as *surface*.

![Fast magnetosonic (solid lines) and magnetoelastic (dashed lines) speed in the outer layers of a magnetar, for different magnetic field strengths ranging from 0 to $10^{16}$G. The neutron star model corresponds to the $1.4 M_\odot$ mass APR+DH model of @Gabler2012. The magnetic field is considered to be constant for simplicity. []{data-label="fig:speeds"}](images/figure1.pdf){width="50.00000%"}

Force-free Electrodynamics {#sec:ff_equations}
==========================

In analogy to @Komissarov2004 and @Parfrey2017 we solve Maxwell’s equations in the force-free limit: $$\begin{aligned}
\frac{\partial {\boldsymbol{{\tilde{B}}}}}{\partial t} = -{\boldsymbol{\nabla}}\times{\boldsymbol{{\tilde{E}}}}\qquad\text{and}\qquad\frac{\partial {\boldsymbol{{\tilde{E}}}}}{\partial t} ={\boldsymbol{\nabla}}\times{\boldsymbol{{\tilde{B}}}} - {\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}\:,
\label{eq:maxwell}\end{aligned}$$ where ${\boldsymbol{{\tilde{E}}}}$, ${\boldsymbol{{\tilde{B}}}}$, and ${\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}$ are the electric field, the magnetic field, and the so-called force-free current, respectively. We place a tilde to distinguish quantities expressed in our Heaviside-Lorentz *geometrised* (HLG) system of units, while the same symbols without tilde express quantities in the Gaussian *non-geometrised* (GNG) system of units (see Table \[tab:unit\_conversions\]). We explicitly include the charge conservation equation $$\begin{aligned}
 \frac{\partial {\tilde{\rho}_{\rm e}}}{\partial t}+{\boldsymbol{\nabla}}\cdot{\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}} = 0\,,
 \label{eq:chargecons}\end{aligned}$$ where ${\tilde{\rho}_{\rm e}}$ represents the charge density. Furthermore, we use a mixed hyperbolic/parabolic correction by the introduction of additional potentials (further discussed in appendix \[sec:augmented\_system\]) in order to numerically ensure the constraints ${\boldsymbol{\nabla}}\cdot{\boldsymbol{{\tilde{B}}}}=0$ and ${\boldsymbol{\nabla}}\cdot{\boldsymbol{{\tilde{E}}}}={\tilde{\rho}_{\rm e}}$ [@Dedner2002; @Palenzuela2009; @Mignone2010].

In the force-free limit 