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The Astrophysical Journal, 626:1116–1130, 2005 June 20 # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

NONLINEAR ENERGETIC CHARGED PARTICLE TRANSPORT AND ENERGIZATION IN ENHANCED COMPRESSIVE WAVE TURBULENCE NEAR SHOCKS J. A. le Roux, G. P. Zank, G. Li, and G. M. Webb Institute of Geophysics and Planetary Physics, University of California, Riverside, CA 92521; [email protected] Received 2005 January 28; accepted 2005 March 4

ABSTRACT During solar maximum, interplanetary coronal mass ejections and associated interplanetary shocks occur frequently. These structures are often accompanied by high levels of low-frequency compressive wave turbulence, which requires a nonlinear extension of standard quasi-linear theory to properly describe energetic particle transport in their vicinities. The same might be true for the solar wind termination shock. We present a nonlinear diffusive kinetic theory for suprathermal particle transport and stochastic acceleration along the background magnetic field in strong compressive dynamic wave turbulence to which small-scale Alfve´n waves are coupled. Our theory shows that the standard cosmic-ray transport equation must be revised for low suprathermal particle energies to accommodate fundamental changes in spatial diffusion (standard diffusion becomes turbulent diffusion), as well as modifications to particle convection and adiabatic energy changes. In addition, a momentum diffusion term, which generates accelerated suprathermal particle spectra with a hard power law, must be added. Such effective first-stage acceleration possibly leads to efficient injection of particles into second-stage diffusive shock acceleration, as described by standard theory. Subject headings: acceleration of particles — interplanetary medium — scattering — shock waves — solar wind — turbulence 1. INTRODUCTION

that both pickup ions preaccelerated at the termination shock and anomalous cosmic rays formed by diffusive shock acceleration of pickup ions at the termination shock might contain enough pressure to significantly mediate the termination shock structure (le Roux et al. 2000a, 2000b). The cosmic-ray pressure gradient is expected to drive ion-acoustic instabilities (Zank & McKenzie 1987), creating a highly turbulent shock (Zank et al. 2004). For corotating interaction region shocks, compressive fluctuations are observed to be important (Crooker et al. 1999). This might well be the case for ICMEs and ICME-driven shocks, since the thermal electron-to-proton temperature ratio Te /Tp > 1, indicating the suppression of Landau damping. For example, a part of ICME structures is associated with proton temperature decreases relative to the electron temperature so that the ratio Te /Tp 3 1 (Richardson & Cane 1995). In the case of magnetic clouds, which are associated with as many as one-third of all ICMEs, a large temperature ratio, Te /Tp  10 20, was observed by Fainberg et al. (1996) inside such a structure. They also noted strong wave activity inside the cloud that is consistent with presence of ionacoustic wave turbulence. In addition, arguments have been presented by Gurnett et al. (1979) for the presence of ion-acoustic wave activity at an oblique interplanetary shock observed with a large temperature ratio, Te /Tp  7. These observations are consistent with considerably enhanced MHD wave turbulence levels that have been identified by the ACE spacecraft in the solar wind near Earth, upstream and downstream of the main oblique interplanetary shock of the Bastille day CME event (Bamert et al. 2004). The large temperature ratios observed suggest, according to kinetic Vlasov theory, that ion-acoustic waves propagate preferentially along the background magnetic field (Gary 1993, p. 101) inside ICMEs and near ICME shocks. Gary (1993) based his results on a Maxwellian distribution for ions and electrons. Note, however, that it has been shown that even a slightly nonMaxwellian plasma with a slight tail on the ion distribution can lead to weak Landau damping of ion-acoustic waves, even when Te /Tp ¼ 1 (Skiff et al. 2002). From these examples, it is clear that one needs to consider both Alfve´n waves and compressive

According to solar wind observations, the (proton) density fluctuations in the solar wind are typically small. A detailed study using high-resolution Voyager data over a 4 year period that includes times of enhanced solar activity shows that most density fluctuations have an amplitude of

/0  0:1 (Matthaeus et al. 1991). Thus, most of the time, the solar wind is nearly incompressible, and compressive wave turbulence in the solar wind is weak at best. This leads to an emphasis on particle scattering by incompressible Alfve´n waves as the standard way of studying energetic particle diffusion along the magnetic field. However, the Matthaeus et al. (1991) solar wind observations also showed that large density fluctuations are present in the solar wind, indicating that there are relatively short periods during the solar cycle when the solar wind becomes highly compressive. Large heliospheric shock waves are almost certainly associated with such high levels of compressive turbulence, as well as strong levels of incompressible magnetic turbulence. For instance, the rate of interplanetary coronal mass ejections (ICMEs) and ICME-driven shocks increases substantially from solar minimum (about one per month) to solar maximum activity (about 1–2 per week) (Richardson & Cane 1995). In addition, the propagation speed of the ICMEs tends to increase with solar activity, so one can expect more and stronger interplanetary shocks during solar maximum, accompanied by stronger density fluctuations. It often happens that a series of closely spaced fast corotating and transient streams (e.g., ICMEs) occur at 1 AU with a duration of
1–3 solar rotation periods characterized by strong magnitude fluctuations in the magnetic field (Burlaga et al. 2003). This suggests that large radial distance intervals on the order of about 6–19 AU in the solar wind might be highly compressive and turbulent. Such systems evolve further out to form turbulent large-scale global merged interaction regions at about 15 AU from the Sun (Burlaga et al. 2003). A second example is the heliospheric termination shock. Some self-consistent kinetic cosmic-ray models suggest the possibility 1116

PARTICLE TRANSPORT IN COMPRESSIVE WAVES fluctuations when studying energetic particle transport in environments of enhanced turbulence near large-scale heliospheric shock waves. 2. MOTIVATION It is generally accepted that standard quasi-linear theory (QLT) is adequate for describing energetic particle transport in the incoherent component of MHD turbulence along the magnetic field (see, for example, Giacalone & Jokipii [1999]) during solar minimum. Given the incompressible nature of the quiet solar wind, the standard approach is to consider diffusive particle transport in terms of gyroresonant particle interactions with Alfve´n wave turbulence propagating along the magnetic field (Bieber et al. 1994). The validity of QLT requires that the parallel mean free path k 3 lc , where lc is the correlation scale of the turbulence (lc  0:024 AU at Earth; Bieber et al. 1995), because of the theoretical assumption of undisturbed gyro-orbits on a spatial scale lc. The parallel mean free path k of suprathermal energetic particles has been determined to vary in the quiet solar wind from 0.02 to 0.5 AU (Dro¨ge 2000), and a typical consensus value is 0.1 AU (Palmer 1982). This indicates that on average lc Tk, so QLT is applicable in the quiet solar wind. However, inside ICMEs, near ICME-driven shocks, and at the termination shock where we expect the incoherent component of MHD turbulence to be enhanced, lc and k might change strongly. Unfortunately, it is not well known, either theoretically or from observations, how lc and k change in these regions. Using incompressible MHD turbulence transport theory as a guide (Zank et al. 1996), it seems that near Earth one can specify the correlation length for Alfve´n wave turbulence as l VcA / 1/A
B VA  1/
UVA for Alfve´n wave turbulence, where A
B ¼
B/B0 , where
B is the rms of the Alfve´nic magnetic field fluctuations,
UVA is the rms of the associated fluid velocity fluctuations, B0 is the magnitude of the background magnetic field, and vA ¼ B0 /(40 )1= 2 is the Alfve´n speed for dispersionless Alfve´n waves, with 0 being the large-scale mass density of the solar wind. Assuming that the correlation scale expression for Alfve´n waves can also be applied to compressive velocity fluctuations, it follows that for
UVs , lcVs / 1/A
U Vs ¼ 1/
UVs , where A
U ¼
U /Vs and Vs ¼ ½(KTe þ p KTp )/M1= 2 is the phase speed of lowfrequency ion-acoustic waves, where Te is the electron temperature, Tp is the proton temperature, K is the Boltzmann constant, p ¼ 5/3 for an adiabatic thermal proton solar wind component, and M is the proton mass. Since VA  Vs near Earth and A
U  A
B , it appears that lcVA  lcVs at 1 AU during solar minimum conditions. By using the expression for lcVA , the QLT parallel mean free path for particles experiencing gyroresonant interactions with the small-scale inertial range Alfve´n waves (Schlickeiser = 1989) can be expressed as k / rg1= 3 /A8
B3 vA2= 3 , where rg is the particle gyroradius. Assuming that A
B  A
U  0:2 during solar minimum but increases to A
B  A
U  1 inside ICMEs or near ICME-driven shocks, we find that while lcVA  lcVs decreases by a factor of at least 5, k decreases much more, by a factor of at least
73. This would mean that lcVs  5 ; 103 AU, while k  1:3 ; 103 AU, so that kTlcVs . This estimate indicates that the conditions for QLT might not be fulfilled in the enhanced turbulence levels found inside ICMEs and near ICME-driven shocks. In this case QLT needs to be extended to a nonlinear level to deal with particle trajectories that may be diffusive on a turbulence correlation scale instead of being undisturbed gyro-orbits. In response, we extend QLT to develop a nonlinear diffusive theory for energetic charged particle transport along B0 in strong compressive wave turbulence (
U  Vs ) with a correlation scale lc. In this turbulent compressive medium, weak small-scale

1117

Alfve´n wave turbulence, representing that part of the inertial range of Alfve´n waves with which energetic particles have gyroresonant interactions, is assumed to be imbedded in and modified by the large-scale density gradients associated with the compressive wave turbulence. As is discussed in detail below, our nonlinear diffusive kinetic theory shows that the standard cosmic-ray transport equation must be revised for low suprathermal particle energies to accommodate fundamental changes in spatial diffusion, as well as modifications to particle convection and adiabatic energy changes. In addition, a momentum diffusion term, which generates accelerated suprathermal particle spectra with a hard power law, must be added. The spectra are qualitatively consistent with solar wind observations of suprathermal ions near ICME shocks (Gloeckler 2003). Such efficient acceleration can possibly serve as a first stage of an acceleration process that leads to the efficient injection of particles into second-stage diffusive shock acceleration, as described by standard theory. 3. PARTICLE TRANSPORT THEORY IN A TURBULENT FLUID In attacking the problem of test particle transport in a turbulent fluid with random fluid velocity fluctuations, the starting point is usually a diffusion-convection equation that, in one spatial dimension in the observer frame, is given by
 @n @ @ @n þ (nU ) ¼  ; @t @z @z @z

ð1Þ

where n (z, t) is the particle density at position z and time t, U is the fluid velocity that contains random fluctuations on all scales less than the turbulence correlation scale lc , and  (z; t) ¼ 13 vk is the small-scale (microscopic) spatial diffusion coefficient with mean free path kTlc . In the case of a collisionless MHD plasma fluid such as the solar wind, one can think of the particles as energetic charged particles, such as cosmic rays or pickup ions, with a microscopic spatial diffusion coefficient determined by resonant wave-particle interactions with small gyroradius-scale, small-amplitude Alfve´n waves described by standard QLT, for example. Let us assume for simplicity that the fluid velocity fluctuations are incompressible. By explicitly separating the plasma flow velocity in equation (1) into a coherent part hU i and a random fluctuating part
U and transforming to the large-scale fluid frame moving with constant speed hU i relative to the observer frame, it follows that U ¼
U , because hU i ¼ 0 and h
U i ¼ 0. The corresponding random density fluctuations are introduced according to
n ¼ hni þ
n, with h
ni ¼ 0. Now one can, by using standard perturbation analysis, derive a new diffusion-convection equation for average density hni that is valid over a large spatial scale L 3 lc . In the limit that  is negligibly small when averaged over large scales (hi  0), the new effective diffusion coefficient is given by a two-point, two-time correlation function of the fluid velocity fluctuations as seen by an observer in the fluid frame (Lagrangian fluid velocity fluctuations) and is given by Z t ¼

1

d
t h
U (z; t)
U ½z 
z(
t); t 
ti:

ð2Þ

0

When assuming a fast exponential decay of the two-point, twotime correlation time so that   h
U (z; t)
U ½z 
z(
t); t 
ti ¼
U 2 e
t=tc ; ð3Þ

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LE ROUX ET AL.

it follows that   t ¼
U 2 (z; t) tc ;

ð4Þ

where tc ¼ lc /(h
U 2 i)1 2 is the turbulence correlation time. The expressions in equations (2) and (4) represent classical Lagrangian turbulent diffusion (see, e.g., Taylor [1921] and Frisch [1989]). As described above, turbulent diffusion of energetic particles follows when hi is assumed to be negligibly small. This implies that the particles should essentially be tied to the background plasma flow through their scattering by small-scale Alfve´n wave– scattering centers embedded in the flow before turbulent diffusion can occur effectively. The scattering centers are embedded in the turbulent flow and are thus convected in a random fashion by the turbulent flow. Because the particles are so strongly coupled to the small-scale scattering centers, they thus act in concert with the small-scale scattering centers and thus are passively convected with the turbulent flow, as described by the turbulent diffusion coefficient. For a more detailed discussion of this point, see Frisch (1989). The microscopic diffusion coefficient on large scales can be small if one considers energetic particles at relatively low energies ( / v4= 3 according to QLT, when particles are interacting with the inertial range of the power spectrum of Alfve´n waves with a Kolmogorov power-law slope). It can also be small because in a nonuniform compressive medium the energy density in Alfve´n waves EA /  , where  is the mass density of the plasma flow. According to standard WKB theory (Hollweg 1974),  ¼ 3/2 when hU i 3 VA , which is appropriate in the superAlfve´nic solar wind, or  ¼ 1/2 when hU iTVA in the subAlfve´nic solar wind. Because in QLT  / 1/EA /  , we see that  is reduced inside compressive regions (enhanced particlescattering rate on Alfve´n waves) but enhanced inside rarefaction regions (reduced scattering rate). However, because particles spend more time in compression regions, where they tend to be trapped, hi is reduced relative to a medium without compressive fluctuations. This reduction could be strong when we have strong compressive fluctuations such as those near large heliospheric shock waves. Near shocks,  can also be reduced strongly, as particles streaming away from the shock produce enhanced levels of ion-cyclotron waves or as wave turbulence gets compressed across the shock ( Lee 1983; Zank et al. 2003). To address the question of how particle energies are affected by their interaction with compressive random fluid velocity fluctuations, one needs a kinetic diffusion-convection equation. In this case the standard cosmic-ray transport equation for near-isotropic particle distributions is commonly used as the initial equation (Toptygin 1985, p. 202; Dolginov & Silantev 1990; Bykov & Toptygin 1993; Webb et al. 2003; Li et al. 2004). This equation in conservation form is given by

 @f @ 1 @ p3 @f @ @f þ (CUf ) þ 2 U ¼  ; ð5Þ @t @z p @p 3 @z @z @z =

where f (z, p, t), a function of momentum p, is the directionaveraged particle distribution function; (z, p, t) is the microscopic spatial diffusion coefficient; and C¼

1 1 @f 3 f @p

ð6Þ

is the Compton-Getting factor for the transformation of a differential current density from the fluid to the observer frame.

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In the case of compressive fluid velocity fluctuations, which is the main focus of this paper, it is useful to write the cosmicray transport equation in the following familiar form, in which the divergence (compressibility) of the fluid velocity field is explicitly tied to particle energy changes:
 @f @f p @U @f @ @f þU  ¼  : ð7Þ @t @z 3 @z @p @z @z By taking the limit that  is small, Toptygin (1985) showed that, in addition to the turbulent diffusion coefficient in equation (4), one can also derive a momentum diffusion coefficient, which for one-dimensional plasma velocity fluctuations is given by   Z 1 @
U @
U (z; t) ½z 
z(
t); t 
t d
t Dpp  p2 @z @z 0 1  p2 ; ð8Þ tc assuming a fast decay of the velocity correlation function. This shows how energetic particles gain energy from the compressive plasma velocity fluctuations while undergoing turbulent diffusion. For further work in this direction, see Dolginov & Silantev (1990), Webb et al. (2003), and Li et al. (2004). 4. UNDERLYING MODEL Our starting point is the more basic Boltzmann equation given by
 0 @f 0 p 0 @f 0 @f 0 @f þ = þ FL = 0 ¼ ; ð9Þ @t sc @t m @x @p which describes the evolution of an energetic charged test particle distribution f 0 ¼ f (x; p 0 ; t) in the observer-frame as a function of the observer frame independent variables position x, momentum p0 , and time t, where m is the particle mass. Particles with net charge q are assumed to be scattered by small-scale MHD fluctuations, which are described by the scattering term on the right-hand side of equation (9). This leads to microscopic particle diffusion. When assuming these fluctuations to be those Alfve´n waves propagating along the magnetic field with which the particles have gyroresonant interactions, the wavelengths of the Alfve´n waves are l  rg , where rg is the particle gyroradius. The microscopically diffusing particles are assumed to encounter random variations in the electromagnetic field with a characteristic scale given by the turbulence correlation scale lc so that lc 3 rg  l. The effect of the electromagnetic field on the particles is described by the Lorentz force FL0 , given by
 p0 < B 0 0 FL (x; p ; t) ¼ q E þ ; ð10Þ m where E is the electric field and B is the magnetic field. In the large-scale solar wind plasma, on scale lengths on the order of the turbulence correlation length lc and larger, away from special discontinuous surfaces such as current sheets or shocks, it can be shown that the most important contribution to the total electric field is from the motional electric field. We therefore specify E ¼ U < B:

ð11Þ

To enable an explicit description of how compressive plasma flow velocity fluctuations with a correlation scale lc associated with ion-acoustic waves affect microscopically diffusing energetic

No. 2, 2005

charged particles, we introduce the plasma or flow velocity U in equation (9). This is achieved by transforming the particle momentum in the observer frame p0 in terms of the momentum p valid in the local plasma fluid frame according to the nonrelativistic transformation [(U /c)2 T1], p 0 ¼ p þ mU:

ð12Þ

The variables r and time t are not transformed. It then follows in the nonrelativistic limit that the distribution function f 0 is approximately invariant when doing the momentum transformation (Gleeson & Webb 1980), so that f (x; p 0 ; t) ¼ f (x; p; t):

ð13Þ

By making use of equations (11)–(13), equation (9) can be transformed, in terms of the mixed-frame variables, to



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PARTICLE TRANSPORT IN COMPRESSIVE WAVES



@f p @f þ þU = @t m @x þ FL  m





 dU @ @f @f  p= ¼ ; = dt @x @p @t sc

ð14Þ

where f is a function of x, p, and t. The Lorentz force simplifies to FL (x; p; t) ¼

q p < B: m

ð15Þ

To simplify the analysis, we make several assumptions and do the following: (1) In a Cartesian coordinate system (x, y, z), the magnetic field is directed along the z-axis so that B ¼ B(x; y; z; t) eB . The Cartesian coordinates ( px , py , pz) of the particle momentum p are measured so that pz is the momentum component along B. Expressed in terms of spherical coordinates, the momentum components become ( p sin  cos , p sin  sin , p cos ), where p is the magnitude of the particle momentum,  is the particle pitch angle, and  is the particle phase angle. (2) The distribution function f is independent of particle phase angle ; hence, @f /@ ¼ 0. This means that cross-field diffusion and drifts of particles are neglected, so only magnetic field–aligned particle transport is considered, and furthermore implies that the particle gyroradius rg Tlc . For typical solar wind parameters at 1 AU (lc ¼ 0:024 AU, B0 ¼ 5 nT), the derivation is valid for energetic particles with rigidity RT2:5 GV in the solar wind, approximately. However, evidence has been presented for largescale cross-field diffusion (mainly because of large-scale turbulent magnetic fields) and for the importance of drifts in the heliosphere (Giacalone & Jokipii 1999) in this rigidity interval, which implies that the theory might be applicable to lower energies than those suggested above. Since we focus on particle transport parallel to the magnetic field, the results should be valid for a wider range of particle energies. (3) Equation (14) is averaged over the particle phase angle . The simplified transport equation is (Skilling 1971; Isenberg 1997)



@f @f 1  2 @Ui 1  3 2 @Ui þ (Ui þ v ) bi bj þ  þ @t @xi 2 @xi 2 @xj





dUi 1 @f 1  2 @bi @Ui   m bi p v þ @p dt p 2 @xi @xi 
 @Ui m dUi @f @f ¼  3 bi bj  2 bi ; p @t sc @xj dt @

ð16Þ

where f is a function of xi, p, , and t, v is the particle speed, Ui (xi, t) is the plasma flow velocity, bi ¼ Bi /jBi j is the unit vector

directed along the magnetic field, and ¼ vi bi /v ¼ cos  is the cosine of the particle pitch angle. Further simplification follows when particle transport is assumed to be one-dimensional (a function of z only), B is assumed to be uniform so that B ¼ B0 eB ¼ B0 , and the plasma flow velocity is one-dimensional and parallel to B0 so that U ¼ U (z; t)ez . Thus, because only particle transport along B0 is considered in our theory, it might only be applicable at quasi-parallel shocks, such as CME shocks close to the Sun at low latitudes and perhaps the termination shock at high latitudes, assuming that the heliospheric magnetic field is a Parker-type field. However, Jokipii & Ko´ta (1989) have argued that the termination shock in the polar regions may be a perpendicular shock most of the time. Evidence for the existence of a Fisk-type heliospheric magnetic field (Fisk 2001) due to differential rotation effects is mounting, which also would suggest that the termination shock is mostly perpendicular in the polar regions. Webb et al. (2003) present a theory valid at all latitudes but with the limitation of weak compressive fluctuations. The assumption of a uniform magnetic field but a nonuniform plasma flow field is probably unrealistic, but our main interest in this paper is to learn, as a first step, how compressive electrostatic velocity and density fluctuations with correlation scale lc 3 rg affect particle transport on large scales. As discussed above, ion-acoustic waves are expected to be electrostatic when propagating along the magnetic field and most unlikely to be damped for this propagation direction. After simplification, equation (16), now valid for a distribution function f (z, p, , t), is given by





@f @f dU @U @f þ (v þ U )  m þ p 2 @t @z dt @z @p


   m dU @U @f @f 1  2 þ 1  2 ¼  ; p dt @z @ @t sc ð17Þ where dU @U @U ¼ þU ; dt @t @z

ð18Þ

and f0 (z; v; t) ¼

1 2

Z

1

d f (z; v; ; t)

ð19Þ

1

is the average of f over in particle momentum space. The particle-scattering term on small-scale MHD fluctuations on the right-hand side of equation (17) is assumed to be
 @f f  f0 ¼ ( f  f0 ); ¼ ð20Þ @t sc

where the characteristic time (z; v; ; t) for the particle distribution to relax toward isotropy is interpreted as the characteristic particle-scattering time, and  ¼ 1/ is the particle-scattering frequency due to particle interaction with small-scale MHD fluctuations. These fluctuations are assumed to be those parallel propagating one-dimensional Alfve´n wave turbulence fluctuations with which particles have gyroresonant interactions. This form of the scattering term is the well-known ‘‘relaxation time’’ or BGK Boltzmann collision term approximation to the Boltzmann scattering integral (Bhatnager et al. 1954). Usually one uses the Fokker-Planck small-angle scattering pitch-angle diffusion term

1120

LE ROUX ET AL.

from QLT when describing particle interaction with small-scale Alfve´n waves rather than the BGK term, which implies equal probability of large-angle and small-angle scattering. However, the BGK term is chosen for its simplicity and for the fact that it is known (Webb et al. 2000) and has recently been shown, by doing an explicit comparison of the small-angle and the BGK models (Kaghashvili et al. 2004), that both kinds of scattering terms give similar spatial diffusion coefficients at late times after many particle-scattering events. For simplicity, we neglected in equation (17) the FokkerPlanck coefficients associated with momentum diffusion (stochastic particle acceleration) and coefficients associated with diffusion in mixed momentum and ordinary space because of particle interaction with small-scale Alfve´n waves (Fokker-Plank diffusion coefficients with mixed momentum and spatial derivatives). The neglect of momentum diffusion might be justified because upstream of a shock, streaming of energetic particles away from the shock leads to the preferential generation of Alfve´n waves propagating in the antishock direction and damping of those waves propagating toward the shock (Lee 1983). Since effective stochastic acceleration requires comparable amounts of Alfve´n waves propagating in both directions (Schlickeiser 1989), this process is expected to be unimportant upstream of shocks. That the FokkerPlanck diffusion coefficients in mixed momentum and ordinary space due to Alfve´n waves are not included is a limitation in our theory, because these coefficients are associated with coherent spatial particle convection by Alfve´n waves, which occur when Alfve´n waves are amplified in one propagation direction (Schlickeiser 1989). This is not a fundamental limitation, however, because Webb et al. (2003) showed that their inclusion does not alter the expressions for the derived transport terms associated with the compressive waves. It only means that the Alfve´n wave–associated convection effects should be added to our final transport equation for completeness. We do derive a momentum diffusion coefficient for ion-acoustic waves as shown below because they accelerate particles even if waves propagate only in one direction at shocks (see discussion in last paragraph of x 6).

d
U @
U @
U ¼ þ
U : dt @t @z

1 d @
U ¼ ;  dt @z

ð21Þ

where  is the mass density of the plasma flow. Thus, a positive plasma velocity gradient implies a rarefaction region, while a negative gradient means a compression region. Similarly, from equation (17) one can express the particle momentum and pitch angle changes in response to velocity gradients associated with random compressive fluctuations as



dz ¼
U : dt

  Z 1 


xi
xj vi (0; 0)vj xj (t 0 ); t 0 dt 0 ; ¼ Dij ¼ 2
t 0

ð25Þ

where the particle equation of motion is given by dxi ¼ vi : dt

ð26Þ

Applied to our equations of motion (eqs. [22] and [24]) in the limit of fast particles (v 3
U ), we obtain  2 Z 1 


z ¼
U ½z(0); 0
U z(t 0 ); t 0 dt 0 ; 2
t 0  2 
¼ 1  2 2 2 ¼ 2
t  Z 1 @
U @
U 0 0  ½z(0); 0 z(t ); t ; dt 0 ; @z @z 0  2  Z 1
p @
U @
U 0 0  ½z(0); 0 z(t ); t ¼ p2 4 ¼ dt 0; @z @z 2
t 0

Dzz ¼ D

Dpp ð22Þ

ð24Þ

From equations (22) and (24) one can construct diffusion coefficients that show how fast particles (v 3
U ) diffuse in real space along B0 (turbulent diffusion), in pitch-angle space, and in momentum space because of their interaction with many random compressive velocity fluctuations in the turbulent medium. According to the Taylor-Green-Kubo formalism (Kubo 1957) for particle motion in stationary homogeneous random fields, we can express the particle diffusion coefficient, valid over large spatial and temporal scales, in terms of the single-particle, twopoint, two-time velocity correlation function given by



dp d
U @
U ¼ m þ p 2 ; dt dt @z    m d
U d @
U 2 ¼ 1 þ ; dt p dt @z

ð23Þ

Assuming that the plasma flow velocity fluctuations have a characteristic spatial scale, called the correlation length lc (in the rest of the paper, lc refers to the correlation length of ionacoustic wave turbulence and not the correlation scale of MHD turbulence), and a typical timescale represented by the correlation time tc ¼ lc /
U , one can show for fast particles with v 3
U that terms with d
U/dt are negligible in equations (22). Thus, in a compressive region of the plasma (@
U /@z < 0), we see that particles experience an increase in particle momentum irrespective of the particle pitch angle, while particles undergo a decrease (increase) in particle pitch angle if > 0( < 0), respectively. In the case of a particle moving in a rarefaction region (@
U /@z > 0), however, particles experience a reduction in particle momentum irrespective of the particle pitch angle, while particles undergo an increase(decrease) in particle pitch angle if > 0( < 0), respectively. For fast particles with small values, the d
U/dt terms in equations (22) become important. Then the dependence of d /dt on the sign of disappears. It also follows from equation (17) that changes in the fluid velocity affect the particle convective propagation speed according to the relation

5. SIMPLE ESTIMATES OF THE FOKKER-PLANCK DIFFUSION COEFFICIENTS DUE TO COMPRESSIVE WAVE TURBULENCE Suppose the plasma flow velocity U ¼ U (z; t) ez is described by a large-scale coherent component and a random fluctuating compressive component so that U ¼ hU i þ
U with h
U i ¼ 0. Viewed from the large-scale or global fluid frame, U ¼
U because hU i ¼ 0. Therefore, the continuity equation for the plasma flow can be written as

Vol. 626

ð27Þ

No. 2, 2005

PARTICLE TRANSPORT IN COMPRESSIVE WAVES

where Dzz is the spatial turbulent diffusion coefficient of particles along B0 in the average large-scale fluid frame that results from random convection in the flow velocity field, D is the pitch-angle diffusion coefficient, and Dpp is the momentum diffusion coefficient for stochastic particle acceleration that arises from compressibility in the flow velocity field. It is assumed that the two-point, two-time correlation fluid velocity functions decay exponentially for particles propagating along B0 on a timescale t pc . Accordingly, 

   0 p
U ½z(0); 0
U z(t 0 ); t 0 ¼
U 2 ½z(0); 0et =t c ;  +   *
@
U @
U 0 0  @
U 2 0 p ½z(0); 0 z(t ); t ½z(0); 0et =t c : ¼ @z @z @z ð28Þ It is straightforward to evaluate the time integrals in the expressions of the diffusion coefficients in equation (19) by using equation (20) and assuming that the characteristic scale of the velocity fluctuations is the correlation scale lc. Thus,   Dzz ¼
U 2 ½z(0); 0t pc ;  2  2 2 2
U D ¼ 1  ½z(0); 0t pc ; lc2  2 2 4
U Dpp ¼ p ½z(0); 0t pc : lc2

ð29Þ

In the following section we show that these simplified estimates of the diffusion coefficients are confirmed by the detailed derivation shown below. The detailed derivation, however, allows one to calculate expressions for tcp based on the physics of the problem. 6. NONLINEAR THEORY OF ENERGETIC CHARGED PARTICLE TRANSPORT IN STRONG COMPRESSIVE WAVE TURBULENCE 6.1. Deriving the Combined BGK Boltzmann Fokker-Planck Equation In order to derive a kinetic equation that describes statistically how microscopically diffusing particles interact with many compressive electrostatic ion-acoustic fluctuations propagating along B0 with correlation scale lc 3 rg  l, we divide U, f, and f0 into slowly evolving, large-scale variations on timescale T and spatial scale L and more rapidly fluctuating random parts associated with the correlation spatial scale lc and correlation timescale tc of the compressive fluctuations so that U ¼ hU i þ
U;

h
U i ¼ 0;

f ¼ h f i þ
f ; h
f i ¼ 0; f0 ¼ h f0 i þ
f0 ; h
f0 i ¼ 0;  ¼ hi þ
;

h
i ¼ 0;

ð30Þ

where angle brackets denote ensemble average quantities. Note in equations (30) that the particle-scattering frequency  due to small-scale Alfve´n waves is expressed in terms of a slowly varying large-scale part and more rapidly fluctuating random parts. Fluctuations in  can be interpreted as variations in the energy of the Alfve´n waves because of density variations associated with

1121

the compressive wave turbulence, which can be explained by standard WKB theory (Hollweg 1974; see x 3). The idea is that random density variations associated with ion-acoustic wave turbulence would bunch together and strengthen MHD small-scale scattering centers in compressive regions (increased ) and spread them out and weaken them in rarefaction regions associated with the ion-acoustic wave turbulence (decreased ) (Hollweg 1974; Bykov & Toptygin 1993; Webb et al. 2003). Since we assumed electrostatic waves,
B ¼ 0 so that B ¼ hBi ¼ B0 eB . Further simplification follows by transforming the space and time coordinates in equation (17) from the observer frame to coincide with the large-scale or global plasma fluid frame assumed to move at a constant velocity hUi ¼ U 0 relative to the observe frame. Thus, it follows from equations (30) that U ¼
U (z; t)eB :

ð31Þ

Because of this transformation, the particle momentum transformation in equation (12) must be reinterpreted as a transformation from the large-scale constant velocity fluid frame to the local noninertial fluid frame according to the transformation p 0 ¼ p þ m
U. The electrostatic assumption
B ¼ 0 implies that there is no magnetic force from the compressive fluctuations on the charged particles. This is the case for ion-acoustic waves propagating along the magnetic field (Gary 1993). In addition, in the largescale fluid frame where hUi ¼ 0, the fluctuating motional electric field can be expressed as
E ¼ 
U <
B 
U < B0 . However, because
B ¼ 0 and
UjjB0 ,
E ¼ 0, so the effect of the compressive velocity fluctuations on the particles in equation (17) is not a consequence of an associated fluctuating motional electric field. Since for simplicity we neglect any electric fields arising from lack of plasma neutrality (from a low-frequency wave turbulence viewpoint, the plasma flow is approximately neutral), energetic particles are affected by compressive velocity fluctuations because these velocity fluctuations cause random variations in the smallscale Alfve´nic scattering centers embedded in the solar wind flow. To the extent that the particles are affected by the small-scale scattering centers, they are coupled indirectly to the compressive plasma flow and respond to plasma velocity variations. We first substitute the expressions of equations (30) modified according to equation (31) in equation (17). After taking an ensemble average of equation (17), terms up to second order in fluctuating quantities are retained, and second-order terms depending on
f are cast in a divergence form. The ensemble-averaged distribution function h f (z, p, , t)i satisfies   @h f i @h f i m  @
U @h f i þ v  1  2
U @t @z p @z @   @
U @h f i  m
U @z @p @ ¼ (h f0 i  h f i)hi þ h
f0
i  h
f
i  h
U
f i @z 
 

@ 1  2 @
U @
U þ p þ m
f @ @t @z p 
 

1 @ @
U @
U p2 þ p 2 þ 2 m
f p @p @t @z  
 m @
U 1
f 1 2 ;  p @t 

ð32Þ

1122

LE ROUX ET AL.

where  is the Lorentz factor given by  2 ¼ 1/(1  v 2 /c 2 )1 2, where v is the particle speed and c is the velocity of light. By assuming nonrelativistic suprathermal particle speeds, the last term on the right-hand side of equation (32) vanishes. By subtracting equation (32) from equation (17) after the expressions of equations (30) are substituted into equation (17), one arrives at an equation for
f given by =

@
f @
f þ v þ hi(
f 
f0 ) @t @z
 @h f i 1  2 @
U @
U @h f i þ þ p m ¼ 
U @z @t @z @ p
 @
U @
U @h f i þ p þ
(h f0 ih f i) þ N
U
f ; þ m @t @z @p ð33Þ where N
U
f represents the nonlinear terms that are second-order or third-order in the fluctuating quantities
U,
f, and
. In standard QLT, the nonlinear terms are neglected and the particles have approximately undisturbed trajectories along the magnetic field that only become diffusive after the particles see many decorrelated fluctuations. However, we are interested in strong compressive wave turbulence such as that found near shocks, for which we assume
U  Vs . By ‘‘strong ion-acoustic wave turbulence’’ we imply weakly nonlinear waves in the sense that the waves maintain their shape close to standard linear periodic waves through a balance of nonlinear steepening and dispersion. This allows us to continue using a standard plasma physics approach to wave turbulence, i.e., a superposition of linear Fourier wave modes in wavenumber space. Within the framework of a Korteweg–De Vries equation for weakly nonlinear lowfrequency ion-acoustic waves (Chen 1984, p. 297), it is clear that the wave phase speed is larger than the ion-acoustic speed Vs for strictly linear waves, and a correction to the phase speed /
UVs is contributed by steepening. It is also assumed that the small-scale Alfve´n waves experience enhanced turbulence levels near shocks. Consequently, in contrast to the case of weak compressive velocity fluctuations in standard QLT, we assume that both the small-scale Alfve´nic fluctuations and the compressive velocity fluctuations are able to scatter particles randomly on the characteristic turbulence correlation length scale associated with the compressive wave turbulence. Such effects are described by the BGK scattering term for small-scale Alfve´n waves on the left-hand side of equation (33) and the nonlinear terms N
U
f for the compressive fluctuations on the right-hand side of equation (33). These terms have to be dealt with in such a way as to achieve a closed large-scale transport equation. This, however, is not a trivial matter and is an enduring problem in this and related fields. The problem is that the exact solution for the disturbance in the distribution function
f when particles are interacting with a compressive fluctuation can formally be expressed as an infinite series, with each successive term being of higher order in terms of fluctuating quantities and so leading to a divergent solution (Kaiser et al. 1973). This suggests that a renormalized theory, similar to the case of quantum field theory, is required. Many different renormalization approaches have been discussed in the literature since the first attempts of Dupree (1966) (for a review, see Krommes [2002], and see also Kaiser et al. [1973], Vo¨lk [1973], and Jones et al. [1978]), and there does not appear to be a consensus on what the best approach should be.

Vol. 626

We followed a closure approach similar to those advocated by Owens (1974), Bykov & Toptygin (1993), and Matthaeus et al. (2003), which give reasonable values for transport coefficients over a wide range of parameters when compared to computer simulations of particle transport (Bykov & Toptygin 1993; Matthaeus et al. 2003) and have the virtue of analytical simplicity. That is, we assume in the BGK scattering term on the left-hand side of equation (33) that the average microscopic scattering frequency should be replaced by an as yet to be determined effective scattering frequency h effi. This scattering frequency is interpreted as representing the net effect of particle scattering arising not only from the nonlinear terms N
U
f because of the compressive velocity fluctuations, but also from the microscopic scattering due to small-scale Alfve´n waves. As is shown below, this simple assumption leads to a closed large-scale transport equation for field-aligned energetic particle transport with a polynomial expression for h effi. The solution of the polynomial equation then reveals explicitly the relative contributions of small-scale Alfve´n waves and compressive fluctuations to effective diffusion along the magnetic field. On this basis we have to solve @
f @
f   þ v þ eA (
f 
f0 ) @t @z
 @h f i 1  2 @
U @
U @h f i þ þ p m ¼ 
U @z @t @z @ p
 @
U @
U @h f i þ p þ
(h f0 ih f i); ð34Þ þ m @t @z @p where the scattering effect of the nonlinear terms N
U
f in equation (33) is embedded in the h effi term on the left-hand side of this equation. The effective scattering time h eA i ¼ 1/heA i, and the effective spatial diffusion coefficient along the magnetic field is now defined as eA ¼ 13 v 2 h eA i. Although this is a useful approach to describe strong wave turbulence effects on particle transport, it is limited in the sense that particle acceleration effects on a particle correlation timescale are neglected. In addition, velocity correlation functions higher than second order are not included, and fast decay of the second-order correlation functions is assumed. Thus, the theory is formally not applicable to situations of sub- or superdiffusion in which decay of the correlation functions is slow or situations in which the MHD fluctuations are intermittent instead of homogeneous. Intermittency usually indicates a mixture of coherent fluctuations with incoherent random fluctuations (Bruno et al. 2003), while we only consider the incoherent fluctuation component of low-frequency ion-acoustic fluctuations. For more on how to deal with intermittent compressive fluctuations, see the review by Bykov & Toptygin (1993). Equation (34) for
f is in the form of a BGK Boltzmann equation with a source term on the right-hand side. Fedorov et al. (1995) showed in detail how one can obtain a Green’s function solution for such an equation. In our case, the solution is (see also Webb et al. 2000)
f (z; t; p; ) ¼ Z 1 Z t dt0 dz0 0 1 Z 1 ; d 0 G(t; t0 ; z; z0 ; p; ; 0 )F(z0 ; t0 ; p; 0 ); 1

ð35Þ

No. 2, 2005

where z0, t0, and 0 are initial values, F(z0, t0, p, 0), given by F(z0 ; t0 ; p; 0 ) ¼
 @h f i 1  2 @
U @
U @h f i þ þ p m 
U @z @t @z @ p
 @
U @
U @h f i þ p þ m @t @z @p

þ
 (h f0 i  h f i) ;

ð36Þ

represents the terms on the right-hand side of equation (34) at the initial values, and G(t, t0, z, z0, p, , 0) is the Green’s function given by 
t=h eA i


(
z  v
t)
(  0 ) G(t; t0 ; z; z0 ; p; ; 0 ) ¼ e 1   e
t=h eA i PV þ 2v eA    

   Z 1 sin (=2)
z=v eA 
t= eA ; d (  )(  0 ) 1 1 ; (
z;
t; ) þ   e
t=h eA i 2v eA    

  
cos (=2)
z=v eA 
t= eA 0 ; (
z;
t; 0 )  0 

       cos (=2)
z=v eA 
t= eA  (
z;
t; )  0 Z =2 1   þ dK e(
t=h eA i)(K cot K1) 2v eA 0 

    2 cos K  0 sin2 K ; cos K
z=v eA 

    þ sin K
z=v eA ( þ 0 ) cos K sin K  1

 ; ð37Þ ; cos2 K þ 2 sin2 K cos2 K þ 20 sin2 K where



1 1þ "

(1=2)½
z=vh eA i(
t=h eA i) 


z
t ; H( )H      v eA

eA

!


t
z  H( )H      v eA

eA

!# :

it is only valid for k K k< /2, while the other terms associated with scattered particles are negligible. Consequently, G(
z;
t; p; ; 0 ) 
(
z  v
t)
(  0 );

z¼z0 ; t¼t0 ; ¼ 0

(
z;
t; ) ¼

1123

PARTICLE TRANSPORT IN COMPRESSIVE WAVES

ð38Þ

suggesting that spatial variation along B0 is associated with undisturbed or ballistic particle motion along B0 over a wavelength q/k. This is the QLT limit of our theory. However, in the case of strong compressive wave turbulence one would expect to be at the limit of strong scattering (diffusive limit), where K T1, whereby one has to assume that
t /h eA i 3 1. Only the last term in the Green’s function of equation (37) remains, and Z =2 1   dK e(
t=h eA i)(K cot K1) 2v eA 0 

    2 ; cos K
z=v eA cos K  0 sin2 K

     þ sin K
z=v eA ( þ 0 ) cos K sin K

  1 ; cos2 K þ 2 sin2 K cos2 K þ 20 sin2 K : ð40Þ

G(
z;
t; p; ; 0 ) ¼

For small K, a series expansion of cot K yields K cot K  1 K 2 /3 to second order in K, while cos K  1 and sin K  0. Consequently, G(
z;
t) ¼ ! Z 1 1
z 2  ( 1=3 )(
t=h

i ) K eA   dK e cos K   2v eA 0 v eA ¼

1 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e(
z =4eA
t) : 4 eA
t

ð41Þ

The Green’s function in equation (41) is recognizable as the Green’s function of the standard heat or diffusion equation. This is consistent with our assumption that particles diffuse spatially along B0 because of the combined effect of both compressive wave turbulence and small-scale MHD scattering centers over a wavelength 1/k of the compressive wave turbulence. After substitution of equation (41) into equation (35), it follows that
f (z; t; p; ) ¼ Z Z t d
t 2 0

In equations (37) and (38), K ¼ kvh eA i, where k is the wavenumber of the particle pulses, h effi is the ensemble-averaged or long-timescale effective particle-scattering time,
t ¼ t  t0 ,
z ¼ z  z0 , and PV is the principal value integral due to singularities along the integration path. For details, see Fedorov et al. (1995). The first term in equation (37) represents the Green’s function of unscattered particles, while the rest of the terms denote the Green’s function associated with scattered particles. The fraction of scattered particles is controlled by the ratio
t/h effi. By assuming large or small values for this ratio, implying the diffusive or weak scattering limit, one can simplify the Green’s function considerably as discussed below. In the case of weak scattering (
U TVs ), where K 3 1, the appropriate assumption is
t /h iT1. For large K the last term of the Green’s function in equation (37) does not exist, because

ð39Þ

"

1

1

d(
z)

1 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e(
z =4eA
t) 4 eA
t

#

; F(z 
z; t 
t; p; ) ;

ð42Þ

where F(z 
z; t 
t; p; ) ¼
 @h f i 1  2 @
U @
U @h f i þ þ p m 
U @z @t @z @ p
 @
U @
U @h f i þ p þ m @t @z @p

þ
(h f0 i  h f i) (z 
z; t 
t; p; ); ð43Þ

1124

LE ROUX ET AL.

after integration over
0 defined as
¼  0 has been performed. The integration was performed by setting
¼ 0 in all terms dependent on
¼ 0 in equation (35). This assumption is consistent with the diffusive particle motion described by the Green’s function because
¼ 0 is a random parameter with a zero average value. To form a closed large-scale transport equation, equations (42) and (43) are substituted into equation (32). Simplified expressions for the transport coefficients are derived by assuming stationary, homogeneous, one-dimensional compressive wave turbulence with random phases so that the two-point, two-time velocity correlation functions in terms of a spatial Fourier transform can be expressed as Z 1   dk e ik½
z! (k )
t
U 2 (k); h
U (0; 0)
U (
z;
t)i ¼   Z 1 1   @
U @
U (0; 0) (
z;
t) ¼ dk k 2 e ik½
z! (k )
t
U 2 (k); @z @z   1 Z 1   @
U (
z;
t) ¼ i dk ke ik½
z! (k )
t
U 2 (k)
U (0; 0) @z  1  @
U (0; 0)
U (
z;
t) ; ¼ @z ð44Þ where k is the magnitude of the wavenumber directed along B0 and !(k) ¼ (k)  i (k), where (k)  Vs k is the approximate dispersion relation for weakly nonlinear waves as discussed above and (k) determines the rate at which the two-time velocity correlation functions decay exponentially for the diffusive particles because of incoherent temporal effects caused by the dynamics of nonlinear wave interactions (see Bykov & Toptygin 1993; Bieber et al. 1994). This contributes a resonance broadening effect in the expressions for the transport coefficients, as shown below. The simplifying assumption that particle velocities along B0 are larger than the phase velocity of the waves (v 3 Vs ) was introduced above. The assumption holds for most pitch angles, except for those in the vicinity of ¼ 0. It was also assumed, in accordance with WKB theory, that density fluctuations associated with compressive fluctuations cause random fluctuations in the energy of small-scale Alfve´n waves and thus by implication cause random variations in the microscopic particle-scattering frequency, as discussed above. Accordingly, when we assume that  / n , with n the plasma number density,
 is modeled as
 ¼ 

hni
n ; n hni

ð45Þ

where, according to WKB theory,  ¼ 3/2 in the supersonic solar wind (Hollweg 1974). The two-point, two-time correlations involving
 or combinations of
 with
U or @
U /@z are defined as in equations (44). The final closed transport equation is in the form of a BGK Boltzmann Fokker-Planck equation given by  

@h f i @h f i m @
U @h f i 2 þ (v  4Uz )  1
U  4U @t @z p @z @
   Z 1 @
U @h f i @h f0 i @h f i þ 2Uz þ  m
U d Uz  4Up @z @p @z @z 1 Z 1 Z 1 @h f i @h f0 i @h f i   2Up  d U d Up @ @p @p 1 1

Vol. 626

  ¼ eA h f0 i  h f i


 @ @h f i @ @h f i 1 @ @h f i 2 Dzz D p Dpp þ þ 2 þ @z @z @ @ p @p @p


 @ @h f i @ @h f i @ @h f i 2Dz 2D z 2Dzp    @z @ @ @z @z @p


 1 @ @h f i @ @h f i 1 @ @h f i 2 2Dpz D p p Dp  2 þ þ 2 ; p @p @z @ @p p @p @ ð46Þ where

  @U 1 @  2 @Uz p 2Up þ 2 2 eA ¼ hi  2D

 þ 2 ; p @p @ @z ð47Þ D

 ¼ Dzz ¼ D ¼ Dpp ¼ D p ¼ Dz ¼ Dzp ¼ Uz ¼ U ¼ Up ¼

Z 1 1 h
 2 i(k) dk ; 2 1 i (k) þ  (k) þ k 2 eA Z 1 1 h
U 2 i(k) dk ; 2 1 i (k) þ  (k) þ k 2 eA  2 Z 2 1
U (k)k 2 1 2 1  2 dk ; 2 i (k) þ  (k) þ k 2 eA 1  2 Z
U (k)k 2 1 2 4 1 p dk ; 2 i (k) þ  (k) þ k 2 eA 1  2 Z 1
U (k)k 2 1 3 p 1  2 dk ¼ Dp ; 2 i (k) þ  (k) þ k 2 eA 1   Z 1
U 2 (k)ik 1  1  2 dk ¼ D z ; i (k) þ  (k) þ k 2 eA 2 1   Z
U 2 (k)ik 1 2 1 p dk ¼ Dpz ; i (k) þ  (k) þ k 2 eA 2 1 Z 1 1 h
U
i(k) dk ; 2 1 i (k) þ  (k) þ k 2 eA Z 1 1  h
U
ni(k)ik 1  2 dk ; 2 i (k) þ  (k) þ k 2 eA Z 1 1 1 2 h
U
i(k)ik p dk : ð48Þ 2 i (k) þ  (k) þ k 2 eA 1

For the sake of analytical progress, the transport coefficients have been derived by assuming that there is an inertial range of compressive fluctuations with a maximum wavelength scale on the order of the correlation scale lc, which contributes the most to nonlinear particle transport. The contribution from other, larger wavenumbers are neglected, assuming that these fluctuations have too little energy to contribute significantly to nonlinear scattering. The existence of an inertial range is usually associated with incompressible fluctuations, but observations (Armstrong et al. 1995) and recent MHD turbulence simulations (Cho & Lazarian 2003; Dastgeer & Zank 2004) suggest that this is also the case for compressive fluctuations. We also made the approximation that h
U 2 itot (lc )  h
U 2 itot , where h
U 2itot is the total amount of kinetic energy in the inertial range of the compressive velocity fluctuations. In the same way, it was assumed that h
U 2 idif (lc )  h
U 2 idif . The same approximation was also made for ensemble averages involving
, because
 /
n (see eq. [45]) and an inertial range is also a feature of the density fluctuations associated with compressive fluctuations (Cho & Lazarian 2003). Consequently, we select the wavenumber

No. 2, 2005

PARTICLE TRANSPORT IN COMPRESSIVE WAVES

k ¼ 1/lc from the inertial range of compressive fluctuations by assuming that D E D E
U 2 (k) ¼
U 2
(k  1=lc ); D E D E
 2 (k) ¼
 2
(k  1=lc ); h
U
i(k) ¼ h
U
i
(k  1=lc );

ð49Þ

whereby the transport coefficients simplify to 1 D 2E p tc ;
 D

 ¼ tot 2 h
U
idif p ˆt c ; Up ¼ p 2 lc h
U
idif p 1  ˆtc ; U ¼ 1  2 2 lc 1 Uz ¼ h
U
itot tcp ; 2 1 D 2E p tc ; Dzz ¼
U tot 2 D E D ¼

Dpp ¼

D p ¼

Dz ¼ Dzp ¼

2 2
U tot p 1 2 1  2 tc ; 2 lc2 D E 2
U 1 2 4 tot p p tc ; 2 lc2 D E 2
U  1 3 tot p p 1  2 tc ¼ Dp ; 2 lc2 D E
U 2 1  dif ˆ p 1  2 tc ¼ D z ; 2 lc D E 2 1 2
U dif p ˆtc ¼ Dpz : p 2 lc

BGK scattering term (first term on the right-hand side of eq. [46]). It is only when the final transport equation in the diffusion approximation is derived that h effi (or eff) acquires its full meaning by also including turbulent diffusion effects from the compressive wave turbulence. The Fokker-Planck coefficients Dij (see the rest of the terms on the right-hand side of eq. [46]) model the diffusion of particles in pitch-angle space (D ), ordinary space (Dzz), momentum space (Dpp), and in combinations of these spaces (D z, Dp , Dzp, etc.) under the influence of the strong compressive wave turbulence. Coherent convection of particles in ordinary, pitch-angle, and momentum spaces also occurs due to terms with Uz, U , and Up, respectively; this is due to the combined effect of compressive velocity fluctuations
U and microscopic scattering frequency variations
 arising from density variations associated with the compressive wave turbulence. These terms are also involved in modifying the average microscopic scattering frequency h effi on large scales. The term D

 also modifies h effi solely due to
 variations. The expressions for Dzz , D , and Dpp from the full-scale transport theory in equation (47) confirm our initial simple estimates (see eqs. [29]). Our nonlinear theory, however, produced a detailed expression for the particle decorrelation time tcp (eqs. [48]). One can derive a simple expression for the effective average microscopic scattering frequency  eff on large scales in the case of a medium with nonpropagating density variations by assuming
U ¼ 0 and Vs ¼ 0 in equations (46)–(48). We find that @h f i @h f i   þ v ¼ eA h f0 i  h f i ; @t @z

ð52Þ

where   eA ¼ hi  2D

 ; 1 D 2E p D

 ¼ tc :
 tot 2 ð50Þ

The timescales t pc and ˆt pc for particles to experience decorrelated compressive fluctuations are  (lc ) þ eA =lc2

2 ; (Vs =lc )2 þ  (lc ) þ eA =lc2 Vs =lc ˆtcp ¼ 2 : 2 (Vs =lc ) þ ½ (lc ) þ eA =lc2

1125

tcp ¼

ð51Þ

The quantity hU 2 itot ¼ hU 2 iþ þ hU 2 i, where hU 2 i  is the average kinetic energy density in compressive wave modes with wavelength lc propagating forward( backward) along B0, respectively, and hU 2 idif ¼ hU 2 iþ  hU 2 i is the energy difference between forward- and backward-propagating waves with wavelength lc. In the same way, expressions are defined for h
 2itot(dif ) and h
U
itot(dif ). In addition,
 signifies random variations in the microscopic scattering frequency associated with small-scale Alfve´n waves because of relatively large-scale density variations associated with compressive wave turbulence, and eA ¼ 13 (v 2 /heA i) represents the effective nonlinear large-scale parallel diffusion coefficient. The effective scattering frequency h effi at this stage of the derivation should be interpreted in a more limited sense as describing the average microscopic scattering frequency on large scales in the presence of strong compressive wave turbulence, resulting in a modified

ð53Þ

From the solution of the equation for h effi we find that 

 eA ¼

hi : 1 þ (3h
 2 itot lc2 )=v 2

ð54Þ

Thus, we see that, effectively, the average microscopic scattering frequency hi is reduced by fluctuations in the microscopic scattering frequency
 because of random density variations in the medium. On the other hand, hi itself should be enhanced with respect to a medium without density variations, because particles spend more time in compression regions, where microscopic scattering is enhanced, compared to rarefaction regions, where microscopic scattering is reduced (see eq. [45]). Due to these competing effects, it is unclear, therefore, whether h effi is enhanced relative to a medium without density variations. 6.2. The Diffusion Approximation Since we are assuming the strong scattering limit, it is to be expected that the particle distribution is nearly isotropic on long timescales. We thus introduce an expansion for the particle distribution in terms of standard Legendre polynomials as a function of particle pitch angle direction. Thus, h f i(z; t; p; ) ¼ h f0 i(z; t; p) þ h f1 i(z; t; p) :

ð55Þ

Because we assume a small anisotropy h f1 iTh f0 i, higher order terms in of the expansion are neglected. After integration

1126

LE ROUX ET AL.

of equation (46) over all pitch-angle directions, we find an equation for the direction-averaged particle distribution function h f0i(z, t, p) (see eq. [19] for definition) given by     @h f0 i 1 @h f1 i 2 m @U 1 @
U @h f1 i þ v 
U h f1 i m
U @t 3 @z 3p @z 3 @z @p

 @ @h f0 i 1 @ 2 @h f i 0 2Dzz p2 Dpp ¼ þ 2 @z @z p @p 5 @p

 1 @ @h f0 i @ 2 @h f0 i 22 p Dpz Dzp  2  ; ð56Þ p @p 3 @z @z 3 @p where 1  2 p t ;
U 2  totc
U 2 tot p 1 Dpp ¼ p2 tc ; 2 l2  2c 1
U dif p ˆtc ¼ Dpz ; Dzp ¼ p 2 lc Dzz ¼

ð57Þ

and tcp and ˆtcp are given by equations (51). Calculation of the first moment of equation (46) in -space yields an equation for h f1i given by   1 @h f1 i 1 @h f0 i 1 @
U @h f0 i þ v  m
U 3 @t 3 @z 3 @z @p 2 @h f1 i 4 2 @h f1 i  Uz þ U h f1 i þ Up 3 @z 15 5 @p
 1 2 1 @ 2 p2 Up h f1 i ¼  hih f1 i þ D

 h f1 i  2 3 3 p @p 5 8 2 @Uz 16 U h f1 i þ D h f1 i h f1 i þ 15
3 @z 105 
 @ 2 @h f1 i 1 @ 2 @h f1 i Dzz p 2 Dpp þ þ 2 @z 3 @z p @p 7 @p
 4 @h f1 i 1 @ 4 @h f1 i D p þ 2 p2 Dp  35 @p p @p 35 @z
 @ 4 4 @h f1 i Dz h f1 i þ D z  @z 15 15 @z

 @ 2 @h f1 i 1 @ 2 @h f1 i Dzp p2 Dpz   2 ; ð58Þ @z 5 @p p @p 5 @z where 1  2 p
 tot tc ; 2  1
U 2 tot p ¼ tc ; 2 lc2

D

 ¼ D

Dpp ¼ p2 D ; 1  2 p Dzz ¼
U tot tc ; 2 D p ¼ Dp ¼ pD ;  2
U dif p ˆtc ; D z ¼ Dz ¼ lc Dzp ¼ Dpz ¼ pD z ; 1 Uz ¼ h
U
itot tcp ; 2

Vol. 626 1 h
U
idif p ˆtc ; 2 lc Up ¼ pU ;

U ¼

ð59Þ

and tcp and ˆtcp are specified by equations (51). To solve equation (58) analytically for h f1i, we first simplify it by comparing the size of the different terms through dimensional analysis and then balancing the term 13 v(@h f0 i/@z) with the term 13 hih f1 i and other terms containing h f1i, which are of the same order of magnitude. This ensures that the final closed transport equation for h f0i is a diffusion equation that includes the effects of microscopic diffusion. This is accomplished by introducing a certain ordering of time and spatial scales valid for strong compressive wave turbulence, where 1= 2  Vs . We assume that suprathermal particles (v 3 Vs ) h
U itot with relatively low energies are effectively undergoing turbulent diffusion in strong compressive wave turbulence, which requires small effective microscopic diffusion according to the 1= 2 Tlc , where lc represents the correlation constraint /h
U 2 itot scale of the compressive wave turbulence and is assumed to be the maximum scale of the inertial range. It is assumed that only wavelengths on the order of lc cause nonlinear transport because there is not enough energy in the fluid velocity fluctuations at = smaller scales. This implies that hki/lc Th
U 2 i1tot2 /v, where hki is the parallel mean free path for microscopic diffusion due to Alfve´n waves averaged over large scales. Thus, we define hki/lc ¼ 2 and h
U 2 i1tot= 2 /v ¼ , where T1. The effective dif1= 2 lc and 1/ (lc ) ¼ fusion coefficient is estimated as eA  h
U 2 itot tc , where tc is the timescale for nonlinear wave interactions given = by tc ¼ lc /h
U 2 i1tot2. It then follows that tcp  ˆtcp  1. Furthermore, we assume that diffusive transport takes place on a spatial scale L 3lc and a timescale T 3tc , so lc /L ¼ , tc /T ¼ , h i/tc ¼

3 , and h i/T ¼ 4 , where h i ¼ hki/v is the microscopic scattering time averaged over large scales. Finally, by requiring that h f1 i/h f0 i ¼ 3 , retaining terms of order 1/ yields the solution to equation (58) as h f1 i ¼ 

v 1 @h f0 i ; hi 1 þ (2=5)(U =hi) @z

ð60Þ

where U is specified in equations (59). Substitution of equation (60) into equation (56) and retaining terms up to order 2 results in a closed diffusive transport equation for the ensemble- and direction-averaged particle distribution h f0i(z, t, p),

 @h f0 i @ @h f0 i 1 @ 1 @h f0 i  eA p 2 Dpz þ 2 @t @z @z p @p 3 @z

 @ 1 @h f0 i 1 @ @h f0 i 21 Dzp p Dpp þ ¼ 2 : @z 3 @p p @p 5 @p ð61Þ This representation, which retains coefficient notation used in the BGK Boltzmann Fokker-Planck equation (eq. [46]), is useful because it reveals the origin of the different terms. Finally, we write equation (61) in the standard conservation form,
 @h f0 i @ @h f0 i þ CU  h f0 i  eA @t @z @z

 1 @ @h f0 i 1 @ 1 @h f0 i p3 U  p2 Dpp þ 2 ¼ 2 ; 3p @p @z p @p 5 @p ð62Þ

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PARTICLE TRANSPORT IN COMPRESSIVE WAVES

and also in a diffusion-convection form, according to which
 @h f0 i 1 @  3  h f0 i @ @h f0 i þ 2 p U  eA @t 3p @p @z @z @z
  2 @U p @h f0 i 2 1 @ 21 @h f0 i  @ h f0 i þ pU ¼ 2 p Dpp ; þ 3 @z@p p @p 5 @p @z 3 @p ð63Þ with 1 p @h f0 i ; 3 h f0 i @p  eA ¼ t þ ; 1 þ (1=5)(U =hi)   t ¼
U 2 tot tcp ; C¼ 

1 v2 1 ¼ vhki; 3 hi 3 h
U
idif p ˆtc ; U ¼ l  2c
U dif p ˆtc ; U ¼ lc  2
U tot p 2 Dpp ¼ p tc ; lc2

7. EFFECT OF COMPRESSIVE WAVE TURBULENCE ON PARALLEL DIFFUSION 7.1. Strong Compressive Wave Turbulence

¼

 (lc ) þ eA =lc2 2 ;

(Vs =lc )2 þ  (lc ) þ eA =lc2 Vs =lc ˆtcp ¼

2 ; 2 (Vs =lc ) þ  (lc ) þ eA =lc2

that the compressive wave field does work on the energetic particles. Note that convection and adiabatic energy changes do not occur if h
U 2 iþ ¼ h
U 2 i . Stochastic particle acceleration, however, also occurs when compressive waves propagate only in one direction. This is different from the standard theory for particle resonant interaction with small-scale incompressible Alfve´n waves, where no stochastic acceleration occurs when Alfve´n waves propagate only in one direction. The reason is compressibility. For waves propagating in one direction, an observer in the Alfve´n wave frame sees no electric field, so particle energization does not occur in that frame, but an observer in the wave frame of the compressive fluctuations sees particles undergoing adiabatic energy changes due to compression or decompression effects.

ð64Þ

tcp ¼

ð65Þ

where C denotes the Compton-Getting factor, eff is the effective nonlinear large-scale parallel diffusion coefficient that is described by a third-order polynomial equation that needs to be solved,  is the microscopic diffusion coefficient averaged over large scales,  is the large-scale averaged microscopic scattering frequency due to resonant interaction with gyroscale Alfve´n waves propagating along B0, U  is an effective speed for coherent particle convection by the compressive fluctuations, and Dpp is the momentum diffusion coefficient for stochastic acceleration of energetic particles by compressive wave turbulence. This transport equation is similar to the equation derived by Webb et al. (2003) for small-amplitude compressive fluctuations and also by Li et al. (2004) for small-amplitude fluctuations in the particle distribution function, but without restriction on the amplitude of the compressive fluctuations. These authors both used the cosmic-ray transport equation as the starting point for the derivation. These publications are more general in the sense that the large-scale solar wind flow speed is included, and in the case of Webb et al. (2003), the convection effect of Alfve´n waves on the particles is included. The advantage of our approach, however, is that it is an explicitly nonlinear approach to particle transport in strong compressive fluctuations starting from a more basic kinetic level. Our transport theory suggests that, in addition to parallel spatial diffusion (third term on the left-hand side of eqs. [61] and [62]) and stochastic particle acceleration (right-hand side of eqs. [61] and [62]), particles also experience convection (second term on the left-hand side of eqs. [61] and [62]) and adiabatic energy changes (fourth term on the left-hand side of eq. [62]). The fourth term on the left-hand side of equation (61) implies

The effective parallel diffusion coefficient eff in equations (64) is the sum of two terms. The first term, t , is a turbulent diffusion coefficient representing the random convective effect of the incoherent flow velocity variations associated with the compressive wave turbulence, while the second term represents the large-scale averaged microscopic diffusion coefficient modified by the term U , which represents the effect of random variations in the microscopic scattering frequency
 and the compressibility of the velocity fluctuations. Equations (64) are a fifthorder polynomial in eff , but no general formula for the roots of a fifth-order polynomial exists. In order to make further analytical progress, we assume that the energy associated with forward- and backward-moving ion-acoustic waves are approximately the same, i.e., h
U 2 iþ  h
U 2 i . The U term is now negligible, and the equation for eff simplifies to a thirdorder polynomial given by h     i 3eA þ 2lc2   2eA þ Vs2 
U 2 tot lc2 þ lc2 lc2  2 eA h i    (Vs lc )2  þ lc4
U 2 tot þ  ¼ 0: ð66Þ Equation (66) can be solved for eff , assuming that for strong compressive wave turbulence h
U 2 itot  V s2 , yielding
  l 2 12 c eA ¼ 3  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !1=3  27 272 4 2 3 þ t 2  þ 27t 2  4 þ t  þ 3 2 4 (=3) 2

; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=3  3 þ (27=2)t 2  þ 27t 2  4 þ (272 =4)t 4  2

þ

ð67Þ where  ¼ 1 þ lc2 / and t ¼ t /, where t ¼ h
U 2 i1tot2 lc is the classical turbulent diffusion expression for passive convection of particles with the turbulent compressive medium. 1= 2 , where tc is the charWe specify 1/(lc ) ¼ tc ¼ lc /h
U 2 itot acteristic timescale (correlation time) for nonlinear wave interactions at the scale lc. Because h
U 2 itot  Vs2 , it follows that !(lc )(lc )  1, which indicates strong dynamic wave turbulence. =

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LE ROUX ET AL.

In addition, by taking the low suprathermal particle energy = 1= 2 , but vT(lc /k)h
U 2 i1tot2 (which limit, that is, v 3 Vs  h
U 2 itot = is equivalent to /h
U 2 i1tot2 Tlc or Tt ), and the high supra1= 2 ( /h
U 2 i1tot= 2 3 thermal particle energy limit, v 3 (lc /k)h
U 2 itot lc or  3 t ), of eff in equation (67) and combining the two limits, one can find a simplified general expression for eff given by eA  49 t þ :

ð68Þ

In the low suprathermal particle energy limit, eA  (4/9)t , and turbulent diffusion dominates microscopic diffusion by Alfve´n waves. The particles are strongly coupled to the Alfve´n waves ( is small), which, according to WKB theory, directly respond in turn to the density variations of compressive velocity fluctuations. Thus, the particles are approximately passively convected with the turbulent flow field via Alfve´n wave mediation. The particles experience turbulent diffusive motion on long timescales, assuming that strong dynamic wave turbulence effects associated with nonlinear wave interactions lead to random diffusive motions of the waves in which the particles are quasi-trapped. In the opposite limit, valid for high-energy suprathermal particles, the particles are weakly coupled to the small-scale Alfve´n waves, so compressive velocity fluctuations have little effect on the particles via Alfve´n wave mediation. Turbulent diffusion is unimportant and microscopic diffusion dominates. Consequently, eA   in this limit. While t is independent of particle speed,  / v4= 3 for standard QLT (Schlickeiser 1989), assuming that suprathermal particles interact resonantly with a Kolmogorov inertial range of the magnetic field fluctuation power spectrum of Alfve´n waves. Thus, at low suprathermal particle energies eff is independent of particle speed because of the dominance of turbulent diffusion, while at high suprathermal energies eA / v4= 3 , for which microscopic diffusion dominates. Note, however, that in the absence of dynamic turbulence, which is valid for strong but noninteracting compressive wave = turbulence ( ¼ 0 in eq. [34]), eA  ( /t )1 3 t in the low suprathermal energy limit, while eA   as before in the highenergy limit. Thus, the combined general expression becomes eA ¼


1=3  t þ : t

ð69Þ

For low suprathermal energies, turbulent diffusion occurs at a reduced level because the particles are quasi-trapped in waves that now exhibit only coherent motion. Some turbulent diffusion still occurs because  is small but finite, allowing particles to eventually cross into neighboring fluid elements in a random way on long timescales. 7.2. Weak Compressive Wave Turbulence By simply replacing eff in the expressions tcp and ˆtcp of equations (65) with , the theory simplifies to its linear limit, because particles are diffusive on a correlation scale by resonant scattering with small-scale Alfve´n waves. This implies that compressive wave turbulence is too weak to contribute. This would be the case for wavelengths l for which lc 3 l 3 rg in the inertial range of compressive fluctuations, where the energy in the fluctuations is much smaller. Thus, we replace lc with l in equations (65). We then find the following approximate general

Vol. 626

expression for a wavelength l after taking low and high suprathermal energy limits as above: " # 
U 2 tot (l ) ð70Þ t þ ; eA ¼ Vs2 1 2 (l )l. Because h
U 2 i1tot2 (l )TVs , turbulent where t ¼ h
U 2 itot diffusion is much weaker, so, when lTlc , one has to consider much lower suprathermal energies before turbulent diffusion dominates microscopic diffusion. This result also supports the assumption mentioned above that compressive fluid velocity fluctuations in the inertial range contribute the most to turbulent diffusion at maximum scales on the order lc. In the absence of dynamic turbulence (no nonlinear wave interaction, so  ¼ 0),  2
U tot (l ) eA ¼  þ   ; ð71Þ Vs2 =

=

and the effect of turbulent diffusion weakens even more, so it is negligible at all particle energies. The U term does not arise in the weak compressive turbulence transport expressions here and below, because it is negligible. 8. EFFECT OF COMPRESSIVE WAVE TURBULENCE ON STOCHASTIC ACCELERATION 8.1. Strong Compressive Wave Turbulence By substituting the expression for eff given by equation (68) into the momentum diffusion coefficient Dpp in equation (64) and taking the low and high suprathermal energy limits, we obtain  2
U tot 2 : ð72Þ Dpp  p (9=4)t þ  In the low suprathermal energy limit,  /h
U 2 i1tot2 Tlc (Tt ) and the expression for Dpp simplifies to Dpp  p2 h
U 2 itot /½(9/4)t , showing that particles gain energy when turbulent diffusion t in strong dynamic compressive wave turbulence dominates the microscopic diffusion . Although it is small, microscopic diffusion  is nonetheless essential for diffusing particles to gain energy stochastically, since it allows particles to randomly sample compressive regions (adiabatic energy gain) and rarefaction regions (adiabatic energy loss) on long timescales. Overall, the particles gain energy because they spend more time in compression regions, where the microscopic scattering rate is enhanced, compared to rarefaction regions, where particles spend less time because scattering is less efficient. = In the high-energy limit /h
U 2 i1tot2 3 lc we find that Dpp  2 2 1= 2 p h
U itot /, because microscopic particle diffusion across compressive fluid elements dominates turbulent diffusion of particles by the compressive fluid elements ( 3 t ). In addition, Dpp / v 2 at low suprathermal energies when turbulent diffusion dominates, while Dpp / v 2= 3 when microscopic diffusion due to Alfve´n waves dominates at high particle energies (assuming that  / v4= 3 according to standard QLT for Alfve´n waves, as discussed above). Thus, unlike eff , Dpp is strongly dependent on particle speed at low suprathermal energies but only weakly dependent on particle speed at high suprathermal speeds. In the absence of dynamical wave turbulence ( ¼ 0), we = find that Dpp ¼ p 2 h
U 2 itot /½t (t /)1 3  at low suprathermal energies, leading to a reduction in the effectiveness of stochastic = acceleration. At high energies, Dpp  p 2 h
U 2 i1tot2 /, as for the =

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PARTICLE TRANSPORT IN COMPRESSIVE WAVES

 6¼ 0 case. By combining the two limits, the general expression for Dpp becomes  2
U tot 2 : ð73Þ Dpp  p ðt =Þ1=3 t þ  The reduction in the effectiveness of turbulent diffusion when  ¼ 0 reduces the effectiveness with which particles can execute random motions across compression and rarefaction regions. The particles are quasi-trapped in fluid elements that execute mainly coherent motions, so particle acceleration by compressive fluctuations loses its stochastic character. 8.2. Weak Compressive Wave Turbulence In the case of weak compressive wave turbulence (
U TVs ), such as that at small wavelengths l in the inertial range where lTlc , we again replace eff with  in the expressions for tcp and tcp (eqs. [65]) before substituting them in the expression for Dpp (eq. [64]). After combining the low- and high-energy limits, we find that  2
U tot (l ) 2 : ð74Þ Dpp  p ½Vs l=(t þ )Vs l þ  In the low-energy suprathermal limit, Dpp is less effective compared to the case of strong compressive turbulence (see eq. [72]). When dynamic turbulence effects are neglected ( ¼ 0 in the expression for tcp ), the combined expression for Dpp becomes  2
U tot 2 ; ð75Þ Dpp  p ðVs l=ÞVs l þ  implying a further reduction in the efficiency of momentum diffusion at low suprathermal energies when compared to equation (74). This expression is in agreement with those derived by Ptuskin (1988) and Bykov & Toptygin (1993). 9. EFFECT OF COMPRESSIVE WAVE TURBULENCE ON COHERENT CONVECTION AND ENERGY CHANGES 9.1. Strong Compressive Wave Turbulence Since we had to assume that h
U 2 iþ  h
U 2 i in order to find a solution for eff , U   0, and no coherent convective and energy change effects on the particles by the compressive wave turbulence occur in this limit. 9.2. Weak Compressive Wave Turbulence After replacing eff by  in tcp given by equations (65), one finds an expression for U  that is valid for weak compressive wave turbulence with wavelengths lTlc in the inertial range, "  # 
U 2 dif  U ¼ ð76Þ Vs : V s2 þ (=lc )2

1129

Clearly, particle convection by weak compressive waves is ineffective because (h
U 2 i  )1= 2 TVs . 10. CONCLUDING REMARKS We have developed a nonlinear kinetic transport equation for particles diffusing along the large-scale background magnetic field in the presence of strong dynamic compressive wave turbulence and small-scale Alfve´n waves. We expect this transport formalism to be applicable near quasi-parallel shocks, such as ICME-driven shocks close to the Sun, and also at the termination shock at high heliolatitudes if the heliospheric magnetic field is a Parker-type magnetic field, where both theory and observations suggest an important role for compressive fluctuations in the form of enhanced ion-acoustic wave turbulence that propagates preferentially along the magnetic field. Our theory as given by equations (62) and (63) shows that the standard cosmic-ray transport equation needs to be modified at relatively low suprathermal particle energies. Strong compressive wave turbulence results in (1) turbulent diffusion, which dominates standard parallel diffusion due to Alfve´n waves so that parallel diffusion is independent of particle energy, (2) modification of particle convection and adiabatic energy changes by the largescale solar wind, and (3) efficient stochastic acceleration so that a momentum diffusion term with coefficient Dpp needs to be included. At low suprathermal energies, the Dpp term in equations (62) and (63) is more important than terms containing eff and U  , so one can, to a first approximation, neglect them. After assuming a steady state, one finds that the accelerated particle spectrum f has a hard power law, so f / v3 . At high suprathermal particle energies, the diffusion term with eff dominates the Dpp term. This domination becomes more pronounced with increasing particle energy. Thus, we expect the particle spectrum to develop an exponential rollover at high particle speeds. This suggests that, in the enhanced compressive turbulence near ICME-driven shocks or the termination shock, a suprathermal accelerated particle spectrum will develop that features a hard power law at low energies followed by an exponential rollover at high energies. This result is qualitatively consistent with observations of suprathermal ion spectra near ICME shocks (Gloeckler 2003). Such efficient stochastic acceleration of suprathermal ions can possibly serve as a first-stage acceleration process, resulting in efficient injection of particles into a secondstage diffusive shock acceleration mechanism. These issues will be explored in more detail in future work, when detailed numerical solutions of the new kinetic transport equation will be presented.

J. A. le Roux and G. P. Zank acknowledge support from NASA grant NAG5-11604 and NSF grant ATM-0112772.

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