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NONLINEAR SHOCK ACCELERATION BEYOND THE BOHM LIMIT

M. A. Malkov and P. H. DiamondUniversity of California at San Diego, La Jolla, CA 92093-0319; [email protected], [email protected]

Received 2005 September 7; accepted 2005 December 12

ABSTRACT

We suggest a physical mechanism whereby the acceleration time of cosmic rays (CRs) by shock waves can be sig-nificantly reduced. This creates the possibility of particle acceleration beyond the knee energy at!1015 eV. The accel-eration results from a nonlinear modification of the flow ahead of the shock supported by particles already acceleratedto the kneemomentum at p ! p". The particles gain energy by bouncing off convergingmagnetic irregularities frozeninto the flow in the shock precursor and not so much by recrossing the shock itself. The acceleration rate is thusdetermined by the gradient of the flow velocity and turns out to be formally independent of the particle mean free path(mfp). The velocity gradient is, in turn, set by the knee particles at p ! p" as having the dominant contribution to theCR pressure. Since it is independent of the mfp, the acceleration rate of particles above the knee does not decreasewith energy, unlike in the linear acceleration regime. The reason for the knee formation at p ! p" is that particles withp > p" are effectively confined to the shock precursor only while they are within limited domains in the momentumspace, while other particles fall into ‘‘loss islands,’’ similar to the ‘‘loss cone’’ of magnetic traps. This structure of themomentum space is due to the character of the scattering magnetic irregularities. They are formed by a train of shockwaves that naturally emerge either from unstably growing and steepening magnetosonic waves or as a result ofacoustic instability of the CR precursor (CRP). These losses steepen the spectrum above the knee, which alsoprevents the shock width from increasing with the maximum particle energy.

Subject headings: acceleration of particles — cosmic rays — shock waves — supernova remnants — turbulence

Online material: color figures

1. INTRODUCTION

The first-order Fermi acceleration, also known as diffusiveshock acceleration (DSA), is regarded as the principal mechanismwhereby Galactic CRs are produced. The physics of the energygain is very simple: particles get kicked sequentially by scatteringcenters that are on the opposite side of the shock front and soapproach each other. Its modern version has been developed byKrymsky (1977), Axford et al. (1977), Bell (1978), Blandford &Ostriker (1978), and others. Soon after that, however, it was re-alized that the mechanism can onlymarginally (if at all; Lagage&Cesarsky 1983) account for the acceleration of the Galactic CRsup to the knee energy at 1015 eV (see also Jones et al. 1998; Kirk&Dendy 2001, for more recent good discussions). Up to the knee,the spectrum is nearly a perfect power law, so it is most probablyproduced by a single mechanism. At the same time, the low par-ticle scattering rate makes the mechanism somewhat slow and theparticle confinement to the shock insufficient to accomplish thistask over the active lifetime of a typical Galactic supernova rem-nant (SNR) shock, which is considered to be the most probablesite of CR acceleration.

Not surprisingly, there have been many suggestions as to howto improve the performance of the DSA. In one way or anotherthey targeted the most important and, at the same time, the mostuncertain parameter that determines the acceleration time, the par-ticle diffusivity !. The acceleration rate can be written roughly as"#1acc $ U 2

sh/! p% &, where Ush is the shock velocity. Therefore, theacceleration rate decreases with particle momentum p, since theparticle diffusion coefficient ! usually grows with it. In the Bohmlimit, which is reasonably optimistic, one can represent ! as ! $crg/3, where rg is the particle gyroradius (substituted in the lastformula as a particle mfp k) and c is the velocity of light. The con-dition k ! rg is justifiedwhen themagnetic field perturbations thatscatter accelerated particles (and are driven by them at the sametime) reach the level of the ambient magnetic field, #B ! B0. Such

a high fluctuation level has been long considered as a firm upperlimit (McKenzie & Volk 1982), and there have been calculationsindicating that the turbulence may saturate at a somewhat lowerlevel due to nonlinear processes (e.g., Achterberg & Blandford1986).

It should be noted, however, that in cases of efficient particleacceleration, i.e., when a significant part of the shock ram pres-sure $U2

sh is converted to the pressure of accelerated particlesPc , there is indeed sufficient free energy of accelerated particlesthat can potentially be transformed to fluctuations with #B3B0.This would decrease k significantly below the particle gyroradius(calculated by the unperturbed field B0), so the acceleration timewould also decrease. From simple energetics principles, one canestimate the maximum fluctuation energy as (McKenzie & Volk1982)

#B2

B20

! MAPc

$U 2sh

; %1&

where MA $ Ush/VA B0% & is the shock Alfven Mach number.Bell & Lucek (2001) suggested that this upper bound may in-deed be reached, corroborating this idea by the numerical sim-ulation (Lucek & Bell 2000). The supporting simulation wasrather limited in the system size and particle energy range, whichin turn truncates the wave spectrum. Therefore, the saturationeffects predicted earlier by McKenzie & Volk (1982), Achterberg& Blandford (1986), and Blandford & Eichler (1987) have notbeen observed in the simulations by Lucek & Bell (2000). In anycase, equation (1) provides only an upper bound to the actualwaveamplitude, whichmay bemuch lower due to the saturation effects,not included in its derivation. On the other hand, Bell (2004) sug-gested that the magnetic field can reach an even higher level if theCR current that drives the instability is fixed. According to thissuggestion, the instability saturationmechanism is due tomagnetic

A

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The Astrophysical Journal, 642:244–259, 2006 May 1# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

tension. Under these circumstances the estimate given by equa-tion (1) acquires an additional factor !MA(Ush/c), which may bequite large in the case of young SNRs,making the turbulence levelsufficient to increase the maximum particle momentum by a factorof 103.

A different approach to the same goal of increasing the turbu-lent component beyond the ambient field level has been pursuedin the recent papers by Ptuskin & Zirakashvili (2003, 2005).These authors studied the DSA in the presence of a Kolmogorov-typemagnetohydrodynamic (MHD) cascade initiated by the strongMHD instability in the CRP. In addition, recently Bell (2004)took up from the Bell & Lucek (2000) and Lucek & Bell (2000)approaches and replaced the time-consuming particle simula-tions by providing an estimate of the growth rate caused by theCR current at the shock. With the current kept fixed, the wavesindeed grow to a level sufficient to increase the maximum par-ticle momentum by a factor of 103. It remains unclear, however,whether the CR trapping by the self-generated turbulence andother wave-particle interactions (e.g., Volk et al. 1984; Achterberg& Blandford 1986) saturates the instability at a lower level. Be-sides, the alleged growth of #B beyond B0 clearly invalidatestreatments of the linear instability that are based on the #BTB0 assumption.

Diamond&Malkov (2004) suggested an alternative picture inwhich a strong #B may be generated as a result of a type of in-verse cascade in k-space. Strong field perturbations driven reso-nantly by already accelerated particles are nonlinearly coupled tolonger scales, not only facilitating the acceleration of higherenergy particles but also providing an ambient field for them asthe longest scale field perturbation. Such a cascade can be drivenby scattering of the Alfven waves on acoustic waves driven, inturn, by the Drury instability of the CRP. Another related routeto large-scale magnetic fluctuations is modulational instability ofthe Alfven waves themselves. The condensation of magnetic en-ergy at long scales allows us to marginally preserve the weakturbulence #B < B0 requirement, where the role of B0 is taken bythe longest scale part of the magnetic energy spectrum (longerthan the particle gyroradius). This could possibly weaken thesaturation factors setting in at #B ! B0, which are noted above.

The above-mentioned works, while aimed at the explanationof acceleration beyond the knee, do not specifically addressmechanisms that could be responsible for the formation of theknee itself. An exception is a paper by Drury et al. (2003), inwhich the authors suggested that the knee can appear as a result ofan abrupt slowing down of the acceleration at the sweep-up phasein combination with the Bell hypothesis about the generation ofthe strongmagnetic field. Some further interesting ideas about theknee origin can be found in a recent paper by Sveshnikova (2003).

In this paper we propose a different scenario of faster thanBohm acceleration that is also intimately related to the kneephenomenon. Odd as it sounds, this mechanism does not requiresuper-Bohm magnetic field fluctuations. The reason for that isthat its rate does not depend on the particle scattering mfp k. Inx 2 we describe the mechanism by comparing it with both the testparticle and nonlinear DSA regimes under the Bohm diffusivity.We analyze the conditions under which the proposed accelera-tion regime is faster than the latter two. This sets the stage forxx 3 and 4, in which a preliminary study of particle dynamics andtransport in the CRP is presented. Section 5 deals with the esti-mates of the maximum energy that can be reached in excess ofthe knee energy. In x 6 an acceleration model that allows us tocalculate the slope of the spectrum between the knee and themaximum energy is presented.We concludewith a summary andbrief discussion.

2. ACCELERATION MECHANISM: A PRIMER

Perhaps the easiest way to understand why and when the sug-gested version of the DSA becomes faster than the standard oneis to consider why the latter is slow. For that, we turn to the in-dividual particle treatment due to Bell (1978). Upon completingone acceleration cycle, i.e., crossing and recrossing the discon-tinuity, a particle gains momentum

!p

p! !U

c; %2&

where !U is the relative velocity between the upstream anddownstream scattering centers!U $ U1 # U2 ! U1. Thus, overthe acceleration time (when the momentum gain !p ! p), thenumber of cycles that the particle needs to make is Ncycl !c/U1 31. Apart from numerical factors and the differences be-tween the upstream and downstream residence time contributions,the particle acceleration time can be estimated as

"acc ’! p% &U 2

1

! kc=U 21 ! "colc

2=U 21 ; %3&

where k and " col are the particle mfp and collision time, respec-tively. It is also useful to note that the acceleration time is of orderthe time needed for the fluid element to cross the diffusion zoneahead of the shock, "acc! Ldif p% &/U1, where Ldif $ ! p% &/U1 isthe particle diffusion length. Using equations (2) and (3), one canwrite the following relation between the acceleration time, thetime needed to complete one cycle, and the collision time:

"acc : "cycl : "col!c2

U 21

:c

U1: 1:

Therefore, out of the c2/U 21 wave-particle collisions needed to

gain a momentum!p ! p, only c/U1 are productive in terms ofthe energy gain. Most of the collisions are wasted.

The situation changes fundamentally when the number andenergy of accelerated particles increase. First of all, the shock getsshieldedwith a cloud of accelerated particles (CRs) diffusing aheadof it, and the plasma flow becomes significantly modified by theirback-reaction. In addition to an abrupt velocity jump (which canbe significantly reduced by this back-reaction), the flow developsan extended CRP of length Lp! ! p"% &/U1 $ Ldif p"% &, where p"is the particle momentum corresponding to the maximum con-tribution to the pressure of accelerated particles. The flow in theCRP gradually slows down toward the main shock (subshock),and the spectrum is flatter than p#4 at high momenta, so that p" ’pmax and thus Lp ’ Ldif p"% &. Let us assume that a particle under-goes subsequent scattering by two scattering centers, approachingeach other at a speed #UTc (Fig. 1). Suppose that immediatelyafter scattering off the right center, the particle has (in that centerreference frame) themomentum p and the cosine of the pitch anglewith respect to the plasma flow (moving to the left), % < 0. Afterscattering off the left center, the particle momentum becomes p0,and the cosine of the pitch angle %0 > 0, i.e., the particle movesback toward the first scattering center. Since scattering is elastic,from particle kinematics we obtain

p0

p’ 1# #U

c%

! "1' #U

c%0

! ": %4&

This result is obviously identical to the momentum gain in theconventional DSA theory (Bell 1978). The left scattering center

NONLINEAR SHOCK ACCELERATION BEYOND BOHM LIMIT 245

can be identified with one of the downstream, and the right onewith one of the upstream, scattering centers. Considering % and%0 as independent stochastic variables evenly distributed overthe intervals (#1, 0) and (0, 1), respectively, one can calculatean average momentum gain,

#ph ip

’ 2#U

c%h i$ 4

3

#U

c; %5&

where

(h i$R 10 (% &% d%R 10 % d%

:

Note that the last result also holds in the case in which there is anintervening scattering that does not reverse the particle direc-tion. This can be seen from equation (4) by noting that in such acase, both %-values would be of the same sign and cancel onaveraging to first order in #U/c. Just as in the conventional( linear) acceleration theory, a pair of scattering centers does notneed (and is unlikely) to be the same at the next acceleration cycle.However, this is quite possible in the scattering environmentconsidered later in this paper. If so, then the momentum gain percycle #p can be obtained from the conservation of the adiabaticinvariant,

Jk $I

pk dlk $ constant;

where the integral runs between the two scattering centers. Ifthe interaction time of a particle with a scattering center is muchshorter than the flight time between collisions, then Jk ’ pkl $constant, where l(t) is the (decreasing) distance between the twocenters. Formula (5) immediately follows from the last relation.

The average time needed to complete the cycle is

#th i$ 2kc%

# $$ 4

kc;

where we substituted k for an averaged distance between scat-tering centers. Now we can write

dp

dt) p ’ #ph i

#th i$ 1

3p#U

k:

As long as kTLp, one can approximate #U as

#U ’ #k@U

@z;

so that the local acceleration rate is

p ’ # 1

3p@U

@z: %6&

We observe that the particle mfp dropped out of the accelerationrate in the smooth part of the shock structure, in contrast to thelinear acceleration regime that occurs at the shock discontinu-ity. The last formula is, of course, consistent with the standarddiffusion convection equation,

@f

@t' U

@f

@z# @

@z!@f

@z$ 1

3

@U

@zp@f

@p; %7&

if one considers its characteristics, ignoring diffusion effects.1

Then in the vicinity of the (abrupt) momentum cutoff pmax, themain terms in equation (7) are the first one on the left-hand sideand the acceleration term on the right-hand side. This clearlyshows that a (front) solution propagates in momentum space atthe speed given by equation (6), basically independent of z and!. It is important to emphasize, however, that the independenceof the acceleration rate of the particle diffusion coefficient (ormfp, as shown above) does not mean that the acceleration isfaster than in the linear case where such dependence is the mainfactor making the acceleration relatively slow. The reason isvery simple. The derivative @U/@z in equations (6) and (7) itselfdepends on !. Indeed, estimating @U/@z as

@U

@z’ U1

Lp’ U 2

1

! p"% &; %8&

one sees that the acceleration time is "acc! ! p"% &/U 21 ; i.e.,

particles with momenta p ! p" are accelerated approximately atthe same rate as in the linear theory (apart from a numericalfactor O(1); see Malkov & Drury [2001]). Clearly, since thelocal (i.e., inside of the CRP) acceleration rate is independent ofmomentum, particles with p < p" accelerate more slowly thanthe linear theory acceleration rate, while particles with p > p"accelerate faster rate than in linear theory and may in principlebe accelerated faster than the Bohm rate. The overall acceler-ation process is illustrated in Figure 2, depending on whetherthe pressure-dominant momentum p* is stopped from growingafter some t $ t". If it is, particle momentum grows exponen-tially; otherwise, it continues to grow linearly in time.There are two problems with realization of this possibility.

First, because in the nonlinear acceleration regime the spectrumat the upper cutoff is flatter than p#4 (it actually flattens to p#3.25

at pmax [Malkov 1997] in the limit of high Mach number and

1 As was shown by Malkov & Drury (2001), there is a reason to do so in cer-tain situations. In particular, under a strong nonlinear shockmodification andBohmdiffusivity (! / p), the velocity profile is linear, i.e., @U /@z ’ constant in a sig-nificant part of the CRP.

Fig. 1.—Flow velocity profile in a CR dressed shock (solid curve). Hatchedcircles represent two scattering centers moving with the flow and approachingeach other at a speed #U. The filled circles depict a CR particle after the first andsecond collision (see text).

MALKOV & DIAMOND246 Vol. 642

pmax 3mc), p" is clearly identifiable with pmax. Therefore, unlessthe spectrum slope changes at p ! p" , it is difficult to accelerateparticles beyond p". This is why the acceleration rates in both lin-ear and nonlinear regimes are similarly slow, although for ratherdifferent reasons. In the linear regime, the acceleration rate slowsdownwith time since the acceleration cycle duration, " cycl , growswith momentum (as the gyroperiod does), so the particle momen-tum grows only linearly with time. In the nonlinear regime, thewidth of the shock grows linearly with p", so that the accelerationrate (8) decreases with p" (not with p) and p" also grows linearlywith time. Clearly, the first requirement would be to stop p" fromgrowing at some point or at least to slow down its growth consid-erably. If p" $ constant, then particles with p > p" increase theirmomentum exponentially (eqs. [6] and [8]).

One simple reason for the saturation of p"(t) is a geometricone. The thickness of the CR cloud ahead of the subshock, Lp( p"),cannot exceed the accelerator size, i.e., some fraction of the shockradius Rs, in a SNR, for example, (e.g., Drury 1983; Berezhko1996). Thus, we may identify p" with the maximum momentumachievable in an SNRbut limited by the geometric constraints, notby the lifetime of SNRs. What is also required here is that, alongwith or, independent of this geometric limit to p", the character ofparticle confinement to the shock changes at p" when p" reachessome critical value. This should lead to a break in the spectrum atp ! p", such that its slope at p > p" is steeper than p#4 and flatterat p < p". Then the main contribution to the particle pressurewould come from particles around p".

For example, the spectral break may be initiated by a strongshock modification by accelerated particles (Malkov et al. 2002).Namely, Alfven wave compression in the CRP shifts the waves tothe short-wavelength end of the spectrum, leaving particles withp > p"without the resonant waves. In fact, some of those parti-cles still remain resonant with the waves with k * r#1

g ( p") ) k"as long as their pitch angles satisfy the condition p% < m!c /k",where % is the cosine of the pitch angle and !c is the gyrofre-quency. At the same time, the number of these resonant particlesrapidly drops with momentum since their pitch-angle distributionis limited by %j j < m!c /k"p, and they can hardly form a good pool

for the acceleration. Rather, the momentum spectrum can beshown to decay exponentially at p > p".

However, the latter example is based on a weak-turbulencepicture inwhich a sharp resonance of particles with waves of ran-dom phases determines particle dynamics. As we demonstratebelow, strong magnetic field perturbations result in a quite differ-ent character of particle confinement. In x 3 we consider a scat-tering environment that (although still simplified) retains someimportant characteristics of a more realistic strongly unstableturbulent CRP. As discussed earlier in this section, this turbu-lence should satisfy the following two conditions: (1) particleswith momenta p > p" should not suffer catastrophic losses, and(2) their spectrum must nevertheless be steeper than p#4. Aswe demonstrate in xx 3 and 4, a gas of relatively weak shocks,which can emerge in a number of ways, provides the requiredscattering.

3. PARTICLE DYNAMICS

The conventional paradigm of particle transport in shock envi-ronment is the diffusive pitch-angle scattering on self-generated,randomphaseAlfven waves. The resulting spatial transport is alsodiffusive with a coefficient that scales with the wave amplitude as! p% &/ #B#2

k , where the fluctuation wavenumber kmust be set inthe resonance relation with the particle momentum, krg p% & ’ 1.While this is themost consistent plasma physics approach that canbe derived from first principles using the quasi-linear theory, pro-vided that #BTB0, there is no guarantee that it holds when thewaves go nonlinear (e.g.,Volk et al. 1984;Achterberg&Blandford1986). The latter is the rule rather than an exception in shock en-vironments, which has been documented via observations, theory,and numerical studies. So the Earth’s bow shock, as well as inter-planetary and cometary shocks, reveals a variety of coherent non-linear magnetic structures. Usually, they originate upstream as aresult of nonlinearly developed instability of the distribution ofions reflected from the shock in the case of oblique propagation. Ifthe shock is quasi-parallel, the unstable ion distribution forms bythermal leakage from the downstream plasma. In cometary plas-ma, the pick-up ions drive the instability. Very often the magneticstructures are observed as an ensemble of discontinuities referredto as shock trains or shocklets (Tsurutani et al. 1987). There is noshortage of theoretical models describing these features, and webriefly return to them below. Standing somewhat outside of thesemodels, but perhaps more specific to the nonlinear CR shocks, isthe acoustic instability driven by the gradient of the CR pressure inthe CRP. It was first studied by Drury (see, e.g., Drury & Falle1986; Zank et al. 1990; Kang et al. 1992, and references therein),and it also evolves into a gas of moderately strong shock wavespropagating toward the main shock (subshock) in the CRP. Inwhat follows in this section, we assume that the shock precursor isfilled with the gas of such shocks separated by some distance L.

Let us consider such a shock train propagating in the CRP.Obviously the relative speed between the individual shocks andthe speed of the bulk flow is small compared to the particle speed,U z% &Tc. Therefore, we can consider the problem of particlescattering by scattering centers (shocks) and particle accelerationcaused by the relative motion of the scattering centers, separately.Note that the acceleration part of the problem has been prelimi-narily considered in x 2, and we return to it in x 6. Turning here tothe scattering part of the problem, we assume for simplicity thatthe shocks are one-dimensional and propagate at the same speedalong the shock normal. Furthermore, we transform to their ref-erence frame, neglecting compression by the main flow U(z) andtheir relative motion. Even under these simplifications, the shocktrain magnetic structures can be still rather complicated. For

Fig. 2.—Schematic representation of acceleration process. Up to the time t"the acceleration proceeds at the standard rate, which is similar during both thelinear and nonlinear phases (solid line). The particle pressure-dominant mo-mentum p"(t) grows as the maximum momentum of the entire spectrum. If fort > t" the pressure-dominant momentum p" remains flat (solid line), particleswith p > p" maintain the same acceleration rate that they had when their mo-mentum was equal to p" (dashed line). The dash-dotted line shows for com-parison how the maximum momentum pmax $ p" would continue to grow aftert $ t", had p"(t) not stopped growing at t $ t". [See the electronic edition of theJournal for a color version of this figure.]

NONLINEAR SHOCK ACCELERATION BEYOND BOHM LIMIT 247No. 1, 2006

example, they have been extensively studied in the frame of theso-called derivative nonlinear Schrodinger equation and its mod-ifications (Medvedev & Diamond 1996). Alfven–ion acousticwave coupling results inwave steepening and gives rise to a shocktrain. A kinetic effect included in this model is particle trapping,which was shown to be important for the formation of the shocktrain and which has been studied numerically (Hada et al. 1990;Medvedev et al. 1997). In another model, the shock train formsfrom a balance of a quasi-periodic driver and the nonlinearity ofthe magnetic field perturbations. The driver may originate fromone or a few unstable harmonics providing the smooth parts of thesolution that steepen into shocks. These shocks also have a com-plicated spatial structure, typically characterized by a fast rotationof the magnetic field vector, caused by dispersion (see Malkov1996, and references therein). However, since the gyroradius ofhigh-energy particles, even if reduced by possible magnetic fieldamplification, is still much larger than both the width of the shocktransition and the dispersive scale c/!pi, we replace each shock bya discontinuity of a coplanarmagnetic field.A different possibilitythat results in a similar behavior of themagnetic field occurswhenthe strong Drury instability discussed earlier evolves into an en-semble of moderately strong shocks.

Both cases are covered by the following representation ofthe magnetic field, which corresponds to a periodic sequence ofshocks, in which only one component of the field varies with thecoordinate in a comoving reference frame,B $ 0; By; 1

% &, where

By z% & $ By ' By sin & z# 1=2% &+ , tanh 1

'cos & z# 1=2% &+ ,

' (:

%9&

Here By is the constant component of the transverse to the shocknormal component of the magnetic field, and By characterizesthe strength of the shocks in the shock train (both normalized tothe constant z-component, B0; Fig. 3). We normalized the coor-dinate z to the distance between shocks L. Of course, in reality,different shocks in the shock train do not have the same strengthso that the coefficient By should be replaced by a stochasticvariable with a probability density function (pdf ), inferred fromthe shock dynamics. We return to this discussion in x 4, but inthis simplified study we assume all the shocks in the shock trainhave the same strength.

Let us consider particle dynamics in the magnetic field givenby equation (9). It is convenient to write the equations of motion

using dimensionless variables in which time is normalized toL/c. Since we are primarily concerned with the dynamics of par-ticles having momenta pk p" 3 mc, we normalize their speedto the speed of light and set the absolute value of particle velocityto unity, V - 1. The remaining variable p is normalized naturallyso that the corresponding gyroradius rg( p) is measured in theunits of L; that is, the particle momentum p is scaled to eB0L /c.We also introduce the cosine of the pitch angle % $ pz /p. Theequations of motion read

p $ n <B; %10&z $ %; %11&

where n $ p/p and p $ constant.Figure 4 shows particle trajectories as a Poincare section that

represents orbit crossings of the shocks, i.e., the planes z $ j(where j is an integer) on the gyrophase–pitch-angle plane, ((,%). To make a connection with the drift approximation, we havetransformed the particle momentum to the coordinate system ro-tated around the x-axis by the angle # $ tan#1By. In this coordi-nate system, particles spiral around the z-axis with % $ constantif By $ 0. The gyroradius is taken to be equal to the distance be-tween the shocks ( p $ 1, resonant case in terms of the linear the-ory), but the shock transitions are broad so that no distinct shockswith the steep magnetic field gradient are present and the parti-cle dynamics remains largely regular. For smaller p the systembecomes almost integrable since it can be reduced to a two-dimensional one, because the particle magnetic moment is con-servedwith good accuracy. In the case shown in Figure 4, however,some of the separatrices are clearly destroyed. Nevertheless, thereis a clear separation of trapped (magnetic mirroring) and pass-ing particles. Based on the conservation of the first adiabatic in-variant I / 1# %2% &/B (magnetic moment), the trapped particlesmust be confined in momentum space to %j j< %crit $ Bmax #%+Bmin&/Bmax,1/2 - 0:27, which is close to what can be seen fromFigure 4. It is important to emphasize here that the trapping areadoes not shrink with p, as was the case in the linear resonancesituation, discussed at the end of x 2. Clearly the spatial transportof these particles is nondiffusive. The trapped particles are con-vected with the shock train. The untrapped particles ( %j j> %crit)propagate ballisitically and escape from the CR cloud. Thus, even

Fig. 3.—The y-component of the magnetic field in the shock train given byeq. (9) for ' $ 0:2, By $ 1, and By $ 2.

Fig. 4.—Particle orbits shown for ' $ 0:5, By $ 2, and By $ 0:3 in eq. (9)and p $ 1 on the ((, %) plane. Each dot corresponds to the intersection ofparticle trajectories with one of the planes located at an integer z and coincidingwith the shock locations (Poincare section). Several particle orbits form invar-iant curves (Kolmogorov-Arnold-Moser tori), as well as stochastic layers aroundsome of the separatrices (see text).


a small-amplitude (coherent) shock train provides a substantiallydifferent confinement of particles than the small-amplitude ran-dom ensemble of Alfven waves does. It ensures perfect confine-ment of trapped particles and no confinement at all of untrappedparticles. This is what is required for the acceleration mechanismoutlined in x 2. When this confinement regime takes over abovesome p $ p", since there are no resonantwaves of the length largerthan rg( p"), the energy spectrummust become steeper for p > p".We come back to this point in x 6.

The dynamics changes evenmore dramatically when the thick-ness of the shocks in the shock train becomes small compared tothe particle gyroradius. Taking the same parameters in equa-tion (9) as those used in Figure 4 except for ' - 0,we show inFig-ure 5 the Poincare section created by a single trajectory. It shouldbe noted, however, that the actual amplitude of the shocks alsoincreases somewhat in this case, even though the amplitude pa-rameter By is fixed. The phase space is now a ‘‘stochastic sea’’with embedded ‘‘islands’’ of quasi-regular motion, quite typicalfor the deterministically chaotic systems (see, e.g., Lichtenberg &Lieberman 1983). The dynamics inside the islands is weakly sto-chastic, so that the particle trajectories there are closer to the nearlyintegrable case, shown in Figure 4, than in the rest of the phaseplane. Therefore, within those islands where the averaged pitchangle % 6$ 0, particles propagate ballistically in the z-direction,similar to the untrapped particles in the previous example. Thereare also islands where % - 0, and particle propagation is sub-stantially suppressed there. The islands are stochastic attractors,and the mapping exhibits cycling around them apart from thequasi-periodic motion inside islands, not shown in Figure 5. Notethat only a single trajectory is shown. The first type of island(% 6$ 0) is responsible for Levy flights (ballistic mode of particlepropagation), whereas the second type of island (% $ 0) repre-sents long rests or traps. The rest of the phase plane is covered bythe region of a global stochasticity where particle propagationappears to resemble that around the islands but with shorter restsand Levy flights. Note that we use the term Levy flights for theabove transport events even though the probability distributionfunction of the jump lengths might have a finite first moment(mean jump length). We investigate particle propagation in thez-direction in more detail in x 4.

4. PARTICLE TRANSPORT

It is useful at this point to recall that in the standard diffusiveshock acceleration the pitch-angle diffusion determines the spa-

tial transport, which is also diffusive. This picture results from thescattering of particles by Alfven waves of small amplitude andrandom phases. By contrast, in x 3 we have considered a coherentmagnetic structure of the finite amplitude shocks, as an alternativeparticle scattering agent at momenta higher than the pressure-containing momentum scale p". Despite the coherence, the par-ticle dynamics, as we have seen, remains strongly chaotic, and theonset of stochasticity depends on both the amplitude of the shocksand their thickness. We thus appeal here to the intrinsic stochas-ticity caused by unpredictable particle motion in a field that can beperfectly regular. The fact that in a real situation the field is alsorandom is often assumed to be of secondary importance.

The resulting spatial transport is not a ‘‘classic’’ diffusion pro-cess. The connection between the particle dynamics, representedby the Poincare section shown in Figure 5 and the particle spatialtransport can be most easily understood by considering equa-tion (11), z $ %, as a stochastic differential equation, in which %is a random process. Some general characteristics of this processcan be inferred from the Poincare section shown in Figure 5. Aswe mentioned above, there are traps represented by islands thattranslate into long flights or long rests depending on whether theaveraged value of %, i.e., % is nonzero or zero, respectively. Herethe averaging is to be taken over an island attractor that is a layerof enhanced phase density around the island.

The particle transport can be conveniently treated as a ran-dom walk on a lattice of shocks located at the integer values ofz. However, in contrast to the classical random walk, this is anon-Markovian process since it is characterized by long rests(Zaslavsky 2002). Particles interact with the same shock repeat-edly while they are trapped near an island with % $ 0. Similarly,they perform a long jump in one direction and cross many shocksin a rowwhile they are trapped in the phase space around an islandwith % 6$ 0.

The particle trajectory that corresponds to the Poincare sectionshown in Figure 5 is demonstrated in Figure 6. The trajectory isindeed nondiffusive. It consists of long ‘‘Levy flights’’’ that con-nect areas (clusters) of a rather slow particle propagation. Suchclustered propagation regime is quite typical for nonlocal (frac-tional) diffusion models (see, e.g., Metzler & Klafter 2000). Thepropagation within a typical cluster is magnified and shown inthe inset of Figure 6. A certain similarity with the dynamics at

Fig. 5.—Same as Fig. 4, but for the narrow shock transitions, ' - 0. In contrastto Fig. 4, however, the entire phase portrait is formed by a single particle orbit thatdoes not visit the ‘‘holes’’ in the phase space.

Fig. 6.—Particle trajectory represented as z(t) that corresponds to the phasespace shown in Fig. 5. It starts close to the islands near % $ 0 so that the orbitstays close to the origin (z $ 0) for a long time. Such long rests will be repeatedmany times at different locations. Overall, particle transport is organized in clus-ters (five such clusters are shown) where the transport is strongly suppressed.The clusters are connected with long jumps where the transport is ballistic.


large can be noted, which is also not unusual for the determin-istically chaotic dynamical systems. However, the intraclusterLevy flights are significantly shorter than the intercluster ones,an effect that is shown to be much stronger for stronger shocks.Besides, long rests at different shocks can be seen inside the clus-ters. Note that it is this intracluster particle dynamics that ensurestheir confinement to a localized region of the flow convectedacross the CRP as described in x 2. As we argued there, everysuch passage through the CRP approximately doubles the par-ticle momentum. The Levy flights, on the other hand, can resultin the escape of a particle from the shock precursor. We quan-titatively describe and compare both possibilities in x 6.2.

Let us come back to the overall dynamics. There are two im-portant characteristics that describe the random walk process.These are the waiting time and the jump length pdf. We have ob-tained these quantities from the trajectory shown in Figure 6. Thewaiting time distribution at different shocks is shown in Figure 7.Due to the gyromotion of particles near the shocks, the pdf isnot smooth, but it has a smooth envelope that decays approxi-mately as "#3, where " is the waiting time. The distribution ofthe lengths of the Levy flights is presented in Figure 8. Becauseof the asymmetry of the shock train (Fig. 3), there is a distinct,

directional asymmetry of the jumps. Otherwise, they exhibit anapproximate power-law distribution up to the jumps havinglengths limited by about 10. The pdf describes both the intra-cluster and intercluster particle jumps, but only the jumps of thefirst type are suitable for the continuous acceleration. The longerjumps, as we estimate in x 5, generally result in the loss of par-ticles. Another way of representing the trajectory clustering is il-lustrated in Figure 9, where the distribution of numbers of visitsof different sites in the shock train is shown.The knowledge of the waiting time and jump distributions is

necessary to derive the formof the adequate operators in the trans-port equation, (e.g., Metzler & Klafter 2000). Due to the non-Markovian character of transport, these operators are integral (alsocalled fractional differential operators, due to the memory in sto-chastic trajectories) rather than conventional diffusion operators.In x 6 we shall take a somewhat simpler approach, which, how-ever, also requires the flight–rest-time pdf to determine the trans-port coefficients.Up to nowwe have numerically examined the particle transport

in shock trains of a period longer or equal to the particles’ gyro-radii. This corresponds to the resonantwave-particle interaction inthe small-amplitude limit. Obviously, even in the case pT1,there are resonant Fourier components in the shock train, since itsspectrum decays as k#2 for k > 2& (recall that the shock trainperiod is set to unity). This is illustrated by Figure 10, where

Fig. 7.—Waiting time distribution in a log-log format. Here n is the numberof times the orbit has spent time " at any shock in the shock train. A somewhatdiscrete character of distribution is related to the gyromotion of particles neara shock so that when they interact with the shock repeatedly, the interactiontime is commensurate with the gyroperiod. [See the electronic edition of theJournal for a color version of this figure.]

Fig. 8.—Distribution of Levy flights shown for forward and backward jumps.[See the electronic edition of the Journal for a color version of this figure.]

Fig. 9.—Distribution of number of visits of different shocks in the shock train.The magnified box shows the distribution inside of one of the clusters (see text).

Fig. 10.—Same as Fig. 6, but for smaller particle momenta, p $ 0:1 and 0.3.[See the electronic edition of the Journal for a color version of this figure.]


particle trajectories with smaller momenta are shown. It shouldalso be noted, however, that particles with smaller gyroradii, suchas pP p", can be efficiently confined due to the resonance withself-generated waves since, as we argued earlier, their population issignificantly more abundant than those with p> p" because of thesteeper spectrum in this momentum range. This result is confirmedlater. It is clear that, nomatter how long the period of the shock trainis, we need to estimate the confinement of particles with a gyro-radius larger than that period, i.e., those with p> 1. One can expectthat their confinement will progressively deteriorate due to thefact that there is only a high-frequency force acting on these par-ticles. To estimate the fraction of p31 particles that can be con-fined, we write equation (10) in the form of the following systemof equations:

($ # 1

p' By z% &%

p)))))))))))))1# %2

p sin (% &; %12&

%$By z% &p

)))))))))))))1# %2

pcos (% &; %13&

p $ constant;

where we have assumed for simplicity that By $ 0. BecauseBy z% & has zero average, ( and % are the particle gyrophase andthe pitch angle, respectively, with respect to the shock normal.Assuming %T1 (which we verify below), from equations (11)and (13), we then obtain

z# 1

pBy z% & cos (% & $ 0: %14&

Considering ( as slowly varying compared to z and %, we canintegrate the last equation once:

p

2z2 ' A z% & cos (% & $ constant' O p#1

% &; %15&

where we have introduced a periodic function A(z) according toBy z% & $ #@A/@z. Since ( is a slowly varying variable (it varieson a timescale p3 1, while z and % vary on a timescale

)))p

p),

equation (15) describes an oscillator lattice with a slowly vary-ing potential as shown in Figure 11. The factor cos (% & in equa-tion (15) slowly inverts the lattice potential. However, trappedparticles do not detrap completely but rather get trapped in one

of the nearby potential wells. The width of the trapping zone inz and thus in % is given by %j j< %s, where

%s $

)))))))))))))))))))))))))))))))2

pAmax # Amin% &

s

T1:

For By z% & given by equation (9), we obtain Amin $ #Amax

and Amax $ By/2&. Particles with %j j> %s are not confinedto the shock train and propagate ballistically.

To conclude this section, we emphasize that when themomen-tum of particles grows and their Larmor radius increases beyondthe distance between the shocks, the fraction of particles that donot escape ballistically shrinks with momentum as %s / 1/

)))p

p(Fig. 12). This is slower than in the standard, weakly turbulentpicture, in which the critical value of % decays as 1/p. On theother hand, the latter dependence is based on the linear cyclotronresonance without broadening, while the resonance broadeningalso improves the confinement (Achterberg 1981). In any case,one has to expect the spectral cutoff at p ! 1 in units used in thissection (the Larmor radius is of order the shock spacing). There-fore, the characteristic distance between shocks determines themaximum momentum of accelerated particles. Of course, in areal situation there is a continuous strength/distance distributionof shocks that should produce a smoother spectrum decay at thecutoff momentum. Here we merely estimate the maximum dis-tance that is equivalent to the minimumwavenumber in the stan-dard acceleration picture. In the framework of the accelerationmechanism considered in this paper, the minimum wavenumberof the randomly phased Alfven waves should be of the order ofr#1g p"% &, while particles with the momenta p" < p < 1 are con-fined via themechanism of interaction with the shock train. In x 5we estimate the distance L between the shocks and thus pmax,since rg pmax% & ’ L, again, similarly to the standard estimaterg pmax% & ’ k#1

min.

Fig. 11.—Potential A z% & cos (% & in eq. (15), corresponding to the magnetic fieldBy $ #@A/@z, given by eq. (9) with ' $ 0 and By $ 0. The function A z% & $By/&% &

cos & z# 1/2% &j j# 1/2+ ,. Solid line: cos (% & $ #1; dashed line: cos (% & $ 1.

Fig. 12.—Phase space of particles in pitch-angle–momentum representation.The shaded area ( pinj < p < p"; %j j < 1) corresponds to the conventional par-ticle confinement via randomly phased Alfven waves. For higher momenta,p > p" [if there were only weakly turbulent waves present with minimumwavenumber, k $ kmin $ 1/rg p"% &], this type of particle confinement would belimited to %j j < %c / 1/p because of the resonance condition krg p% &% $ 1. Ashock train with the spacing rg p"% & < L < rg 1% & confines particles similar to themirroring-type confinement %j j < %c $ !B/B% &1/2 (hatched area), except forthe phase-space fragmentation (see Fig. 5 and text). Beyond p $ 1 the particleconfinement deteriorates to %j j < %s / 1/

)))p

p(see text).


5. ESTIMATE OF THE MAXIMUM MOMENTUM

In this section we estimate the maximum distance between theshocks in the shock train, which, according to x 4, is directlyrelated to the maximum momentum. In what follows we assumethat the shocks are formed due to the development of an acousticinstability driven by the CR pressure gradient in the CRP. Thelatter has the scale height Lp as being created by particles accel-erated in a standard DSAmanner and having maximummomen-tum p". Specifically, Lp ’ ! p"% &/U1. As was demonstrated byDrury & Falle (1986), the linear growth rate of this instabilitycan be written as

).D $ # )c Pc

$!. Pc

Cs$1' @ ln !

@ ln $

! ": %16&

Here Pc and Pc are the CR pressure and its gradient (whichis antiparallel to the shock normal), respectively; )c is theiradiabatic index; $ is the plasma mass density; and Cs is the soundspeed. For efficiently accelerating shocks, one can assume )c -4/3. The first term in equation (16) represents the wave dampingcaused by CR diffusion, calculated earlier by Ptuskin (1981). Thesecond term is positive for the waves propagating along the pres-sure gradient,while the oppositely propagatingwaves are damped.The factor in the parentheses can reduce or even completely elim-inate the instability if @ ln ! /@$$ #1. However, there are nophysical grounds on which this particular selection should bemade (Drury & Falle 1986). It should also be noted here thatparticle diffusivity ! refers here to particles with pP p", while thetransport of higher energy particles is nondiffusive.

Within the approximation leading to the growth rate given byequation (16), there is no dependence on the wavenumber. Amore thorough investigation can be found in Kang et al. (1992),which shows that the wavenumber dependence is indeed notstrong. Under these circumstances it is reasonable to assume thatthe largest seed waves are the most important ones. Furthermore,we assume that the latter are related to the cyclotron instabilityof the Alfven and magnetosonic waves (which can provide thecompressional seed component) that have excited by the accel-erated particles with momenta pP p", since particles of highermomenta have a steeper spectrum and do not contribute signif-icantly to the growth rate of the cyclotron instability. We shouldfocus primarily on the farthest part of the precursor, since per-turbations starting to grow there have the best chances to developfully while convected with the flow toward the subshock. Thispart of the precursor is accessible only to the particles near themaximum momentum achievable within the standard accelera-tion scenario, i.e., p". Hence, we assume that the typical wave-length of the Drury instability is kD! rg p"% &.

While these perturbations grow to a nonlinear level and prop-agate with the flow toward the subshock, they also steepen intothe shocks. This behavior is seen in both two-fluid simulations ofDrury & Falle (1986) and, what is particularly relevant to ourstudy, in the kinetic simulations of Kang et al. (1992). Due to thesignificant difficulties in the numerical realization of the kineticmodel, this study is limited in the maximum particle momentum,and thus in the length of the CRP, Lp. Besides that, a mono-chromatic wave has been chosen as a seed for the instability. As aresult, no significant interaction between the shocks has beenobserved. In a more realistic situation with a much longer pre-cursor and shocks of varying amplitudes, strong interactions be-tween shocks are to be expected (e.g., Gurbatov et al. 1991). Inparticular, stronger shocks overtakeweaker ones and absorb them,so that the interaction has a character of inelastic collisions. Time

asymptotically, such decaying shock turbulence is characterizedby decreasing spatial density of the shocks.In the context of the present study, we must consider driven,

rather than decaying, turbulence. A simple version of the drivenBurgers model that reveals shocks merging and creation of newshocks has been considered by Malkov et al. (1995). In the caseof driven turbulence, shocks not only merge, but also new shocksare formed in between them. The new shocks merge with theirneighbors and so on. In a steady state, some statistically stationaryensemble of shocks can be assumed with a certain average dis-tance L between the shocks and a shock strength characterized byan average Mach number M. For a systematic study of particlepropagation in such a gas of shocks, the pdfs of Mach numbersand distances between shocks are clearly required (Bykov &Toptygin 1981; Webb et al. 2003). As we mentioned above, wemake here only rough estimates of these quantities. In any event,for the generation of a statistically stationary shock ensemble, theinstability should act faster than the convection of a fluid elementacross the precursor does, which requires)DLp /U131. The lattercondition can be transformed to the following,

Pc

$U1Cs31: %17&

Note that the last condition does not contain the precursor scaleLp and can also be represented as *M 31, where * $ Pc /$1U 2

1 .With Pc referring here to its subshock value, * is the accelerationefficiency, andM $U z% &/Cs is an acoustic Mach number of theflow. Not surprisingly, the criterion *M 31 differs from thecondition #B2/B2

0 31 only by a factor ofM /MA $ 1/)))+

p, which

follows from equation (1). Indeed, both instabilities share thesame source of free energy, the CR pressure gradient.Now that we can assume that the Drury instability has suffi-

cient time to fully develop to a nonlinear level, we need to ex-amine whether the nonlinear stage of the wave steepening lastslonger than the linear one. For the wave breaking time beingshorter than the linear growth, it is sufficient to fulfill )D <!v/L,where !v is the characteristic magnitude of the velocity jumpacross the shocks and L is the characteristic distance betweenthem. The last condition can be rewritten as

L

Lp<

!v

U1

1

*M: %18&

Under these circumstances, in a steady state shock mergingmust be equilibrated with their growth due to the instability, i.e.,)D! #v/L, where #v is the characteristic variation of the shockstrength, i.e., the rms difference between neighboring values of!v. The latter condition can be written as

L

Lp! #v

U1

1

*M: %19&

Among the three parameters entering the last expression, #v/U1,*, andM, only the local Mach numberM(z) needs special atten-tion. Indeed, the far upstream M is considered to be prescribedand may be very large, but because of the multiple shock for-mation in the CRP, strong heating should occur and drive thisparameter down considerably.Let us estimate the local Mach numberM. There are two major

factors determining this parameter. After a fluid element passesthrough a shock in the shock train, it gets heated, but it cools adi-abatically before it crosses the next shock. For this rough estimatewe can assume that all shocks are of the same compression ratio


and that the overall flow pattern is L-periodic. For this order-of-magnitude type of estimate, we neglect the gradual variation of theflow at the scale Lp > L. We thus assume that the flow speed infront of each shock is v1 and is equal to v2 $ v1/r behind it, where ris the shock compression ratio. Similarly, the densities are relatedthrough $2 $ r$1.We estimate the shock heating from the standardRankine-Hugoniot relations. Let us denote the gas pressure in frontof a shock in the shock train asPg1. Then, the gas pressure behind it is

Pg2 $2)M 2 # ) ' 1

) ' 1Pg1:

The subsequent decompression phase decreases the pressure sothat it becomes equal to P 0

g1 $ r#)Pg2 in front of the next shock.Combining this with the above equation, we obtain

P 0g1

Pg1$ ) # 1' 2=M 2

) ' 1

! ")2)M 2 # ) ' 1

) ' 1:

Therefore, the total pressure change after passing trough theshock and the smooth flow behind it is zero only for M $ 1.However, the pressure does not really change for a broad rangeof Mach numbers around M $ 1 (where the right-hand side ofthe last expression is close to unity). This result means that Mcannot be large, since it would lead to strong plasma heatingdriving its temperature to a state with M ’ 1. However, in thissituation the calculation of heating using the Rankine-Hugoniotrelations is not correct, since shocks are likely to be subcritical.Note that in our estimates, we also ignored the increase of mag-netic energy behind the shocks. Since the above estimates showthat shocks cannot be highly supercritical also, we concludethat a reasonable estimate forM to substitute in equation (19) isa critical value of the Mach number, i.e., M ’ M", which is oforder a few (M" ’ 3, for example), depending on plasma + and#nB, the angle between the shock normal and the magnetic field(see Sagdeev 1966; Papadopoulos 1985, for a review of colli-sionless shock physics).

The efficiency *may then be calculated once the heating rate isknown (see, e.g., Malkov & Drury 2001). We note that the effectof the precursor heating on the shock structure and thus on * isvery sharp for a certain range of the heating rate parameter. Thelatter is difficult to quantify, so that we assume here that it is be-low the critical value at which a strong effect on the shock struc-ture is expected. As for the parameter #v/U1, the first obviousconstraint is #v <!v, which obviously verifies the consistencyof equations (19) and (18). In fact, based on the studies of thedriven Burgers turbulence (Cheklov & Yakhot 1995; Gotoh &Kraichnan 1998), we can assume that the shock strength vari-ance #v and its mean !v are related by #v P!v.2 At the sametime!vmay be assumed to be not so much weaker than the sub-shock itself (Drury & Falle 1986; Kang et al. 1992). The sub-shock strength can be calculated from nonlinear accelerationtheory (Malkov 1997; Blasi 2002; Blasi et al. 2005), along withthe acceleration efficiency. Given the uncertainty of the heatingrate, one can estimate * ! 1

513, so that the estimate L/Lp! 1

5 doesnot seem to be totally unreasonable.

As follows from x 4, themaximumparticlemomentumcan be es-timated from the relation rg pmax% &! L. Bearing in mind that Lp ’rg p"% &c/U1, the maximum momentum pmax can be estimated as

pmaxp"

’ c

U1

L

Lp:

Since L/Lp, as we argued, can be not very small and c/U1 is alarge parameter, the suggested mechanism can produce a sig-nificant additional acceleration beyond the break momentumat p $ p". This stage of acceleration lasts for approximately"acc pmax% &! ! p"% &/U 2

1

* +ln pmax/p"% &.

6. PARTICLE SPECTRUM BETWEEN THE BREAKAND MAXIMUM MOMENTUM

Up to nowwe have calculated the particle energy gain by theirscattering on unspecified scattering centers carried by the con-verging flow in the CRP. Independently of that, we then consid-ered the scattering of particles on an ensemble of shocks. Thelatter process did not lead to energy gain, since no relative motionbetween the scatterers was assumed. Recall that we considered thesimple shock ensemble as a magnetic structure traveling at aconstant speed. Nevertheless, each shock in the shock train maypossess its own flow structure with upstream and downstreamregions and thus can, in principle, accelerate particles via the stan-dard DSA mechanism. However, as we pointed out earlier, theseshocks do not have sufficient resonant turbulence upstream anddownstream with k < 1/rg p"% &. Therefore, there is no sufficientcoupling of particles withmomenta p > p" to the converging flowaround each such shock in order to gain momentum.

In the situation we consider further in this section the mo-mentum gain results from the combination of particle scatteringoff the scattering centers (shocks, localized magnetic structures)and gradual flow compression, leading to the convergence of thecenters. In fact this is similar to what we have considered in x 2 atan elementary level. An essential difference is that the scatteringin momentum space is not homogeneous as the results of x 3suggest. Therefore, we relax the assumption, also made in x 2,that particle transport in momentum space is diffusion thatevenly covers isoenergetic surfaces. This affects the formula forthe momentum gain.

6.1. Particle Momentum Gain

To calculate the rate at which particles gain momentum, let uswrite down the equations of particle motion in the subshockreference frame and apply them in the area ahead of the subshockwhere the velocity of the flow is U z; t% & $ U z; t% &ez (with thez-axis pointing in the shock normal direction, ez),

dp

dt$ e

cv# U% & < B: %20&

In this approximation, we have replaced the electric field byE $ #c#1U <B, assuming that due to the perfect conductivitythe electric field vanishes in the local plasma frame. In contrastto x 3, we do not specify the distance between the scatteringcenters here, so it is convenient to normalize the length to c/!ci ,time to !#1

ci , p to mc, and v and U(z) to c.Introducing the cosine of the particle pitch angle to the shock

normal, %, as well as the particle gyrophase, (, as in x 4, it isuseful to rewrite the equations of motion given by equation (20)as

p $)))))))))))))1# %2

pU Im be#i(

% &; %21&

% $ 1# %U

pUp; %22&

( $ # 1

p' %# U)))))))))))))

1# %2p Re be#i(

% &; %23&

z $ %: %24&2 This rough estimate should be taken with caution. Both quantities are really

distributions, often with a power-law pdf, and may not even posses finite means.


We have introduced the following complex variable instead ofthe transverse component of the magnetic field by b $ Bx '%iBy&/B0. The above equations are written in the shock frame.Now we need to transform particle momentum p to the localplasma frame, i.e., the frame moving with respect to the shockwith the velocityU(z, t), since it is thismomentum the convection-diffusion equation [eq. (7)] refers to. This transformation can bewritten in the approximationUT1 as p0 $ 1# U%% &p. Nowwedifferentiate the both sides of the last formula with respect to time,taking into account that to this order of approximation dU /dt -z@U /@z. Making also use of equations (22) and (24), for theacceleration rate in the local plasma frame we obtain

p0 $ #%2 @U

@zp0: %25&

Recall that the precursor scale height Lp that determines @U/@zshould be calculated using the spectral break momentum at p",i.e., Lp $ ! p"% &/U1. We also note that the acceleration rate inequation (25) does not contain the dynamical variable ( and isactually almost independent of z. This is because U, to a goodapproximation, can be considered as a linear function of z in asignificant part of the shock precursor (in the case of the Bohmdiffusion). As we mentioned earlier, due to the large gyroradius,we can neglect any small-scale variation ofU. This includes thestructure of individual shocks in the shock train, so that only thegradual variation of U across the CRP is important. In addition,we are really interested in the averaged value of the accelerationrate p0/p0. For example, if the particle transport in pitch angle isa small step diffusion, % can be regarded as a random variableevenly distributed over the interval (#1, 1). In this simple case,the average %2 $ 1

3 , and we recover the standard accelerationrate given by equation (6). In the case of structured phase spaceconsidered earlier in x 3, one can assume that during most ofan acceleration cycle, a particle is bound to the island having% $ 0. If % 6$ 0, the particle performs the Levy flights, whichwe discuss below. In the case of small islands (i.e., with thewidth %0T1), we can replace %2 by %2(for % 6$ 0) and by %2

0 /3(for % $ 0).

The situation becomes somewhat more complicated in thecase of an oblique shock, for which the phase space shown, forexample, in Figure 3, implies that the cosine of the pitch angle %is to be measured with respect to the average magnetic field andnot with respect to the shock normal as in equation (25). Thetransformation to the angles % and (, i.e., to the reference framehaving the z-axis aligned with the averaged magnetic field, isgiven by

% $ % cos ##)))))))))))))1# %2

psin # sin(; %26&

where # $ tan#1By (see x 3). Therefore, the acceleration rategiven by equation (25) does include the gyrophase (. However,the dynamics on the (; %

% &plane in the case of trapping we are

interested in here, and that is exemplified in Figure 5, is mostly ro-tation around the origin, so that one can expect that % sin( - 0.Thus, the acceleration rate can be represented as

p

p- #K

@U

@z; %27&

where

K $ 1

3%20 cos

2#' 1

21# 1

3%20

! "sin2#

, -%28&

and %0 denotes the width of the island in which the particle or-bit is trapped. We also omitted the primes in the momentum p.Using this formula it is easy to calculate particle momentum

gain assuming that the particle is trapped by the flow at the dis-tance z from the subshock in the CRP where the local flow speedis U(z) and is convected with the flow to the subshock where theflow speed is#U0. Suppose that at themoment t and coordinate zthe particle has the momentum p0. From equation (27) we havefor the particle momentum at z $ 0 where U $ #U0,

p$ p0 exp #K

Z t0

t

@U

@zdt

, -$ p0 exp K ln

U z% &#U0

, -$ p0

U

#U0

! "K

:

%29&

Note that in the conventional diffusive acceleration, in whichparticles ergodically cover the entire isoenergetic surface, i.e.,%0 $ 1, the acceleration constant K $ 1

3 (independent of #), andthe last result signifies the effect of adiabatic compression of theCR gas by the converging flow. On the other hand, if the particledynamics is such that %0 - 0 and # - &/2, which correspondsto two-dimensional compression across the magnetic field, oneobtains K - 1

2. In strong CR-modified shocks, the shock pre-compression R $ U1/U0 scales as M 3/4 (Kazanas & Ellison1986; Berezhko et al. 1996). As was shown by Malkov (1997),the latter scaling is only valid for M <M" ) 'inj p"/pinj

% &4/3,

and the shock precompression R saturates at R ! 'inj p"/pinjwhen M > M" (see also Blasi et al. 2005). Here the injectionparameter 'inj ’ cpinj/mU

21

% &nc/n1 and pinj is the injection mo-

mentum, while nc and n1 are the number density of acceler-ated particles and that of the background plasma far upstream,respectively. In both Mach number ranges the shock precom-pression can be quite significant unless the gas heating in theCRP is strong. At the same time, strong reduction of the shockprecompression by plasma heating would diminish the heatingitself by weakening the acoustic (Drury) instability of the CRP.The situation may settle at some critical level at which a mod-erate, but still quite significant, precompression is accompaniedby the CRP heating caused by the acoustic instability (Malkovet al. 2000; Malkov & Drury 2001). Further interesting analysisof the nonlinear shock acceleration and its bifurcation has beenpublished recently by Blasi et al. (2005).

6.2. Acceleration Model: Details

The picture that emerges from the above study of particle dy-namics in the modified shock precursor can be described as fol-lows. There are three groups of particles. One group is made upby particles that are locally trapped in the plasma flow by a struc-ture in the shock train. They are convected with the flow andeither do not propagate at all or propagate very slowly with re-spect to the flow. They are clearly seen in, e.g., Figure 10, as thequasi-horizontal portions of the stochastic trajectory of a singleparticle. The remaining two groups of particles are particles per-forming Levy flights, i.e., those propagating ballistically in posi-tive and negative directions at the speedsU+ andU#, respectively.These are characterized by the steep portions of the trajectory inFigure 10, with positive and negative slopes, respectively. Let usdenote the phase-space density of the above three groups of par-ticles by F0(t, z, p) and F.(t, z, p). Furthermore, we introduce theprobabilities of transition in units of time by denoting by ,. theprobability of trapping of particles performing Levy flights in pos-itive and negative directions, respectively. We denote the rates ofthe reverse transitions by +.. Now we can write the balance of


particles in a phase-space cell between z1, p1 and z2 $ z1 ' dz,p2 $ p1 ' dp as follows:

@

@tF0 t; z; p% &dzdp$ F0U z1# F0Uj jz2

. /dp' pF0 p1 # pF0

00 00p2

. /dz

# +'' +#% &F0 dzdp' ,'F' ' ,#F#% & dzdp;%30&

or after retaining only the linear terms in dz and dp and usingequation (27), we obtain the following equation for the phase-space density of trapped particles F0,

@F0

@t' @

@zUF0#K

@U

@z

@

@ppF0$# +''+#% &F0','F'' ,#F#:

%31&

Here the constant K is given by equation (28). Proceeding in asimilar manner, we obtain two equations for F. , i.e., for thephase-space density of the particles in the Levy flight state,

@F.

@t' @

@zU.F. # K.

@U

@z

@

@ppF. $ #,.F. ' +.F0:

%32&

In contrast to equation (31), the acceleration constants K. aredefined by

K. $ %2. cos2#' 1

21# %2

.% &

sin2#

, -; %33&

where %. are the averaged values of the cosines of the particles’pitch angles (%. $ %) performing Levy flights in the positive andnegative direction, respectively, while U. $ c%. cos #3U aretheir speed components along the shock normal (cf. eq. [28]).Note that we ignore direct transitions between the opposite Levyflights as comparatively rare events. This can be inferred fromFigure 10, for example.

Equations (31) and (32) form a closed system that must besupplemented with the boundary conditions. Far upstream fromthe shock, at z $ 1, it is natural to impose the following bound-ary conditions:

F0 1% & $ F# 1% & $ 0; %34&

which simply mean that accelerated particles can only leave thesystem in that direction, but do not enter from it. Note thatF' 1% & 6$ 0 and should be determined from equations (30) and(32). The second boundary condition, that at the subshock lo-cation z $ 0, is somewhat more ambiguous. In the standardDSA scheme it is fixed by the flux of particles convected withthe flow downstream. It can be expressed through the particlephase-space density and the downstream flow speed since theparticle distribution is isotropic. In the case of high momenta( p > p") considered here, particles are not bound to the flow sostrongly as to be able to make an isotropic distribution. Rather,according to equations (30) and (32), particles convected withthe flow at the speed U(z) are being converted into particlespropagating at high speed,U. ! c, and can either leave the sys-tem upstream (unless they are retrapped) or can penetrate deeplydownstream and reach the contact discontinuity. The latter pos-

sibility for the high-energy particles has been already discussed inthe literature (Berezhko 1996; Blondin & Ellison 2001). The factthat the contact discontinuity and the forward shock may indeedbe closer to each other due to the back-reaction of accelerated par-ticles on the shock dynamics, as clearly follows from the nonlin-ear DSA, seems to find impressive observational confirmation inWarren et al. (2005). The strong magnetic field at the contact dis-continuity may very well reflect particles with p > p", and theycan return to the shock. In a simple form this requirement can beobviously formulated by mathematically setting the total particleflux through z $ 0 equal to zero:

#F0 0% &U0 ' F# 0% &U# ' F' 0% &U' $ 0: %35&

In other words, the negative part of the total particle flux, rep-resented by the first two terms (U# < 0), is reflected by the strongmagnetic turbulence downstream and ultimately reappears at theshock in the form of the untrapped particles propagating upstreamat the speed U' > 0. The strong turbulence downstream may beformed by the magnetic perturbation convected and compres-sively amplified from the upstream media and by the Raleigh-Taylor instability of the contact discontinuity (Jun et al. 1996).

Theway particle momentum enters equations (31) and (32), aswell as the homogeneity of the boundary conditions given byequations (34) and (35), suggest the power-law solution:

F0; . / p#q: %36&

Assuming for simplicity U. $ constant (which is close to thetruth, as, e.g., Fig. 10 suggests), we can rewrite these equationsas follows:

d

dzUF0 $ )0F0' ,'F'' ,#F#; %37&

U.dF.

dz$ ).F. ' +.F0: %38&

Here we introduce the following notation:

)0 $# K q# 1% & @U@z

# +;

). $ # K. q# 1% & @U@z

# ,. ;

where

+ $ +' ' +#: %39&

From equations (37) and (35), we have

F0 z% & $# 1

U

Z 1

z

exp #Z z 0

z

)0U

dz00

!,'F' z 0% &',#F# z 0% &+ ,dz0:

%40&

To zeroth-order approximation in +L/UT1, U /U.T1, fromequation (38) we have F' - constant and F# - constant, whilefrom the boundary condition given by equations (34) and (35), weobtain

F' $ U0

U'F0 0% &


and F# - 0. Upon substitution of F. into equation (40) andtaking it at z $ 0, we finally obtain the following equation forthe spectral index q:

Z 1

0

U z% &#U0

, -K q#1% &exp

Z z

0

+

Udz 0

! ",' z% & dz

U'$ 1: %41&

In fact, the last formula allows a very simple physical inter-pretation, making it essentially equivalent to the well-knowngeneral Fermi (1949) result. Indeed, Fermi showed that if agroup of particles undergoes continuous acceleration at the rate"#1acc and have certain probability to escape that can be charac-terized by the escape time " esc, then the energy [momentum dis-tribution normalized above as F0( p)] has the following spectralindex:

qF $ 1' "acc"esc

: %42&

At least for some short period of time, by definition of thesetimescales, one can state that the momentum gain p t% &/p 0% & )p/p0 $ exp t/"acc% &. The probability of staying in the accelerator ifescape is a Poisson process is Pconf $ exp #t/"esc% &. Then equa-tion (42) can be recast as (cf. Bell 1978; Blandford&Eichler1987;Achterberg et al. 2001):

qF $ 1'ln 1=Pconf

% &

ln p=p0% & : %43&

Note that t can be regarded here as the duration of an acceler-ation cycle and it does not enter the spectral index explicitly. Itshould also be clear that determination ofPconf and the momen-tum gain (or in other words, "esc and "acc) requires in our modelmore information about particle transport, in particular, theparticle trapping and untrapping rates , and +.

To demonstrate that the index q in our formula (41) must beinterpreted in the same way as in the classical Fermi problem, letus analyze various factors in the integrand in equation (41) sep-arately. Assume that a particle enters an acceleration cycle at z $ 0as a Levy flight in a positive direction with a speedU+ against theplasmaflow of speedU(z) in the shock precursor, located in a half-space z > 0. During its flight, the particle has a possibility to betrapped in the flow that is characterized by the rate ,+. In otherwords, during time dt the probability to be trapped between z andz' dz equals ,' t% &dt $ ,' z% &dz/U'. We see this probabilitydensity in the integral given by equation (41). Assume that thetrapping event actually occurred at some z. Then the particle isconvected (with the flow) back to the subshock at the speed U(z).During this convection it has a certain probability to escape,whichis characterized by the rate +. This means that if the particlehas a probability of being in the flow P(t), then the probabilityto remain there at t ' dt is P t ' dt% & $ P t% & 1# +dt% &, or if theparticle is certainly in the flow at t $ 0 (just trapped there, forexample), then P t% & $ exp #

R t

0 + dt% &

. Noting that dt $ dz/U ,we can write the probability of returning with the flow from z toz $ 0 as

exp #Z 0

z

+ dz=U

! ":

Combining with the above probability of being trapped between zand z' dz, we can express the probability of completing an ac-celeration cycle of the length between z and z' dz as3

dPcycl z% & $ exp

Z z

0

+

Udz0

! ",' z% & dz

U':

The total probability of completing one such cycle of any possiblelength 0 < z < 1 is

Pcycl $Z 1

0

dPcycl

dzdz:

We can now rewrite equation (41) as follows:

Z 1

0

U z% &#U0

, -K q#1% &dPcycl

dzdz $ 1: %44&

Here the quantity G z% & $ U z% &/U0j jK is equal to a particle mo-mentum gain pf /pi after it is convected with the flow from z to theorigin (eq. [29]). Using themean value theorem, equation (44) canbe represented as

pfpi

# $q#1

Pcycl $ 1;

where pf /pi1 2

$G z% & is the mean momentum gain per cycleand Pcycl is, as stated above, the probability of completing thisacceleration cycle. From the last formula we obviously have

q $ 1'ln%1=Pcycl&ln pf =pi1 2 ; %45&

which is formally equivalent to the classical Fermi result.The quantities ,+ and +, which the index q depends on, can be

inferred from the particle stochastic trajectories shown, e.g., inFigures 6 and 10. Indeed the probability for a particle to be in thetrapped state can be written as "tr/ "tr ' "'L ' "#L

% &, where " tr and

"L are the average trapping and Levy flight times, respectively,that can be calculated from the particle trajectory. Then the par-ticle trapping rate ,+ can be written as

,' - "tr"tr ' "'L ' "#L% &

1

"'L;

similarly,

+. - ".L

"tr ' "'L ' "#L% &1

"tr:

For the case of sufficiently strong shocks in the shock precursor,the trapping process is quite efficient, so we can assume that

Z Lp

0

+

UdzT1: %46&

3 Particle behavior during a Levy flight is non-Markovian, since it bears ele-ments of deterministic motion (this is also true for the trapped particles). How-ever, the joint probability of two such events is multiplicative, since they evolveindependent of each other.


Here the shock precursor length Lp can be related to the trappingprobability as

Lp!R10 ,' z% &z dzR1

0 ,' z% & dz:

The requirement for the inequality in equation (46) is then

"p"L"2tr

T1; %47&

where the precursor crossing time is

"p $Z Lp

0

dz

Uj j:

An example of a particle trajectory with such long trappingtimes is shown in Figure 13. Note that when a particle is bounc-ing between two to three neighboring shocks it can also be con-sidered as ‘‘trapped’’ in the flow, as is the case for a particle stuckto a single shock. These kind of events are clearly seen in Fig-ure 13. Using equation (41), we present in x 6.3 a simplifiedcalculation of the power-law index q.

6.3. Estimate of the Power-Law Indexbetween the Knee and Cutoff

The power-law index q in the momentum range beyond theknee ( p" < p < pmax) essentially depends on the flow profileU(z)(eq. [41]). For the Bohm-type diffusivity!( p) in the region p< p"[i.e., the pressure-dominated part of the spectrum, thus importantfor determination of U(z)] the flow profile has been calculated byMalkov (1997). Assuming here that the integral in equation (41)cuts off by the exponential factor rather than by,+(z) at suchvaluesof z [where U(z) is still not very different from its subshock value#U0, i.e., U ' U0j jTU1], one can represent U(z) as follows:

#U z% &=U0 - 1' -))z

p:

Here

- - 2.U0 # U2

U0

)))))))))p0p"

p ))))))))))))))))))& U0j j=!0

p;

where . is the particle injection rate and p0 and p" are the in-jection and the knee momenta, respectively, in units of mc and

the particle diffusivity !0 ) ! p0% & (see also Fig. 2). In the lastexpression for - we omitted an additional factor that is close tounity. Equation (41) for q can be rewritten as follows:

2,'%#k#%#2

U'-2

Z 1

0

x' %% &k'%e#xx dx $ 1; %48&

where instead of the power-law index q we use

k $ K q# 1% &

and where

% ) +!0

2&. 2p0p" U0 # U2% &2:

The latter parameter contains the important quantity .2p0 p*,which was shown to be the control parameter of nonlinear shockmodification (Malkov 1997). Namely, this parameter should belarge in the case of strong shock modification by accelerated par-ticles. The remaining parameter +!0/ U0 # U2% &2 represents thedetrapping probability of a high-energy particle ( p > p") over thetime of crossing the diffusion length of a freshly injected particle( p ! p0). It is implied here thatU0 # U2 ! U0. Therefore, we canassume that%T1. In general, equation (48) can be represented interms of an incomplete gamma function, but since %T1, it takesthe following simple form:

" k' 2% & $ '%k'1; %49&

where " is the gamma function. Here we introduced a new pa-rameter ' according to

' $ U'+

U0,'’ U'

U0

"L"'L

" 2tr

! c

U031:

Since %T1, equation (49) has a solution at kP 1. It depends onthe parameters ' and %, but taking the limit of large ', one canapply the Stirling formula for the gamma function, which leads tothe following estimate of the power-law index q $ 1' k /K:

q $ 1' 1

K

ln '%% &ln 1=%% &

:

Fig. 13.—Stochastic particle trajectory represented in the same format as inFig. 10, but for p $ 0:3, By $ 1, and By $ 2.

Fig. 14.—Particle power-law index q for p > p" as a function of % for dif-ferent values of ' and K $ 1

3 (see text). The test particle result (q $ 2) and anindex corresponding to a strongly modified shock (q $ 1:5) are also shown.


The ratio of logarithms is typically below unity, while the constantK is in the range of 1

312. Thus, the power-law index is (almost)

definitely well above 2 (the particle pressure clearly converges),but it can hardly be significantly higher than 3 for strongly non-linear fast shocks (see Fig. 14 for a numerical solution of eq. [49]).For a more accurate determination of the power-law index in thep > p" range, further study of particle scattering in the shock pre-cursor is needed. Fortunately, the index q only weakly (logarith-mically) depends on the most uncertain parameters of the presentmodel, namely, on % and ' or, equivalently, on the trapping anduntrapping rates , and +.

7. SUMMARY AND CONCLUSIONS

The principal conclusion of this paper is that particle accelera-tion inside the CR shock precursor may very well be faster andproceed to higher energies during SNR shock evolution than theconventional DSA theory suggests. The requirement for suchenhanced acceleration is some limitation on the momentum ofparticles that contribute the most to the CR pressure in orderto prevent further inflation of the shock precursor. This can beachieved by spatial limitations of momentum growth, as dis-cussed in detail by, e.g., Berezhko (1996), by the wave compres-sion and blueshifting of the wave spectrum in the CRP (Malkovet al. 2002), and by the change of acceleration regime after theend of the free expansion phase (Drury et al. 2003). As a resultof these limitations, a break on the particle spectrum is formedand maintained at p $ p" beyond which particles do not contrib-ute to the pressure significantly but are still accelerated at the samerate as particles with momenta p < p".

The fundamental acceleration mechanism is essentially iden-tical to the nonlinear DSA. The difference from the linear DSA isthat frame switching that results in the energy gain occurs pre-dominately between the scatterers convected with the graduallyconverging upstream flow in the precursor and not between theupstream and downstream scattering centers, as usually as-sumed. The maximum momentum is estimated to be

pmax !c

U1

L

Lpp"; %50&

with the maximum value of L/Lp P 15. Therefore, this mecha-

nism provides up to !0.2(c/U1) enhancement to the maximummomentum, based on the standard spatial constraints (e.g.,(Berezhko 1996). The spectral index between p" and pmax de-pends on both the ratio of the particle trapping to flight time " tr /"Lin the upstream turbulent medium and on the shock precompres-sion R. This is in deep contrast with the spectrum below p", whichtends to have a form quasi-independent of R (e.g., Malkov &Drury 2001). The acceleration time is

"acc pmax% &! "NL p"% & ln pmax

p"; %51&

where "NL p% & ’ 4! p% &/U 21 is the nonlinear acceleration time,

which only slightly differs from the upstream contribution tothe standard linear acceleration time (Malkov & Drury 2001).

The overall temporal development of the acceleration processcan be described as follows. It starts as the standard DSA mech-anism from some slightly suprathermal momentum pinj and pro-ceeds at the rate "#1

acc pmax% &! U21 /! pmax% &, where ! p% &! crg p% &

and rg pmax% &kmin! 1. The minimum wavenumber kmin decreaseswith growing pmax(t), since the MHD waves confining particleswith momenta pinj < p < pmax to the shock are resonantly gener-ated by the accelerated particles themselves. After pmax reaches a

certain value (that can be calculated analytically, e.g., Malkov &Drury 2001) and is clearly seen in numerical time-dependent solu-tions (e.g., Berezhko et al. 1996), a CRP forms, and the accel-eration then proceeds mostly in the CRP. The acceleration rate,however, behaves with time approximately as it does at the pre-vious (linear) stage, since dU/dz decreaseswith the growingwidthof the CRP, Lp ! ! pmax% &/U1. Note that dU/dz sets the acceler-ation rate. The next acceleration-boosting transition occurs whenLp reaches one of its above-mentioned ‘‘natural’’ limits, such asthe accelerator size [a significant fraction of an SNR radius, for ex-ample, i.e., ! pmax% &/U1! RSNR ]. Assume pmax ’ p" at this point.Then particles with momenta p > p" cannot be confined to theshock diffusively, since their diffusion lengthLdif p% & $ ! p% &/U1>Lp ’ ! p"% &/U1. Fortunately, by this time instabilities of the CRPproduce an ensemble of shocks or similar nonlinear magneticstructures. These ensembles of structures turn out to be capable ofconfining particles with p > p", since the mean distance betweenthe structures (which is the scale that replaces the wavelength ofparticle-confining turbulence in the quasi-linear picture) is muchlonger than the particle gyroradius, L3 rg p"% &. Particle confine-ment is, however, selective in terms of particle location in phasespace. This is similar to, but more general than, particle confine-ment in magnetic traps, in which only particles having pitch an-gles satisfying %j j< !B/B% &1/2 are trapped while others leavethe system freely. The price to be paid for this is a spectrum atp > p" that is even steeper than p#4. This is, in fact, more a bless-ing than a curse since these particles do not contribute to the CRpressure significantly and Lp does not grow, thus maintaining theacceleration rate at the same level U 2

1 /! p"% & for all particles withmomenta p" < p < pmax.The total acceleration time up to the maximum momentum

pmax (given by eq. [51]) is thus only logarithmically larger thanthe acceleration time to reach the spectral break (knee) at p ’ p",and the maximum momentum itself is also pinned to the breakmomentum p" through equation (50). The break momentum issimply the maximum momentum in the standard DSA scheme,whatever physical process stops it from growing further. Therealization of the above scenario, which obviously produces asignificant acceleration enhancement in terms of both the max-imum energy and acceleration time, depends crucially on thepresence of the ensemble of shocks or similar scattering mag-netic structures in the CRP. Their existence in various (evenCR-unmodified) collisionless shocks is perhaps beyond anyreasonable doubt, as we discussed in x 3, but the real challengeis to calculate the statistical characteristics of their spacing L andamplitude parameter !v. A simple approach that we pursued inx 5 leads to the conservative estimate of L/Lp! 1

10 , so that tak-ing U1/c ! 103, the maximum momentum exceeds its standardvalue by a factor of 100 (eq. [50]).At this point it is worthwhile to recapitulate the major differ-

ences between our approach and other recent models discussedin x 1 that are also aimed at the enhancement of acceleration ef-ficiency. They primarily seek to increase the turbulent magneticenergy by tapping the free energy of (already) accelerated par-ticles. This would result in decreasing the particle dwelling time(more frequent shock crossing and thus faster acceleration) and,concomitantly, the particle diffusion length (better confinement,higher maximum energy). Our approach is based on the appre-ciation of the following:

1. In a nonlinear regime, the acceleration time is reduced be-cause of a steeper velocity profile in the smooth part of the shocktransition (where acceleration mostly occurs). The steep gradientis maintained due to the formation of a break on the particle


spectrum.Without the break, the gradient would flattenwith grow-ing particle energy slowing down the acceleration. Apart from thisgradient, the acceleration rate does not depend on the mfp ex-plicitly, as long as the latter is smaller than the shock transition.

2. Particle confinement (maximum energy) is enhanced byincreasing the wavelength (shock spacing in a shock train insidethe CRP, for example) of turbulence that is in a distinctivelystrong regime. Departure from the weak turbulence is marked bymodification of particle confinement in the CRP. This steepensthe momentum spectrum beyond the spectral break and ensuresits existence, as required in (1).

It remains unclear, whether the realization of the first (i.e.,strong magnetic field amplification) hypothesis would precludeor otherwise strongly influence the acceleration scenario sug-gested in this paper. It would almost certainly not, if the magneticenergy increases predominantly at longest scales as suggested byDiamond & Malkov (2004) . This should merely scale the esti-mates provided in this paper by renormalizing such basic quanti-ties as Lp and "NL, thus making acceleration improvement evenstronger. This seems to be necessary for the successful expla-nation of the Galactic CR spectrum between the knee (at -4 ;1015 eV) and the second knee (at-1018 eV), e.g., Horandel et al.(2005). The knee position can thus be interpreted as p" corre-sponding to the interstellar magnetic field moderately amplified(by a factor of 10 or so), whereas the spectrum between the kneesis formed by the mechanism described in xx 6.2 and 6.3. It is also

worthwhile to recall in this regard that due to the scale invarianceof the equations of motion of the accelerated particles (in whichparticle momentum and charge enter in combination p/Z, where Zis the charge number), the high-energy spectra obtained for theprotons remain valid for other particles with the same value of p/Z.This is what observations actually suggest (e.g., Ellison et al.1997; Horandel et al. 2005).

One disadvantage of acceleration enhancement with signifi-cant field amplification, however, is that the increased magneticfield inevitably leads to increased radiative losses with a reduc-tion of the particle maximum energy as an implication. This im-portant acceleration constraint has been studied in detail byAharonian et al. (2002), including applications to many popularacceleration sites. Naturally, the most prominent impact is ex-pected on the acceleration of the extremely high energy CRs(EHECR;k1020 eV). Detailed models of such acceleration havebeen proposed by Miniati et al. (2001), Kang & Jones (2005),and others. Clearly, the acceleration improvement without sig-nificant increase of the magnetic field potentially provides a sig-nificant edge on the field amplification scenario in the generalcontext of the problem of EHECR acceleration.

Support by NASA grant ATP 03-0059-0034 and under USDepartment of Energy grant FG 03-88ER53275 is gratefullyacknowledged.

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