Fig. 4.29.3. Accelerated particle spectra at the parallel shock wave in the shock rest frame for (a) the flat (Eq. 4.29.1) and (b) the Kolmogorov (Eq. 4.29.2) wave spectrum of magnetic field perturbations. The upstream perturbation amplitude ¿B/£0,1 is given near the respective results. Linear fits to the power-law parts of the spectra are presented and values of the phase space distribution function spectral indices a are given in parentheses. Particles in the energy range indicated by arrows can effectively interact with the magnetic field inhomogeneities (kmin < kres < kmax ). For upstream particles probabilities of reflection from the shock, PT , are presented in the bottom panels as a function of particle energy for the respective particle spectra above (the transmission probability P12 = 1 - PT ). According to Niemiec and Ostrowski (2003b).
From Fig. 4.29.3 it can be seen that the particle spectral indices deviate from the small amplitude results of the pitch angle scattering model (Kirk and Schneider, 1987a,b; Heavens and Drury, 1988; Kirk and Heavens, 1989). In addition, the increasing magnetic field perturbations can produce non-monotonic changes of the particle spectral index - the feature which has not been discussed for parallel shocks so far. Analogously to oblique shock waves (Niemiec and Ostrowski, 2003a; see Section 4.29.2), the particle spectra obtained are non power-law in the full energy range and the shape of the spectrum varies with the amplitude of turbulence and the wave power index. The non-monotonic variation of the spectral index with the turbulence amplitude results from modifications of the particle acceleration process at the shock. The long-wave finite-amplitude perturbations produce locally oblique magnetic field configurations and lead to the occurrence of particles reflected from the compressed field downstream of the shock. The probability of reflection depends on the turbulence amplitude and the amount of field perturbations with wavelengths larger than the resonance wavelength for a given particle, as presented in the bottom panels (c) and (d) in Fig. 4.29.3. For ¿B/Bq,1 = 1.0 the reflection probability is higher compared to the other perturbation amplitudes considered and the particle spectrum is flatter. For smaller (¿B/B01 = 0.3) and larger (SBJBq1 =
3.0) turbulence amplitudes the reflection and transmission probability do not differ considerably, which results in the similar values of the spectral indices.
From Fig. 4.29.3 can be seen also that the spectra obtained for the Kolmogorov case seem to exhibit a continuous slow change of inclinations. Thus the fitted power-laws depend to some extent on the energy range chosen for the fit. One can also note a steep part of the spectrum at low energies for ¿B/Bq,1 = 0.3 in panel (b).
The reflection (transmission) probabilities presented decrease (increase) at high particle energies owing to a limited dynamic range of the magnetic field turbulence. The locally oblique field configurations are mainly formed by long-wave perturbations (k < kres) in accordance with Ostrowski (1988b). For high energy particles with kres < km^n there are no corresponding long waves and the upstream particles can be only transmitted downstream of the shock. In these conditions the acceleration process would converge to the 'classic' parallel shock acceleration model, but in the simulations considered particles move far to the introduced escape boundary forming a cut-off (Niemiec and Ostrowski, 2003b).
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