Shock Acceleration of Ions in the Heliosphere
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- Lee, M.A., Mewaldt, R.A. & Giacalone, J. Space Sci Rev (2012) 173: 247. doi:10.1007/s11214-012-9932-y
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Energetic particles constitute an important component of the heliospheric plasma environment. They range from solar energetic particles in the inner heliosphere to the anomalous cosmic rays accelerated at the interface of the heliosphere with the local interstellar medium. Although stochastic acceleration by fluctuating electric fields and processes associated with magnetic reconnection may account for some of the particle populations, the majority are accelerated by the variety of shock waves present in the solar wind. This review focuses on “gradual” solar energetic particle (SEP) events including their energetic storm particle (ESP) phase, which is observed if and when an associated shock wave passes Earth. Gradual SEP events are the intense long-duration events responsible for most space weather disturbances of Earth’s magnetosphere and upper atmosphere. The major characteristics of gradual SEP events are first described including their association with shocks and coronal mass ejections (CMEs), their ion composition, and their energy spectra. In the context of acceleration mechanisms in general, the acceleration mechanism responsible for SEP events, diffusive shock acceleration, is then described in some detail including its predictions for a planar stationary shock, shock modification by the energetic particles, and wave excitation by the accelerating ions. Finally, some complexities of shock acceleration are addressed, which affect the predictive ability of the theory. These include the role of temporal and spatial variations, the distinction between the plasma and wave compression ratios at the shock, the injection of thermal plasma at the shock into the process of shock acceleration, and the nonlinear evolution of ion-excited waves in the vicinity of the shock.
KeywordsParticle accelerationSolar energetic particlesDiffusive shock acceleration
Populations of energetic ions and electrons are major constituents of the heliospheric plasma environment. They are ubiquitous, extending from sites in the lower corona to the interface of the heliosphere with the interstellar medium. The energies of these particles extend from solar wind energies up to ∼10 GeV for ions and ∼100 MeV for electrons. In addition to these heliospheric populations are the galactic cosmic rays (GCR) that penetrate the heliosphere from interstellar space; these particles do not originate in the heliosphere and are described in the companion papers by Helder et al. (2012, this issue) and Schure et al. (2012, this issue). The majority of the observed populations are associated with heliospheric shock waves. However, other populations have no obvious association with shock waves and their acceleration mechanisms are less certain.
The focus of this Chapter is the acceleration of these populations. The basic acceleration mechanisms are few. Nevertheless, the observed features of a particular population generally depend on complexities such as geometry, magnetic field orientation, temporal dependence, specific parameters, seed particle origin, and variability. Although we shall describe the basic acceleration mechanisms with some generality, we shall limit our detailed discussion of the observations and their interpretation to ESP and large SEP events. The other populations are described elsewhere in this volume: the diffuse ions and related field-aligned-beams at Earth’s bow shock are described by Burgess et al. (2012, this issue); the smaller so-called impulsive SEP events presumably accelerated by magnetic reconnection processes in flares are presented by Raymond et al. (2012, this issue) and Cargill et al. (2012, this issue); the quiet-time suprathermals are discussed by Fisk and Gloeckler (2012, this issue); finally, the ACRs and their spatial distribution in the boundary regions of the heliosphere are described by Giacalone et al. (2012, this issue).
In Sect. 2 we review the observations of SEP and ESP events, in Sect. 3 we present the basic theory of particle acceleration relevant for the heliosphere, and in Sect. 4 we address extensions of the simplest theory of diffusive shock acceleration that are required to account for interplanetary observations of events associated with shocks.
2 Observations of SEPs and ESP Events
2.1 Solar Energetic Particle Composition and Energy Spectra
Although it has long been known that the composition of SEP events is highly variable, Solar Cycle 23 was the first cycle for which there were composition studies of heavy ions in SEP events with good statistical accuracy over a broad energy interval, including resolution of individual elements, isotopes, and ionic charge states. These new capabilities have led to several revisions in understanding the origin of SEP compositional variations.
This behavior can be explained if the spectra break at the same value of the diffusion coefficient as SEP ions escape the shock during acceleration (Cohen et al. 2005; Mewaldt et al. 2005). For typical rigidity-dependent diffusion coefficients, Fe spectra will break at lower energy/nucleon than lighter ions because Fe has a higher rigidity than lighter ions at the same energy/nucleon. Li et al. (2005) predicted that for quasi-parallel shocks the break-energy per nucleon would scale as (Q/M)s with s≈2, which is close to the observed dependence in Fig. 7 (right). However, Li et al. (2009) later showed that values between s≈0.2 (for a quasi-perpendicular shock) and s≈2 were possible depending on the magnetic obliquity of the shock and other conditions. It is hoped that multi-spacecraft studies during Solar Cycle 24 with STEREO and L1 spacecraft that include ACE, SOHO and Wind will test ideas about the role of shock geometry in shaping SEP composition and energy spectra.
The Fe-rich SEP events in Fig. 6 typically also have other unusual properties, including enrichments in 3He and highly-ionized charge states of Fe and other heavy elements (Cohen et al. 1999a; Mewaldt et al. 2006; Desai et al. 2006a). There have been several proposed explanations for these compositional features, including the acceleration of remnant seed-particles from earlier small 3He-rich “impulsive” events (Mason et al. 1999b; Tylka et al. 2005; Mewaldt et al. 2006; Tylka and Lee 2006), mixtures of flare and shock-accelerated ions (e.g., Cane et al. 2003, 2006); shock acceleration of a mixture of solar wind and escaping flare particles (Li and Zank 2005a), and the acceleration of a mixture of suprathermals and ICME material (Mewaldt et al. 2007; Li et al. 2012). While all of these may occur, in our opinion there is more direct evidence to support the first of these explanations (acceleration of remnant suprathermal ions) as evidenced by the ubiquitous presence of suprathermal 3He in interplanetary space at 1 AU (Mason et al. 1999b; Wiedenbeck et al. 2003) and by the frequent overabundance of 3He in gradual SEP events (Cohen et al. 1999a; Desai et al. 2006a; Mewaldt et al. 2006).
2.2 Solar Energetic Particles, CMEs, and Flares
A variety of evidence has shown that the Sun can accelerate particles to high energy in at least two ways. In large solar flares X-ray and γ-ray data show that particles can be quickly accelerated to GeV energies in the low corona (see, e.g., reviews by Lin 2011 and Raymond et al. 2012, this issue). The energy released in large flare events that is ultimately responsible for particle acceleration is understood to be due to the reconnection of opposing magnetic fields. It appears that most of the flare-accelerated particles in these events lose their energy in the solar atmosphere and do not escape the Sun. Reconnection events also happen on much smaller scales and it is estimated that there are ∼10,000 small particle-acceleration events per year that can be observed at 1 AU in sub-MeV electrons and ions (Wang et al. 2012; Raymond et al. 2012, this issue). On the other hand, most of the largest SEP events observed at Earth are associated with large, fast CMEs, and are widely (but not universally) believed to be due to shock acceleration processes. The number of these events observed near Earth per year at solar maximum is ∼10 to 20 (see, e.g., the list in Cane et al. 2006).
2.3 Solar Energetic Particles and CME Properties
Several authors have shown that there is a correlation between the peak intensity of SEPs and CME speed (Reames 2000; Kahler 2001), as might be expected from the theory of shock acceleration (see Sect. 3.2). However, the peak intensity of SEPs associated with a given CME speed varies over ∼4 orders of magnitude, suggesting that there are also other important variables that determine accelerated particle intensities.
In an interdisciplinary study, Emslie et al. (2004) combined a wide range of imaging and in situ data to measure the energy budget of two large SEP events, April 21, 2002 and July 23, 2002. Included were estimates (and uncertainties) for the CME, thermal plasma at the Sun, hard X-rays produced by accelerated electrons, γ-ray producing ions, and solar energetic particles (a later paper by Emslie et al. 2005 gave revised estimates of flare thermal energies and also estimated the total irradiated energy in the two events). The best estimates of the various components indicated that the CME contained the greatest fraction of the released energy in both events.
2.4 What Do CME-Driven Shocks Accelerate?
Although it had earlier been assumed that CME-driven shocks accelerate mainly solar wind, it was shown that the composition of large SEP events differs in systematic ways from the solar wind composition, including, e.g., depletions of C, Ne and S (Mewaldt et al. 2006). In addition, many large SEP events are found to have large admixtures of species that are rare in the solar wind but present in the suprathermal pool, including 3He (Cohen et al. 1999a; Mason et al. 1999b), He+ (Kucharek et al. 2003), and highly-ionized charge states of Fe (Mazur et al. 1999; Cohen et al. 1999b; Tylka et al. 2001; Klecker et al. 2007). Theoretical studies also show that suprathermal ions are much more easily injected into the acceleration process (see Sect. 4.3).
2.5 Pre-conditioning by Previous CMEs
Li et al. (2012) suggested a two-CME scenario associated with a pseudo-streamer-like configuration in which reconnection between closed magnetic field lines that drape the preceding CME and open field lines draping the subsequent CME lead to an enhanced seed population and higher turbulence levels. Other suggested explanations for the observations include differences in the open and closed field-line geometry, and a lowering of the Alfven velocity, leading to the formation of a stronger shock (Gopalswamy et al. 2004). All of these suggestions could benefit from additional modeling of the processes involved, and from in situ observations of conditions closer to the Sun in the wake of CMEs, as expected from Solar Probe Plus and Solar Orbiter near the end of this decade.
2.6 Energetic Neutral Atoms—A New Window into SEP Acceleration Processes
After considering alternatives, it was concluded that the precursor was made up of energetic neutral H atoms (ENAs). The measured kinetic energies were used to calculate the ENA emission profile from the Sun as shown in Fig. 12. Note that the ENA emission profile is consistent with the soft X-ray time profile, suggesting that the ENAs were due to the charge exchange of flare-accelerated particles. This is certainly one possibility, but it requires that a significant fraction of the flare-accelerated protons make it into the high corona, or else the ENAs would be stripped before leaving the Sun. This conclusion is based in part on the estimated number of flare-accelerated protons derived from RHESSI γ-ray observations.
A second possibility illustrated in the right panel of Fig. 12 is that the ENAs could be produced by charge exchange of protons accelerated by the CME-driven shock (assuming CME properties typical of large SEP events—see Mewaldt et al. 2010). Note that there was a type-II radio burst in this event, indicating that a shock formed, but unfortunately there were no coronagraph observations to further test this possibility.
The discovery of ENAs associated with large solar events opens up a new window into SEP acceleration and transport that can reveal when, where, and how low-energy solar protons are accelerated, interact with coronal material, and escape from the Sun.
2.7 Ground Level Solar Energetic Particle Events
2.8 Future Prospects
The combination of STEREO and near-Earth spacecraft are now providing a 360∘ view of the Sun and interplanetary medium. This combination should be able to address the questions of the relative contributions of flare and shock-accelerated particles in large SEP events, the importance of shock geometry, and issues of how particles are transported in longitude.
In 2017 and 2018 the Solar Orbiter and Solar Probe Plus missions will be launched and begin their journeys to the inner heliosphere, providing the opportunity to explore particle acceleration in situ within the prime acceleration region of CME-driven shocks (down to 9.5 solar radii), and close to the source of many of the populations of suprathermal seed particles. These new missions, aided by imaging and modeling, will directly or indirectly address most, if not all, of the issues presented here.
3 Basic Physical Processes and Theories of Particle Acceleration
The first term on the right hand side of Eq. (1) is the electric field due to the bulk motion of the plasma. If the bulk velocity is nearly uniform then we may transform to that frame of reference in which V=0, and therefore, due to this term, E=0. In order to obtain particle acceleration (or deceleration) from this term V must vary. Thus, acceleration by this term is intimately connected to the spatial and temporal structure of V as recognized by Fermi (1949, 1954). V(x,t) may vary coherently and systematically or stochastically, leading to both first-order and second-order Fermi acceleration. The second term on the right hand side of Eq. (1) is the electric field due to the MHD fluctuations δV and δB. A broad spectrum of waves or turbulence leads inherently to a stochastic vector field δE(x,t) and a version of stochastic or second-order Fermi acceleration.
Of course, the actual acceleration process depends not only on the structure of V(x,t) or δE(x,t), but also on the transport of the accelerating particles. For example, even if V(x,t) is inhomogeneous, acceleration according to the first term on the RHS of Eq. (1) is impossible if the particles do not have access to the regions of inhomogeneity. In principle transport is governed by the Vlasov (or collisionless Boltzmann) equation. In practice, however, even with the help of modern computers, progress is limited without further simplifications. Most of the particle populations described in Sect. 1 are characterized by speeds v≫|V|, by nearly isotropic distributions, and by spatial scales larger than pitch-angle scattering mean free paths. There are of course exceptions; for example, early in SEP events, particles are observed to stream away from the Sun with high anisotropy. As another example, the velocities of particles extracted and accelerated out of the solar wind cannot satisfy v≫|V|. Nevertheless, these generally observed simplifications provide a valuable place to start in describing the transport of energetic particles in the heliosphere.
3.1 The Transport Equation of Parker
A systematic expansion [(∇⋅V)>0] gives rise to systematic deceleration. A heliospheric example is the adiabatic deceleration of GCRs in the solar wind, a process that motivated Parker to develop Eq. (2) in the first place. Similarly, an isolated compression [(∇⋅V)<0] gives rise to systematic acceleration. If the spatial scale of the convective-diffusive gradient is smaller than the spatial scale of the compression, then the energy gain is adiabatic (Drury et al. 1982). However, if the reverse is true, as is generally the case at a shock, then spatial diffusion allows some particles to traverse the shock many times. These particles sample the compression many times and are accelerated to high energies. This is the essence of diffusive shock acceleration (DSA) (Krymsky 1977; Axford et al. 1977; Bell 1978; Blandford and Ostriker 1978).
3.2 Planar Stationary Diffusive Shock Acceleration
3.3 Shocks Modified by the Energetic Particles
Equation (17) illustrates the growth of P(z) through the shock from zero to, for this case of infinite Mach number, the value required for a strong shock. The pressure gradient decelerates the flow to the downstream density and velocity required by the Rankine-Hugoniot relations. In fact, shock modification is generally a small effect in the heliosphere; for example, the deceleration of the solar wind upstream of Earth’s bow shock due to the accelerated proton pressure is ∼5 %. Nevertheless shock modification may be more important in large SEP/ESP events in which a previous shock provides a population of advected energetic particles that are reaccelerated by the following shock wave as described in Sect. 2.5 (Eichler 1981; Gopalswamy et al. 2002). There is also evidence that the solar wind termination shock is modified by the low-energy ACRs (Florinski et al. 2009).
3.4 Wave Excitation by the Energetic Particles
3.5 Stochastic Acceleration
The last term on the LHS of Eq. (2) describes the process known as second-order Fermi acceleration, or more generally, stochastic acceleration. As it appears in Eq. (2) it describes the random walk of particles in p-space as they sample a stochastic sequence of electric field vectors δE due to MHD fluctuations (e.g. Lee and Völk 1975; Bogdan et al. 1991). The diffusion in p-space requires a broad spectrum of fluctuations and is dramatically enhanced if the particles and waves are in resonance. A stochastic acceleration term identical in form is also recovered from the first four terms on the LHS of Eq. (2) if it is assumed that V fluctuates (Jokipii and Lee 2010). In this case the process describes the random walk of particles in p-space due to the spatial random walk of particles through regions of compression and expansion where the particles are accelerated and decelerated, respectively. The velocity diffusion equation results from the spatial random walk of the particles in the presence of either a coherent variation of V(x,t) (Ptuskin 1988; Zhang and Lee 2011) or a stochastic sequence of δV(x,t) (Jokipii and Lee 2010). Although the stochastic diffusion coefficient D depends on the type of fluctuations contributing to δE, this process generally suffers from small values of D∝[Mω/(kp)]2≪1 and limited energy density in the fluctuations. An exception may be the stochastic acceleration of SEPs in an impulsive solar flare event in the lower corona where the Alfven speed can be large so that D may be substantial. For the other energetic particle populations described in Sect. 1 shock acceleration is the more promising acceleration mechanism.
4 Complexity of Shocks and Their Associated Particles in Space
In addition, at interplanetary shocks adiabatic deceleration due to the expansion of the solar wind can compete with DSA and may also produce an exponential rollover in the particle energy spectra (Fisk and Lee 1980; Reames et al. 1997c).
4.1 The Dependence of the Power-Law Index −β on Compression Ratio
4.2 Mechanisms of Ion Escape Upstream of the Shock
Although Eqs. (5)–(7) describe the loss of ions upstream of the shock by invoking a rapidly increasing spatial diffusion coefficient with increasing z, the actual cause of the ion loss and the exponential rollover of the distribution function is more complex. The time dependence of shock acceleration can lead to the evolution of the foreshock ion distribution and energy spectrum toward the form given by Eqs. (5) and (6) for ζ∞→∞. Since interplanetary travelling shocks tend to weaken with time, a more likely possibility is that the proton-excited wave enhancement decays with increasing heliocentric radial distance so that Kzz increases with time. In addition, the escape of ions is enhanced by anti-sunward magnetic mirroring in the field upstream of the shock (Lee 2005). This process is included in the focused transport equation (e.g. Roelof 1969; Earl 1976; Isenberg 1997). A similar effect is also obtained in solutions of the telegrapher’s equation (modeling the diffusive escape of particles which have a finite speed), which show that some particles are able to race ahead of the diffusive wake (Morse and Feshbach 1953). Planetary bow shocks are sufficiently small that drift transport across field lines, or advection of a given flux tube along the shock surface, can transport the accelerating particles to a weaker portion of the shock where acceleration is slower or ions may escape upstream through the weaker turbulence. The advection of flux tubes along the shock surface is also thought to play an important role at the solar wind termination shock where the ACRs are primarily accelerated on the flanks and towards the heliospheric tail rather than near the nose (McComas and Schwadron 2006; Schwadron et al. 2008; Kota 2008).
The upstream escape of particles at shocks in the solar wind is important in order to account for the observed temporal structure of SEP events. The escaping particles are first to arrive at Earth orbit, they forecast the arrival of higher SEP intensities at later times, and, since they were accelerated closer to the Sun where acceleration rates are higher, they usually contain the highest energy particles. As is evident in Eq. (11), the acceleration rate is inversely proportional to the spatial diffusion coefficient. Since the scattering mean free path generally scales with a power of the particle gyroradius, the acceleration rate is controlled by the gyrofrequency Ω, which is proportional to B. Within the orbit of Earth B∝r−2, which is much larger close to the Sun. In the model described by Eqs. (4)–(7) the escaping particle flux is given by expression (7). According to Eqs. (19) and (20), if I∝k−α, then Kzz∝v(p/Q)2−α, which implies the scattering mean free path λ∝(p/Q)2−α. Since the parameter α is generally observed to lie in the range 1<α<2, Kzz generally increases with increasing p. Therefore, the denominator on the RHS of Eq. (7) becomes ζ∞ at high energies and the escaping flux is harder than the intensity at the shock. An interesting consequence of the dominance of proton-excited waves in the foreshock is evident in Eq. (22). Since ζ∞∼f0−f∞, the escaping flux at high energies is approximately independent of the accelerating particle intensity as long as it is large enough to produce wave intensities larger than that in the ambient solar wind. This feature greatly reduces the variability of the escaping flux and leads to the “streaming limit” of the escaping proton intensity identified and interpreted by Reames and colleagues (Reames 1990; Ng and Reames 1994; Reames and Ng 1998, 2010).
4.3 Ion Injection at the Shock Front
Perhaps the most complex and controversial issue within the theory of shock acceleration is the rate at which thermal plasma is injected at the shock front into the process of DSA. Equation (2) formally requires near isotropy of the particle distribution function. This condition is clearly not satisfied in the frame of the shock for the bulk of the upstream thermal plasma, which impinges on the shock front with Mach number greater than unity. Even though Eq. (2) is not satisfied, a small fraction of the incident ions is reflected at the shock front, drifts parallel to the motional electric field given by the first term on the RHS of Eq. (1), and is sufficiently mobile and energetic to propagate back upstream. These ions have begun the process of first-order Fermi/shock acceleration even though they may not formally satisfy near isotropy and v≫V as required for the validity of Eq. (2). This injection process has been observed in detail at Earth’s bow shock as a solar wind flux tube is swept into the bow shock. When the angle between the upstream magnetic field and the shock normal, ψ, is reduced to ψ≈65∘–70∘ (Paschmann et al. 1981; Bonifazi and Moreno 1981), field-aligned beams of protons begin to stream along the upstream magnetic field away from the bow shock with energies of a few keV/nucleon. A similar process presumably occurs at interplanetary shocks although their nearly stationary structure with turbulent upstream conditions adjacent to the shock must produce a more diffuse injection at the shock front.
It is evident from this discussion that the magnetic obliquity of a shock is critical for injection of thermal plasma. This issue is complicated by the meandering of magnetic field lines on scales much larger than the energetic particle gyroradii so that the shock obliquity varies not only in response to shock warps but also due to field line meandering. This variation contributes to the spatial and temporal variation of the energetic particle intensity along the shock front as injection rates and acceleration rates vary [see Eqs. (11) and (18)] since Kzz depends on obliquity. The meandering also contributes in general to K⊥, which describes spatial diffusion perpendicular to the average magnetic field (Jokipii 1971).
An interesting question is whether there is an effective energy threshold for injection as a function of ψ. A reasonable answer is “yes” since a particle can only access the upstream plasma if ν>Vsecψ, unless transport normal to the average field is effective, as would be expected if there is significant field-line meandering. It is important to note that such a particle may have originated in the upstream thermal distribution and subsequently accelerated at the shock front to exceed the threshold. Thus, this threshold does not provide an injection rate since it is unclear what fraction of the incident thermal distribution attains the threshold. In any case, if this condition is satisfied for an ion reflected upstream, then the first-order Fermi acceleration process is underway.
However, we emphasize that field-line meandering can strongly enhance cross-field transport of low-energy ions (e.g. Giacalone and Jokipii 1999). In fact, it can be shown that for very low-energy ions moving within magnetic turbulence that is typical of that in the solar wind, the perpendicular diffusion coefficient K⊥ can exceed KA. In this case, the injection threshold at a perpendicular shock is only slightly larger than that at a parallel shock. In fact, self-consistent hybrid simulations have shown that even thermal plasma can be accelerated by a perpendicular shock moving through a plasma containing pre-existing large-scale magnetic fluctuations (Giacalone 2005).
In spite of the uncertainty in the physics of the injection process, Eq. (11), together with Eq. (18) and K∥≫K⊥, show that the acceleration timescale at nearly perpendicular shocks is much shorter than that at a parallel shock (Jokipii 1987). If a particular shock event has quasi-parallel and quasi-perpendicular phases along the observer’s magnetic field line, then particles accelerated in the quasi-parallel phase should have higher intensity (due to higher injection rates out of the solar wind plasma) and lower energy (due to smaller acceleration rates). Similarly, the particles accelerated in the quasi-perpendicular phase should have lower intensity (due to lower injection rates of solar wind plasma, but straightforward injection of remnant suprathermal or energetic particles) and higher energy (due to higher acceleration rates). Since the solar wind and remnant energetic ions have markedly different composition (e.g. Mason et al. 1999a; Desai et al. 2006b; Mewaldt et al. 2007; the remnant particles contain higher fractions of heavy ions, notably Fe, and 3He), one expects compositional fractionation between high and low energies. Tylka and Lee (2006) have found that the fluence ratio Fe/O as a function of energy supports this scenario. Papers that emphasize the importance of SEP contributions from the concomitant flare include Cane et al. (2003, 2006), and Masson et al. (2009). This issue is also discussed in Mewaldt et al. (2007) and Tylka et al. (2005). Clearly the energetic particles observed by a particular observer in a specific event arise from a superposition of sources and transport trajectories, each of which experienced acceleration at a sequence of varying shock configurations in space and time. Attempting to infer the source distribution, and the temporal and spatial evolution of the shock, from the measured energetic particle time profiles at one (or even two or three locations) is a challenge indeed.
4.4 Ion Shock Acceleration Close to the Sun
A common criticism of the shock origin of gradual events is the prompt arrival of ∼100 MeV/nucleon ions in some events, implying escape from the shock when the CME is only ∼4 Rs (solar radii) from the Sun. When combined with the time required to form the shock wave, accounting for such acceleration efficiency is indeed a challenge. However, it must be recognized that B∝r−2 within the inner heliosphere. Since λ∥∝rg∝r2, the timescale for acceleration τa satisfies τa∝r2, implying extremely efficient acceleration close to the Sun. In addition, although the CME-driven shock can be as rapid as 2000–3000 km/s, the Mach number, and therefore also the compression ratio, may not be large and in any case is difficult to determine. The Alfven speed dominates the MHD fast speed in the corona and depends on the complex magnetic field structure at the flaring active region. One popular radial profile of the Alfven speed is presented by Mann et al. (2003), which increases from ∼400 km/s to ∼700 km/s between 1.5 and 4 Rs from Sun center, and then gradually decreases as VA∝r−1 out to ∼20 Rs before approaching a constant value at large heliocentric radial distances. Although the Mach number and compression ratio are not large at the peak of the Alfven speed, acceleration to high energy could be efficient at either r<4 Rs if the shock has formed in this low region of the corona, or in the range r>4 Rs before the CME has begun to decelerate. Clearly a detailed model of the shock driven by the CME is required for a quantitative assessment of particle acceleration near the Sun. Nevertheless shock acceleration close to the Sun appears to be the most promising acceleration mechanism for the SEP phase of gradual events. It will be very exciting to observe SEP events with Solar Orbiter and Solar Probe close to the Sun.
4.5 Long-Time Behavior of SEPs
Another interesting feature of SEP events is their long-time behavior. Multi-spacecraft studies have shown that large events spread nearly uniformly throughout the inner heliosphere in their decay phase with “invariant” power-law energy spectra for all ion species (Reames et al. 1997a, 1997b). The events decay, presumably by adiabatic deceleration and slow escape to large heliocentric radial distances. The escape rate is presumably reduced by the Archimedes spiral magnetic field, which creates a confining “reservoir” in the inner heliosphere. As these events extend in volume they may coalesce with other events to form “super events” (McKibben 1972). Some coalesced events may still be modified by a large shock remaining from a coalescence of individual shock events (Pyle et al. 1984). In other events the shocks decay while their accelerated particles escape the decaying turbulence adjacent to the shock and fill the inner heliosphere. To our knowledge a detailed theoretical description of events transitioning to, or in, the decay phase does not exist. During solar maximum conditions these events may ensure a supply of remnant energetic particles for reacceleration at new shock waves.
4.6 Upstream Nonlinear Waves
Finally, we emphasize that shocks, including particle acceleration as one important channel of dissipation, are nonlinear structures involving coupled large-amplitude changes in the particle parameters and electromagnetic fields. Although the simplest versions of DSA describe the energetic particles using Eq. (2) as a linear equation in which the coefficients are independent of the energetic particle distribution, Eqs. (13)–(16) show that the energetic particles may influence the plasma bulk flow and Eq. (22) shows that the energetic particles excite waves that modify the diffusion coefficients. Other nonlinearities occur when the excited waves grow to large amplitude. Nonlinear wave-wave interactions can modify the excited wave intensity, and therefore the spatial diffusion coefficient. Upstream of Earth’s bow shock the compressive large-amplitude magnetosonic waves are observed to steepen into “shocklets” with whistler precursors (Hoppe et al. 1981; Hada et al. 1987). Even more dramatic are the Short Large-Amplitude Magnetic Structures (SLAMS) observed at Earth’s bow shock (Schwartz and Burgess 1991; Lucek et al. 2008) that grow to magnetic field magnitudes several times that of the ambient field, presumably due to their effective reflection of the accelerating ions and the effective transfer of energy from the ions to the waves. A related phenomenon that may occur at a very strong shock with a turbulent foreshock was predicted by Bell and Lucek (2001) and further developed by Bell (2004) (see also Schure et al. 2012, this issue): The foreshock turbulence and the discrepancy between the energetic ion and plasma electron gyroradii hinder the cancellation of the energetic ion and plasma currents. This results in a Lorentz force on the bulk plasma and a nonresonant instability that in turn leads to an enhancement of the average foreshock magnetic field strength. The increased field strength reduces the spatial diffusion coefficient and the acceleration timescale [according to Eq. (11)] with important consequences for cosmic ray acceleration at supernova remnant shocks.
With a variety of sites and plasma configurations, and measurements by many spacecraft in different locations, the heliosphere is an ideal space plasma environment in which to investigate particle acceleration. Based on these detailed observations, and the theories and understanding they generate, inferences may be drawn about the acceleration processes that operate at various sites throughout the Galaxy and beyond. In this review we have focused on SEP events and their acceleration at CME-driven coronal/interplanetary shock waves. Although the observed temporal/spatial behavior, energy spectra and composition of these events are broadly consistent with the theory of diffusive shock acceleration in its simplest form, there remain puzzles, discrepancies, and therefore challenges. The broad range of SEP intensities observed with similar CME speeds is one well-advertised puzzle. The lack of a theory of thermal plasma injection at the shock front as a function of shock strength and magnetic obliquity is another. The rate of ion escape upstream of the foreshock, which affects the form of the energy spectral rollover, is yet another, as is the nonlinear evolution and magnetic field amplification in the foreshock that dictates the acceleration timescale. Some of these puzzles hinge on our inadequate understanding of particle propagation in turbulent electromagnetic fields and the limitations of diffusive spatial transport. Others depend on the restrictions of the Parker transport Eq. (2) to high particle speeds and small anisotropy in velocity space. All of these issues require interesting modifications and extensions of the simplest theory guided by current and future detailed measurements. Solar Cycle 24 will provide new events and in 2017/2018 Solar Orbiter and Solar Probe Plus will provide a completely new view of SEP events close to the Sun.
The authors wish to acknowledge the hospitality of the International Space Science Institute, the stimulating program of the Workshop on Particle Acceleration in Cosmic Plasmas, and the patience of Dr. Andre Balogh while awaiting this typescript. The contribution of M.A.L. was supported, in part, by NASA grants NNX08AJ13G, NNX11AO97G and NNX12AB32G. The contribution of R.A.M. was sponsored by NASA under grants NNX8AI11G, NNX06AC21G, and under subcontract SA2715-26309 from UC Berkeley under NASA contract NAS5-03131. The contribution of J.G. was supported, in part, by NASA grants NNX11AO64G and NNX10AF24G.