Magnetic Field Amplification by the Weibel Instability at Planetary and Astrophysical Shocks with High Mach Number

Collisionless shocks are ubiquitous in the Universe and often associated with a strong magnetic field. Here, we use large-scale particle-in-cell simulations of nonrelativistic perpendicular shocks in the high-Mach-number regime to study the amplification of the magnetic field within shocks. The magnetic field is amplified at the shock transition due to the ion-ion two-stream Weibel instability. The normalized magnetic field strength strongly correlates with the Alfvénic Mach number. Mock spacecraft measurements derived from particle-in-cell simulations are fully consistent with those taken in situ at Saturn’s bow shock by the Cassini spacecraft.

Figure 1

Density and magnetic field in run B2: (a) ion density in logarithmic units; and (b) profile of normalized ion density (red line), profile of normalized magnetic field (green line), $B_x/B_0$ (magenta line), $B_y/B_0$ (dark blue line), and $B_z/B_0$ (light blue line). Profiles are calculated in the shock reference frame and averaged over the shock reformation cycle. The shock region is marked by dashed lines, and $x_{\mathrm{sh}}$ is the shock position.

Figure 2

Normalized magnetic field strength (a) and magnetic energy density normalized by upstream ion energy density (b), both evaluated in the shock region defined in Fig. 1. Blue and red colors correspond to left ($\beta =5\times {10}^{-4}$) and right ($\beta =0.5$) shocks, respectively. The green dotted line in Fig. 2 reflects $|B_{\mathrm{sh}}|/B_0=2\sqrt{M_A}$, and that in Fig. 2 shows $U_{\mathrm{sh},\mathit{B}}/U_{\mathrm{sh},\mathit{i}}=4M_A^{-1}$.

Figure 3

Growth rate of Weibel modes for five mass ratios (a) and three shock speeds (b).

Figure 4

Cassini measurements [25] indicated by gray crosses and PIC simulation data displayed with blue and red dots for left ($\beta =5\times {10}^{-4}$) and right ($\beta =0.5$) shocks, respectively. The yellow dash-dotted line is an earlier prediction, $B_{\mathrm{over}}/B_0\approx 0.4M_A^{7/6}/1.26$ [cf. Eq. (1)], corrected for shock reformation. The green dashed line is the behavior found in our PIC simulations, $B_{\max }/B_0=5.5(\sqrt{M_A}-2)$.