As discussed by Ostrowski (1988b) for non-relativistic shocks, the presence of finite-amplitude magnetic field perturbations modifies the character of the diffusive particle acceleration at the shock wave with the mean field parallel to the shock's normal. The effect arises owing to locally oblique field configurations formed by long-wave perturbations at the shock front and the respective magnetic field compressions. As a result the mean particle energy gains may increase and the particles reflected from the shock front may occur. The same phenomena should occur at relativistic shocks (Ostrowski, 1993).

In the simplified numerical simulations of the first-order Fermi acceleration at parallel mildly relativistic shocks the acceleration time scale reduces with increasing turbulence level, but no spectral index variation occurs (Bednarz and Ostrowski, 1996, 1998). However, the mentioned acceleration models apply very simple modeling of the perturbed magnetic field effects by introducing particle pitch-angle scattering. The purpose of the Niemiec and Ostrowski (2003b) work is to simulate the first order Fermi acceleration process at mildly relativistic shock waves propagating in more realistic perturbed magnetic fields, including a wide wave vector range of turbulence with the power-law spectrum. The magnetic field is continuous across the shock, according to the respective jump conditions. This feature leads to substantial modifications of the acceleration process at parallel shocks: as usually the upstream (downstream) quantities are labeled, as usual, with the index '1' ('2').

In Niemiec and Ostrowski (2003b) the simulations trajectories of ultra-relativistic test particles are derived by integrating their equations of motion in the perturbed magnetic field. A relativistic shock wave is modeled as a plane discontinuity propagating in electron-proton plasma. The magnetic field is defined upstream of the shock. It consists of the uniform component, By, parallel to the shock normal and finite-amplitude perturbations imposed upon it. The perturbations are modeled as a superposition of 294 sinusoidal static waves of finite amplitudes (Ostrowski, 1993). They have either a flat (Eq. 4.29.1) or a Kolmogorov (Eq. 4.29.2) wave power spectrum in the wide wave vector range (kmin, kmax) with kmax/kmin = 105. The shock moves with the velocity U1 with respect to the upstream plasma. The downstream flow velocity u2 and the magnetic field structure are obtained from the hydrodynamic shock jump conditions. Derivation of the shock compression ratio as measured in the shock rest frame, R = U1/U2 , is based on the approximate formulae derived in Heavens and Drury (1988). In the analysis of the acceleration process the particle radiative (or other) losses are neglected. In Fig. 4.29.3 are presented particle spectra for the parallel shock wave with U1= 0. 5c. The shock compression ratio is r = 5.11.

In Fig. 4.29.3 are reflected the particle spectra measured at the shock for three different magnetic field perturbation amplitudes and the flat, panel (a) or the Kolmogorov wave power spectrum, panel (b).

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(a) |
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