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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 sameactive region (AR 13088) and observed by Solar Orbiter (SolO) on August 31 2022and September 05 2022, are modelled. While both events originated from the sameactive region, they exhibited notable differences, including: 1) the August ESPevent 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-overupstream of the shock where the intensity of higher energy protons exceedsthose of lower energy protons, leading to positive (``reverse'') spectralindices prior to the shock passage. For both events, our simulations replicatethe observed duration of the shock sheath, depending on the decelerationhistory of the CME. Imposing different choices of escaping length scale, whichis related to the decay of upstream turbulence, the modelled time intensityprofiles prior to the shock arrival also agree with observations. Inparticular, the cross-over of this time profile in the September event is wellreproduced. We show that a ``reverse'' upstream spectrum is the result of theinterplay between two length scales. One characterizes the decay of upstreamshock accelerated particles, which are controlled by the energy-dependentdiffusion coefficient, and the other characterizes the decay of upstreamturbulence power, which is related to the process of how streaming protonsupstream of the shock excite Alfv\'{e}n waves. Simulations taking into accountreal-time background solar wind, the dynamics of the CME propagation, andupstream turbulence at the shock front are necessary to thoroughly understandthe ESP phase of large SEP events.

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