Nonlinear electron scattering by electrostatic waves in collisionless shocks

We present a theoretical analysis of electron pitch-angle scattering by ion-acoustic electrostatic fluctuations present in the Earth's bow shock and, presumably, collisionless shocks in general. We numerically simulate electron interaction with a single wave packet to demonstrate the scattering through phase bunching and phase trapping and quantify electron pitch-angle scattering in dependence on the wave amplitude and wave normal angle to the local magnetic field. The iterative mapping technique is used to model pitch-angle scattering of electrons by a large number of wave packets, which have been reported in the Earth's bow shock. Assuming that successive electron scatterings are not correlated, we revealed that the long-term dynamics of electrons is diffusive. The diffusion coefficient depends on the ratio $\varPhi _0/W$ between the wave packet amplitude and electron energy, $D\propto (\varPhi _0/W)^{\nu }$. A quasi-linear scaling ($\nu \approx 2$) is observed for sufficiently small wave amplitudes, $\varPhi _0\lesssim 10^{-3}W$, while the diffusion is nonlinear ($1<\nu <2$) above this threshold. We show that pitch-angle diffusion of ${\lesssim }1$ keV electrons in the Earth's bow shock can be nonlinear. The corresponding diffusion coefficient scales with the intensity $E_{w}^{2}$ of the electrostatic fluctuations in a nonlinear fashion, $D\propto E_{w}^{\nu }$ with $\nu <2$, while its expected values in the Earth's bow shock are $D\sim 0.1\unicode{x2013}100$ $(T_{e}/W)^{\nu -1/2}\,{\rm rad}^{2}\,{\rm s}^{-1}$. We speculate that in the Earth's quasi-perpendicular bow shock the stochastic shock drift acceleration mechanism with pitch-angle scattering provided by the electrostatic fluctuations can contribute to the acceleration of thermal electrons up to approximately 1 keV. The potential effects of a finite perpendicular coherence scale of the wave packets on the efficiency of electron scattering are discussed.