Issue 
A&A
Volume 553, May 2013



Article Number  A129  
Number of page(s)  15  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201220804  
Published online  28 May 2013 
Online material
Appendix A: Time dependency of the pitchangle cosine
The interaction of a charged particle with the fluctuation δB of an Alfvén wave forces the particle to leave its gyro orbit. This procedure is referred to pitchangle scattering. Consequently the change of parallel momentum is connected to this process via . Then the time derivative of p_{∥} is given by the Lorentz force (A.1)where E_{p} is the energy of the particle, which is assumed to be constant as the scattering is purely elastic. Since the parallel momentum remains constant during an unperturbed gyration, only the perpendicular momentum changes due to the circular movement p_{⊥} = p_{x} ± ip_{y} = p_{⊥ 0}exp [ ∓ i(φ_{0} − Ωt)]. This leads to a total change of μ for a particle with intial phase φ_{0} to the magnetic field with (A.2)(Lee & Lerche 1974). A similar approach is given by Schlickeiser (2002) using Eq. (19). In this case the solution along the characteristics of the generalised force term is given by Eq. (12.2.4b) Schlickeiser (2002). This result reduces to Eq. (A.2) by using the magnetostatic approximation, which assumes the electric field fluctuations to be negligible, δE = 0. The coordinates used are still Eqs. 20. The fluctuation δB, i.e. of an Alfvén wave as used in the presented case, is given in Fourier space by (A.3)The exponential function can be described by the generating Bessel functions J_{n}(A.4)where the argument of J_{n} is (A.5)and Ψ = cot^{1}(k_{x}/k_{y}), . The total time derivative of the pitchangle cosine, separated in parallel and perpendicular interactions, reads then (A.6)For Alfvén waves δB is aligned towards k × e_{z}B_{0} and consequently δB_{l,r}(k) = δB(k)( ∓ i)exp( ± iΨ). The time integration and the identity (A.7)(Abramowitz & Stegun 1965) then gives the time dependency of the pitchangle cosine with (A.8)The application of a purely parallel propagating wave, as shown in our toy model, will then simplify this equation because z(k_{⊥} = 0) = 0. Thus, all Bessel functions vanish, except J_{0}(0) = 1. This is the case for n = ± 1. Furthermore, the k integration reduces by the assumption of a single wave , which leads to the presented form in Sect. 4.1.
Appendix B: Derivation of the pitchangle diffusion coefficients
In the simulations, the turbulence consists of Alfvén and pseudo Alfvén waves, thus the pitchangle diffusion coefficient must be calculated separately for the two modes. The two modes are decomposed using the method presented Maron & Goldreich (2001). For the pitchangle diffusion coefficient for Alfvén waves, Schlickeiser (2002) gives (B.1)where μ, v and Ω again are the particle’s pitchangle cosine, speed, and gyrofrequency, , P_{xx,A} the xxcomponent of the Alfvén mode turbulence power spectrum tensor, and B_{0} the background magnetic field. For the resonance function ℛ(k,ω) we assume no damping, which gives the delta function ℛ(k,ω) = π δ(k_{∥}v_{∥} − ω + nΩ). Thus, in the magnetostatic limit (ω = 0) we have (B.2)For the magnetosonic waves, Schlickeiser (2002) gives for the fast mode on the cold plasma limit (B.3)for high particle velocities, v_{A} ≪ v. As the polarisation of the fast mode on the cold plasma limit is the same as for the pseudo Alfvén wave, it is straightforward to show that the same form also holds for the pseudo Alfvén waves, with the difference in the dispersion relation. However, by using the magnetostatic limit with ω = 0 and again assuming no damping, we get for the resonance function ℛ(k,ω) = πδ(k_{∥}v_{∥} + Ω), thus giving for the diffusion coefficient (B.4)Unlike for the Alfvén waves, the Cherenkov resonance, n = 0, is nonzero for the pseudo Alfvén waves and has to be considered separately. In this case, the resonance function is (B.5)This results in (B.6)which has a singularity at v_{∥} = 0, and equals zero elsewhere. Consequently, this term is not used by our model.
For other terms in the sum over n, we again use the magnetostatic approximation ω = 0, thus giving the resonance condition (B.7)and the pitchangle diffusion coefficient (B.8)
Appendix C: Discretisation of the pitchangle diffusion coefficients
For the numerical calculation of the pitchangle diffusion coefficient, we consider the spectrum to be continuous in parallel direction, but discrete in the perpendicular direction, i.e. P_{xx}(k_{x,}k_{y},k_{z}) = P_{xx}(hΔk,iΔk,k_{z}), with h, i = −M,...,M. In this manner, the integrals over k_{x} and k_{y} can be written as a sum, while the integral over k_{z} is evaluated using the delta function. Then, the diffusion coefficient for the PseudoAlfvén waves can be written as (C.1)In this equation, (nΩ)/(μv) represents the parallel wavenumber at which δ(k_{∥}v_{∥} + nΩ) is nonzero. Consequently, as turbulence data is available only at discrete wavenumbers, we define (nΩ)/(μv) = lΔk. Thus, with l restricted to integer values, we find the values of (v_{∥} = μv) at which the value of D_{μμ} can be solved. For a particle with v = Ω/(mΔk), with m ≤ l ≤ M, the pitchangle is given as μ = m/l. Applying this discretisation, we have
and thus (C.3)For the Alfvén waves, using the same discretisation, we get (C.4)
Appendix D: 512^{3} results
In this section we present the results of the investigation of the resolution by using a spatial grid of 512^{3} cells. Again, the MHD simulations were performed, first for the background turbulence, afterward with the peaked modes at and 24. The results are in conformance with the other setups. In particular, we observed the generation of higher harmonic wave modes again.
Fig. D.1
Twodimensional magnetic energy spectra of the decay stage for both peaks in the simulation with G and higher resolution with a grid of 512^{3} cells. The left figure shows the state for the peak at at t = 14.45 s. The right figure shows the peak at t = 3.4 s. The larger Fourier space grid reveals the higher harmonics of the peak. 

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These harmonics are also visible for the peak at within the higher resolved simulation with a 512^{3} grid. In this simulation setup the number of active modes is eight times greater, which means the antialiasing edge is shifted by a factor two to k′ = 2π·86. Consequently, the higher harmonics at and 72 are visible. The evolution of the peaks in both simulations is comparable to the 256^{3} grid simulations. A dominant energy transport towards high perpendicular wavenumbers is observed.
Fig. D.2
Scatter plots for the low gridsize, peak position , decay stage, t = 25 gyration periods. The higher resolution of the grid causes slightly stronger scattering, due to the higher amount of active modes. The resonance patterns are comparable to the 256^{3} grid. 

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The test particle simulations led to comparable resonance patterns. As presented in Fig. D.2 the resonant interactions at μ_{R} = 0,1 and − 0.95 are strongly tilted and a significant amount of particles reach the maximum Δμ, as indicated by the sharp thresholds. The scattering coefficient is in this case again without any structure and hence not shown here. The stronger scattering is primarily caused by the higher wave modes, which are not truncated by the antialiasing anymore and hence contribute to the waveparticle interactions. The test particle simulations were performed for the decay stage of the peaks only because of their huge computational effort.
Fig. D.3
Scatter plots for the low gridsize, peak position , decay stage, t = 25 gyration periods. Because of the higher resolution, higher harmonics of the peak have also developed and interact with the particle. This leads to more diffuse scattering. The resonances are barely visible. 

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Additionally, an increase of the scattering rate at the peak within the 512^{3} grid was observed. Consequently, the resonances are not as significant as in the smaller grid. When comparing Fig. 9 and D.3 it is harder to recognize the resonant structures.
Fig. D.4
Comparison between SQLT approach and particle simulation of the scattering coefficients D_{αα} for the low gridsize, peak position . The results are similar to Fig. 15. 

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Fig. D.5
Comparison between SQLT approach and particle simulation of the scattering coefficients D_{αα} for the low gridsize, peak position . Because of the increased active modes the scattering became stronger and less structured (see Fig. D.3). Consequently, the D_{αα} of the simulation cannot be compared well to the QLT approach. 

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As discussed previously, the 512^{3} gridsize particle simulation within the decay stage led to strong effectively tilted resonances and consequently a very unstructured D_{αα} curve. This can be observed in both Figs. D.4 and D.5. Thus, the results differ greatly from those given by SQLT.
© ESO, 2013
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