Modelling two Energetic Storm Particle Events Observed by Solar Orbiter Using the Combined EUHFORIA and iPATH Models

By coupling the EUropean Heliospheric FORcasting Information Asset (EUHFORIA) and the improved Particle Acceleration and Transport in the Heliosphere (iPATH) model, two energetic storm particle (ESP) events, originating from the same active region (AR 13088) and observed by Solar Orbiter (SolO) on August 31 2022 and September 05 2022, are modelled. While both events originated from the same active region, they exhibited notable differences, including: 1) the August ESP event lasted for 7 hours, while the September event persisted for 16 hours; 2) The time intensity profiles for the September event showed a clear cross-over upstream of the shock where the intensity of higher energy protons exceeds those of lower energy protons, leading to positive (``reverse'') spectral indices prior to the shock passage. For both events, our simulations replicate the observed duration of the shock sheath, depending on the deceleration history of the CME. Imposing different choices of escaping length scale, which is related to the decay of upstream turbulence, the modelled time intensity profiles prior to the shock arrival also agree with observations. In particular, the cross-over of this time profile in the September event is well reproduced. We show that a ``reverse'' upstream spectrum is the result of the interplay between two length scales. One characterizes the decay of upstream shock accelerated particles, which are controlled by the energy-dependent diffusion coefficient, and the other characterizes the decay of upstream turbulence power, which is related to the process of how streaming protons upstream of the shock excite Alfv\'{e}n waves. Simulations taking into account real-time background solar wind, the dynamics of the CME propagation, and upstream turbulence at the shock front are necessary to thoroughly understand the ESP phase of large SEP events.