Electron Acceleration in a Nonrelativistic Shock with Very High Alfvén Mach Number

Electron acceleration associated with various plasma kinetic instabilities in a nonrelativistic shock with very high Alfvén Mach number ($M_A\sim 45$) is revealed by means of a two-dimensional fully kinetic particle-in-cell simulation. Electromagnetic (ion Weibel) and electrostatic (ion-acoustic and Buneman) instabilities are strongly activated at the same time in different regions of the two-dimensional shock structure. Relativistic electrons are quickly produced predominantly by the shock surfing mechanism with the Buneman instability at the leading edge of the foot. The energy spectrum has a high-energy tail exceeding the upstream ion kinetic energy accompanying the main thermal population. This gives a favorable condition for the ion-acoustic instability at the shock front, which in turn results in additional energization. The large-amplitude ion Weibel instability generates current sheets in the foot, implying another dissipation mechanism via magnetic reconnection in a three-dimensional shock structure in the very-high-$M_A$ regime.

Figure 1

Overview of the two-dimensional shock structure at ${\Omega }_{gi}T=4$. (a) Two-dimensional ($X$-$Y$) profile of the ion number density ($N_i$) normalized by the upstream value. The inset shows the profile along the $x$ axis at $Y=6.3{\lambda }_i$. (b) Strength of the electrostatic field (${\mathbf{E}}_{\mathrm{est}}$) at the leading edge of the foot in the zoomed-in region in panel (a). (c) $B_z$ in the foot. (d) The $y$ component of the electrostatic field around the overshoot. The fields are normalized by the upstream magnetic field $B_0$.

Figure 2

(a) Typical trajectories of accelerated electrons (black) superposed on the two-dimensional profile of the electrostatic field strength normalized by $B_0$. Open squares indicate the positions of the particle following the trajectory back for $15{\Omega }_{ge}^{-1}$. The number next to each open square is $\gamma$. The snapshot is taken at ${\Omega }_{ge}T=957.25$. (b) Energy spectra of the electron in the foot (gray) and downstream (black) regions. Dashed lines are fitted relativistic Maxwell distributions to the downstream spectrum. The lower and upper $x$ axes refer to $\gamma$ and the kinetic energy normalized by the upstream ion kinetic energy, respectively.

Figure 3

(a) Fourier power spectrum of $B_z$ in the foot. The wave number on each axis ($k_x$ and $k_y$) is normalized by the ion inertia length. (b) Ion four velocity distribution sampled in $46.5{\lambda }_i\le X\le 48.5{\lambda }_i$ and $2.4{\lambda }_i\le Y\le 4.4{\lambda }_i$. The axes show the $x$ and $y$ components of the four velocity ($u_x$-$u_y$) normalized by the speed of light. The number of particles in each bin is color coded on a logarithmic scale.

Figure 4

(a) Fourier power spectrum of the $y$ component of the electric field ($|E_{yk}|$) averaged in the downstream region ($43.3{\lambda }_i\le X\le 45.0{\lambda }_i$). The wave number $k_y$ is normalized by the average electron inertia length. (b) Distribution functions of the ion (solid line) and electron (dashed line) sampled in the downstream region. The $x$ axis shows the $y$ component of the four-velocity ($u_y$) normalized by the speed of light. (c) Ion velocity distribution in the comoving frame of the shock front in the same format as Fig. 3b.