Issue 
A&A
Volume 535, November 2011



Article Number  A34  
Number of page(s)  6  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201117507  
Published online  27 October 2011 
Heavyion acceleration and selfgenerated waves in coronal shocks
^{1} Department of Physics and Astronomy, University of Turku, Finland
email: markus.battarbee@utu.fi
^{2} Jeremiah Horrocks Institute for Astrophysics and Supercomputing, University of Central Lancashire, UK
^{3} Department of Physics, University of Helsinki, Finland
Received: 17 June 2011
Accepted: 26 September 2011
Context. Acceleration in coronal mass ejection driven shocks is currently considered the primary source of large solar energetic particle events.
Aims. The solar wind, which feeds shockaccelerated particles, includes numerous ion populations, which offer much insight into acceleration processes. We present first simulations of shockaccelerated minor ions, in order to explore trapping dynamics and acceleration timescales in detail.
Methods. We have simulated diffusive shock acceleration of minor ions (^{3}He^{2+}, ^{4}He^{2+}, ^{16}O^{6+} and ^{56}Fe^{14+}) and protons using a Monte Carlo method, where selfgenerated Alfvénic turbulence allows for repeated shock crossings and acceleration to high energies.
Results. We present the effect of minor ions on wave generation, especially at low wavenumbers, and show that it is significant. We find that maximum ion energy is determined by the competing effects of particle escape due to focusing in an expanding flux tube and trapping due to the amplified turbulence. We show the dependence of cutoff energy on the particle charge to mass ratio to be approximately (Q/A)^{1.5}.
Conclusions. We suggest that understanding the acceleration of minor ions at coronal shocks requires simulations which allow us to explore trapping dynamics and acceleration timescales in detail, including evolution of the turbulent trapping boundary. We conclude that steadystate models do not adequately describe the acceleration of heavy ions in coronal shocks.
Key words: acceleration of particles / turbulence / Sun: coronal mass ejections (CMEs)
© ESO, 2011
1. Introduction
Particle acceleration by coronal and interplanetary shocks driven by coronal mass ejections (CMEs) is widely accepted as the primary source of strong solar energetic particle (SEP) events. Particles scatter off plasma waves, cross the shock front repeatedly and gain energy on each crossing. In large events, turbulence sufficient for extended trapping can be generated by the streaming of the accelerated particles themselves, as plasma waves in the upstream are amplified by scattering particles. This diffusive shock acceleration mechanism, as presented by, e.g., Bell (1978), has recently been studied quantitatively as a model of the acceleration of SEPs by, e.g., Lee (2005), Vainio & Laitinen (2007, 2008) and Ng & Reames (2008). Other noteworthy studies of shockaccelerated SEP events include, e.g., Ng et al. (1999), Tylka et al. (2005), Tylka & Lee (2006) and Sandroos & Vainio (2007, 2009a,b).
Although the most abundant ion species in the solar wind, proton, is likely to dominate wave generation in shocks, minor ions are scattered off the same turbulence and accelerated as well. If the turbulence is protongenerated, the maximum rigidity obtained by the protons determines a lowwavenumber cutoff in the spectrum of plasma waves. It has been suggested that this would prevent any ions from being accelerated beyond the same rigidity in a quasiparallel shock wave (e.g., Zank et al. 2007). This would then lead to a dependence of the maximum (nonrelativistic) energy (per nucleon) of the ions on the charge to mass ratio of the form (Q/A)^{2}.
In this paper, we present the first simulations of particle acceleration in selfgenerated waves with minor ions included as particles contributing to the wave generation process. Instead of using a full spectrum of ion species in the solar wind, we limit ourselves to selected interesting populations, namely ^{3}He^{2+}, ^{4}He^{2+}, ^{16}O^{6+} and ^{56}Fe^{14+}. By simulating a propagating coronal shock for an extended period of time, we investigate the spectra and maximum attained energy of each particle population, as well as gauge the effect each population has on wave generation at different wavenumbers.
2. Model
In our simulation model, we numerically solve the particle transport equation through propagating representative particles using the guiding centre approximation. We approximate the quasilinear theory by employing a pitchangle independent resonance condition (1)where u_{sw} is the solar wind speed, v_{A} is the Alfvén speed, v is the particle speed, γ is the Lorentz factor, f_{ci} = (1/2π)q_{i}B/m_{i}c is the ion cyclotron frequency, B is the magnetic flux density and q_{i} = Qe and m_{i} = Am_{p} are the charge and mass of the ion in question, and e and m_{p} are the charge and mass of a proton. We scatter representative particles isotropically off plasma waves with scattering frequency (2)where P(f) is the wave power at frequency f. The initial wave power is scaled to give an ambient 100 keV proton mean free path of λ_{0} = 1 R_{⊙} at r_{0} = 1.5 R_{⊙}, where R_{⊙} is the solar radius.
Once particles have been swept up by the shock, they are propagated using a Monte Carlo method, focused along the mean magnetic field due to adiabatic invariance and traced in a superradially expanding flux tube with the solar wind speed inferred from mass conservation. For additional details of our simulation model, we refer the reader to Vainio & Laitinen (2007), Vainio & Laitinen (2008) and Battarbee et al. (2010).
2.1. Particleshock interactions
As particles encounter the propagating parallel stepprofile shock, they scatter off the compressed plasma and may return upstream with a momentum boost. The plasma compression ratio is solved from the RankineHugoniot jump conditions. The analytical return probability from a shock encounter for an isotropic particle population is given as (3)where v_{w} is the speed of the particle in the downstream plasma frame and u_{2} the downstream plasma speed in the shock frame. However, the downstreamtransferred population is no longer isotropic, unless v_{w} ≫ u_{2}. Thus, following Vainio et al. (2000), we propagate and scatter particles in the downstream plasma frame up to a distance of 2λ behind the shock, where λ is the particle mean free path. At this distance, the particle has encountered enough scatterings to warrant the assumption of isotropy and its representative weight is multiplied by the isotropic return probability P_{ret}, sending it back towards the upstream from the distance of 2λ with a randomized shockbound pitchangle. Particles propagate and experience smallangle scatterings in the downstream until they either return to the shock front or their cumulative return probability drops below 0.1%. Simulation time is not advanced while the particles are in the downstream region. This corresponds to a situation where the downstream scattering is so intense that the mean residence time downstream is negligible compared to the time between subsequent shock encounters. Using this assumption, we do not propagate the shape of turbulence into the downstream.
2.2. Injected populations
In our simulation, particles swept up by a shock propagating through the solar corona are accelerated and traced up along a grid extending to a distance of 300 R_{⊙}. In addition to a proton population n_{p}(r) based on the solar wind density model of Cranmer & van Ballegooijen (2005), we inject fully ionized helium (^{3}He^{2+} and ^{4}He^{2+}) and partially ionized heavier elements (^{56}Fe^{14+} and ^{16}O^{6+}) according to estimated solar wind abundance values. This results in H^{+}relative abundances for ^{4}He^{2+}, ^{16}O^{6+}, ^{56}Fe^{14+} and ^{3}He^{2+} of 4.0 × 10^{2}, 8.0 × 10^{4}, 1.0 × 10^{4} and 1.6 × 10^{5}, respectively.
As a large portion of the solar wind consists of thermal particles, and our strong steplike shock, with the assumed extremely intense downstream turbulence (see Sect. 2.1), injects an unrealistically large proportion of thermal particles, we model the solar wind as consisting of two kinetic populations. The majority, 99% of particles, represents a thermal core and is passed directly downstream. A minority, 1% of particles, is considered a partially suprathermal halo and follows a κdistribution (Prested et al. 2008). This population, which has a continuous spectrum, is encountered by the shock in our simulation. The parameter κ receives values of 6...2 going from 1.5 R_{⊙} to 3.0 R_{⊙} respectively. It should be noted, though, that the composition of the seed population, and its dependence on distance from the sun, is not known and these parameters are arbitrary.
The average thermal speed (4)is based on the radial temperature profile given in Cranmer & van Ballegooijen (2005), where T is the proton temperature and k_{B} is the Boltzmann constant. All minor ion populations are initialised using the same distribution function (5)where Γ denotes the gamma function.
2.3. Wave evolution
The requirement for particle trapping in front of a shock is strong enough turbulence. This can be attained by wave amplification via the scattering of energetic particles upstream of the shock. A large shocknormal velocity results in large amounts of kinetic energy deposited into accelerated particles and thus larger amounts of energy deposited into upstream waves. Also, as particles reach higher velocities, they become resonant with waves of lower frequencies.
Fig. 1 Left: evolution of wave power spectra. Right: wave amplification factor Γ_{w,i} integrated over 640 seconds of simulation, at 2 × 10^{5} R_{⊙} and 6 × 10^{4} R_{⊙} from the shock, where V_{s} = 1500 km s^{1}. 

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In steadystate upstream solar wind, the evolution equation for a normalized wave power spectrum can be written as (6)Here, P(r,f,t) is the Alfvén wave power spectrum as a function of radial distance (r), frequency (f) and time (t), V = u_{sw} + v_{A} is the group speed of the Alfvén waves and Γ_{w} is the wave growth rate. is an adhoc diffusion coefficient. This coefficient is chosen so that an unenhanced spectrum tends towards the form of Kolmogorov turbulence P ∝ f^{ − 5/3} at r_{⊕} = 1 AU above the breakpoint frequency f_{b} = 1 mHz, as suggested by observations (e.g., Horbury et al. 1996). Turbulence and diffusion magnitudes are normalized to result in an 100 kev proton having mean free paths of 1 R_{⊙} at r = 1.5 R_{⊙} and 54 R_{⊙} at r = 1 AU.
Our simulation coordinates are attached to the propagating coronal shock, allowing for good numerical accuracy in the nearshock region. Wave amplification is calculated over time intervals of 1.6 ms at grid cell boundaries. We simulate the effect of diffusion using a CrankNicholson method, and advect wave power along the moving grid with a LaxWendroff scheme utilizing a Van Leer flux limiter.
For our simulations, we have extended the wave growth approximation of Vainio (2003) to accommodate for different particle masses and charges. Energy deposited into parallelpropagating Alfvén waves per particle scattering can be written as ΔE_{w} = −v_{A}pΔμ, where p = γmv = γAm_{p}v is the particle momentum and Δμ is the change in particle pitchangle. Integrating all particles of a given ion population i at a set position gives the rate of change of the wave energy density as (7)where F(r,v,t) is the particle distribution function.
Fig. 2 Particle populations for H^{+} and ^{56}Fe^{14} along with the critical contour where the focusing velocity V/L exceeds the shock velocity. Results are shown for the simulations where V_{s} = 1500 km s^{1}, with colour contours at one magnitude intervals. A radial 3cell boxcar smooth function has been applied. 

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The pitchangle diffusion coeffient, according to quasilinear theory, is (8)where ω_{ci} = Qω_{cp}/Aγ = QeB/Am_{p}cγ is the angular ion cyclotron frequency, k_{r} is the resonant wavenumber and W(r,k,t)dk is the energy density of waves propagating parallel to the mean magnetic field with wavenumber in the range from k to k + dk. This can be written as (9)which, because ⟨ Δμ ⟩ /Δt = ∂D_{μμ}/∂μ, yields the wave growth rate (10)As in Vainio (2003), we neglect the μdependence of the resonance condition and replace δ(k + ω_{ci}/vμ) with (1/2)δ(k − ω_{ci}/v). Now, using partial integration in μ, we get (11)which can be further represented as (12)where is the particle streaming per unit velocity in the frame of the Alfvén waves evaluated at the resonant particle speed v_{r} = ω_{ci}/k. We monitor particle streaming by counting whenever a particle crosses a grid cell boundary of the shockattached tracking grid. To correct for the difference of particle streaming between the frame travelling with the shock and the frame of the Alfvén waves, we use an additional weighing factor of (v_{w}μ_{w})/(v_{s}μ_{s}), where v_{w}, μ_{w}, v_{s}, and μ_{s} are the particle speed and pitchangle in the Alfvén wave frame and shockattached frame, respectively.
3. Results
We simulate three parallel, constant velocity shocks in the corona starting from 1.5 R_{⊙}. The shocknormal velocities V_{s} are 1250 km s^{1}, 1500 km s^{1} and 1750 km s^{1}. We use zero crosshelicity for the turbulence in the downstream, while in the upstream waves propagate away from the Sun in the plasma frame. Turbulence is tracked on a logarithmic grid reaching out to 300 R_{⊙} in front of the shock.
3.1. Wave turbulence and its generation
The left panel of Fig. 1 shows the normalized wave power spectra (multiplied with the frequency f) in front of the shock after 160 and 640 s of simulation. In the right panel we show the wave amplification, , for each particle population i, integrated over the whole simulation time. The value z is the distance from the shock front and r_{shock} is the position of the shock. We display wave amplification for the V_{s} = 1500 km s^{1} run in both the measurement cell nearest to the shock (z = 2 × 10^{5} R_{⊙}) and further out (z = 6 × 10^{4} R_{⊙}). Although minor ions are much less abundant than protons, they have higher charges and are resonant with lower frequencies, allowing them to generate a significant amount of turbulence close to the shock. Such dominance of minor ions has been reported by Lee (1982) for Helium ions at low frequencies in relation to Earth’s bow shock. In our simulations, close to the shock, heavier ions surpass protons in wave generation at a narrow frequency range below 7 Hz. Between 7 Hz and approximately 70 Hz, displays wave generation equal to ~ 50% of that of H^{ + }.
For the V_{s} = 1250 km s^{1} shock, powered wave generation is equal to H^{ + }powered generation in the region below 100 Hz, with dominating below 30 Hz. As the shocknormal velocity increases, the dominance of protons on turbulence amplification increases, with the takeover moving to 3 Hz. In all cases, however, protons are responsible for the bulk of turbulence amplification, as abundant lowenergy protons generate a great deal of turbulence above 100 Hz.
3.2. Accelerated particle populations
In Fig. 2 we demonstrate the evolution of two particle populations, H^{+} and ^{56}Fe^{14+}. Particles within the expanding flux tube are efficiently accelerated up to a maximum energy at which the turbulence can no longer trap the particles, and instead the focusing effect of the diverging magnetic field allows them to escape.
The focused diffusion model of particle transport has been examined in detail by Kocharov et al. (1996). Here, we start with Parker’s equation, which in the fixed frame, upstream of our coronal shock reads (13)where f_{0} is the isotropic part of the distribution function, D = (1/3)λv is the spatial diffusion coefficient, λ = v/ν is the particle mean free path and is the fluxtube crosssectional area related to the focusing length L by . Parker’s equation can be expressed, using the linear density , as a FokkerPlanck equation (14)which shows that the effect of focusing in the particle motion is twofold: it contributes to the advection velocity by ṙ_{foc} = D/L and to the adiabatic energy changes by ṗ_{foc} = −(p/3) V/L. The addition to the advection velocity at large distances from the Sun is large, since there the waves have not yet grown to make D small. It is clear that particles will, on average, move away from the shock in the upstream region in areas where V + D/L > V_{s}. This facilitates the escape of particles from the shock to the upstream and the distance where V + D/L = V_{s} can be regarded as the boundary of the turbulent trapping region ahead of the shock. This boundary, displayed in Fig. 2 as a dashed line, outlines an escaping population further away from the shock. In addition, the energy at which the boundary intersects the shock surface is representative of the maximum energy that the particles can be accelerated to at a given time.
Fig. 3 Particle spectra after 80 s (left panel) and 640 s (right panel) of simulation, where V_{s} = 1500 km s^{1}. A powerlaw and an exponentional cutoff has been fitted to each population, ignoring the enhancement at the lowest energies. 

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In the latter stages of the simulation, we find decreased wave generation due to lower particle densities and thus less sweptup particles. This, along with wave diffusion, results in the turbulent trapping boundary at the shock moving to lower energies, which causes high energy particles to escape instead of experiencing further acceleration.
3.2.1. Spectral indices
The accelerated particle populations were integrated over the whole upstream. We found the spectral index α for the powerlaw part of the particle spectrum by fitting a line to a chosen section of the loglog representation of data points. Observing the one magnitude contours in Fig. 2, we see that the spatial distributions of iron and protons differ for both the escaping population and particle populations within the turbulent trapping boundary. For iron, this results in much harder particle spectra than what the steadystate model of Bell (1978) suggests. Spectra along with more complete parameter fits are exemplified in Fig. 3.
Fig. 4 Temporal evolution of spectral index α for H^{+} and ^{56}Fe^{14+} populations, as fitted to the powerlaw section of the particle spectra. 

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As the simulation continues the spectrum for heavy ions softens. Figure 4 displays the evolution of the spectral indices α for H^{+} and ^{56}Fe^{14+}. Protons, being the dominant particle population, do not exhibit a softening of the spectral index, whereas the effect is exceedingly prominent in the case of iron accelerated by a V_{s} = 1250 km s^{1} shock.
3.2.2. Attained maximum energies
When attempting to gauge the maximum energy attained by a particle population, we attempted to fit a power law with an exponential cutoff to the simulated spectrum. First, we found the spectral index α for the powerlaw part of the particle spectrum, as in Sect. 3.2.1. We then used this as the basis for fitting an exponential cutoff energy E_{c}. The form used is where C is a fitted constant and ϵ is chosen to fit the sharpness of the cutoff. In our work, we used values of ϵ = 4...2.5, with the value decreasing over simulation time. Figure 5 displays how the cutoff energy follows a ratio of masstocharge to the power of 1.5–1.6, where the exponent is significantly smaller than the theoretical estimation of 2.
Fig. 5 Ratios of particle cutoff energy to the charge/mass number as a loglogplot. 

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4. Discussion and conclusions
Having studied the acceleration of multiple particle populations through selfgenerated turbulence with three different coronal shocks, we find that during early phases of the acceleration process, very hard, even flat spectra can be seen for highmass ions. At all but the highest frequencies, the effect of minor ions on wave generation is nonnegligible, especially in the region directly in front of the shock. As the shocknormal velocity increases, the deduced spectra become harder and the maximum energy attained increases. It is also seen that the maximum energy dependence (Q/A)^{β} does not exhibit β = 2, as suggested by Zank et al. (2007). Rather, the behaviour of cutoff energies is between β = 1.5 and β = 1.6.
At high energies, accelerated particles stream away from the shock due to focusing, as scattering particles supply insufficient wave amplification power to trap highenergy particles to the shock. This causes the turbulent trapping boundary to approach and intersect the shock at the maximum ion energy. As the shocknormal velocity increases, the particle spectra become harder and the energy at which the turbulent trapping boundary intersects the shock increases.
To gauge the effect of focusing versus trapping on particle energy, we note that out of V + D/L = V_{s} only the diffusion coefficient D ∝ v^{2}/ (f_{ci}f_{res}P(f_{res})) depends on the particle species. Thus, at the maximum energy edge of the trapping boundary, D must be same for all particles. If at low frequencies the wave spectrum increases from ambient to amplified levels with a power law P ∝ f^{b}, as shown in Fig. 1, we find (15)This gives the (nonrelativistic) cutoff energy per nucleon a dependency of (Q/A)^{2(b + 2)/(b + 3)}. For a sharp cutoff, i.e. a purely rigiditylimited case, this results in a (Q/A)^{2}dependence for the cutoff energy, while a smoother transition results in a significantly weaker dependence. Due to the weak dependence of β on b, and the dynamic evolution of turbulence, care should be taken when inferring the turbulence spectrum shape from ion cutoff energies.
To examine particle acceleration dynamics, we can calculate the time τ_{R} required to accelerate a particle from an injection rigidity R_{0} to a given rigidity R = p/q_{i} ∝ Av/Q, where R ≫ R_{0}. Assuming zero residence time in the downstream, we find (16)where is a constant and G(R) is a function of rigidity based on the shape of the wave power spectrum. If P(f) ∝ f^{b} where b is constant, the acceleration time to a given rigidity R is directly proportional to Q/A. Another item of interest is the time required to accelerate a heavy ion from injection speed v_{0} to the maximum speed v_{ion} = (Q/A)^{(b + 2)/(b + 3)}v_{p}, where v_{p} is the proton speed at the turbulent trapping boundary. This gives (17)which, using previous assumptions for b, yields an acceleration time independent of the chargetomass ratio. Thus, it is clear that minor ions are accelerated to the maximum rigidity of protons much faster than the protons themselves, after which the
ions slowly continue to gain energy until they reach the turbulent trapping boundary. This results in minor ions gaining harder, even flat spectra, especially in early phases of the simulation. In latter phases of the simulation, the value of b increases at low frequencies, which leads to an increase in minor ion acceleration time and thus softer minor ion spectra.
At later stages of the simulations, wave amplification rates decay in line with the decay of injection efficiency. Thus, accelerated particles can stream away from the shock at lower energies, and further acceleration to higher energies ceases. Particles reaching the turbulent trapping boundary propagate in space, forming a plateau which is not consistent with Bell’s steadystate result.
In conclusion, diffusive shock acceleration of protons and minor ions cannot realistically be represented by a steadystate approximation, but instead requires numerical simulations to reveal the full dynamics of the acceleration process and the various particle populations.
Acknowledgments
The authors would like to thank the IT Centre for Science Ltd (CSC) for computational services and the Academy of Finland (AF) for financial support of projects 122041 and 121650. T.L. acknowledges support from the UK Science and Technology Facilities Council (standard grant ST/H002944/1).
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All Figures
Fig. 1 Left: evolution of wave power spectra. Right: wave amplification factor Γ_{w,i} integrated over 640 seconds of simulation, at 2 × 10^{5} R_{⊙} and 6 × 10^{4} R_{⊙} from the shock, where V_{s} = 1500 km s^{1}. 

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In the text 
Fig. 2 Particle populations for H^{+} and ^{56}Fe^{14} along with the critical contour where the focusing velocity V/L exceeds the shock velocity. Results are shown for the simulations where V_{s} = 1500 km s^{1}, with colour contours at one magnitude intervals. A radial 3cell boxcar smooth function has been applied. 

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In the text 
Fig. 3 Particle spectra after 80 s (left panel) and 640 s (right panel) of simulation, where V_{s} = 1500 km s^{1}. A powerlaw and an exponentional cutoff has been fitted to each population, ignoring the enhancement at the lowest energies. 

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In the text 
Fig. 4 Temporal evolution of spectral index α for H^{+} and ^{56}Fe^{14+} populations, as fitted to the powerlaw section of the particle spectra. 

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In the text 
Fig. 5 Ratios of particle cutoff energy to the charge/mass number as a loglogplot. 

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In the text 
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