Continuation of Force-Free Electrodynamics upon the loss of magnetic dominance

Abstract

Force-Free Electrodynamics (FFE) describes the evolution of the electromagnetic field in magnetically dominated plasmas, but ceases to be hyperbolic once the magnetic dominance condition $F^{ab}F_{ab}>0$ is lost. We demonstrate that, after the loss of magnetic dominance, FFE may be replaced by a theory of null fields, $F^{ab}F_{ab}=0=F^{ab}\tilde{F}_{ab}$, characterized by the condition that its principal null direction is tangent to a geodesic congruence. In flat spacetime, this theory may be equivalently stated as the condition that the field satisfies $\vec{B}^2-\vec{E}^2=0=\vec{E}\cdot\vec{B}$ and the integral curves of the drift velocity $\vec{E}\times\vec{B}/\vec{B}^{2}$ are straight lines. We develop the general structure and the properties of this theory and test it against 1D PIC simulations using the collision of planar symmetric Alfv\'en waves. We find that the force-free combined with the null continuation shows remarkable macroscopic agreement with PIC simulations, including the birth and evolution of the null region ($F^{ab}F_{ab}=0$) and the formation of a current sheet. As an independent test, we apply the null continuation to Adhikari's type-changing solution, an exact FFE solution exhibiting finite-time loss of magnetic dominance, and also find macroscopic agreement with 1D PIC simulations.