Modelling two energetic storm particle events observed by Solar Orbiter using the combined EUHFORIA and iPATH models

Context. By coupling the EUropean Heliospheric FORcasting Information Asset (EUHFORIA) and the improved Particle Acceleration and Transport in the Heliosphere (iPATH) models, we 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 5, 2022.Aims. By combining numerical simulations and SolO observations, we aim to better understand particle acceleration and the transport process in the inner heliosphere.Methods. We simulated two coronal mass ejections (CMEs) in a data-driven, real-time solar wind background with the EUHFORIA code. The MHD parameters concerning the shock and downstream medium were computed from EUHFORIA as inputs for the iPATH model. In the iPATH model, a shell structure was maintained to model the turbulence-enhanced shock sheath. At the shock front, assuming diffuse shock acceleration, the particle distribution was obtained by taking the steady state solution with the instantaneous shock parameters. Upstream of the shock, particles escape, and their transport in the solar wind was described by a focused transport equation using the backward stochastic differential equation method.Results. While both events originated from the same active region, they exhibited notable differences. One notable difference is the duration of the events, as the August ESP event lasted for 7 h, while the September event persisted for 16 h. Another key difference concerns the time intensity profiles. The September event showed a clear crossover 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 crossover 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 the accelerated particles upstream of the shock, 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 Alfven waves.Conclusions. The behavior of solar energetic particle (SEP) events depends on many variables. Even similar eruptions from the same AR may lead to SEP events that have very different characteristics. 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.