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

Abstract

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 $β_e$ and Alfven Mach number in the de Hoffmann-Teller frame $M_A/\cosθ_{bn}$ are sufficiently large, where $M_A$ is \Alfven 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 condition is roughly given by $M_A/\cosθ_{bn}\gtrsim50$. Since such shocks are super-critical 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.

Figures (9)

• Figure 1: Electron VDFs in the plasma rest frame at the upstream edge ($x = -3.0$) for (a) no diffusion ($D_{\mu\mu} t=0$) and (b) weak diffusion ($D_{\mu\mu} t=0.01$). The black dashed lines represent the accessibility boundaries given by Eq.(\ref{['eq:upstream_accessible']}) (the lower ellipse) and Eq. (\ref{['eq:downstream_accessible']}) (the upper hyperbola).
• Figure 2: Linear growth rates in the $(k_{\parallel}, k_{\perp})$ plane driven by the electron VDF with $D_{\mu\mu} t=0.01$ (corresponding to Fig. \ref{['fig:diffusion_eff']}b). From top to bottom, the panels show: the total growth rate, and the individual contributions from the normal cyclotron ($n=-1$), Landau ($n=0$), and anomalous cyclotron ($n=+1$) resonances. The black contours indicate resonance velocities $v_{\mathrm{res},n}$, where the solid and dashed lines represent positive and negative values, respectively.
• Figure 3: $\omega-k$ diagram illustrating the conditions for wave growth and damping. The black curve represents the whistler-mode dispersion relation for propagation angles $\theta_{Bk}=0^\circ$ (solid) and $\theta_{Bk}=60^\circ$ (dashed). The shaded areas indicate regions of strong damping by the background electrons: gray for Landau damping ($n=0$) and blue for normal cyclotron damping ($n=-1$). The red and blue lines represent the resonance conditions for beam electrons with $n=-1$ (normal) and $n=+1$ (anomalous), respectively.
• Figure 4: Schematic diagrams illustrating the wave generation mechanism driven by a reflected electron beam in the plasma rest frame for (a) the downstream-directed cyclotron-driven whistler (DCW) instability and (b) the upstream-directed anomalous-driven whistler (UAW) instability. The black curves denote contours of the electron VDF, the green dashed curves indicate iso-energy contours in the plasma frame, and the orange dashed curve represents the diffusion path imposed by resonant wave–particle interactions. The orange arrow marks the preferred direction of electron diffusion driven by the local gradient of the VDF. Note that the vertical black dashed lines indicate the shock speed in HTF. In both panels, electrons diffusing inward (inside the isoenergy contours) lose energy to the wave, thereby driving the instability.
• Figure 5: Evolution of the electron VDF and resulting instabilities for increasing diffusion strength. The left panels (a, c, e) show the electron VDFs, while the right panels (b,d,f) show the corresponding growth rate maps. From top to bottom, the degree of diffusion is increased: $D_{\mu\mu} t=0.05$ (top panel), $D_{\mu\mu} t=0.2$ (middle panel) and $D_{\mu\mu} t=0.5$ (bottom panel).