it is necessary to guarantee that there are either no forces acting on the system or, more generally, that the forces of the system balance each other. This is equivalent to a vanishing net Lorentz force on the charges ${\tilde{\rho}_{\rm e}}$ [see, e.g. @Camenzind2007]: $$\begin{aligned}
{\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}&=0\label{eq:electromagnetic_work}\\
{\tilde{\rho}_{\rm e}}{\boldsymbol{{\tilde{E}}}}+{\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}\times{\boldsymbol{{\tilde{B}}}}&=0\label{eq_lorentz_force}\end{aligned}$$ From equation (\[eq\_lorentz\_force\]) one readily obtains the degeneracy condition $$\begin{aligned}
{\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}&=0.\label{eq:force_free_crossfield}\end{aligned}$$ Additionally, force-free fields are required to be magnetically dominant, the magnetic field being always stronger than the electric one, such that the following condition must hold: $$\begin{aligned}
{\boldsymbol{{\tilde{B}}}}^2-{\boldsymbol{{\tilde{E}}}}^2&\geq 0.\label{eq:force_free_dominance}\end{aligned}$$ Conditions (\[eq:force\_free\_crossfield\]) and (\[eq:force\_free\_dominance\]), as well as the conservation condition $\partial_t\left({\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}\right)=0$ can be combined in order to obtain an explicit expression for ${\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}$ [cf. @Komissarov2011; @Parfrey2017]: $$\begin{aligned}
{\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}=\left[{\boldsymbol{{\tilde{B}}}}\cdot{\boldsymbol{\nabla}}\times{\boldsymbol{{\tilde{B}}}}-{\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{\nabla}}\times{\boldsymbol{{\tilde{E}}}}\right]\frac{{\boldsymbol{{\tilde{B}}}}}{{\tilde{B}}^2}+{\tilde{\rho}_{\rm e}}\frac{{\boldsymbol{{\tilde{E}}}}\times{\boldsymbol{{\tilde{B}}}}}{{\tilde{B}}^2}\label{eq:ff_current}\end{aligned}$$ Across the literature [e.g. @Komissarov2004; @Alic2012; @Parfrey2017] we find various modifications in the definition of ${\boldsymbol{{\tilde{J}}}}_{\rm \text{\tiny FF}}$ in order to drive the numerical solution of the system of partial differential equations (\[eq:maxwell\]) towards a state which fulfills equation (\[eq:force\_free\_crossfield\]) by introducing a suitable cross-field conductivity. In the numerical setup, we choose to combine the prescription of @Komissarov2004 with the force-free current given above. This strategy effectively minimises the violations of equations(\[eq:force\_free\_crossfield\]) and (\[eq:force\_free\_dominance\]) by 