Self-consistent Alfvén-wave transmission and test-particle acceleration at parallel shocks

The transmission of Alfvén waves at parallel shocks is treated self-consistently, i.e., taking the pressure and energy flux exerted by the waves into account when determining the shock’s gas compression ratio. The resulting test-particle acceleration and transport parameters at the shock are shown to be functions of five parameters: the Alfvénic Mach number of the shock, the upstream plasma beta, and the cross helicity, magnetic amplitude, and power-spectral index of the upstream waves. In a large region of parameter space, the scattering-center compression ratio is significantly larger than the gas compression ratio. For some conditions, it may get very large values that result in an extremely hard test-particle-energy spectrum of the cosmic-rays accelerated at the shock. This result is qualitatively the same as that obtained by the same authors under the “testwave” approach, i.e., assuming that the waves have no impact on the shock’s dynamics. However, the present analysis gives limits to the application of the previous results, and widens the scope of the previous analysis by removing a mathematical singularity in shocks with squared Mach number approaching the test-wave-theory gas compression ratio.