J.
6 (2004)
J. Plasma
Plasma Fusion
FusionRes.
Res.SERIES,SERIES,Vol.
Vol.
6 (2004)28–34
000–000

Generation of Mesoscale Magnetic Fields and the Dynamics
of Cosmic Ray Acceleration
DIAMOND Patrick H. and MALKOV Mikhail A.
University of California at San Diego, La Jolla, California 92093-0319, USA(Received: 11 February 2003 / Accepted: 22 March 2004)

Abstract
The problem of the cosmic ray origin is discussed in connection with their acceleration in supernova remnant
shocks. The diffusive shock acceleration mechanism is reviewed and its potential to accelerate particles to the
maximum energy of (presumably) galactic cosmic rays (1018 eV) is considered. It is argued that to reach such energies,
a strong magnetic field at scales larger than the particle gyroradius must be created as a result of the acceleration
process, itself. One specific mechanism suggested here is based on the generation of Alfven wave at the gyroradiusscale with a subsequent transfer to longer scales via interaction with strong acoustic turbulence in the shock precursor.
The acoustic turbulence in turn, may be generated by Drury instability or by parametric instability of the Alfven
waves.
Keywords:
acceleration of particle, cosmic ray, magnetic field, shock wave, MHD, turbulence

1. Introduction
Cosmic rays (CRs) were discovered almost a century
ago, yet their origin is unknown. There is sufficient evidence
that at least part of their spectrum (E < 1018 eV) originates in
the Galaxy, while particles of higher energies are thought tocome to us from extragalactic sources. One simple theoreticalargument is that particles with such high energies (usuallyreferred to as ultra high energy CRs, UHECR) could not be
confined to the Galaxy, on account of their large gyroradius.
Second, their (power law- E –2.7) spectrum is harder than thatof (presumably) galactic particles at E < 1018 eV (E –3.1), whichis consistent with their extra-galactic origin. The last point
becomes clear if we turn to the other break on the overall CR
spectrum, the one at E ~ 1015 eV, commonly known as the“knee”, shown in Fig. 1. The spectrum above this energy
steepens (from E –2.7 to E –3.1), so that the premise of their
extragalactic origin would require an explanation of why thegalactic part of the spectrum terminates, while the extragalactic part appears exactly at the knee energy. As we shallsee, to explain the cosmic ray power law spectrum between
the “knee” at 1015 eV and the “ankle” at 1018 eV in terms ofacceleration within the Galaxy is one of the most serious
challenges of contemporary acceleration mechanisms and oneof the main motivations of this study.
The explanation of the spectrum beyond the ankle (thehighest energy event observed so far is 3 · 1020 eV) poses amajor challenge to fundamental physics. Given the distanceof possible accelerators (at least a few tens of Mpc), particles

Fig. 1 Cosmic Ray spectrum.

of such high energy should have experienced significant
losses through their interaction with the cosmic microwave
background radiation (the so called Greisen, Zatsepin, Kusmin
or GZK cut-off) while propagating over long distances. Note©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: pdiamond@ucsd.edu
128

Diamond P.H. and Malkov, M.A., Generation of Mesoscale Magnetic Fields and the Dynamics of Cosmic Ray Accelerationthat UHECR are sub-atomic particles with the energy of awell-hit baseball. This energy exceeds (by at least three ordersof magnitude) that achievable by all existing terrestrial
accelerators (e.g., Large Hadron Collider, LHC-1017 eV).
Another fundamental aspect of the problem of CR origin isthat they are the ultimate receptacle of a significant portionof the gravitational energy in the Universe. Indeed, star formation from the gravitational collapse of the primordial gas
with subsequent SN (supernova) explosions and their blast
waves results in CR acceleration. On the observational side,
there is an evidence [1-3] (in the form of both synchrotron
and inverse Compton radiation) that electrons of energies up
to 100 TeV are accelerated in the supernova shock waves.

mechanism passes this test at best only marginally. If the
turbulence level is lower, then the maximum energy should
be reduced proportionally. There are indeed a number ofphenomena which may either reduce the turbulence level [9],or which can shift the turbulence spectrum (in wave number)away from resonance with the high energy particles andtherefore cause their losses [10].
Another reason for concern about this mechanism, atleast in its standard (Bohm limit) version, is its seeming
inability to explain acceleration of particles beyond 1015 eV.
As was discussed above, the cosmic ray spectrum has only abreak at this energy, and continues to about 1018 eV where
the extragalactic component is believed to start dominating
the spectrum.
One approach to this problem is to generate a fluctuating component d B significantly exceeding the unperturbed
field B0 [11]. Physically, such generation is possible since thefree energy source is the pressure gradient of accelerated
particles, which in turn may reach a significant fraction of
the shock ram energy. Specifically, the wave energy density(d B/B0)2 may be related to the partial pressure Pc of CRs that
resonantly drives these waves through the relation [12]2. Acceleration mechanism
The leading CR acceleration mechanism, namely the
diffusive shock acceleration (DSA, also known as the 1st-order Fermi mechanism) was proposed originally by Fermi
in Ref. [4], and in its modern form by a number of authors in
the late seventies [5-8]. The mechanism is basically simple –particles gain energy by bouncing between converging
upstream and downstream regions of the flow near a shock
wave such as that from an SNR (supernova remnant) shock.
This mechanism requires magnetic fields. First, the field binds
particles to the accelerator (shock wave), in the direction
perpendicular to the field. Confinement in the direction along
the field lines is, in turn, ensured by particles themselves
through the generation of Alfven waves by accelerated
particles streaming ahead of the shock. This occurs via
Doppler resonance, W = k (p/m)m, where W and p are the(nonrelativistic) gyrofrequency and momentum, k is the wave
number, m is the particle mass, and m is the cosine of its pitch
angle. These waves, in turn, scatter particles in pitch angle(at the rate, n ~ W (mc/p) (d B/B0)2, where c is the speed oflight) back and force so that they can gain energy by repeatedly crossing the shock. Particle self-confinement along the fieldis thus diffusive and the diffusivity k ~ c2/n is inversely
proportional to the fluctuation energy dB2, as the fluctuating
field is responsible for pitch-angle scattering. However, themean field B0 ultimately determines the acceleration rate andthe particle maximum energy since it sets the work done by
the induced electric field on the particles. Indeed, Emax ~ (e/c)usB0Rs, where us and Rs are the speed and typical size ofthe shock wave, such as the radius of the SNR shock. The
fluctuating part, d B, is typically assumed to be saturated atmost at the level d B ~ B0, which provides pitch-angle
scattering at the rate of gyrofrequency, and thus limits theparticle mean free path (m.f.p.) along the field to a distanceof the order of gyroradius (the so called Bohm diffusion
limit). An important thing to keep in mind is that, due to theresonance condition kp = const, confinement of particles ofhigher energies requires that longer waves need be excited.
The most critical test of this mechanism is the requirement that it accelerates galactic CRs to the energy of 1015 eVover the life-time of supernova remnant shocks. Even withthe above “optimistic” estimates of the turbulence level, the(dB / B0 )2 ~ M A Pc / rus2

(1)

where MA = us/VA >> 1is the Alfven Mach number and ru2s isthe shock ram pressure. Of course, when d B/B0 exceeds unity,
particle dynamics, and thus their confinement and acceleration
rates, are very difficult to assess if the turbulence spectrum issufficiently broad. The numerical studies by [11] showed thatat least in the case of an MHD (magnetohydrodynamic)
description of the background plasma and rather narrow wave
(and particle energy) band, the amplitude of the principal
mode can reach a few times that of the background field.
Moreover, the authors of Ref. [13] argue that in the case of
efficient acceleration, field amplification may be evenstronger, reaching a mG (10 –3 Gauss) level from thebackground of a few mG (10 –6 Gauss) ISM field, thus
providing acceleration of protons up to 1017 eV in SNRs.
Recently, the authors of Ref. [14] approached thisproblem from a different perspective. They considered a
Kolmogorov turbulent cascade to small scales assuming thewaves are generated by efficiently accelerated particles on thelong-wave part of the spectrum. They obtained a particle
maximum energy similar to that of [13].
Apart from the excitation of magnetic fluctuations during
acceleration process, there is yet another aspect of the CRmagnetic field connection discussed in the literature. Zweibel[15] points out that since CRs were already present in young
galaxies (observed at high redshifts), magnetic fields ofappreciable strength must also have been there at that time.
She emphasizes, however, that the approximate equipartition
between the CR and magnetic field energy established by thecurrent epoch in our Galaxy is not required for the acceleration mechanism and presumably results from the fact that theyboth have a common energy source, namely the supernovae.
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Diamond P.H. and Malkov, M.A., Generation of Mesoscale Magnetic Fields and the Dynamics of Cosmic Ray Acceleration

are the flow profile U (x) and the particle diffusivity k (x, p).
The first one is coupled to the particle distribution f throughthe equations of mass and momentum conservation

Indeed, as we discussed already, the magnetic field strength
merely determines the maximum energy of accelerated
particles, given the time available for acceleration and the sizeof the accelerator. If the latter are sufficient then the totalenergy of accelerated particles can, and in most of the DSA
models does, exceed that of the magnetic field. The latter, inturn, remains unchanged, apart from the conventional
compression at the shock and the MHD fluctuations discussed
earlier.
In this paper we discuss the possibility of a different
scenario, in which the magnetic field may absorb a significant
part of the shock energy as a result of the acceleration
process, which may in fact be strongly enhanced. The
mechanism of such enhancement is based on the transfer ofmagnetic energy to longer scales, which we call inverse
cascade for short, even though specific mechanisms of suchtransfer may differ from what is usually understood as acascade in MHD turbulence. This transfer is limited only bysome outer scale Lout such as the shock precursor size k (pmax)/
us ~ rg(pmax) c/us >> rg(pmax). This approach is in contrast to
the above discussed models [13,14], which operate on thegenerated magnetic fields with the scale lengths of the orderof the Larmor radius rg(pmax) of the highest energy particles
and smaller. The advantage of the inverse cascade for theacceleration is that the turbulent field at the outer scaled B(Lout)  Brms (which necessarily must have long autocorrelation time) can be obviously regarded as an “ambient field”
for accelerated particles of all energies. If Brms >> B0, then theacceleration can be enhanced by a factor Brms /B0. Note thatthe resonance field d B(rg) may remain smaller than Brms, sothat standard arguments about Bohm diffusion apply, and it
is less likely that this field will be rapidly dissipated by
nonlinear processes, such as induced scattering on thermal
protons, not included in the enhanced acceleration model[11,13].
As it should be clear from the above, an adequate
description of the acceleration mechanism must include bothparticle and wave dynamics on an equal footing. In fact thesituation is even more difficult, since the acceleration process
turns out to be so efficient that the pressure of accelerated
particles markedly modifies the structure of the shock (boththe overall shock compression and the flow profile).

(3)



rU + ÊË rU 2 + Pc + Pg ˆ¯ = 0
t
x

(4)

where

Pc ( x ) =

•
4p
mc 2
pinj
3

Ú

p 4 dpp2 + 1f ( p, x )(5)is the pressure of the CR gas and Pg is the thermal gas
pressure. The lower boundary pinj in momentum space
separates CRs from the thermal plasma that is not directly
involved in this formalism but rather enters the equations
through the magnitude of g at p = pinj, which specifies theinjection rate of thermal plasma into the acceleration process.
The particle momentum p is normalized to mc. The spatial
diffusivity k, induced by pitch angle scattering, prevents
particle streaming away from the shock, thus facilitating
acceleration by ensuring the particle completes several shock
crossings.
The system (2-5) indicate a marked departure from thetest particle theory. Perhaps the most striking result of thenonlinear treatment is the bifurcation of shock structure (inparticular shock compression ratio) in the parameter space
formed by the injection rate, shock Mach number and particle
maximum momentum [16].

4. Wave dynamics in the CR shock precursor
The transformation of magnetic energy to longer scales,
while bearing certain characteristics of the conventional
turbulent dynamo problem, is still rather different from it, in
its conventional form. First, this process should take place inthe strongly compressible fluid near the shock. Second, theAlfven wave turbulence is generated by accelerated particles
via Cerenkov emission, and thus is strongly coupled to them.
Third, the shock precursor itself is unstable to emission of
acoustic waves. The latter phenomenon is known as the Drury
instability and will be discussed later. Acoustic waves, in turn,interact with particle generated Alfvenic turbulence,
stimulating decay instability (i.e., “inverse cascade”).
The spatial structure of an efficiently accelerating shock,i.e., the shock that transforms a significant part of its energy
into accelerated particles, is very different from that of theordinary shock, Fig. 2. Most of the shock structure consistsof a precursor formed by accelerated CRs diffusing ahead of
the shock. If the CR diffusivitity k (p) depends linearly on
particle momentum p (as in the Bohm diffusion case), then,at least well inside the precursor, the velocity profile U (x) is
approximately a linear function of x, where x points along
the shock normal [17]. Ahead of the shock precursor, the flow

3. Accelerated particles and plasma flownear the shock front
The transport and acceleration of high energy particles(CRs) near a CR modified shock is usually described by the
diffusion-convection equation. It is convenient to use a distribution function f (p) normalized to p2dp.

f
f
 f 1 U f
=
p
+U
- k
t
x x x 3 x p
r + rU = 0
t
x

(2)

Here x is directed along the shock normal, which for simplicity, is assumed to be the direction of the ambient magnetic
field. The two quantities that control the acceleration process
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Diamond P.H. and Malkov, M.A., Generation of Mesoscale Magnetic Fields and the Dynamics of Cosmic Ray Acceleration

Fig. 2

Schematic representation of nonlinearly accelerating shock. Flow profile with a gradualdeceleration upstream is also shown at the top.

velocity tends to its upstream value, U1, while on the downstream side it undergoes a conventional plasma shock transition to its downstream value U2 (all velocities are taken in
the shock frame). This extended CR precursor (of the size
LCR ~ k (pmax)/U1) is the place where we expect turbulence isgenerated by the CR streaming instability and where it
cascades to longer wavelengths.

parametric phenomena [18]. We will concentrate on the
acoustic type perturbations (which may be induced by Drury
instability), so that we can write for the density r and velocity
Ur = r 0 + r˜ ; U = U0 + U˜
The variation of the Alfven velocity ṼA = VA – VA0

4.1 Alfvenic turbulence
The growth rate of the ion-cyclotron instability is
positive for the Alfven waves traveling in the CR streaming
direction i.e., upstream, and it is negative for oppositely propagating waves. The wave kinetic equation for both types ofwaves can be written in the form

For simplicity, we assume that the plasma b < 1 (which is
not universally true in the shock environment) and neglect
the variation of U compared to that of VA in eq. (6). The above
perturbations of VA in turn induce perturbations of Nk±, so we
can write

N k± w ± N k± w ± N k±
+
t
k x
x k
= g k± N k± + C ± ÏÌÓ N k+ , N k- ¸N k± = N k± + N˜ k±(6)
Our goal is to obtain an evolution equation for the averaged
number of plasmons ·Nk±Ò. Averaging eq. (7) we have

Here Nk± denotes the number of quanta propagating in the±

upstream and downstream directions, respectively. Also, w
are their frequencies, w ± = kU ± kVA, where VA is the Alfven
velocity. The linear growth rates g ± are nonzero only in theresonant part of the spectrum, krg (pmax) > 1. In the mostgeneral case, the last term on the r.h.s. represents nonlinear–
interaction of different types of quanta Nk+ and Nk and, if compressibility effects are present, also interactions between thesame type. As seen from this equation, the coefficients in the
wave transport part of this equation (l.h.s.) depend on theparameters of the medium through U and VA, which in turn,
may be subjected to perturbations. This usually results inN k± + (U ± VA )N k± - kU x
N k±
k
x
t
˜
r

+
kVA x N˜ k± = g k± N k± + C ÊË N k± ˆ¯
k
r0

(7)

Here the index x stands for the x-derivatives. To calculate the
˜correlator rr0x N˜ k± in the last equation, we expand the r.h.s.
of eq. (6) retaining only the main linear part in Ñ

431

Diamond P.H. and Malkov, M.A., Generation of Mesoscale Magnetic Fields and the Dynamics of Cosmic Ray Acceleration

2

(8)

Dk =

The time scale separation between the l.h.s. and r.h.s. of eq.
(6) suggests that to lowest order, the linear growth g + rate is+approximately balanced by the local nonlinear term C . Likewise, the linear damping of the backward waves g – may bebalanced by their nonlinear growth and conversion of the–forward waves C . Generally, the Dwk± in eq. (8) is a 2 ¥ 2matrix operator. If the wave collision term is quadratic in N,
then Dwk± @ g k±. This balance can be established only for the
resonant waves (g ±  0), whereas our primary focus will beon the extended longwave interval k < 1/rg(pmax) for which gª 0. In this domain, cascading from the generation region k >1/rg(pmax) takes place and the refraction (last) term on the l.h.s.
of eq. (7) plays a dominant role along, with the nonlinear
term on the r.h.s.
To calculate the refraction term we write eq. (6), linearized with respect to Ñk±, as:

L± N˜ k± = - kVAr̃ x 
N k±
2 r 0 k

rq
1 2 2
k VA
q2
¬L-k 1,q
r
2
0
q

Â

(13)

The last (wave refraction) term on the r.h.s. of eq. (12) canbe estimated as U 21/k (pmax), which is the inverse acceleration
time and can be neglected as compared to the frequencies±–1
for we have:qVA and Dw. Hence, ¬Lk,q
1¬L±k ,q ª

Dw k±
q 2 VA2 + Dw k±2

For further convenience, we introduce here the number ofphonons

Nqs =

Wq

w qs

where Wq is the energy density of acoustic waves (with wqs =qCs as their frequency).

Wq = Cs2

(9)

where

r q2
r0

For Dk in eq. (10) we thus finally have
L = + (U ± VA ) - kU x
+ Dw k±
t
x
k
±

Dk =

Solving eq. (9) for Ñk±, from eq. (7) we thus have the following
equation for ·Ñk±ÒN k± + U
N k± - kU x
N k±
t
x
k


- D
N k± = g k± N k± + C ÊË N k± ˆ¯k k

1 2 2 r˜ x -1 r˜ x
k VA
L
r0
r0
4

(10)

Âr e

q

Dw k±
Nqs
q 2 VA2 + Dw k±2

4.2 Acoustic turbulence
Unlike the Alfvenic turbulence that originates in theshock precursor from accelerated particles, there are two
separate sources of acoustic perturbations. One is related toparametric [18] processes undergone by the Alfven waves in
the usual form of a decay of an Alfven wave into another
Alfven wave and an acoustic wave. The other source is thepressure gradient of CRs, which directly drives instability.
The latter leads to emission of sound waves due to the Drury
instability. By analogy with eq. (7) we can write the following
wave kinetic equation for the acoustic waves:(11)

Dk is an example of the well-known phenomenon of induced
diffusion. Transforming to Fourier space, we first represent r̃
as

r̃ =

Â

q 2w qs

Note that Dk represents the rate at which the wave vector of
the Alfven wave random walks due to stochastic refraction.
Of course, such a random walk necessarily must generatelarger scales (smaller k), thus in turn facilitating the confinement (to the shock) of higher energy particles. Thus, confinement of higher energy particles is a natural consequence of
Alfven wave refraction in acoustic wave generated density
perturbations.

Here we introduced a diffusion operator for the Alfven waves
in k space due to random refraction by the acoustic perturbations r̃ (via the density dependence of VA), i.e.,Dk =k 2 VA2
2Cs2 r 0

iqx - iW q t

q

q




Nq + UNq - qU x
Nq
t
x
k

and note that due to the local Galilean invariance of L, we
can calculate its Fourier representation in the reference frame
moving with the plasma at the speed U (x) as:

L±k ,q = ± iqVA + Dw k± - kU xk= ÊË g qd + g D ˆ¯ Nq + C ÏÌÓ Nq ¸Here g D is the Drury instability growth rate and g qd is that ofthe decay instability. We first consider the decay instabilityof Alfven waves. Note, however, that the combination ofDrury instability and decay instability can lead to generation(12)

Then, eq. (11) can be re-written as:

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Diamond P.H. and Malkov, M.A., Generation of Mesoscale Magnetic Fields and the Dynamics of Cosmic Ray Acceleration

accelerating shock g C ª 4/3. Note that we have omitted a term
–Ux which is related to a simple compression of wave number
density by the flow and should be generally incorporated into
the r.h.s. of eq. (14) (see [19]). This term is smaller by a factor
Cs /U than the second (destabilizing) term. The first term is
damping caused by CR diffusion, calculated earlier by Ptuskin[23].

of mesoscale fields at a faster than – exponential rate, bycoupling together the Drury and decay instability processes.

4.2.1 Decay instability
The mechanism of this instability is the growth of thedensity (acoustic) perturbations due to the action of theponderomotive force from the Alfven waves. This force canbe regarded as a radiative pressure term appearing in thehydrodynamic equation of motion for the sound waves(written below in the comoving plasma frame)

5. Mechanisms of transfer of magnetic
energy to larger scalesV
1  Ê 2
=Cs r̃ + Prad ˆ¯tr 0 x Ë

As it follows from the above considerations, there are avariety of nonlinear processes that can lead to the transfer
of magnetic energy (generated by accelerated particles inform of the resonant Alfven waves) to longer scales. First,
as it can be seen from eq. (10) (last term on the l.h.s.)scattering of the Alfven waves in k-space due to acoustic
perturbations transfers magnetic fluctuations away from theresonant excitation region to smaller (but also to larger) k.
Second, the nonlinear interaction of Alfven waves and magnetosonic waves represented by the wave collision term on
the r.h.s. can be responsible for such process. It is also wellknown that in the presence of nonzero magnetic helicity
there is a strong inverse cascade of magnetic energy [24,25].
Finally, even in the frame work of the weak turbulence,induced scattering of Alfven waves on thermal protons leads
to a systematic decrease in the energy of quanta which, given
the dispersion law, means again transformation to longer
waves.
Returning to the wave refraction process on acoustic perturbations generated by the Drury instability, it is important
to emphasize the following. As is seen from the instability
growth rate, eq. (15), it is proportional to the gradient of Pc.
As the latter should be increased as a result of instability,
through a better confinement and faster acceleration, this willreinforce the instability, possibly triggering “explosive”
growth. This can significantly contribute to mechanisms ofregulation of Pc discussed earlier in [16].

Eliminating velocity by making use of continuity equation

r̃
V
+ r0
= 0,
t
x
we obtain

 2 r˜2 Ê 2
Cs r˜ + Prad ˆ¯2 =tx 2 Ë

(14)

The Alfven wave pressure can be expressed through theirenergyPrad =

Â w ( N˜ + N˜ )
+
k

k

k

k

Using the relation (9) between the density perturbations andthe Alfven waves and separating forward and backward
propagating sound waves r± , we can obtain from eq. (14) the
following dispersion relation for the acoustic branch

w 2 - q 2 Cs2 = q 2

wkÂ 2r iqkV L±-1
A k ,q

0

k


N k±
k

or on writing w = ± qCs + ig ±, we have the following growthrate of acoustic perturbationsg± =q 2 VA4 r 0 Cs

Â kw L±-1k k ,q

k


N k±
k

6. Conclusions and discussion
We have considered a number of possible mechanisms
for generation of large scale magnetic field in front of strong
astrophysical shocks. All these mechanisms are immediate
results of the particle acceleration process. Such generationis necessary to further accelerate particles well beyond the“knee” energy at 1015 eV, as is suggested by observations of
the CR background spectrum and the wide consensus on their
SNR shock origin. The fact that accelerated CRs constitutean ample reservoir for the turbulence which is required tofurther accelerate them provides a logical basis for ourapproach. Indeed, the scenario proposed here may be viewed
as a self-regulating enhanced acceleration process, which
ultimately forces the energetic particle pressure gradient to
its marginal point for Drury instability.
A new description of the instability of Alfven wave
spectrum to acoustic modulations is given. Along with theDrury instability this is shown to provide an efficient

Note that the instability requires an inverted population of
Alfven quanta. As they are generated by high energy resonantparticles in a finite domain of k space, such an inversion
clearly can occur.

4.2.2 Drury instability
This instability also leads to efficient generation of
acoustic waves and it is driven by the pressure gradient of
the CRs in the shock precursor. The growth rate has been
calculated in Ref. [19] (see also [20-22]), and can be written
in the form:g D± = -g C PC PCx Ê lnk ˆ
±
Á1 +
˜
rk
Cs r Ë
lnr ¯(15)

Here PC and PCx are the CR pressure and its derivative,
respectively, and g C is their adiabatic index. For an efficiently
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Diamond P.H. and Malkov, M.A., Generation of Mesoscale Magnetic Fields and the Dynamics of Cosmic Ray Acceleration

mechanism for transformation of magnetic energy to longer
scales. It should be also emphasized that the theoreticalanalysis of CR acceleration is a challenging problem inplasma wave physics, particle kinetics and shock hydrodynamics. It should, and indeed must, include a self-consistentdescription of particle transport, wave dynamics, and shock
structure.

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429 (2001).
[18] R.Z. Sagdeev and A.A. Galeev, Nonlinear Plasma
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Acknowledgements
This work was supported by the U.S. DOE under Grant
No. FG03-88ER53275.

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