Electron Acceleration at Shock Ripples: Role of Pitch-angle Diffusion

Suprathermal electrons are routinely observed in interplanetary space. At higher energies, there are in-situ evidences that shocks, both interplanetary shocks, often driven by fast coronal mass ejections, and terrestrial bow shocks, can accelerate electrons up to transrelativistic energies (∼MeVs). The acceleration mechanism responsible for these energetic electrons is still under debate. In this work, we study the effects of large-scale shock ripples on electron acceleration at a quasi-perpendicular shock in a 2D system. For tractability of the numerical simulation, we consider the scenario where the magnetic field line contains ripples, and the shock is assumed planar and piecewise. The propagation of gyrophase-averaged electrons is governed by the focused transport equation, where the effect of the turbulent magnetic field is modeled by the pitch-angle diffusion, described by the quasi-linear theory. A Monte Carlo simulation on the equivalent time-forward Itô stochastic differential equation is performed within a periodic box to obtain the phase-space distribution function of the accelerated electrons. Our model predicts power-law energy spectra with a cutoff at high-energy ends, whereas their spectral indices are softer than those predicted by the diffusive shock acceleration theory. We demonstrate that, with a suitable choice of pitch-angle diffusion strength, a small fraction of electrons can experience magnetic traps in multiple ripples along the shock surface, boosting their energies to ∼MeVs. Our results therefore provide a framework for a better understanding of relativistic electron events associated with shocks within the heliosphere.