Generation of whistler waves by reflected electrons and their self-confinement at quasi-perpendicular shocks

We investigate the mechanism of whistler-mode wave generation by shock-reflected electrons at quasi-perpendicular collisionless shocks. By employing Liouville mapping to construct the electron velocity distribution function in the shock and performing linear instability analysis, we explore whistler wave generation by the mirror-reflected electrons near the upstream edge of the shock transition layer. We find that the reflected electrons can excite two distinct instabilities with different propagation directions when both the upstream electron beta βe and Alfvén Mach number in the de Hoffmann–Teller frame MA/ cos θBn are sufficiently large, where MA is Alfvén Mach number and θBn is the angle between the upstream magnetic field and the shock normal. In the parameter regime of Earth's bow shock, the instability threshold condition is roughly given by MA/ cos θBn≳50. Since such shocks are supercritical with respect to the whistler critical Mach number, the generated waves cannot propagate upstream and will accumulate in the transition layer. Furthermore, we find that the pitch-angle scattering by the generated waves may trigger secondary instabilities on the same branch. We suggest that the sequence of instabilities likely happening within the shock transition layer can efficiently scatter the reflected electrons over a broad range of pitch angles. Consequently, the reflected electrons may be confined within the shock by the waves generated by themselves. The self-confinement provides the necessary ingredient of stochastic shock drift acceleration, which then offers a plausible mechanism for the electron injection into diffusive shock acceleration.