Resistive relativistic magnetohydrodynamics without Amperes Law

TL;DR

Resistive MHD in relativistic plasmas is challenged by stiffness when Ampère's law is used at high conductivity. The authors propose a dual one-form MHD formulation where magnetic-field flux is carried by a conserved two-form current $J^{\mu\nu}$ and resistivity appears as a diffusive correction, augmented by a second-order relaxation term to ensure causality via the BDNK framework, effectively turning diffusion into a Telegrapher's equation. They implement a BHAC-based numerical scheme with constrained transport and two primitive-variable recovery methods, validating it on a Telegrapher's equation benchmark, shock tubes, a cylindrical explosion, and the Orszag-Tang vortex; the results show stability, causal diffusion, and compatibility with traditional resistive MHD in the low-resistivity limit. This work establishes a practical, causal, higher-form MHD formulation and outlines future directions for full dissipative transport and applications to extreme astrophysical phenomena such as black-hole accretion and neutron-star magnetospheres.

Abstract

Resistive magnetohydrodynamics is thought to play a key role in transient astrophysical phenomena such as black hole flares and neutron star magnetospheres. When performing numerical simulations of resistive magnetohydrodynamics, one is faced with the issue that Amperes law becomes stiff in the high conductivity limit which poses challenges to the numerical evolution. We show that using a description of resistive magnetohydrodynamics based on higher form symmetry, one can perform simulations with a generalized dual Faraday tensor without having to use Amperes Law, thereby avoiding the stiffness problem. We also explain the relation of this dual model to a traditional description of resistive magnetohydrodynamics and how causality is guaranteed by introducing second order corrections.

Figures (9)

• Figure 1: Boosted 1D telegraph solution comparing numerical and analytic solutions. The numerical solution is obtained for 128 gridpoints and a CFL-number of 0.3.
• Figure 2: Convergence for the boosted and rotated 2D telegraph solution after simulating for $t=0.8$.
• Figure 3: Shocktube problem for a range of resistivities $\eta$ compared with the ideal MHD case. For the one-form MHD case we set $r=\eta$. Shown are the out-of-plane magnetic field component (corresponding to $J^{ty}$ for one-form MHD) and the gas pressure in the top and bottom panel respectively. Differences between the standard resistive MHD and one-form MHD are most noticeable near the tangential discontinuity for the high resistivity case $\eta=10^{-2}$. Both schemes converge to the ideal solution which is closely realized at $\eta=10^{-4}$.
• Figure 4: Strong cylindrical explosion test according to Komissarov2007 with $r=0.018,\, \tau=0.09$. The top panels show the magnetic field components ($J^{tx},J^{ty}$) along with their fieldlines. The bottom panels show $\log_{10}p$ (left) and the Lorentz factor $\Gamma$ (right). In the latter panel we indicate the adaptive mesh blocks which consist of $16^2$ cells each.
• Figure 5: Density evolution in the relativistic Orszag-Tang vortex problem, comparing ideal, one-form and traditional resistive MHD for equivalent initial conditions.