Beltrami Equations of extended MHD models for special relativistic plasmas

Relativistic magnetohydrodynamics (RMHD) has been extensively used to study various astrophysical phenomena. However, conventional RMHD omits key two-fluid effects such as the Hall term and electron inertia. To address this limitation, Koide (2009) proposed the relativistic extended MHD (RXMHD), which incorporates both effects. Building on this, Yoshino et al. (2024) clarified the applicability of various reduced models, showing that relativistic Hall MHD (RHMHD) becomes particularly important in highly relativistic regimes.

While force-free equilibria under the Beltrami condition—where the Lorentz force vanishes—have been well studied in non-relativistic extended MHD, their relativistic counterparts remain poorly understood. In non-relativistic Hall MHD, Mahajan and Yoshida (1998) showed that the Beltrami condition leads to a double-curl equation with two eigenvalues.

In this study, we derive Beltrami-type equilibrium equations for RHMHD by imposing stationarity and the force-free condition. We find that the resulting equations also take a double-curl Beltrami form with two eigenvalues, as in the non-relativistic case. However, special relativistic effects introduce significant mathematical complexity, rendering analytic solutions intractable. These results suggest the possibility of rich equilibrium structures in relativistic jets from high-energy objects, where a force-free state is considered as a physically appropriate model.