Generation of Whistler Waves by Reflected Electrons and Their Self-Confinement at Quasi-Perpendicular Shocks

Diffusive shock acceleration is widely accepted as the primary mechanism to generate high-energy particles in supernova remnant shocks but faces challenges with efficiently accelerating low-energy particles, known as the injection problem. Shock stochastic drift acceleration presents a promising pre-acceleration mechanism, in which the whistler waves in shock transition region can be essential in scattering and energizing low-energy electrons, aligning well with observations (Amano et al., 2022). However, the physical origin of these waves within the shock transition layer has not been fully understood.

In our study, we investigate the generation of whistler-mode waves by shock-reflected electrons at quasi-perpendicular collisionless shocks. Using Liouville mapping, we construct the electron velocity distribution function in the shock, which allows us to explicitly capture the phase-space features of mirror-reflected electrons near the upstream edge of the shock transition region. Based on the constructed distribution, we perform a linear instability analysis using a semi-analytical method (Kennel & Wong, 1967) to examine the whistler wave generation by the mirror-reflected electrons.

We find that the reflected electrons can excite two distinct instabilities with different propagation directions when both the upstream electron beta β_e and Alfvén Mach number in the de Hoffmann-Teller frame M_A/cosθ_Bn are sufficiently large, where M_A is Alfvén Mach number and θ_Bn is the angle between the upstream magnetic field and the shock normal. In the parameter regime of Earth's bow shock, the instability threshold is approximately M_A/cosθ_Bn>∼50. Since such shocks are super-critical with respect to the whistler critical Mach number, the excited waves cannot propagate upstream and instead accumulate within the shock 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. We propose that this sequence of self-generated instabilities enables the confinement of the reflected electrons within the shock transition region. Such self-confinement provides the key ingredient of stochastic shock drift acceleration, which then offers a plausible mechanism for the electron injection into diffusive shock acceleration.