Electron Heating in 2D Particle-in-Cell Simulations of Quasi-Perpendicular Low-Beta Shocks

We measure the thermal electron energization in 1D and 2D particle-in-cell (PIC) simulations of quasi-perpendicular, low-beta ($\beta_p=0.25$) collisionless ion-electron shocks with mass ratio $m_i/m_e=200$, fast Mach number $\mathcal{M}_{ms}=1$-$4$, and upstream magnetic field angle $\theta_{Bn} = 55$-$85^\circ$ from shock normal $\hat{\boldsymbol{n}}$. It is known that shock electron heating is described by an ambipolar, $\boldsymbol{B}$-parallel electric potential jump, $\Delta\phi_\parallel$, that scales roughly linearly with the electron temperature jump. Our simulations have $\Delta\phi_\parallel/(0.5 m_i {u_\mathrm{sh}}^2) \sim 0.1$-$0.2$ in units of the pre-shock ions' bulk kinetic energy, in agreement with prior measurements and simulations. Different ways to measure $\phi_\parallel$, 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 $\phi_\parallel$ in our low-$\beta_p$ shocks. We further focus on two $\theta_{Bn}=65^\circ$ shocks: a $\mathcal{M}_s=4$ ($\mathcal{M}_A=1.8$) case with a long, $30 d_i$ precursor of whistler waves along $\hat{\boldsymbol{n}}$, and a $\mathcal{M}_s=7$ ($\mathcal{M}_A=3.2$) case with a shorter, $5d_i$ precursor of whistlers oblique to both $\hat{\boldsymbol{n}}$ and $\boldsymbol{B}$; $d_i$ is the ion skin depth. Within the precursors, $\phi_\parallel$ has a secular rise towards the shock along multiple whistler wavelengths and also has localized spikes within magnetic troughs. In a 1D simulation of the $\mathcal{M}_s=4$, $\theta_{Bn}=65^\circ$ case, $\phi_\parallel$ shows a weak dependence on the electron plasma-to-cyclotron frequency ratio $\omega_{pe}/\Omega_{ce}$, and $\phi_\parallel$ decreases by a factor of 2 as $m_i/m_e$ is raised to the true proton-electron value of 1836.