$m_{\mathrm{i}}u_{\mathrm{sh}}^{2}/2$, like the overplotted parallel potential
$\pm\tilde{\phi}_{\parallel}(x)$. Electron energy is also multiplied by
$\mathrm{sgn}(v_{\parallel})$ to show the direction of travel along $x$. All
other panels are unchanged from Figure 13.

Figure 24 shows electron kinetic energy, multiplied by
$\mathrm{sgn}(v_{\parallel})$, as an alternative to $x$-$v_{\parallel}$ phase
space in Figure 13. The electron kinetic energy so plotted has a direct
mapping to the “Trapped” and “Backstreaming” fractions in Figure 24(i)–(l);
recall that trapped electrons have $\varepsilon<0$, and backstreaming
electrons have $\varepsilon>0$ and $v_{\parallel}>0$. In Figure 24(e)–(h),
electrons with signed kinetic energy between the $\pm\tilde{\phi}_{\parallel}$
curves are trapped, while those above the $+\tilde{\phi}_{\parallel}$ curve
are backstreaming.

##  Appendix D Numerical Convergence of Shock Structure

Figure 25:  Particle per cell (PPC) convergence for four case-study
simulations: 1D $\mathcal{M}_{\mathrm{s}}=4$ (left column), 2D
$\mathcal{M}_{\mathrm{s}}=4$ (second from left column), 1D
$\mathcal{M}_{\mathrm{s}}=7$ (third from left column), and 2D
$\mathcal{M}_{\mathrm{s}}=7$ (right column). Upstream angle
$\theta_{Bn}=65^{\circ}$ for all shocks. In all panels, black curves are
fiducial simulations with 2048 (1D) and 128 PPC (2D). Colored curves are
simulations with varying PPC: 1024–8192 (1D) and 32–256 (2D). All quantities
are $y$ averaged. (a–d): Magnetic field strength $B/B_{0}$. (e–h): Electron
temperature jump $T_{\mathrm{e}}/T_{0}-1$. (i–l): Parallel potential
$\phi_{\parallel,\mathrm{amb}}$, in units of
$0.5m_{\mathrm{i}}u_{\mathrm{sh}}^{2}$. (m–p): Trapped electron fraction as
defined for Figure 13. (q–t): Backstreaming electron fraction as defined for
Figure 13.

We use our case study shocks to assess numerical convergence with respect to
the upstream number of particles per cell (PPC), both ions and electrons.
Recall that we adopted 2048 and 128 PPC for 1D and 2D simulations
respectively.

The overall shock structure is well converged for all of our case studies
(Figure 25(a)-(d)). In 2D, the shock ramp and precursor waves are not matched
in phase for simulations with different PPC; the 1D simulations, constrained
to a single wave train along $x$, are more coherent. The electron energy gain,
as measured by both $T_{\mathrm{e}}$ and $\phi_{\parallel,\mathrm{amb}}$,
appears converged to within $10\%$ (Figure 25(e)-(l)). For the 2D
$\mathcal{M}_{\mathrm{s}}=7$ shock, the upstream to downstream jumps in
$T_{\mathrm{e}}$ and $\phi_{\parallel,\mathrm{amb}}$ do not vary monotonically
with PPC, which we interpret as statistical fluctuation rather than a lack of
convergence.

The trapped and backstreaming electron fractions (Figure 25(m)-(t)) also
appear converged in all of our case study shocks. The trapped fraction, in
particular, serves as a coarse proxy for local phase mixing and scattering of
both $v_{\parallel}$ and pitch angle within the whistler precursor. The
electrostatic power in numerical noise is not small, being within an order of
magnitude of the power in electron holes and small-scale structures (Figure
17(i)–(l)). It is not clear if the _relative_ contributions of different
particle scattering mechanisms are converged. But, at least, Figure 17(i)–(l)
suggests that the _total_ phase-space flow in our shocks is insensitive to PPC
sampling.

##  Appendix E Simulation Parameters

Table E provides simulation input parameters and some derived parameters for
the main parameter sweep of 1D and 2D shocks, as well as a set of 1D
simulations with varying $\omega_{\mathrm{pe}}/\Omega_{\mathrm{ce}}$ and
$m_{\mathrm{i}}/m_{\mathrm{e}}$ presented in Section 7.2. The parameters
$\sigma$, $\Theta_{\mathrm{e}}$, and $u_{0}$ are actual code inputs, from
which the shock parameters $\mathcal{M}_{\mathrm{s}}$,
$\mathcal{M}_{\mathrm{A}}$, $\mathcal{M}_{\mathrm{ms}}$, $\beta_{\mathrm{p}}$,
$u_{\mathrm{sh}}$, $u_{\mathrm{sh}}/r$ can be derived assuming a single-fluid
adiabatic index $\Gamma=5/3$. The columns of Table E not already defined in
the main text are as follows.

  * •

$t\Omega_{\mathrm{ci}}$ is the simulation duration.

  * •

$\sigma=B_{0}^{2}/\left[4\pi\left(\gamma_{0}-1\right)\left(m_{\mathrm{i}}+m_{\mathrm{e}}\right)n_{0}c^{2}\right]$
is a ratio of upstream magnetic and kinetic energy density, with
$\gamma_{0}=1/\sqrt{1-u_{0}^{2}/c^{2}}$.

  * •

$u_{y}$ is the post-shock transverse drift along $y$ predicted from the R-H
conditions and used to set the electromagnetic fields at the simulation’s left
boundary.

  * •

$u_{\mathrm{right}}$ is the manually-chosen expansion speed of the domain’s
right-side boundary.

Table 1:  Simulation input parameters.

$m_{\mathrm{i}}/m_{\mathrm{e}}$ | $\mathcal{M}_{\mathrm{s}}$ | $\mathcal{M}_{\mathrm{A}}$ | $\mathcal{M}_{\mathrm{ms}}$ |  $\theta_{Bn}$ (∘) | $t\Omega_{\mathrm{ci}}$ | $\sigma$ | $\Theta_{\mathrm{e}}$ | $u_{0}/c$ | $u_{\mathrm{sh}}/c$ | $u_{\mathrm{sh}}/r/c$ | $u_{y}/c$ | $u_{\mathrm{right}}/c$
---|---|---|---|---|---|---|---|---|---|---|---|---
200 | 3 | 1.37 | 1.25 | 85 | 40.0 | 1.76E+1 | 1.01E$-$2 | 9.56E$-$3 | 3.89E$-$2 | 2.93E$-$2 |  $-$5.89E$-$4 | 4.09E$-$2
200 | 3 | 1.37 | 1.25 | 75 | 40.0 | 1.71E+1 | 1.01E$-$2 | 9.72E$-$3 | 3.89E$-$2 | 2.91E$-$2 |  $-$1.81E$-$3 | 6.11E$-$2
200 | 3 | 1.37 | 1.25 | 65 | 40.0 | 1.60E+1 | 1.01E$-$2 | 1.00E$-$2 | 3.89E$-$2 | 2.88E$-$2 |  $-$3.17E$-$3 | 7.42E$-$2
200 | 3 | 1.37 | 1.25 | 55 | 40.0 | 1.45E+1 | 1.01E$-$2 | 1.05E$-$2 | 3.89E$-$2 | 2.83E$-$2 |  $-$4.76E$-$3 | 1.13E$-$1
200 | 4 | 1.83 | 1.66 | 85 | 40.0 | 2.98E+0 | 1.01E$-$2 | 2.33E$-$2 | 5.18E$-$2 | 2.86E$-$2 |  $-$1.10E$-$3 | 4.42E$-$2
200 | 4 | 1.83 | 1.66 | 75 | 40.0 | 2.93E+0 | 1.01E$-$2 | 2.34E$-$2 | 5.18E$-$2 | 2.84E$-$2 |  $-$3.33E$-$3 | 7.21E$-$2
200 | 4 | 1.83 | 1.66 | 65 | 40.0 | 2.83E+0 | 1.01E$-$2 | 2.38E$-$2 | 5.18E$-$2 | 2.80E$-$2 |  $-$5.62E$-$3 | 8.11E$-$2
200 | 4 | 1.83 | 1.66 | 55 | 40.0 | 2.70E+0 | 1.01E$-$2 | 2.44E$-$2 | 5.18E$-$2 | 2.74E$-$2 |  $-$8.00E$-$3 | 1.21E$-$1
200 | 5 | 2.28 | 2.08 | 85 | 40.0 | 1.28E+0 | 7.08E$-$3 | 2.97E$-$2 | 5.42E$-$2 | 2.45E$-$2 |  $-$1.10E$-$3 | 3.54E$-$2
200 | 5 | 2.28 | 2.08 | 75 | 40.0 | 1.26E+0 | 7.08E$-$3 | 2.99E$-$2 | 5.42E$-$2 | 2.43E$-$2 |  $-$3.29E$-$3 | 7.06E$-$2
200 | 5 | 2.28 | 2.08 | 65 | 40.0 | 1.23E+0 | 7.08E$-$3 | 3.03E$-$2 | 5.42E$-$2 | 2.40E$-$2 |  $-$5.45E$-$3 | 8.47E$-$2
200 | 5 | 2.28 | 2.08 | 55 | 40.0 | 1.18E+0 | 7.08E$-$3 | 3.09E$-$2 | 5.42E$-$2 | 2.34E$-$2 |  $-$7.56E$-$3 | 1.36E$-$1
200 | 7 | 3.20 | 2.91 | 85 | 30.0 | 4.77E$-$1 | 5.06E$-$3 | 4.11E$-$2 | 6.41E$-$2 | 2.31E$-$2 |  $-$9.71E$-$4 | 3.48E$-$2
200 | 7 | 3.20 | 2.91 | 75 | 30.0 | 4.73E$-$1 | 5.06E$-$3 | 4.13E$-$2 | 6.41E$-$2 | 2.29E$-$2 |  $-$2.88E$-$3 | 8.64E$-$2
200 | 7 | 3.20 | 2.91 | 65 | 30.0 | 4.64E$-$1 | 5.06E$-$3 | 4.16E$-$2 | 6.41E$-$2 | 2.25E$-$2 |  $-$4.67E$-$3 | 1.12E$-$1
200 | 7 | 3.20 | 2.91 | 55 | 30.0 | 4.52E$-$1 | 5.06E$-$3 | 4.22E$-$2 | 6.41E$-$2 | 2.20E$-$2 |  $-$6.25E$-$3 | 1.67E$-$1
200 | 10 | 4.56 | 4.15 | 85 | 20.2 | 1.99E$-$1 | 3.04E$-$3 | 4.93E$-$2 | 7.09E$-$2 | 2.18E$-$2 |  $-$6.69E$-$4 | 3.24E$-$2
200 | 10 | 4.56 | 4.15 | 75 | 20.2 | 1.98E$-$1 | 3.04E$-$3 | 4.94E$-$2 | 7.09E$-$2 | 2.16E$-$2 |  $-$1.96E$-$3 | 9.24E$-$2
200 | 10 | 4.56 | 4.15 | 65 | 20.2 | 1.96E$-$1 | 3.04E$-$3 | 4.97E$-$2 | 7.10E$-$2 | 2.13E$-$2 |  $-$3.12E$-$3 | 1.53E$-$1
200 | 10 | 4.56 | 4.15 | 55 | 20.2 | 1.92E$-$1 | 3.04E$-$3 | 5.01E$-$2 | 7.09E$-$2 | 2.09E$-$2 |  $-$4.04E$-$3 | 1.84E$-$1
200 | 4 | 1.83 | 1.66 | 65 | 40.0 | 2.83E+0 | 2.02E$-$2 | 3.38E$-$2 | 7.33E$-$2 | 3.96E$-$2 |  $-$7.95E$-$3 | 1.13E$-$1
200 | 4 | 1.83 | 1.66 | 65 | 40.0 | 2.84E+0 | 5.06E$-$3 | 1.68E$-$2 | 3.66E$-$2 | 1.98E$-$2 |  $-$3.97E$-$3 | 5.79E$-$2
200 | 4 | 1.83 | 1.66 | 65 | 40.0 | 2.84E+0 | 2.53E$-$3 | 1.19E$-$2 | 2.59E$-$2 | 1.40E$-$2 |  $-$2.81E$-$3 | 4.11E$-$2
200 | 4 | 1.83 | 1.66 | 65 | 40.0 | 2.84E+0 | 1.26E$-$3 | 8.41E$-$3 | 1.83E$-$2 | 9.90E$-$3 |  $-$1.98E$-$3 | 2.91E$-$2
400 | 4 | 1.83 | 1.66 | 65 | 40.1 | 2.84E+0 | 1.01E$-$2 | 1.69E$-$2 | 3.67E$-$2 | 1.98E$-$2 |  $-$3.98E$-$3 | 8.11E$-$2
800 | 4 | 1.83 | 1.66 | 65 | 40.1 | 2.84E+0 | 1.01E$-$2 | 1.19E$-$2 | 2.60E$-$2 | 1.40E$-$2 |  $-$2.81E$-$3 | 8.11E$-$2
1836 | 4 | 1.83 | 1.66 | 65 | 40.1 | 2.84E+0 | 1.01E$-$2 | 7.87E$-$3 | 1.71E$-$2 | 9.27E$-$3 |  $-$1.86E$-$3 |

Note. — Table E is available in a machine-readable format in the online
journal.

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