Electron Heating in 2D Particle-in-cell Simulations of Quasi-perpendicular Low-beta Shocks

Abstract

We measure the thermal electron energization in 1D and 2D particle-in-cell simulations of quasi-perpendicular, low-beta ( β _p = 0.25) collisionless ion–electron shocks with mass ratio m _i / m _e = 200, fast Mach number ${{ \mathcal M }}_{\mathrm{ms}}=1$ –4, and upstream magnetic field angle θ _Bn = 55°–85° from the shock normal $\hat{{\boldsymbol{n}}}$ . It is known that shock electron heating is described by an ambipolar, B -parallel electric potential jump, Δ ϕ _∥ , that scales roughly linearly with the electron temperature jump. Our simulations have ${\rm{\Delta }}{\phi }_{\parallel }/(0.5{m}_{{\rm{i}}}{{u}_{\mathrm{sh}}}^{2})\sim 0.1$ –0.2 in units of the pre-shock ions’ bulk kinetic energy, in agreement with prior measurements and simulations. Different ways to measure ϕ _∥ , including the use of de Hoffmann–Teller frame fields, agree to tens-of-percent accuracy. Neglecting off-diagonal electron pressure tensor terms can lead to a systematic underestimate of ϕ _∥ in our low- β _p shocks. We further focus on two θ _Bn = 65° shocks: a ${{ \mathcal M }}_{{\rm{s}}}\,=\,4$ ( ${{ \mathcal M }}_{{\rm{A}}}\,=\,1.8$ ) case with a long, 30 d _i precursor of whistler waves along $\hat{{\boldsymbol{n}}}$ , and a ${{ \mathcal M }}_{{\rm{s}}}\,=\,7$ ( ${{ \mathcal M }}_{{\rm{A}}}\,=\,3.2$ ) case with a shorter, 5 d _i precursor of whistlers oblique to both $\hat{{\boldsymbol{n}}}$ and B ; d _i is the ion skin depth. Within the precursors, ϕ _∥ has a secular rise toward the shock along multiple whistler wavelengths and also has localized spikes within magnetic troughs. In a 1D simulation of the ${{ \mathcal M }}_{{\rm{s}}}\,=\,4$ , θ _Bn = 65° case, ϕ _∥ shows a weak dependence on the electron plasma-to-cyclotron frequency ratio ω _pe /Ω _ce , and ϕ _∥ decreases by a factor of 2 as m _i / m _e is raised to the true proton–electron value of 1836.

Keywords

• Shocks
• Interplanetary shocks
• Planetary bow shocks