All temporal and spatial units are normalized correspondingly to the timestep $\Delta t$ and the cell size $\Delta x$; the fields are normalized to the fiducial value $B_0$, and the densities are normalized to $n_{\rm ppc}$. We use the following equations
\begin{equation*} d_e = \frac{c}{\omega_{\rm p}},\,\,\omega_{\rm p}^2=\frac{4\pi n_e e^2}{m_e},\,\,r_L=\gamma\beta \frac{m_e c^2}{|e|B},\,\,\sigma=\frac{B^2}{4\pi n_e m_e c^2}. \end{equation*}
For all the details on how the code units are defined see the following section.
$c=\mathrm{CC}\frac{\Delta x}{\Delta t}$
$d_e=\mathrm{COMP}\Delta x$
$n_{\rm ppc}=\mathrm{PPC}\Delta x^{-D}$
$\mathrm{CC}=$
$\mathrm{COMP}=$
$\mathrm{PPC}=$
$\sigma=$
$\omega_{\rm p}^{-1}=~$$~\Delta t$
$\omega_B^{-1}=(\gamma\beta)^{-1}~$$~\Delta t$
$r_L=\gamma\beta~$$~\Delta x$
Pulsar setup
Pulsar setup is initialized with a conducting sphere of radius $R_*$ in the middle of the simulation box. The field is initially dipolar with an enforced strength $B_*$ at the surface. The conductor rotates with a period $P$.
We use the following equations (all the quantities are in code units)
\begin{equation*} \Omega = \frac{2\pi}{P},\,\,R_{ LC} = \frac{c}{\Omega},\,\,B_{ LC}\approx B_*\left(\frac{R_{ LC}}{R_*}\right)^{-3}, \end{equation*}
\begin{equation*} n_{GJ} = \frac{\Omega B}{2\pi c|e|},\,\,\frac{\Delta V_{\rm pc}}{m_e c^2}=\omega_B^0\frac{B_*}{B_0}\frac{R_*}{c}\left(\frac{R_*}{R_{LC}}\right)^2,\,\,s_{ LC}=\frac{d_e(n_{GJ}^{ LC})}{R_{ LC}}. \end{equation*}
$R_*=\,$$\,{\color{gray}\Delta x}$
$B_*=\,$$\,{\color{gray}B_0}$
$\frac{\Delta V_{\rm pc}}{m_e c^2}=\,$
$d_e^*=\,$${\color{gray}\langle\gamma\rangle^{1/2} \left(\frac{n}{n_{GJ}^*}\right)^{-1/2} \Delta x}$
$n^*_{GJ}=\,$$\,{\color{gray}\Delta x^{-3}}$
$\sigma^*=\,$${\color{gray}\left(\frac{n}{n_{GJ}^*}\right)^{-1}\left(\frac{B}{B_*}\right)^2}$
$P=\,$$\,{\color{gray}\Delta t}$
$R_{ LC}=\,$$\,{\color{gray}\Delta x}$
$B_{ LC}=\,$$\,{\color{gray}B_{0}}$
$d_e^{LC}=\,$${\color{gray}\langle\gamma\rangle^{1/2} \left(\frac{n}{n_{GJ}^*}\right)^{-1/2} \Delta x}$
$r_L^{LC}={\color{gray}\gamma\beta}\,$${\color{gray}\left(\frac{B}{B_{LC}}\right)^{-1} \Delta x}$
$s_{LC}=\,$
$n^{LC}_{GJ}=\,$$\,{\color{gray}\Delta x^{-3}}$
$\sigma^{LC}=\,$${\color{gray}\left(\frac{n}{n^{LC}_{GJ}}\right)^{-1}\left(\frac{B}{B_{LC}}\right)^2}$