z and E x field components (lower panels). In the upper panel thesolid lines denote electron and positron kinetic energies (labeled by ‘elec.’ and ‘posi.’),transverse magnetic field energy (‘ B ⊥ ’), and transverse and longitudinal electric field energies(‘ E ⊥ ’ and ‘ E x ’), respectively. The dashed line indicates the total energy, which is wellconserved during the run. Although rather long system evolution (up to Ω t = 1616) iscalculated, the system is still far from the so-called equilibrium state. In the lower panelswave amplitude spectra for positive and negative helicity (corresponding to positive andnegative wavenumbers respectively) modes of the B z component are shown in the left ( B Rz )and middle ( B Lz ) panels by using a technique of Fourier decomposition (Terasawa et al. 4 –1986). Note that in the middle panel the actual sign of the wavenumber is negative. Sincemost of the daughter waves are right-hand polarized, the left (middle) panel shows waveintensity mainly of positively (negatively) propagating waves. At the beginning only theparent wave has significant intensity at kc/ Ω = 6 .
21 in the left panel, although it is notparticularly outstanding in the figure because of the narrow spectrum. In the early stage,50 < Ω t < B Rz spectrum. Couplings among the daughter and the parent waves result in compressionalelectrostatic fluctuations with wavenumber smaller than that of the parent wave as seen inthe right panel. In a decay instability, on the other hand, daughter magnetic fluctuationshave negative wavenumbers which appear in the B LZ spectrum, while daughter electrostaticfluctuations have wavenumbers larger than that of the parent wave in the right panel. Most ofthe parent wave energy is wasted in this stage. However, the intensities of the daughter wavesare still large so that successive parametric instabilities occur thereafter. These successiveprocesses are sustained mainly by decay instabilities which can be confirmed because peakwavenumbers of the E x spectrum is always larger than those of the B Rz and B Lz spectra. Theprevious simulation study by Matsukiyo & Hada (2003) confirmed occurence of up to thesecond decay instability for the case of B p /B = 0 . B z component at the corresponding times are plotted in Fig.2b. Rapidheating occurs after the first instabilities developed (Ω t = 95: black dashed line). Up to thisstage, most of the electrons show semi-stochastic motions in a noizy system with a primarywave. Here we refer the primary wave and the noize to the parent wave and superpositionof the daughter waves, respectively. This results in rapid scattering in pitch angle as shownin Fig.2c, where an average pitch angle of electrons with respect to B and its standarddeviation are plotted as a function of time. The standard deviation just after its rapidgrowth at Ω t ∼
70 is ∼ .
5, which roughly coincides with an analytical estimate of themaximum pitch angle width of an electron resonating with the monochromatic parent wave(see Appendix). After the parent wave disappears, most of the electrons are detrapped bythe parent wave and start to wander in the phase space. As time passes, a high energy tailappears in the electron distribution function (Fig.2a), which gradually approaches a power-law spectrum with index α ∼ − . t = 316: gray solid line, 1616: black solid line). Atthe same time the wave amplitude spectrum develops the power-law type spectrum also,with the index α ∼ − .
0, while the wave spectral peak shifts toward lower wavenumbers.A behaviour of the most efficiently accelerated electron is shown in Fig.3. Fig.3a 5 –indicates energy time history of the electron. Spatial trajectory of the electron during0 < Ω t <
600 (800 < Ω t < B ⊥ ≡ p B y + B z . Horizontal arrows in Fig.3c(d)and Fig.3a correspond to each other denoting time intervals where strong accelerations oc-cur. Fig.3e(f) shows a snapshot of the wave profile at the time denoted as the dashed linein Fig.3c(d). The red, blue, and black lines represent B y , B z , and an envelope, respectively.The dotted line indicates the position of the electron. It is seen that strong accelerationoccurs when the electron is trapped in a trough of the magnetic envelope. Such sharp enve-lope troughs are temporarily observed in various regions of the system throughout the run.Hence, the electron experiences similar acceleration processes several times. In this exampleone may recognize four such acceleration periods. Only the second and the third periods aremarked (the first (fourth) one is 160 < Ω t <
190 (Ω t > B as seen in Fig.3b which shows a trajectory of the electron inthe u ⊥ − u k space, where u ⊥ and u k denote the four-velocities perpendicular and parallel to B . The red markers indicate positions of the electron in the phase space at the same timeas Fig.3e (lower marker) and Fig.3f (upper marker). We checked a hundred most efficientlyaccelerated electrons’ trajectories and confirmed that all of them show essentially the samefeatures as mentioned above, i.e., trapping within the envelope troughs and successive per-pendicular acceleration. In the next section the acceleration process is modeled and analyzedin detail.
3. Acceleration of High Energy Electrons3.1. Coherent waves observed in the simulation: modeling
In the PIC simulation shown above sharp magnetic envelope troughs are locally formedthroughout the period of strong electron accelerations. In a successive decay process a num-ber of daughter Alfv´en waves propagating both parallel and antiparallel to B are excited.Depending on their phases, amplitudes of some wave modes are sometimes locally canceledout each other, or they are simply less dominant in amplitude. Then, there appear someregions where two oppositely propagating waves dominate. Here, the envelope structuresobserved in the PIC simulation are modeled by a superposition of such two oppositely prop-agating waves as follows. (cid:18) B yw B zw (cid:19) = B w cos kx (cid:18) cos ωt sin ωt (cid:19) (1) (cid:18) E yw E zw (cid:19) = − ωkc B w sin kx (cid:18) cos ωt sin ωt (cid:19) (2) 6 –Eq.(1) is equivallent to B w = B w (cid:18) cos( kx − ωt ) − sin( kx − ωt ) (cid:19) + B w (cid:18) cos( − kx − ωt ) − sin( − kx − ωt ) (cid:19) . Hereafter, we assume ω and k are both positive without losing generality. Typical waveformsat two different time domains are shown in Fig.4a and b. The black solid and dashed linesdenote y and z components of rotating carrier waves, and the gray lines represent envelopeswhich is independent of time. In the neighborhood of a trough, it is confirmed that eqs.(1)and (2) give a reasonable model of a waveform observed in the PIC simulation shown inFig.4c. In the following motion of a test particle in this system is analyzed. In this section we consider motion of an electron in the electromagnetic waves given byeqs.(1) and (2). The equation of motion is d u dt = − em c (cid:18) E + u γ × B (cid:19) , (3) dxdt = u x γ c, (4)where u ≡ γ v /c , γ = √ u , e and m indicate charge and rest mass of the electron, B = B x + B w , and E = E w , respectively. When we write u = ( u k ( t ) , u ⊥ ( t ) cos φ ( t ) , u ⊥ ( t ) sin φ ( t ))and introduce normalized variables as ξ = x Ω /c , τ = Ω t , κ = kc/ Ω , ν = ω/ Ω , b w = B w /B , and v ph = ν/κ , eqs.(3) and (4) are written as follows.˙ u k = b w u ⊥ γ cos κξ sin ψ (5)˙ u ⊥ = b w (cid:18) v ph sin κξ cos ψ − u k γ cos κξ sin ψ (cid:19) , (6)˙ ψ = − b w u ⊥ (cid:18) v ph sin κξ sin ψ + u k γ cos κξ cos ψ (cid:19) − (cid:18) ν − γ (cid:19) . (7)˙ ξ = u k γ . (8)Here, ψ = φ − ντ , and the dot denotes time derivative in terms of τ , respectively. Note thata variation of normalized particle energy, or the Lorentz factor, is given as˙ γ = b w v ph u ⊥ γ sin κξ cos ψ. (9)In the following behaviours of the electron is discussed by using eqs.(5)-(9). 7 – The above set of equations clearly has several fixed points as listed in Table 1. Here, u denotes a value of u ⊥ at a corresponding fixed point and γ = p u . Apparently, the fixedpoints I - III (IV - VI) correspond to troughs (crests) of the magnetic envelope. As far as theacceleration is concerned, it is easily inferred that the fixed points IV - VI are not importantsince the electric field strength is very weak around there. Actually, efficient accelerationobserved in the PIC simulation always occurs around the troughs of the magnetic envelope.Therefore, only the fixed points I - III are focused here. Stability of these fixed points isdiscussed in the following. Expanding eqs.(5)-(9) around the fixed point I and retaining only the first order terms,we obtain δ ˙ u k = − b w u γ κδξ, (10) δ ˙ u ⊥ = − b w v ph δψ, (11) δ ˙ ψ = b w u v ph δu ⊥ − γ δγ, (12) δ ˙ ξ = δu k γ (13) δ ˙ γ = − b w u γ v ph δψ = u γ δ ˙ u ⊥ , (14)where δ denotes small first order quantities.From eqs.(10) and (13), we have δ ¨ u k = − b w u γ κδu k . (15)The above expression represents a trapping oscillation with trapping frequency ω trap =( b w u κ/γ ) / , which is rewritten with the original parameters as ω trap = ( B w kcu /B Ω γ ) / .From eqs.(11), (12), and (14), on the other hand, we obtain δ ¨ u ⊥ = b w u v ph γ (cid:18) − b w v ph γ u (cid:19) δu ⊥ . (16)Here, the first term in the parenthesis (or the second term in the right hand side of eq.(12))arises from the variation of the Lorentz factor, i.e., due to the relativistic effect. In the 8 –non-relativistic limit, therefore, eq.(16) represents a trapping motion in the perpendicularmomentum space with frequency ω trap = b w v ph /u = ( B w /B )( ω/kc )(1 /u ). However, inthe relativistic case with b w v ph γ /u < δu ⊥ (and δψ ) diverges in time. When u is smallenough so that γ ≈
1, the above inequality is hardly satisfied. In such a case, the fixedpoint is stable. But if u becomes large and the inequality is satisfied, such an electron gainstransverse energy while it keeps being trapped in the x − direction. We consider this solutionlater more in detail.For the fixed point II, equations corresponding to eqs.(15) and (16) are obtained by for-mally changing the sign of b w . Therefore, the system is unstable for parallel fluctuations whileit is stable for perpendicular fluctuations. This fixed point is actually conjugate to the rela-tivistic fixed point discussed above. Another fixed point (III), conjugate to the nonrelativisticone, which should be a saddle point in u ⊥ − ψ phase space, appears at ( u ⊥ , ψ ) = (0 , nπ ) assingular points of du ⊥ /dψ , where n = 0 , ± , ± , · · · . u ⊥ − ψ phase space In order to study perpendicular dynamics let us first consider a reduced system in whichparallel quantities are fixed at the fixed point, i.e., u k = 0 and κξ = π/
2. Then we only haveto consider the following two equations.˙ p = 2 √ pb w v ph cos ψ (17)˙ ψ = − b w √ p v ph sin ψ − (cid:18) ν − √ p (cid:19) (18)Here, p = u ⊥ has been introduced. This system has a Hamiltonian defined as H ( p, ψ ) = 2 √ pb w v ph sin ψ + νp − p p, (19)where ˙ p = ∂H/∂ψ and ˙ ψ = − ∂H/∂p are satisfied. Contours of the Hamiltonian for b w = 1 . v ph = 0 .
17, and ν = 0 .
11 are represented as gray lines of the upper (linear scale) and lowerpanels (logarithmic scale) in Fig.5a. The above parameters are chosen so that the wavesinteracting with the accelerated electron in the second acceleration stage observed in thePIC simulation (Fig.3e) is appropriately reproduced. It is easily confirmed that a center at ψ = π/
2, which is clearly recognized in the lower panel, corresponds to the nonrelativisticstable fixed point of the fixed point I and another center at ψ = 3 π/
2, which can be seenboth in the upper and lower panels, to the relativistic fixed point II discussed above.The nonrelativistic center should satisfy the following condition from Table 1. − b w v ph u = ν − γ (20) 9 –Although ν − /γ = 0 in the small amplitude limit, it is never satisfied in the nonrelativisticcase since ν <
1. Therefore, the nonrelativistic center appears only when wave amplitudebecomes finite. The closed trajectories around this center essentially coincides with thenonresonant trapping discussed by Kuramitsu & Krasnoselskikh (2005).At the relativistic center, the resonance condition ν − /γ = 0 should be satisfied inthe ultrarelativistic limit, since b w v ph /u is negligible. In such a case relativistic decrease ofthe gyro frequency allows another resonance with low frequency waves. This resonance canalso be present in small amplitude limit. Relativistic linear resonance conditions between anelectron and two oppositely propagating waves ( γν ∓ κu k − u k = 0 where ν − /γ = 0 and γ = p u ⊥ are satisfied,while there never appears such an intersection in nonrelativistic limit. This indicates thatan electron can resonate simultaneously with two waves in the relativistic case. By usingeq.(19), the maximum width of the separatrix in u ⊥ is estimated as∆ u ⊥ = 4 r b w κν . (21)Furthermore, the maximum u ⊥ on the separatrix is given by u ⊥ ,max = r b w κ + r ν ! = r B w B Ω kc + r Ω ω ! . (22)For instance, u ⊥ ,max ≈ . δu k = 0. If we allow δu k to slightly deviated from zero, it should satisfy δ ¨ u k = − b w u γ κ sin ψδu k (23)around the fixed point in the original system eqs.(5)-(8). Hence, the system is stable for0 < ψ < π and is unstable for π < ψ < π in terms of parallel fluctuations. The black solidline in Fig.5a and 5b shows a numerical solution of eqs.(5)-(8). A trajectory of the electroninitially positioned near the separatrix at ψ = 0 (indecated by a small arrow in Fig.5a) isrepresented. The electron moves alomost along the separatrix when it is in 0 < ψ ( < π ). Butwhen the electron enters in π < ψ < π , its trajectory starts deviating from the separatrix.And finally it is clearly detrapped when it approaches ψ = 3 π/
2. The trajectory in themomentum space shown in Fig. 5b is very similar to what is observed in the PIC simulationwhich is again plotted as a gray line. 10 –
Let us briefly discuss the power-law like distribution of electorns in the current accel-eration process. It is shown in Fig.2a that a high energy tail with power-law index ∼ − . t > ∼ − . kc/ Ω >
3, is theremnant of initial parent wave and side band waves generated by the modulational instabil-ities in the early stage of the run. In order to confirm correlations between the wave andelectron energy spectra, the following test particle simulation with periodic boundary con-ditions is performed. Waves are given by superposition of a number of right hand polarizedAlfv´en waves with power-law spectrum shown in Fig.6 (power-law index α is an externalparameter), which models the lower wavenumber part of Fig.2b. Both positive and negativewavenumber modes are evenly distributed. The wave form is given by (cid:18) B y B z (cid:19) = X kc/ Ω = − b w B (cid:12)(cid:12)(cid:12)(cid:12) kK (cid:12)(cid:12)(cid:12)(cid:12) α (cid:18) sin( k ( x − v ph t ) + Φ k )cos( k ( x − v ph t ) + Φ k ) (cid:19) , (24)where v ph /c = 0 .
17 and { Φ k } are initial wave phases which are randomly distributed at t = 0, and corresponding transverse electric fields are given by k × E = ( | k | v ph /c ) B . Thepower-law index α is fixed to 1.0 for k min ≤ k ≤ K , where k min ≡ π/L and K is thecoherence wavenumber. The system size L = 80 πc/ Ω is common for all the following runs,and b w = 0 .
75 and K c/ Ω = 0 . electrons. An initialdistribution function is given as a spatially homogeneous gyrotropic ring distribution with v ⊥ /v ph = 0 .
1. Fig.7 shows distributions in (a) v ⊥ − v k and (b) X − W kin phase spaces, and(c) energy distribution functions at two different times, Ω t = 1000 and 3000, respectively,where v = ( v k , v ⊥ ) denotes particle velocity and W kin = v / v ph is normalized kinetic energy.The solid gray lines in Fig.7b denote envelope profiles of given magnetic fluctuations. Theparticles are spatially bunched around the troughs of the magnetic envelopes because ofmirror effect. In the velocity space the particles rapidly pitch angle diffuse (Fig.7a) whilethe averaged particle energy slowly increases in time (Fig.7c). These are the features ofa stochastic or a second order Fermi acceleration. In comparison, drastic changes occurif relativistic effects are taken into account. The simulation results with the same initial 11 –conditions as in the nonrelativistic case are plotted in Fig.8 with the same format as Fig.7.Here, the normalized kinetic energy is defined as W kin = ( γ − c /v ph . Compared with thenonlerativistic case, the maximum particle energies at each corresponding time are extremelyhigher, and they are at the same order as that obtained from eq.(22) with k = K and b w = 0 .
75 (the black solid and black dashed lines in Fig.8c). Spatial bunching of particlessimilar to the nonrelativistic case is found in Fig.8b, while a fraction of the bunched particlesare accelerated to extremely high energies. Such high energy particles have rather large pitchangles, denoting perpendicular acceleration (Fig.8a). Interestingly, a power-law like energydistribution appears only at Ω t = 1000 in Fig.8c, although the corresponding energy rangein the distribution function is small (the black solid line). The power-law index does notchange even in the case with a different initial ring velocity, v ⊥ /v ph = 5 . α = − . K c/ Ω = 0 .
4, the bulk electronsare accelerated and the energy distribution is no longer power-law at Ω t = 1000 (the graydotted line in Fig.8c). It is noted from eqs.(21) and (22) that the minimum u ⊥ of theseparatrix of the relativistic resonance of the dominant wave mode at k = K decreases withincreasing K . Hence, most of electrons which initially distribute below the separatrix canenter inside the separatrix through stochastic motions at rather early stage, and they canbe perpendicularly accelerated within a short time in the similar way seen in the previoussection (cf. Fig.5). The peak energy is roughly consistent with the u ⊥ ,max ( ∼ b w = 0 .
75 and κ = 0 . b w = 0 . t = 1000 is much smaller than the value obtained from eq.(22)(not shown). The reason may be that in this run the minimum u ⊥ of the separatrix is ratherhigh because of its narrow width so that particles which initially distribute far below theseparatrix have not entered in it until this time.In Fig.8c the power-law like nature appears as a transient state in the system atΩ t = 1000 for α = − .
0. In this run, at later time (Ω t = 3000), high energy end ofthe particle distribution is so enhanced that the energy distribution does not fit the power-law spectrum(black dashed line). It is confirmed that the hump of the high energy partgrows at least till Ω t = 6000. After sufficiently long time, it probably results in bulk accel-eration as seen in the case of small K . In the PIC simulation the wave spectrum is not timestationary but cascading through the successive decay instabilities as already mentioned. Inother words, a wave with a certain wavenumber has finite life time. This may be why thehigh energy tail evolves without extra accumulation at high energy end of the distributionfunction in the late stage, Ω t > K c/ Ω ∼ (Ω t ) − . which gives a reasonable 12 –fit in the late stage of the PIC simulation. The particle distribution and waves at Ω t = 1000in the run corresponding to the black solid line in Fig.8c are chosen as initial conditions.Then, in the later time Ω t = 3000 high energy tail extends roughly obeying the power-lawand some low energy particles remain unaccelerated, as shown in Fig.8c as the gray solidline.
4. Summary and Discussions
In the present paper an efficient particle acceleration process in the course of successiveparametric instabilities of large amplitude Alfv´en waves was investigated. The accelerationtakes place as a result of interactions between coherent waves in the developing Alfv´enturbulence and relativistic particles. An important point is that relativistic wave-particleinteractions allow simultaneous resonance between a particle and two different waves. Themaximum attainable energy through this acceleration process was analytically estimated.In this acceleration process a high energy particle is preferentially accelerated. Dur-ing the successive decay instabilities, a peak of the wave Fourier spectrum shifts in timetoward a lower frequency (longer wavelength) regime. Because of relativistic decrease ofparticle’s gyro frequency, low frequency waves preferentially resonate with and accelerateparticles with large energy. Therefore, if once a particle is accelerated and its effective massis increased, in later time such a particle easilt satisfies the resonance condition with lowerfrequency waves generated by successive decay processes. Fig.3a shows an example of suchan electron’s energy time history in which one can recognize four acceleration phases aroundΩ t ∼ , , ω ∼ Ω /nγ , where n is an integer, shown in their Fig.8 might be related with therelativistic resonance. In our test particle simulation the power-law like energy distributionfunction similar to what was observed in the PIC simulation is reproduced by assuming thetime evolving Fourier spectrum of Alfv´en turbulence. Finite life time of a wave mode indeveloping turbulence may contribute to creation of such a distribution function. However,details of the acceleration process including analytical estimate of the power-law index arestill unclear and will be investigated in the near future. 13 –The acceleration process discussed in this paper is essentially different from some of therecently studied coherent acceleration processes, i.e., electron surfing acceleration and wake-field acceleration. In these processes electrostatic field plays essential roles. Furthermore,the processes mainly affect electrons since generated electrostatic waves have rather highfrequencies, while ion acceleration may also occur in a very late stage of nonlinear evolutionof a system (Hoshino 2008). In the acceleration process discussed here roles of electrostaticfields are subdominant, although they are necessary for the decay instabilities. It should benoted further that the process may be able to act on ions too when a left hand polarizedwave is introduced as an initial parent wave. The Alfv´en waves are essentially incompressibleso that they may survive for rather long period compared with electrostatic Langmuir or ionacoustic waves. Hence, the process may last for long time, and may also follow the abovementioned electrostatic acceleration processes in some occasions.In the present study all the multidimensional effects have been excluded. In higher di-mensional cases daughter waves propagating in oblique to the ambient magnetic field can alsohave finite growth rates (Vi˜nas & Goldstein 1991). It is confirmed by MHD simulation thatthese waves destroy the planar structure assured in a one dimensioal simulation (Ghosh et al.1993, 1994; Del Zanna et al. 2001). However, in low frequency regime the decay instabilityof parallel progagation is dominant (Vi˜nas & Goldstein 1991). Therefore, at least in such aregime the acceleration process observed here are expected to work, while acceleration ratemay decrease to some degree because of appearance of nonplanar structures. This is similarto what is discussed for electron surfing acceleration in which the acceleration takes placeeven in two dimensional cases despite decrease of acceleration rate (Amano & Hoshino 2008).At all events, further investigations are necessary for estimate efficiency of this accelerationprocess in a more realistic situation.We finally give a comment on applications of this acceleration process. Since the situ-ation simulated in section 2 is rather general, there may be several fields of application likea pulsar wind nebula, a pulsar magnetosphere, an outflow of GRB, and an AGN jet, andso on. Vicinity of a relativistic shock is probably one of the candidates. Similary to theearth’s foreshock, large amplitude Alfv´en waves may be generated by beam-plasma interac-tions between backstreaming ions and an upstream plasma, and the waves nonlineary evolvevia parametric instabilities. Since the acceleration is efficient and locally takes place, it maycontribute to the so-called injection process into the DSA. As another possible case, cosmicray-plasma interactions upstream of a supernova remnant shock are now extensively stud-ied after the pioneering works by Lucek & Bell (2000), Bell & Lucek (2001), and Bell (2004,2005). Most of such studies pay attention to amplification of upstream magnetic fluctuationsthat can be scatterers of cosmic rays. And that is expected to result in increase of maximumattainable energy in the DSA process. On the other hand, amplified magnetic fluctuations 14 –may lead to the local and the coherent acceleration process discussed here. In other words,some particles may be accelerated through this process without crossing a shock. In thissense the process is similar to the so-called second order Fermi acceleration, although anefficiency of the acceleration process discussed here is much higher than that of the secondorder Fermi process as shown by the test particle simulation.The authors thank Victor Mu˜noz for useful discussions. The PIC simulation was per-formed by the super computer in ISAS/JAXA Sagamihara. This work was supported in partby Incentive aid to the prominent research, Interdisciplinary graduate school of engineeringsciences, Kyushu University 2007. A. Motion of a relativistic particle in a monochromatic wave
According to Kuramitsu & Krasnoselskikh (2005), an equation of motion of an electronin a monochromatic circularly polarized wave in a wave frame is reduced as˙ µ = − ∂H∂ψ , ˙ ψ = ∂H∂µ , (A1)where µ is the pitch angle cosine, ψ the particle gyrophase with respect to the wave phase,and H = K (cid:18) µ + 1 K (cid:19) + b w p − µ cos ψ (A2)is Hamiltonian. The above expressions are held even when relativistic effects are takeninto account by putting K = ukc/ Ω and b w = ( B w /B ) q − v ph /c . Electron trajectriesinteracting with the parent wave ( B w /B = 1 and kc/ Ω = 6 .
21) for different values of H are plotted in Fig.9. Here, u = 0 .
64 is assumed since the value is derived as average one atΩ t = 70 in the PIC simulation. The factor of v ph /c is neglected because of its smallness.Fig.9 is essentially the same as Fig.5 in Kuramitsu & Krasnoselskikh (2005). The maximumvalue of half the width of the separatrix is ∼ . ψ = π ), which roughly coincides withthe standard deviation of pitch angle observed in the PIC simulation at Ω t = 70 (Fig.2c). REFERENCES
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18 –Table 1: Fixed points κξ ψ u k u ⊥ constraintI ± π ± π − b w v ph u = ( ν − γ )II ± π ∓ π b w v ph u = ( ν − γ )III ± π π ) 0 0 sin ψu → ∓ ( ν − γ ) /b w v ph IV 0( π ) 0( π ) 0 ν − γ = 0V 0( π ) π (0) 0 ν − γ = 0VI 0( π ) ± π sin κξu → ∓ ( ν − γ ) /b w v ph
19 –Fig. 1.— Time evolution of wave power spectra. The right and left hand helicity modesof B z component are Fourier decomposed in left and middle panels, respectively. The rightpanel shows E x component. See the electronic edition of the Journal for a color version ofthis figure. 20 – -7 -5 -3 -1 F ( γ e ) γ e -1 Ω t = 0 95 316 1616 α = -2.5 -5 -4 -3 -2 -1 | B z ( k ) | / B kc/ Ω Ω t = 0953161616 α = -2.0 (a)(b) Fig. 2.— Electron energy distribution functions at various times. 21 –Fig. 3.— Behaviours of an accelerated electron. (a) Time history of the electron energy.(b) Trajectories of the electron in u ⊥ − u k space and (c)(d) in X − t space. The arrows in(c) and (d) indicate time domains where strong acceleration occur as shown by the arrowsin (a). Background color scale in (c) and (d) denotes amplitude of magnetic fluctuations.(e)(f) Spatial profiles of magnetic fluctuations in two acceleration phases represented bydashed lines in (c) and (d). The red, blue, and black solid lines show B y , B z componentsand envelope, respectively. The dotted lines show positions of the particle at corresppondingtimes. See the electronic edition of the Journal for a color version of this figure. 22 – -101 B -101 E kx -101 B -101 E kx -101 B / B -101 E / B Ω ) (a) 0 < ω t < π /2 (b) 3 π /2 < ω t < 2 π (c) Ω t = 1170 Fig. 4.— (a), (b) Modeled wave forms at different time domains ( B w = 1 , ω/kc = 0 . t = 1170. 23 – u ⊥ ψ π π u ⊥ u ⊥ -20 0 20 u || u ⊥ -2 0 2 u || (a) (b) (c) Fig. 5.— Electron trajectories in (a) u ⊥ − ψ , (b) u ⊥ − u k phase space, and (c) linear resonanceconditions of wave-particle interactions in relativistic (solid line) and nonrelativistic (dashedline) plamsas. Solutions of eq.(19) are plotted as gray lines in (a). In (b) a gray linedenotes a trajectory of the electron shown in Fig.3. A numerical solution of eqs.(5) - (8) forΩ w / Ω = 1 , kc/ | Ω | = 0 .
65, and ω/ | Ω | = 0 .
11 is indicated as black lines in (a) and (b). 24 –
Log ( B w ( k ) / B ) Log(kc/ Ω ) α K k min k max = 3 b w Fig. 6.— Wave power spectrum used in the test particle simulation. v ⊥ / v ph W k i n v ⊥ / v ph -40 0 40 v || / v ph W k i n X / (c/ Ω ) -5 -3 -1 F ( W k i n )
10 100 1000 W kin Ω t = 1000 3000 (a) (b) (c) Ω t = 1000 Ω t = 1000 Ω t = 3000 Ω t = 3000 Fig. 7.— Results of nonrelativistic test particle simulation. Electron distributions in (a) v ⊥ − v k and (b) X − W kin phase spaces, and (c) energy distribution functions at Ω t = 1000and 3000, respectively. The solid gray lines in (b) denote profiles of magnetic field envelopesat the corresponding times. 25 – u ⊥ W k i n u ⊥ -160 0 160 u || W k i n X / (c/ Ω ) -5 -3 -1 F ( γ )
10 100 1000 W kin -2.5 (a) (b) (c) Ω t = 1000 Ω t = 1000 Ω t = 3000 Ω t = 3000 α =-2.0, Ω t=1000 α =-2.0, Ω t=3000 α =-3.0, Ω t=1000 K c/ Ω =0.4K variesin time Fig. 8.— Results of relativistic test particle simulation in the same format as Fig.7. Anumber of energy distribution functions are plotted in (c). See datails in the text. P i t c h A ng l e ϕ
00 2 ππ /2 ππ Fig. 9.— Trajectories of relativistic electrons interacting with the monochromatic circularlypolarized parent wave. B w /B = 1, kc/ Ω = 6 .
21, and u = 0 ..