exponentially damping the (numerically induced) components of the electric field parallel to ${\boldsymbol{{\tilde{B}}}}$ and suitably adjusting the electric field in magnetospheric current sheets in order to obtain ${\boldsymbol{{\tilde{B}}}}^2-{\boldsymbol{{\tilde{E}}}}^2\rightarrow 0$ at these locations.

Throughout the literature, the magnetic dominance condition (\[eq:force\_free\_dominance\]) condensates to a necessary condition of FFE [e.g. @Uchida1997; @McKinney2006]. For some authors [e.g. @McKinney2006] the breakdown of the magnetic dominance implies the invalidity of the numerical model. Others [e.g. @Uchida1997] claim that some physical processes (e.g. radiation losses) taking place in the regions where condition (\[eq:force\_free\_dominance\]) is breached may restore the magnetic dominance condition. Indeed, [@Uchida1997] explicitly allows for transient phases violating condition (\[eq:force\_free\_dominance\]) - these regions are then interpreted as abandoning the freezing of magnetic flux onto the flux of matter, being necessarily accompanied by dissipation. Following @Uchida1997, the force-free regime continues to be a valid approximation as long as the dissipative effects are only a small fraction of the total energy. The violation of the perpendicularity condition (\[eq:force\_free\_crossfield\]) is an additional source of (Ohmic) dissipation [studied for example in the context of Alfvén waves in force-free electrodynamics by @Li2018]. In practice, this channel of dissipation occurs when ${\boldsymbol{{\tilde{E}}}}\cdot{\boldsymbol{{\tilde{B}}}}\neq 0$ such that ${\boldsymbol{{\tilde{J}}}}\cdot{\boldsymbol{{\tilde{E}}}}\neq 0$. Currently used force-free codes 