General-relativistic Resistive Magnetohydrodynamics with Robust Primitive-variable Recovery for Accretion Disk Simulations

Abstract

Recent advances in black hole astrophysics, particularly the first visual evidence of a supermassive black hole at the center of the galaxy M87 by the Event Horizon Telescope, and the detection of an orbiting "hot spot" nearby the event horizon of Sgr A * in the Galactic center by the Gravity Collaboration, require the development of novel numerical methods to understand the underlying plasma microphysics. Non-thermal emission related to such hot spots is conjectured to originate from plasmoids that form due to magnetic reconnection in thin current layers in the innermost accretion zone. Resistivity plays a crucial role in current sheet formation, magnetic reconnection, and plasmoid growth in black hole accretion disks and jets. We included resistivity in the three-dimensional generalrelativistic magnetohydrodynamics (GRMHD) code BHAC and present the implementation of an implicit-explicit scheme to treat the stiff resistive source terms of the GRMHD equations. The algorithm is tested in combination with adaptive mesh refinement to resolve the resistive scales and a constrained transport method to keep the magnetic field solenoidal. Several novel methods for primitive-variable recovery, a key part in relativistic magnetohydrodynamics codes, are presented and compared for accuracy, robustness, and efficiency. We propose a new inversion strategy that allows for resistive-GRMHD simulations of low gas-to-magnetic pressure ratio and highly magnetized regimes as applicable for black hole accretion disks, jets, and neutron-star magnetospheres. We apply the new scheme to study the effect of resistivity on accreting black holes, accounting for dissipative effects as reconnection.

Magnetic Reconnection and Hot Spot Formation in Black Hole Accretion Disks

“…In both cases the magnetic field strength is set such that β = 2p max /b 2 max = 100. Plasma-β and the magnetization σ = b 2 /(ρc 2 ) for a cold (i.e., p ρ) plasma are defined using the magnetic field strength b 2 co-moving with the fluid (see Ripperda et al 2019c for a definition). We set an atmospheric rest-mass density and pressure as ρ atm = ρ min r −3/2 and p atm = p min r −5/2 where ρ min = 10 −4 and p min = 10 −6 /3.…”

“…We model dissipation in the accretion flow by assuming a small, uniform, and constant resistivity η = 5 × 10 −5 in the set of GRRMHD equations (Ripperda et al 2019c), resulting in a large Lundquist number S = Lc/η ∼ 2 × 10 5 that is well above the plasmoid threshold, where we assume the typical length scale to be the length of a current sheet L = L sheet ∼ O(10r g ), and the speed of light c as the typical velocity. A small resistivity allows us to capture both the global near-ideal accretion dynamics (Ripperda et al 2019c) and the fast reconnection resulting in plasmoid formation in localized thin current sheets (Ripperda et al 2019b).…”

“…Extreme resolutions are necessary to resolve current sheets that are thin enough to be liable to the plasmoid instability, while also capturing the global accretion flow dynamics. A combination of an implicit-explicit (IMEX) timestepping scheme (Ripperda et al 2019c) to capture fast reconnection dynamics, together with adaptive mesh refinement (AMR) capabilities of the Black Hole Accretion Code (BHAC, Porth et al 2017;Olivares et al 2019) to accurately resolve the smallest scales in the system allows us to study resistive reconnection and plasmoid formation in black-hole accretion disks for the very first time. We employ the GRRMHD module of BHAC to investigate whether a hot spot can form and grow nearby the event horizon.…”

“…A detailed outline of the numerical method used in this work, including a comprehensive description of the GRRMHD equations can be found in Ripperda et al (2019c). From here onward we adopt Lorentz-Heaviside units where a factor of 1/ √ 4π is absorbed into the elec- tromagnetic fields and velocities are measured in units of the speed of light c. We employ a 3 + 1 decomposition of the GRRMHD equations based on the Arnowitt-Deser-Misner formalism (Arnowitt et al 1959) with a (−, +, +, +) signature for the spacetime metric where Roman indices run over space only i.e., (1,2,3).…”

MPI-AMRVAC: A parallel, grid-adaptive PDE toolkit

“…A recent code comparison project [68] between the most modern software efforts to simulate GRMHD conditions as suitable in the vicinity of black holes showed that BHAC meets all standards of merit for guiding and interpreting contemporary astrophysical research. The BHAC code has recently been extended with an IMEX scheme to handle the extension to general relativistic, resistive MHD equations [69], where all covariant Maxwell equations, especially also those for electric field evolutions, enter. The IMEX scheme then treats the stiff resistive source terms, and can use a staggered representation of the GR(R)MHD variables, to ensure that magnetic monopoles only occur at machine precision [70].…”