In this paper, the effects of wave–wave interactions of the lowest order, i.e., three-wave interactions, on parallel-propagating Alfvén wave spectra on a closed magnetic field line are considered. The spectra are then used to evaluate the transport parameters of energetic particles in a coronal loop. The wave spectral density is the main variable investigated, and it is modelled using a diffusionless numerical scheme. A model, where high-frequency Alfvén waves are emitted from the two footpoints of the loop and interact with each other as they pass by, is considered. The wave spectrum evolution shows the erosion of wave energy starting from higher frequencies so that the wave mode emitted from the closer footpoint of the loop dominates the wave energy density. Consistent with the cross-helicity state of the waves, the bulk velocity of energetic protons is from the loop footpoints towards the loop apex. Protons can be turbulently trapped in the loop, and Fermi acceleration is possible near the loop apex, as long as the partial pressure of the particles does not exceed that of the resonant waves. The erosion of the Alfvén wave energy density should also lead to the heating of the loop.

<p>Fluxes of solar energetic particles (SEPs) are associated with solar flares and coronal/interplanetary shock waves. In the case of shocks, particles are thought to get accelerated to high energies via the diffusive shock acceleration mechanism. In order to be efficient, this mechanism requires an enhanced level of magnetic turbulence in the vicinity of the shock front, in particular, in the so-called foreshock region upstream of the shock. This turbulence enhancement can be produced self-consistently, i.e., by the accelerated particles themselves via streaming instability. This idea underlies the SOLar Particle Acceleration in Coronal Shocks (SOLPACS) Monte-Carlo simulation code, which we developed earlier to simulate acceleration of protons in coronal shocks. In the present work, we apply SOLPACS to model an energetic storm particle (ESP) event measured by the STEREO A spacecraft on November 10, 2012. All but one main SOLPACS input parameters are fixed by the in-situ plasma measurements from the spacecraft. Comparison of a simulated proton energy spectrum at the shock with the observed one then allows us to fix the last simulation input parameter related to efficiency of particle injection to the acceleration process. Subsequent comparison of simulated proton time-intensity profiles in a number of energy channels with the observed ones shows a very good correspondence throughout the upstream region. Our results give support for the quasi-linear formulation of the foreshock. This research has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 870405 (EUHFORIA 2.0).</p>