monte carlo simulations of coronal diffusive shock acceleration in self

The Astrophysical Journal, 658:622 Y 630, 2007 March 20 # 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.

MONTE CARLO SIMULATIONS OF CORONAL DIFFUSIVE SHOCK ACCELERATION IN SELF-GENERATED TURBULENCE R. Vainio Department of Physical Sciences, University of Helsinki, Finland

and T. Laitinen Department of Physics, University of Turku, Finland Received 2006 July 27; accepted 2006 October 17

ABSTRACT We report on Monte Carlo simulations of solar energetic particle (SEP) acceleration at quasi-parallel coronal shocks under the influence of self-generated Alfve´n waves. The results indicate that the accelerated particles amplify ambient Alfve´n waves efficiently and that the solution close to the shock can be qualitatively described with the results from quasi-steady theories of diffusive shock acceleration, provided that the acceleration and injection parameters do not change rapidly. The escape of the first particles to the interplanetary medium occurs before the waves have grown appreciably to trap the particles in the vicinity of the shock wave. The escape process is well described by the analytical model developed by Vainio, at least for the promptly escaping component. In addition to the compression ratio and speed of the shock wave, the rate of injection of low-energy particles to the acceleration process is a key factor for the acceleration efficiency of shocks that are driven by coronal mass ejection. Quasiparallel coronal shocks seem to be capable of accelerating suprathermal protons up to 100 MeV and beyond after some number of minutes. Extrapolations of our simulation results indicate, however, that the wave intensities may reach nonlinear values before acceleration to GeV energies occurs in the corona. This may mean that the quasi-linear approach has to be replaced by a more general theory to describe particle acceleration at quasi-parallel coronal shocks in the largest SEP events. Subject headingg s: acceleration of particles — shock waves — Sun: particle emission — turbulence

1. INTRODUCTION

wave growth in the solar corona at the heliocentric distance r can be given as (Vainio 2003) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R
2 n
vA
vA
1 1033 d 2N n
vA
p ’ ; ’ dp d

!cp
vA (r) p 2 ; 108 cm3 vA (r)

Particle acceleration in gradual solar energetic particle (SEP) events is one of the open questions on the nature of solar eruptions. These particles are commonly thought to be accelerated at shocks driven by coronal mass ejections (CMEs). The mechanism responsible for particle acceleration is believed to be diffusive shock acceleration (Axford et al. 1977; Krymsky 1977; Bell 1978; Blandford & Ostriker 1978), which requires a turbulent coronal medium to exist ahead of the CME-driven shock. It is well known that such conditions can be created ahead of the shock by the accelerated particles themselves through the instability of Alfve´n waves resonant with the streaming accelerated particles ( Bell 1978). Quantitative models employing this scenario have so far relied on quasi-stationary descriptions of the coupled particle acceleration and wave-generation process ( Zank et al. 2000; Li et al. 2003; Rice et al. 2003; Lee 2005) with the exception of some important work in progress by Ng and coworkers (e.g., Ng 2004). In the quasi-stationary models, the wave and particle spectra in the vicinity of the shock wave are modeled using the assumption that they quickly adjust to a steady state in which the maximum energy of the particles and, correspondingly, the lowest wavenumber of the waves are determined by the dynamical timescale of the shock evolution. However, it was shown by Vainio (2003) that time dependence is a more complicated factor of the acceleration and wavegeneration process: a finite number of accelerated particles has to stream across a point in space before waves in that location have grown appreciably. The momentum spectrum of injected particles (per steradian at the solar surface) required for appreciable

ð1Þ where vA is the Alfve´n speed, !cp is the (angular) proton gyrofrequency, n is the electron density in the corona, and the subscript
refers to quantities measured at the solar surface, r ¼ R
. This threshold spectrum is inversely proportional to particle momentum and at high energies (>10 MeV ) constitutes a nonnegligible fraction of the total number of accelerated particles injected from the corona even in the largest SEP events. Because the quasi-stationary modeling typically assumes the waves to be available for scattering the particles immediately, it is unclear whether such modeling can correctly describe the acceleration process. Thus, (numerical) models including full time dependence are urgently needed to verify the validity of the assumption. The process responsible for the escape of particles from the vicinity of the CME shock is also still under discussion. At least two possibilities have been investigated to explain this. In the quasi-stationary modeling, the process responsible for the escape is magnetic focusing. One can show that particle escape from the shock occurs at a distance where the self-generated wave intensities are low enough for the focusing process to win over scattering and force the particles to leave the shock (Lee 2005). In the time-dependent modeling, one finds another way for the particles to escape upstream. Particles that are accelerated before the waves have grown substantially can escape without problems from the 622

CORONAL SHOCK ACCELERATION

623

shock, leading to a spectrum of escaping particles given by equation (1) multiplied at most by a factor of a few, but as mentioned above, it is not clear whether the acceleration is efficient before substantial wave growth has occurred. In this paper we take the first steps toward a fully time-dependent coronal shock acceleration model including self-generated waves. We have tried to keep the model as simple as possible to achieve maximum numerical efficiency, but at the same time tried to include the essential features of the model to study the effects of full time dependence on shock acceleration in self-generated waves. Our aim in this first study is not to perform a complete parametric study of the model, but we do concentrate on the basic requirements for efficient particle acceleration at CME-driven shocks and check the crucial assumptions of steady state modeling. The structure of the paper is as follows. In x 2 we briefly describe the model (a detailed description will be published elsewhere), in x 3 we present the results, and in x 4 we discuss the results and present our conclusions. 2. MODEL Our model consists of a background solar wind with speed u(r), electron density n(r), and a radial coronal/interplanetary magnetic field B(r), all predetermined functions of the heliocentric radial distance r, a shock wave propagating at constant speed Vs parallel to the mean magnetic field, and a population of accelerated particles and outward-propagating Alfve´n waves traced on top of the background. The particles are simulated using the Monte Carlo technique in modeling the pitch-angle scattering caused by the Alfve´n waves. The scattering frequency (v) is determined by the power spectrum P( f ; r) of the Alfve´n waves resonant with the particles. The particle distribution is used in turn to calculate the growth rate  ( f ; r) of the Alfve´n waves. The waves are modeled using Wentzel-Kramers-Brillouin ( WKB) theory complemented by wave growth and a diffusion term in frequency space. For the solar wind parameters we use the model of Vainio et al. (2003). In this ad hoc model, the phase speed of the outwardpropagating Alfve´n waves, V ¼ u þ vA ;

ð2Þ

is taken to be spatially constant, and the magnetic field is assumed to be radial with a magnitude modeled as "  6 # r  2 R
 1 þ bf ; ð3Þ B ¼ B0 r r corresponding to an overexpanding field geometry (see Fig. 1), where vA ¼ B/ð0 mp nÞ1/2 is the Alfve´n speed, the plasma is assumed to be fully ionized hydrogen, r ¼ 1 AU, and bf is the flux tube expansion coefficient, which we fix at bf ¼ 1:9. The field magnitude at 1 AU is taken as B0 ¼ 2:90 nT, and the density at 1 AU is fixed by taking the Alfve´n speed there to be 20 km s1. The solar wind speed at 1 AU is taken to be 380 km s1. Using these assumptions together with the conservation of mass, nu /B ¼ const, gives a solar wind model that agrees well with observations of the electron densities in the quiet-Sun equatorial regions ( Vainio et al. 2003) with a coronal base density of n
 4:54 ; 108 cm3. The coronal model has a monotonically decreasing Alfve´n speed, which does not agree with some previous semiempirical models (Vrsˇnak et al. 2002; Mann et al. 2003; Vainio & Khan

Fig. 1.—Coronal parallel shock wave (bottom) propagating with constant speed Vs on a radial field line B. The flux tube has a superradial expansion. The background solar wind model has a constant phase speed V ¼ u þ vA of outward-propagating Alfve´n waves. The speed profiles (top) resulting in our model are plotted as well. Particles are not traced downstream of the shock (gray area), but the shock is treated as a boundary condition.

2004), where a local maximum in the high corona has been found. Typical values of the maximum Alfve´n speed in the references cited above are in the range 450Y750 km s1. However, given the uncertainties of the measured coronal densities and especially of the magnetic fields, we still regard our model to be a reasonable approximation. Considering a coronal plasma with proton and electron temperatures fulfilling Tp þ Te ¼ 3 MK gives a plasma  of about 0.3. Such a corona would be able to host both slow-mode and fast-mode shock waves. As we are modeling particle acceleration, we are restricting the discussion to super-Alfve´nic shocks, which are all of fast mode. The assumption of radial shock propagation at constant speed is also a restrictive idealization and is clearly best satisfied near the leading edge of a fast, impulsively accelerated CME. A large number of particles is simultaneously traced under the guiding-center approximation on a field line upstream of the shock wave. The transport processes included are the ones employed in the focused transport theory ( Kocharov et al. 1998; Vainio et al. 2000): streaming along the magnetic field lines, focusing in the diverging mean magnetic field, and scattering off the Alfve´n waves. Focusing conserves the particle energy in the inertial frame, and scattering in the frame comoving with the waves. We do not trace the particles downstream of the shock wave, but instead employ a probability of return from the downstream region (Jones & Ellison 1991), P ret ¼ (v 0  V2 ) 2 /(v 0 þ V2 ) 2 , where V2 ¼ (Vs  V )/rsc is the effective speed of the downstream Alfve´n waves as measured in the shock frame, rsc is the scatteringcenter compression ratio at the shock ( Vainio & Schlickeiser 1998), and v 0 is the particle speed as measured in the frame of the downstream waves. Each time the particle hits the shock wave

624

VAINIO & LAITINEN

from the upstream region, we multiply its weight in the simulation by P ret and return it to the upstream region assuming complete isotropization in the downstream wave frame and applying flux weighting of the returning particles by the shock-frame pitchangle cosine (Jones & Ellison 1991; Vainio et al. 2000). During this process, v 0 is assumed to be conserved, so when returning to the upstream plasma, the particle has increased its energy in the frame of the upstream waves because of the compression of the scattering-center speed at the shock. Thus, the shock wave acts as the lower boundary of the particle simulation box, and the boundary condition employed in the particle code is equivalent to the assumptions employed in the diffusive shock acceleration theory. Low-energy particles are injected into the simulation at a constant rate (per unit area of the flux tube) at the shock with an exponential shock-frame velocity spectrum dNinj /dv ¼ (Ninj /v1 )H(v  u1 )e(vu1 )/v1 , where u1 ¼ Vs  u and v1 is a model parameter, which we set to be 375 km s1. This injection spectrum corresponds to a slightly suprathermal particle population being picked up by the shock. The injected particles are also flux weighted. In quasi-linear theory, the resonance condition between protons and Alfve´n waves is f ¼ fcp V/(v); where f is the oscillation frequency of the Alfve´n wave, fcp ¼ (2
)1 eB/mp is the proton cyclotron frequency, and v is the particle speed and  its pitchangle cosine, both measured in the frame comoving with the waves. In these simulations we neglect the dependence of the resonant wavenumber on pitch angle and use a simplified form of the resonance condition, f ¼ fcp

V : v

ð4Þ

Consequently, the scattering frequency employed in our simulations (Vainio et al. 2003),  ¼
2 fcp

f P( f ) ; B2

ð5Þ

is a function of proton momentum only. Since scattering is isotropic, the corresponding mean free path is k ¼ v/. Likewise, using the approximated resonance condition, the wave growth rate is conveniently simplified from its full quasi-linear form and can be written as ( Vainio 2003)  ¼
2 fcp

pSp ; nvA

ð6Þ

R þ1 where Sp ¼ 2
p 2 1 v F(r; p; ; t) d is the streaming per unit momentum of protons as measured in the frame of the Alfve´n waves and F(r; p; ; t) is the distribution function of the accelerated protons. Equations (5) and (6) thus provide the nonlinear coupling between the waves and the particles responsible for the precursor instability ahead of the shock wave. Note that in a low- plasma the effect of wave damping by thermal particles in gyroresonance occurs at wavenumbers in the ion-cyclotron range and thus can be neglected in our model. From all possible wave modes scattering energetic particles, we include only the unstable outward-propagating slab-mode Alfve´n waves. The equation for the power spectrum of outwardpropagating waves for a time-independent background solar wind is @ P˜ @ P˜ @ þV ¼ P˜ þ @t @r @f



 @ P˜ D ; @f

Vol. 658 TABLE 1 Parameters of the Simulation Runs

Run

dNinj /d
(1034 sr1)

Vs ( km s1)

rsc

E0 ( keV )

Ec ( MeV )



1....................... 2....................... 3....................... 4....................... 5....................... 6....................... 7....................... 8....................... 9....................... 10..................... 11..................... 12..................... 13..................... 14..................... 15.....................

3 10 30 3 10 30 3 10 30 3 10 30 3 10 30

800 800 800 1500 1500 1500 1500 1500 1500 1500 1500 1500 2200 2200 2200

4 4 4 2 2 2 3 3 3 4 4 4 4 4 4

7.2 7.2 7.2 20 20 20 19 19 19 18 18 18 35 35 35

0.13 1.7 7.4 0.29 0.50 0.94 0.50 1.7 5.4 0.54 5.4 26 1.3 9.4 44

0.33 0.43 0.31 1.21 0.85 0.73 0.49 0.39 0.31 0.30 0.33 0.25 0.31 0.28 0.21

Notes.—The effective injection energies and simulated cutoff energies in the accelerated particle spectra are given. The last column gives the value of  (see the Appendix) required to fit the value of Ec with eq. (A28).

˜ f ; r) ¼ (V 2 /BvA )P, the left-hand side contains the where P( WKB transport terms, the first term on the right-hand side implements wave growth, and D is the diffusion coefficient used to spread the wave power in frequency space. This is an ad hoc term inserted to provide stability by dissipating numerical noise and by preventing very large wave intensities from accumulating in narrow frequency ranges as a result of the artificially narrow resonance condition used in our model. The numerical method solving the wave transport equation uses a Lagrangian grid, xi ¼ ri  Vt, in the spatial direction. (This automatically implements wave propagation at constant speed V.) The grid is logarithmically spaced in x and f, i.e., xi ¼ r0 exp (i
x ) and fj ¼ f0 exp ( j
f ). The wave growth is implemented by computing the growth rate ( fj ; xi ; t) from the flux of the simulated ˜ f j ; xi ) particles across xi and multiplying the powerR spectrum P( tþ
t ( fj ; xi ; t) dt. after each time step from t to t þ
t by exp½ t After this, frequency diffusion is performed using a CranckNicholson scheme (e.g., Morton & Mayers 1994). The initial state of the waves is obtained by injecting a wave spectrum P(r0 ; f ) ¼ (B0 ) 2 /f from the innermost grid level and tracing it outward under the influence of diffusion with  ¼ 0. We choose the diffusion coefficient as D ¼ (V /r ) f 8/3 fb2/3 , where fb is a model parameter. As a result, the diffusion process produces erosion of the spectrum at the high-frequency end and a broken power-law wave spectrum at r ¼ 1 AU is obtained, with P / f 1 at f < fb and P / f 5/3 at f > fb , which is in qualitative agreement with observations ( Horbury et al. 1996 ). We, however, set fb ¼ 1 mHz in our simulations, which is a couple of orders of magnitude larger than the typical turnover frequency in the slow solar wind, to keep the effect of this ad hoc frequency diffusion at a low level. The value (B0 ) 2 is fixed so that it, at t ¼ 0, gives a mean free path of k ¼ k0 for 100 keV protons at r ¼ r0 , where k0 is a model parameter. These parameters are related to each other by k0 ¼

2 0:0146c B02 :
!cp (r0 ) (B0 ) 2

ð8Þ

3. RESULTS ð7Þ

We have performed a set of simulations with the code, addressing the following questions: (1) what is the main factor

No. 1, 2007

CORONAL SHOCK ACCELERATION

625

Fig. 2.—Particle intensities (left) and Alfve´n-wave spectra (right) of simulation run 11 at the end of the simulation (at t ¼ 640 s). The vertical axis gives the position outward from the Sun measured from the shock. The contours are half an order of magnitude apart in both panels. Dark means high and light means low values in the ( logarithmic) gray coding.

determining the maximum energy of coronal shock acceleration, and (2) what is the form and escape mechanism of the proton spectrum measured far upstream. The simulation time in each case is tmax ¼ 640 s, and the ambient 100 keV proton mean free path is set to k 0 ¼ 1 R
at r0 ¼ 1:5 R
, corresponding to a rather weak initial Alfve´n wave intensity (per logarithmic bandwidth) of (B0 ) 2 ¼ 6:0 ; 107 B02

ð9Þ

at the base of the simulation box. Other parameters of the simulation runs are given in Table 1. Figure 2 gives a snapshot of the simulated particle intensities and wave power spectrum at the end of simulation run 11. The particle intensities in the left panel are scaled by the inverse flux tube cross-sectional area to 1 AU. The wave power spectrum is scaled by the WKB transported-wave power spectrum, P WKB ( f ; r) ¼ P( f ; r0 )

BvA ; B0 vA0

ð10Þ

which means that the contours in the right panel show deviations from the WKB transport. (The unit level of the plotted quantity corresponds to the leftmost contour at z ¼ 0:01 R
.) At large ra-

dial distances and frequencies, the values of P/PWKB are below unity because of diffusion eroding the spectrum. At small distances from the shock, proton-amplified waves are clearly visible in the spectrum. At the end of the simulation, the shock is at 2.88 R
, and the accelerated particles extend out to about 10 R
. The spectral cutoff energy attained in this simulation is only about Ec ¼ 5:4 MeV, so the parameters used in this simulation are not able to generate a strong SEP event. (See below for the definition of the cutoff energy Ec .) Despite the modest particle acceleration efficiency of the shock, the Alfve´n-wave spectrum has a maximum value exceeding the WKB-transported initial solar spectrum by more than 2.5 orders of magnitude, yielding values of f P/B 2 approaching 0.001 near f ¼ 7 Hz, r ¼ 2:88 R
, and t ¼ 640 s. In more efficient simulations (like runs 12 and 15), the maximum wave intensities have values of f P/B 2  0:007Y0:009, indicating that the limits of the quasi-linear theory are approached by stronger injections. We compare our simulated particle intensity as a function of position and energy to the steady state theory of Bell (1978) in Figure 3. The theory, as applied to our simulations, is presented in the Appendix. The agreement of the simulations with the simple theory is not perfect, but it shows that reasonable estimates of the particle intensities can be made using the steady state approximation at distances close to the shock and at energies well below

Fig. 3.—Proton energy spectrum at the shock (left) and the normalized proton intensity as a function of position at E ¼ 189 keV (right). In the left panel we show results for Vs ¼ 1500 km s1 and rsc ¼ 4 for different injection strengths from simulation runs 10 (long-dashed line), 11 (thick solid line), and 12 (short-dashed line) along with the theoretical prediction of Bell (1978) for the parameters of run 11 (thin solid line). In the right panel we show the simulated result (thick solid line) and the theoretical prediction (thin solid line) for run 11.

626

VAINIO & LAITINEN

Vol. 658

Fig. 4.—Time-integrated net proton flux vs. energy (left) and the net proton flux at E ¼ 189 keV vs. time (right) of simulation run 11 measured at a fixed radial distance of r ¼ 4:94 R
, corresponding to z ¼ 2:06 R
at the end of the simulation (cf., Figs. 2 and 3).

the cutoff energy in the spectrum. The right panel also shows a population of particles escaping ahead of the shock wave. These particles are propagating in a region where the waves have not yet grown appreciably and produce the population of particles escaping in the ambient interplanetary medium. The time-integrated net flux of these particles (see Fig. 4) to the ambient medium agrees well with the prediction of Vainio (2003), i.e., with equation (1), at energies below the high-energy cutoff in the spectrum. The low-energy cutoff in the spectrum of escaping particles occurs due to velocity dispersion, which is clearly visible in Figure 2 as well. The simulated spectra at the shock are typically power laws with a rapid (faster than exponential in energy) steepening at high energies (although in the cases with the weakest acceleration, the power-law part is almost absent). The power-law part of the spectrum, when it is visible, is reasonably well described by Bell’s theory, and the cutoff energy is increasing with increasing simulation time. We did not find any ‘‘universal’’ form of the spectrum to be used as a fitting function of the simulated spectra at high energies. Therefore, the cutoff energy Ec given in Table 1 is defined as the point of intersection between the simulated particle spectrum at the shock (at the end of the simulation) and the

theoretical power-law spectrum of Bell (1978) divided by 10. This, of course, is a completely ad hoc definition, but it nevertheless gives a reasonably robust method of deriving a ‘‘maximum energy’’ in the spectrum without any assumptions about the spectral shape at the highest energies. Table 1 also gives a dimensionless parameter  (see the Appendix), which relates the acceleration rate to the diffusion coefficient at the shock. It should have values below unity, and this is seen to be the case in all but one of our simulations; the one with  ¼ 1:21 is very inefficient in accelerating particles, and the estimated value of the cutoff energy is, therefore, uncertain. The cutoff energy is plotted in Figure 5 as a function of the number of injected particles. For shocks with a high scattering-center compression ratio, the dependence of the cutoff energy on the injection rate is very strong, as predicted by the steady state theory (see eq. [A28]). We note, however, that a common approximation for evaluating the acceleration rate using the spatial diffusion coefficient  at the shock (i.e., setting  ¼ 1) is generally too optimistic. Our simulations show that this approximation can overpredict the cutoff energies by more than an order of magnitude. The reason is that the diffusion coefficient increases as a function of distance and the average value of  felt by the particles during their stay in the upstream region is much larger than  at the shock. 4. DISCUSSION AND CONCLUSIONS

Fig. 5.—Final cutoff energy in the simulated energetic particle spectrum at the shock as a function of the number of injected particles ( per steradian at the solar surface). Simulation results with the same values of Vs and rsc are connected by lines. The symbols denote ½Vs ðkm s1 Þ; rsc  ¼ (2200; 4) (circles); (1500; 4) ( filled squares); (800; 4) (crosses); (1500; 3) (open squares); and (1500; 2) (asterisks).

We have presented a numerical model to simulate the acceleration of solar energetic particles in quasi-parallel coronal shock waves. Our model takes self-consistently into account the generation of waves in the upstream region by the accelerated protons. The results indicate that particle acceleration and wave growth near the shock can be described by the quasi-stationary-state analytical models, but that simple estimates of the maximum energy of the accelerated protons, taking account of the scattering conditions only close to the shock, are overestimations by even more than an order of magnitude in some of our simulations. In a more dynamical scenario with rapidly varying injection and acceleration parameters, however, the particle spectrum may experience other important dynamical effects, which cannot be described by a quasi-stationary modeling. These effects can, of course, be implemented in our model as well. Particle escape from the shock in our simulations seems to be well described by the analytical model of Vainio (2003), which predicts a hard (/p1 ) power-law spectrum of particles escaping from the inner corona before an appreciable wave growth has

No. 1, 2007

CORONAL SHOCK ACCELERATION

occurred, followed by a phase of trapping by turbulence and gradual leakage. The first particles that the shock accelerates to a given energy are most prone to escape from its vicinity to the far upstream region, because the mean free path is still large and adiabatic focusing helps the decoupling of the escaping particles from the shock. Another component of gradual injection of particles to the ambient medium is likely to occur, because as the shock propagates to the region of lower Alfve´n speed, the wave growth and thus turbulent trapping are less efficient and more and more particles can escape toward upstream (Vainio 2003). This regime, however, is not yet accessible to our simulation because of computational limitations restricting the maximum simulation time. We report here the first set of simulations, in which we concentrate on the first 10 minutes of the shock acceleration process. The number of injected (suprathermal) particles per steradian at the solar surface is limited to 3 ; 1035 during the time of the propagation. Both limitations are made because of the numerical resources: increasing the simulation time obviously leads to longer simulations and to a requirement of a larger grid, and increasing the number of the injected particles requires a smaller spatial grid cell size, which forces one to use a smaller time step. The maximum number of injected particles is still small compared to the number of thermal protons overtaken by the shock, which is 8:2 ; 1040 sr1 at the solar surface for Vs ¼ 1500 km s1 and t max ¼ 640 s. Thus, it would seem possible to increase the injection efficiency, even considerably. However, the model would soon lead to nonlinear wave amplitudes, and more physics would be needed to describe the nonlinear damping and wave-wave interactions occurring in the system. Nevertheless, we feel that it is safe to extrapolate our results and predict that proton acceleration at quasi-parallel CME-driven shocks up to 100 MeV can occur in a few minutes, but predictions about acceleration up to the GeV range are difficult to obtain from the present simulation model. It may well be that the quasi-linear approach has to be replaced by a more general theory of wave-particle interactions to describe coronal shock acceleration in the largest SEP events.

627

Our model is still in a rather qualitative stage. It is restricted to quasi-parallel shocks. Oblique shocks would be rather easy to include in the model if the adiabatic approximation was employed when describing the interaction of the particles with the compressed magnetic field of the shock. The description of the resonance between the particles and the Alfve´n waves is a simplified one, and the use of the full pitch-angle-dependent resonance condition may affect the acceleration rate and the spectrum of escaping particles, as the waves generated by lower energy particles may then scatter particles of higher energies, and vice versa. The model does not yet include a complete description of the downstream region of the shock either. Instead, we treat the shock as a boundary condition under the assumption that the downstream plasma waves quickly and elastically isotropize the particles there in a frame of reference moving at a single average wave speed. This means (1) that the time spent by the accelerated particles in the downstream region is neglected; (2) that the effects of stochastic acceleration in the downstream region are neglected; (3) that the possible storage of the accelerated particles by turbulent trapping in the downstream region is neglected; and (4) that the scattering-center compression ratio of the shock, as determined by the cross helicity of the downstream Alfve´n waves (Vainio & Schlickeiser 1998), is not described self-consistently. All these effects are potentially important and should be included in a more comprehensive numerical model, and this is our long-term goal. In conclusion, our simulations demonstrate the capability of the CME-driven shocks to accelerate protons to tens of MeV in 10 minutes, and reasonable extrapolation of the simulation results indicates the ability of the shocks to accelerate particles beyond 100 MeV after some number of minutes. The simulations also show that simplified analytical models can give a qualitatively correct picture of the acceleration process (Bell 1978; Zank et al. 2000; Li et al. 2003; Rice et al. 2003; Lee 2005) and of the subsequent escape of the particles to the interplanetary medium (Vainio 2003). Accurate quantitative modeling, however, requires numerical simulations for which the tools should become available in the foreseeable future.

APPENDIX DIFFUSION APPROXIMATION IN THE STEADY STATE In this appendix we consider our model under the steady state diffusion approximation. The spatial diffusion coefficient in our model reads ¼

1 1 v2 2 v 2 1 0 kv ¼  ¼ ; 3 3  3
!cp P P

ðA1Þ

where P¼

f P( f ; r) ; B2

0 ¼

2 v 2 : 3
!cp

ðA2Þ

The wave growth rate is ¼

R þ1 2
p3 1 v  F d pSp

!cp ¼ !cp ; 2 2 nvA nvA

ðA3Þ

where !cp ¼ eB/mp ¼ 2
fcp is the angular proton-cyclotron frequency. Since the particle speeds in our model are clearly faster than the shock speed relative to the upstream scattering centers, particles are quasi-isotropic close to the shock, v

@F0 @  @F1 @F0 (1   2 ) ¼ j F1 ¼ k : @ 2 @r @ @r

ðA4Þ

628

VAINIO & LAITINEN

Vol. 658

Thus, streaming is diffusive, Sp ¼ 2
p

2

Z

þ1

vF d ¼ 4
p 2 

1

@F0 4
p 2 0 @F0 ¼ ; @r P @r


0 4
p3 @F0
0 @N ;  ¼  !cp   !cp 2 2 vA P n @r vA P @r

ðA5Þ ðA6Þ

where N (z; p) ¼ 4
p3 F0 /n. Note that here  is positive and r grows in the direction of wave propagation (i.e., outward from the Sun). Next, consider particles and waves in the shock frame, where the spatial coordinate z grows inward and has z ¼ 0 at the shock. Thus, @/@r ! @/@z in the above equations. Close to the shock, particles obey @N @ 0 @N ¼ : @z @z P @z

ðA7Þ

@P
0 @N @N ¼ P ¼ a ; @z 2 vA =!cp @z @z

ðA8Þ

(u1  vA ) Neglecting spectral diffusion, waves obey (u1  vA )

where u1 ¼ Vs  u is the upstream bulk speed in the shock frame, so u1  vA ¼ Vs  V is a constant in our model. Thus, @N u1  vA @P ¼ ; @z @z a

ðA9Þ

and plugging equation (A9) into equation (A7) gives (u1  vA )

@P @ 0 @P ¼ : @z @z P @z

ðA10Þ

Integrating once from 1 to z, we get (u1  vA )P 2 ¼ 0

@P : @z

ðA11Þ

This results in P¼

P 0 z0 ; z0  z

N ¼

N 0 z0 ; z0  z

ðA12Þ

where P 0 , z0 , and N 0 are functions of momentum related to each other by P0 ¼

0 ; (u1  vA )z0

N0 ¼

u1  v A P0: a

ðA13Þ

At the shock ( Bell 1978),  3 p N (0; p) ¼ N 0 ( p) ¼ " ; p0

ðA14Þ

where  ¼ 3rk /ðrk  1Þ (>3) and " is the fraction of thermal particles injected into the acceleration process at p ¼ p0 . Thus,  3  3 0 0 p p z0 ( p) ¼ ¼ ¼h ; ðA15Þ p0 a N 0 a" p0 h¼

0 2 1 vA ¼ ; a"
" !cp

ðA16Þ

consistent with the results of Bell (1978). In the case of a distribution of injection momenta, dNinj /dpinj , one can define an effective injection momentum p0 by requiring that the shock-processed momentum distribution,   Z p dNinj dN 1 p 2 ¼ dpinj ; ðA17Þ dp pinj 0 dpinj (  3)pinj

No. 1, 2007

CORONAL SHOCK ACCELERATION

629

approaches, at large momenta, the momentum obtained from a monoenergetic injection at p0 with the same number of injected particles, i.e.,   2 Ninj dN p ; p3h pinj i; ðA18Þ  dp (   3)p0 p0 giving p3 0

1 ¼ Ninj

Z 0

1

dNinj 3 p dpinj : dpinj inj

ðA19Þ

In our model, the injection rate is Q¼

Ninj ¼ "u1 n; A(r)t max

ðA20Þ

so "¼

Ninj dNinj =d
B dNinj =d
¼ 2 ¼ A(r)u1 t max n r
u1 t max n B
r
2 u1 t max n




vA vA


2

!cp
; !cp

ðA21Þ

giving 2 r
2 u1 t max n
vA
vA
:
 dNinj =d
vA !cp


ðA22Þ

rk  1 V3 t3 vA
5:0 ; 103 km; rk N35 vA

ðA23Þ

h¼ Numerically, h¼

where dNinj /d
¼ N35 ; 1035 sr1, Vs ¼ V3 ; 103 km s1, and t max ¼ t3 ; 103 s, and u1  Vs and n
¼ 4:54 ; 108 cm3 have been assumed. The grid size in our simulations has to be smaller than this, which sets a stringent limit on the maximum simulation time we can reasonably achieve in our model for a large injection strength. In the steady state, the residence time of the accelerated particles in the upstream region is R0   4z0 z1 z1 N (z; p) dz ¼ ln 1  ; ðA24Þ
t ¼ v N (0; p)=4 v z0 where z1 < 0 is a boundary introduced in the far upstream, where the density of the returning particles is assumed to vanish. The mean gain of momentum per interaction with the shock is
p ¼ (4/ )( p/v)(u1  vA ). Thus, diffusive shock acceleration rate is   1 u1  v A u1  vA p 3 p˙  ¼ ; ðA25Þ  z0 ( p) p0 p h where  ¼ 1/ ln (1  z1 /z0 ) < 1 is a parameter logarithmically dependent on momentum, which we take to be a constant for simplicity. In our model, h is also approximately constant in the inner corona, where uTV . Thus, Z pc (t)  4 p dp Vs  V t; ðA26Þ ¼ p0 p0 h p0 giving   pc (t)   3 Vs  V 1=( 3) t ¼ 1þ  : p0  h Thus, at the end of the simulation, t ¼ tmax , the cutoff energy in the spectrum scales like   (Vs  V )t max 2(rk 1)=3 Ec (t max ) ¼ E0 1 þ  rk h   2(rk 1)=3 1  M 1 N35  E0 1 þ 200 rk  1

ðA27Þ

ðA28Þ

at nonrelativistic energies, where M ¼ Vs /V , E0 ¼ p02 /2mp , and p0 is given by equation (A19); otherwise, the same notation and assumptions from equation (A23) have been used.

630

VAINIO & LAITINEN

REFERENCES Axford, W. I., Leer, E., & Skadron, G. 1977, in Proc. 15th Int. Cosmic Ray Morton, K. W., & Mayers, D. F. 1994, Numerical Solution of Partial DifferConf. ( Plovdiv), 132 ential Equations (Cambridge: Cambridge Univ. Press) Bell, A. R. 1978, MNRAS, 182, 147 Ng, C. K. 2004, in First Ann. Meeting Asia Oceania Geosciences Society Blandford, R. D., & Ostriker, J. P. 1978, ApJ, 221, L29 (Singapore: AOGS), 57-OSP-M190 Horbury, T. S., Balogh, A., Forsyth, R. J., & Smith, E. J. 1996, A&A, 316, 333 Rice, W. K. M., Zank, G. P., & Li, G. 2003, J. Geophys. Res., 108, 1369 Jones, F. C., & Ellison, D. C. 1991, Space Sci. Rev., 58, 259 Vainio, R. 2003, A&A, 406, 735 Kocharov, L., Vainio, R., Kovaltsov, G. A., & Torsti, J. 1998, Sol. Phys., 182, Vainio, R., & Khan, J. I. 2004, ApJ, 600, 451 195 Vainio, R., Kocharov, L., & Laitinen, T. 2000, ApJ, 528, 1015 Krymsky, G. F. 1977, Akad. Nauk SSSR Dokl., 243, 1306 Vainio, R., Laitinen, T., & Fichtner, H. 2003, A&A, 407, 713 Lee, M. A. 2005, ApJS, 158, 38 Vainio, R., & Schlickeiser, R. 1998, A&A, 331, 793 Li, G., Zank, G. P., & Rice, W. K. M. 2003, J. Geophys. Res., 108, 1082 Vrsˇnak, B., Magdalenic´, J., Aurass, H., & Mann, G. 2002, A&A, 396, 673 Mann, G., Klassen, A., Aurass, H., & Classen, H.-T. 2003, A&A, 400, 329 Zank, G., Rice, W. K. M., & Wu, C. C. 2000, J. Geophys. Res., 105, 25079