Niemiec and Ostrowski (2003a) simulate the first order Fermi acceleration process at mildly relativistic shock waves propagating in more realistic perturbed magnetic fields, taking into account a wide wave vector range turbulence with the power-law spectrum and continuity of the magnetic field across the shock, involving the respective matching conditions at the shock: below the upstream (downstream) quantities are labeled, as usual, with the index '1' ('2'). In the simulations the trajectories of ultra-relativistic test particles are derived by integrating their equations of motion in the perturbed magnetic field. Niemiec and Ostrowski (2003a) consider a relativistic planar shock wave propagating in rarefied electron-proton plasma. Upstream of the shock the field consists of the uniform component, B01, inclined at some angle y/1 to the shock's normal and finite amplitude perturbations imposed upon it. The perturbations are modeled as a superposition of 294 sinusoidal static waves of finite amplitudes (Ostrovski, 1993) which have either a flat
or a Kolmogorov
wave power spectrum in the wide wave vector range ( kmin ,kmax ) and kmax/kmin = 10 . (4.29.3)
The shock moves with the velocity u\ with respect to the upstream plasma. The downstream flow velocity U2 and the magnetic field structure are obtained from the hydrodynamic shock jump conditions, so that the field is continuous across the shock. Derivation of the shock compression ratio defined in the shock rest frame as r = uj/ U2 is based on the approximate formulae derived in Heavens and Drury (1988). The acceleration process in Niemiec and Ostrowski (2003a) is considered in the particle energy range where radiative (or other) losses can be neglected. In Fig. 4.29.1 there are presented particle spectra for the oblique sub-luminal ub,1 = Uj/cos^i < c (4.29.4)
shock wave with U1= 0.5c and y/1= 45° (the shock velocity along the mean magnetic field is then Ub 1= 0.71c, and the shock compression ratio is r = 5.11).
The particle spectra are measured at the shock for three different magnetic field perturbation amplitudes and the flat wave power spectrum - panel (a) and the Kolmogorov spectrum - panel (b) in Fig 4.29.1.
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