  Electron acceleration to non-thermal energies in low Mach number (M<5) shocks
is revealed by radio and X-ray observations of galaxy clusters and solar
flares, but the electron acceleration mechanism remains poorly understood.
Diffusive shock acceleration, also known as first-order Fermi acceleration,
cannot be directly invoked to explain the acceleration of electrons. Rather, an
additional mechanism is required to pre-accelerate the electrons from thermal
to supra-thermal energies, so they can then participate in the Fermi process.
In this work, we use two- and three-dimensional particle-in-cell plasma
simulations to study electron acceleration in low Mach number shocks. We focus
on the particle energy spectra and the acceleration mechanism in a reference
run with M=3 and a quasi-perpendicular pre-shock magnetic field. We find that
about 15 percent of the electrons can be efficiently accelerated, forming a
non-thermal power-law tail in the energy spectrum with a slope of p~2.4.
Initially, thermal electrons are energized at the shock front via shock drift
acceleration. The accelerated electrons are then reflected back upstream, where
their interaction with the incoming flow generates magnetic waves. In turn, the
waves scatter the electrons propagating upstream back toward the shock, for
further energization via shock drift acceleration. In summary, the
self-generated waves allow for repeated cycles of shock drift acceleration,
similarly to a sustained Fermi-like process. This mechanism offers a natural
solution to the conflict between the bright radio synchrotron emission observed
from the outskirts of galaxy clusters and the low electron acceleration
efficiency usually expected in low Mach number shocks.
