Giacalone and Jokipii (2005), Giacalone (2005) consider then non-diffusive test-particle numerical simulations to better address the physics of acceleration at low energies. In these calculations, the trajectories of an ensemble of test particles are integrated by numerically solving the Lorentz force on each particle using pre-specified electric and magnetic fields. The mean magnetic field makes an angle

0Bn with respect to the shock-normal direction. Superimposed on this is a fluctuating component that is determined from a pre-specified power spectrum that resembles the usual Kolmogorov spectrum. The correlation scale of the turbulent magnetic field is taken to be 2000 ui/Qj, where ui is the upstream flow speed and Qj is the ion cyclotron frequency. Both components satisfy Maxwell's equations. Test particles (protons) are released with an energy of 3 times the plasma-ram energy in the local fluid frame just behind the shock front. Each particle's trajectory is integrated until it escapes downstream by convection (based on a probability of return criterion), or reaches an arbitrary high-energy cutoff (taken to be 2 X105 times the plasma-ram energy). Fig. 4.26.3 shows the steady-state energy spectra downstream of the shock for 7 numerical simulations in which the only varying parameter is dBn . Note that the spectra for the cases of dBn = 0°, 15°, 30° all lie on top of one another indicating that there is no dependence on this parameter at all for quasi-parallel shocks.

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