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Self-consistent Monte Carlo simulations have been a fruitful approach to model particle acceleration dynamically coupled with the foreshock development in quasi-parallel shocks. There is the global Coronal Shock Acceleration (CSA) Monte Carlo simulation code that is capable of modelling self-consistent shock acceleration from the inner corona to the solar wind. However, in CSA, the resonant interactions of particles with the foreshock Alfvén waves are not modelled using the full resonance condition. The used simplified condition implies that particles of a given energy interact with only one particular spectral component of the wave spectrum. In contrast, the exact (within quasi-linear theory) treatment implies that such particles due to scattering in pitch angle can interact with different wave spectrum components. This changes the modelled particle acceleration efficiency of the shock and the wave spectrum evolution in the foreshock. We have developed a new self-consistent Monte Carlo simulation code, in which we overcome the previous simplification, and applied the two codes to model acceleration of protons in a parallel coronal shock. We also used the new code to simulate proton acceleration in an interplanetary shock. Due to the choice of the plasma and shock parameters, this simulation is applicable to quasi-parallel bow shock of the Earth. Comparison of the results shows that the resonant wave-particle interactions governed by the full resonance condition yield less efficient particle acceleration at the shock and the opposite energy dependence of the proton mean free path in the foreshock than the simplified treatment. Moreover, the Alfvén wave intensity spectrum resulting from the new code exhibits a k−2 dependence at large wavenumbers, characteristic for both the coronal shock and the interplanetary one. This result is in agreement with that of a hybrid-Vlasov simulation (Vlasiator) of the foreshock evolution of the Earth’s bow shock.

Particle acceleration and foreshock evolution in heliospheric shocks from self-consistent Monte Carlo simulations