As of August 2023 the horizon project is complete. We will continue to update and maintain the list of codes on the codes page. The list includes the GPU-enabled GRMHD code KHARMA, developed by Ben Prather, which is developed in the open and includes mesh refinement and nonideal effects.

# Author: gammie@illinois.edu

# Global Radiation Magneto-hydrodynamic Simulations of Sub-Eddington Accretion Disks around Supermassive Black Holes

Jiang, Yan-Fei; Blaes, Omer; Stone, James; Davis, Shane W.

We use global three dimensional radiation magneto-hydrodynamic simulations to study the properties of inner regions of accretion disks around a 5\times 10^8 solar mass black hole with mass accretion rates reaching 7% and 20% of the Eddington value. This region of the disk is supported by magnetic pressure with surface density significantly smaller than the values predicted by the standard thin disk model but with a much larger disk scale height. The disks do not show any sign of thermal instability over many thermal time scales. More than half of the accretion is driven by radiation viscosity in the optically thin corona region for the lower accretion rate case, while accretion in the optically thick part of the disk is driven by the Maxwell and Reynolds stresses from MRI turbulence. Coronae with gas temperatures > 10^8 K are generated only in the inner \approx 10 gravitational radii in both simulations, being more compact in the higher accretion rate case. In contrast to the thin disk model, surface density increases with increasing mass accretion rate, which causes less dissipation in the optically thin region and a relatively weaker corona. The simulation results may explain the formation of X-ray coronae in Active Galactic Nuclei (AGNs), the compact size of such coronae, and the observed trend of optical to X-ray luminosity with Eddington ratio for many AGNs.

# A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport

A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport

Felker, Kyle Gerard; Stone, James M.

We present a fourth-order accurate finite volume method for the solution of ideal magnetohydrodynamics (MHD). The numerical method combines high-order quadrature rules in the solution of semi-discrete formulations of hyperbolic conservation laws with the upwind constrained transport (UCT) framework to ensure that the divergence-free constraint of the magnetic field is satisfied. A novel implementation of UCT that uses the piecewise parabolic method (PPM) for the reconstruction of magnetic fields at cell corners in 2D is introduced. The resulting scheme can be expressed as the extension of the second-order accurate constrained transport (CT) Godunov-type scheme that is currently used in the Athena astrophysics code. After validating the base algorithm on a series of hydrodynamics test problems, we present the results of multidimensional MHD test problems which demonstrate formal fourth-order convergence for smooth problems, robustness for discontinuous problems, and improved accuracy relative to the second-order scheme.

# Shearing-box simulations of MRI-driven turbulence in weakly collisional accretion discs

Shearing-box simulations of MRI-driven turbulence in weakly collisional accretion discs

Kempski, Philipp; Quataert, Eliot; Squire, Jonathan; Kunz, Matthew W.

We present a systematic shearing-box investigation of magnetorotational instability (MRI)-driven turbulence in a weakly collisional plasma by including the effects of an anisotropic pressure stress, i.e. anisotropic (Braginskii) viscosity. We constrain the pressure anisotropy (∆p) to lie within the stability bounds that would be otherwise imposed by kinetic microinstabilities. We explore a broad region of parameter space by considering different Reynolds numbers and magnetic-field configurations, including net vertical flux, net toroidal-vertical flux, and zero net flux. Remarkably, we find that the level of turbulence and angular-momentum transport are not greatly affected by large anisotropic viscosities: the Maxwell and Reynolds stresses do not differ much from the MHD result. Angular-momentum transport in Braginskii MHD still depends strongly on isotropic dissipation, e.g. the isotropic magnetic Prandtl number, even when the anisotropic viscosity is orders of magnitude larger than the isotropic diffusivities. Braginskii viscosity nevertheless changes the flow structure, rearranging the turbulence to largely counter the parallel rate of strain from the background shear. We also show that the volume-averaged pressure anisotropy and anisotropic viscous transport decrease with increasing isotropic Reynolds number (Re); e.g. in simulations with net vertical field, the ratio of anisotropic to Maxwell stress (α_{A}/α_{M}) decreases from ̃0.5 to ̃0.1 as we move from Re ̃ 10^{3} to Re ̃ 10^{4}, while 〈4π∆p/B^{2}〉 → 0. Anisotropic transport may thus become negligible at high Re. Anisotropic viscosity nevertheless becomes the dominant source of heating at large Re, accounting for {≳ } 50 {{ per cent}} of the plasma heating. We conclude by briefly discussing the implications of our results for radiatively inefficient accretion flows on to black holes.

# Magneto-immutable turbulence in weakly collisional plasmas

Magneto-immutable turbulence in weakly collisional plasmas

Squire, J.; Schekochihin, A. A.; Quataert, E.; Kunz, M. W.

We propose that pressure anisotropy causes weakly collisional turbulent plasmas to self-organize so as to resist changes in magnetic-field strength. We term this effect `magneto-immutability’ by analogy with incompressibility (resistance to changes in pressure). The effect is important when the pressure anisotropy becomes comparable to the magnetic pressure, suggesting that in collisionless, weakly magnetized (high-> β >) plasmas its dynamical relevance is similar to that of incompressibility. Simulations of magnetized turbulence using the weakly collisional Braginskii model show that magneto-immutable turbulence is surprisingly similar, in most statistical measures, to critically balanced magnetohydrodynamic turbulence. However, in order to minimize magnetic-field variation, the flow direction becomes more constrained than in magnetohydrodynamics, and the turbulence is more strongly dominated by magnetic energy (a non-zero `residual energy’). These effects represent key differences between pressure-anisotropic and fluid turbulence, and should be observable in the > β\gtrsim 1 > turbulent solar wind.

# Accretion of magnetized stellar winds in the Galactic centre: implications for Sgr A* and PSR J1745-2900

Ressler, S. M.; Quataert, E.; Stone, J. M.

The observed rotation measures (RMs) towards the Galactic centre magnetar and towards Sagittarius A* provide a strong constraint on MHD models of the Galactic centre accretion flow, probing distances from the black hole separated by many orders of magnitude. We show, using three-dimensional simulations of accretion via magnetized stellar winds of the Wolf-Rayet stars orbiting the black hole, that the large, time-variable RM observed for the pulsar PSR J1745-2900 can be explained by magnetized wind-wind shocks of nearby stars in the clockwise stellar disc. In the same simulation, both the total X-ray luminosity integrated over 2-10 arcsec, the time variability of the magnetar’s dispersion measure, and the RM towards Sagittarius A* are consistent with observations. We argue that (in order for the large RM of the pulsar to not be a priori unlikely) the pulsar should be on an orbit that keeps it near the clockwise disc of stars. We present a two-dimensional RM map of the central 1/2 parsec of the Galactic centre that can be used to test our models. Our simulations predict that Sgr A* is typically accreting a significantly ordered magnetic field that ultimately could result in a strongly magnetized flow with flux threading the horizon at ̃10 per cent of the magnetically arrested limit.

# Large-Scale Poloidal Magnetic Field Dynamo Leads to Powerful Jets in GRMHD Simulations of Black Hole Accretion with Toroidal Field

Liska, M. T. P.; Tchekhovskoy, A.; Quataert, E.

Accreting black holes launch relativistic collimated jets, across many decades in luminosity and mass, suggesting the jet launching mechanism is universal, robust and scale-free. Theoretical models and general relativistic magnetohydrodynamic (GRMHD) simulations indicate that the key jet-making ingredient is large-scale poloidal magnetic flux. However, its origin is uncertain, and it is unknown if it can be generated in situ or dragged inward from the ambient medium. Here, we use the GPU-accelerated GRMHD code HAMR to study global 3D black hole accretion at unusually high resolutions more typical of local shearing box simulations. We demonstrate that accretion disc turbulence in a radially-extended accretion disc can generate large-scale poloidal magnetic flux in situ, even when starting from a purely toroidal magnetic field. The flux accumulates around the black hole till it becomes dynamically-important, leads to a magnetically arrested disc (MAD), and launches relativistic jets that are more powerful than the accretion flow. The jet power exceeds that of previous GRMHD toroidal field simulations by a factor of 10,000. The jets do not show significant kink or pinch instabilities, accelerate to γ∼10 over 3 decades in distance, and follow a collimation profile similar to the observed M87 jet.

# Two-temperature GRRMHD Simulations of M87

Two-temperature GRRMHD Simulations of M87

Ryan, Benjamin R.; Ressler, Sean M.; Dolence, Joshua C.; Gammie, Charles; Quataert, Eliot

We present axisymmetric two-temperature general relativistic radiation magnetohydrodynamic simulations of the inner region of the accretion flow onto the supermassive black hole M87. We address uncertainties from previous modeling efforts through inclusion of models for (1) self-consistent dissipative and Coulomb electron heating (2) radiation transport (3) frequency-dependent synchrotron emission, self-absorption, and Compton scattering. We adopt a distance D = 16.7 Mpc, an observer angle θ = 20°, and consider black hole masses M/{M}_{☉ }=(3.3× {10}^{9},6.2× {10}^{9}) and spins a _{⋆} = (0.5, 0.9375) in a four-simulation suite. For each (M, a _{⋆}), we identify the accretion rate that recovers the 230 GHz flux from very long baseline interferometry measurements. We report on disk thermodynamics at these accretion rates (\dot{M}/{\dot{M}}_{Edd}}̃ {10}^{-5}). The disk remains geometrically thick; cooling does not lead to a thin disk component. While electron heating is dominated by Coulomb rather than dissipation for r ≳ 10GM/c ^{2}, the accretion disk remains two-temperature. Radiative cooling of electrons is not negligible, especially for r ≲ 10GM/c ^{2}. The Compton y parameter is of order unity. We then compare derived and observed or inferred spectra, millimeter images, and jet powers. Simulations with M/M _{☉} = 3.3 × 10^{9} are in conflict with observations. These simulations produce millimeter images that are too small, while the low-spin simulation also overproduces X-rays. For M/{M}_{☉ }=6.2× {10}^{9}, both simulations agree with constraints on radio/IR/X-ray fluxes and millimeter image sizes. Simulation jet power is a factor 10^{2}-10^{3} below inferred values, a possible consequence of the modest net magnetic flux in our models.

# Hydrodynamic simulations of the inner accretion flow of Sagittarius A* fuelled by stellar winds

Hydrodynamic simulations of the inner accretion flow of Sagittarius A* fuelled by stellar winds

Ressler, S. M.; Quataert, E.; Stone, J. M.

We present ATHENA++ grid-based, hydrodynamic simulations of accretion on to Sagittarius A* via the stellar winds of the ̃30 Wolf-Rayet stars within the central parsec of the galactic centre. These simulations span ̃4 orders of magnitude in radius, reaching all the way down to 300 gravitational radii of the black hole, ̃32 times further than in previous work. We reproduce reasonably well the diffuse thermal X-ray emission observed by Chandra in the central parsec. The resulting accretion flow at small radii is a superposition of two components: (1) a moderately unbound, sub-Keplerian, thick, pressure-supported disc that is at most (but not all) times aligned with the clockwise stellar disc, and (2) a bound, low-angular momentum inflow that proceeds primarily along the southern pole of the disc. We interpret this structure as a natural consequence of a few of the innermost stellar winds dominating accretion, which produces a flow with a broad distribution of angular momentum. Including the star S2 in the simulation has a negligible effect on the flow structure. Extrapolating our results from simulations with different inner radii, we find an accretion rate of approximately a few ×10^{-8} M_{☉} yr^{-1} at the horizon scale, consistent with constraints based on modelling the observed emission of Sgr A*. The flow structure found here can be used as more realistic initial conditions for horizon scale simulations of Sgr A*.

# PIC Simulations of Velocity-space Instabilities in a Decreasing Magnetic Field: Viscosity and Thermal Conduction

PIC Simulations of Velocity-space Instabilities in a Decreasing Magnetic Field: Viscosity and Thermal Conduction

Riquelme, Mario; Quataert, Eliot; Verscharen, Daniel

We use particle-in-cell (PIC) simulations of a collisionless, electron-ion plasma with a decreasing background magnetic field, {\boldsymbol{B}}, to study the effect of velocity-space instabilities on the viscous heating and thermal conduction of the plasma. If | {\boldsymbol{B}}| decreases, the adiabatic invariance of the magnetic moment gives rise to pressure anisotropies with {p}_{| | ,j}> {p}_{\perp ,j} ({p}_{| | ,j} and {p}_{\perp ,j} represent the pressure of species j (electron or ion) parallel and perpendicular to B ). Linear theory indicates that, for sufficiently large anisotropies, different velocity-space instabilities can be triggered. These instabilities in principle have the ability to pitch-angle scatter the particles, limiting the growth of the anisotropies. Our simulations focus on the nonlinear, saturated regime of the instabilities. This is done through the permanent decrease of | {\boldsymbol{B}}| by an imposed plasma shear. We show that, in the regime 2≲ {β }_{j}≲ 20 ({β }_{j}\equiv 8π {p}_{j}/| {\boldsymbol{B}}{| }^{2}), the saturated ion and electron pressure anisotropies are controlled by the combined effect of the oblique ion firehose and the fast magnetosonic/whistler instabilities. These instabilities grow preferentially on the scale of the ion Larmor radius, and make {{∆ }}{p}_{e}/{p}_{| | ,e}≈ {{∆ }}{p}_{i}/{p}_{| | ,i} (where {{∆ }}{p}_{j}={p}_{\perp ,j}-{p}_{| | ,j}). We also quantify the thermal conduction of the plasma by directly calculating the mean free path of electrons, {λ }_{e}, along the mean magnetic field, finding that {λ }_{e} depends strongly on whether | {\boldsymbol{B}}| decreases or increases. Our results can be applied in studies of low-collisionality plasmas such as the solar wind, the intracluster medium, and some accretion disks around black holes.