Electron acceleration at nearly perpendicular collisionless shocks: 2. Reflection at curved shocks

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Abstract
Electrons can be efficiently energized at interplanetary shocks and planetary bow shocks. The acceleration and reflection process is extremely sensitive to the angle θBn between the upstream magnetic field and the shock normal, and is most prominent at θBn ∼ 90°. The mechanism has been investigated by theoretical and simulation means, and can be interpreted as a fast Fermi process or as gradient drift acceleration. Previous work has been carried out for plane shocks only, and is expanded here to take into account the global curvature of a shock. Simple estimates suggest that this curvature may have a strong limiting effect on the acceleration to high energies, i.e., above several keV in case of the Earth's bow shock. We perform two‐dimensional test particle calculations to address this question, and evaluate the reflected electron flux as a function of θBn at the shock surface. The shock profile is derived from hybrid code simulations, and modified to include the first order effects of a global curvature in the vicinity of θBn = 90°. At low energies, the calculated fluxes exhibit a cut‐off and a maximum, which can give rise to observed bump‐on‐tail distributions in the electron foreshock. Results at high energy show that while individual electrons gain less energy in a curved shock, concerning the flux this fact is largely offset by two‐dimensional focusing effects. Electrons that drift into the shock over a wide area converge and stream out within a narrow spatial area, thus greatly enhancing the flux of reflected electrons. A κ distribution of suprathermal solar wind electrons (of index κ = 6) is capable of producing the observed large fluxes of reflected electrons at the Earth's bow shock up to energies of 10 to 15 keV, even when the global shock curvature is accounted for. Beyond this energy range observed spectra are harder, as has been found previously for a plane shock. As a likely reason, the solar wind seed population may be denser than modeled above several keV.