The Acceleration of Electrons at Collisionless Shocks Moving Through a Turbulent Magnetic Field

We perform a numerical-simulation study of the acceleration of electrons at shocks that propagate through a prespecified, kinematically defined turbulent magnetic field. The turbulence consists of broadband magnetic fluctuations that are embedded in the plasma and cover a range of wavelengths, the smallest of which is larger than the gyroadii of electrons that are initially injected into the system. We find that when the variance of the turbulent component of the upstream magnetic field is sufficiently large -- $\sigma^2 \sim$ 10 $B_0^2$, where $B_0$ is the strength of the background magnetic field -- electrons can be efficiently accelerated at a collisionless shock regardless of the orientation of the mean upstream magnetic field relative to the shock-normal direction. Since the local angle between the incident magnetic-field vector and the shock-normal vector can be quite large, electrons can be accelerated through shock-drift acceleration at the shock front. In the upstream region, electrons are mirrored back to the shock front leading to multiple shock encounters. Eventually the accelerated electrons are energetic enough that their gyroradii are of the same order as the wavelength of waves that are included in our description of the turbulent magnetic field. Our results are consistent with recent \textit{in situ} observations at Saturn's bow shock. The study may help understand the acceleration of electrons at shocks in space and astrophysical systems.