magnetars ($P\sim 1$ - $10$s) the Goldreich-Julian particle density [@GJ1969] for a magnetar magnetosphere has the typical value $$\begin{aligned}
n_{\textsc{gj}} = 7\times 10^{12} \,\text{cm}^{-3}\,\left ( \frac{B_{\rm pole}}{10^{15} \,{\rm G}}\right) \left( \frac{R_{*}}{r}\right )^3 \left ( \frac{10 \,{\rm s}}{P}\right),\end{aligned}$$ where $B_{\rm pole}$ is the magnetic field strength at the magnetar pole, $R_*$ the magnetar radius and $r$ the distance to the center of the star. This limits the magnetospheric current density close to the surface to $J<e\,c\,n_{\textsc{gj}} \approx 3\times 10^8$A/m$^2$, much below the typical values needed to support currents in strongly twisted magnetospheres of magnetars, of the order of $$J\sim \frac{B c}{4 \pi r}\sim 8.2\times 10^{12}\,\text{A}\,\text{m}^{-2}\,\left( \frac{B_{\rm pole}}{10^{15}\,\text{G}}\right ) \left( \frac{R_*}{10\,\text{km}}\right )^{-1}\label{eq:GJcurrent}.$$ In general, magnetospheric currents in magnetars cannot be supported neither by Goldreich-Julian charges nor by charges lifted from the surface. @Beloborodov2007 proposed that the currents are supported by $e^+$-$e^-$ pairs generated in the magnetosphere in an intermittent discharge process that can be sustained for voltages along magnetic field lines of about $10^8 -10^9$V. This voltage can be maintained by self-induction in untwisting magnetospheres [@Beloborodov2009]. This untwisting is driven by the effective resistivity of the magnetosphere; the thermal photons from the magnetar’s surface scatter resonantly off the charges supporting the magnetospheric 