General Relativistic Magnetohydrodynamic Equations Research Articles

A model for general relativistic magnetohydrodynamic spine jets

We study jets using a semi-analytical model of the general relativistic magnetohydrodynamic (GRMHD) equations in the Kerr metric that describes them near the rotation axis, assuming a steady state, and axisymmetry. The goal is to model the inner spine of a relativistic jet in order to solve for the bulk acceleration and the shape of the jet and understand how these quantities depend on the enthalpy and the magnetic field. The model is constructed by expanding the rotating black hole metric and forces with respect to the polar angle about the rotation axis. This results in a system of ordinary differential equations that determine the dependence on the radial distance. The difference with previous semi-analytical models that expand the metric around the rotation axis is that the flow is governed by a polytropic equation of state. The solutions in this work start from a stagnation surface very close to the event horizon and become highly relativistic, achieving large Lorentz factors at large distances.

GRaM-X: a new GPU-accelerated dynamical spacetime GRMHD code for Exascale computing with the Einstein Toolkit

We present GRaM-X (General Relativistic accelerated Magnetohydrodynamics on AMReX), a new GPU-accelerated dynamical-spacetime general relativistic magnetohydrodynamics (GRMHD) code which extends the GRMHD capability of Einstein Toolkit to GPU-based exascale systems. GRaM-X supports 3D adaptive mesh refinement (AMR) on GPUs via a new AMR driver for the Einstein Toolkit called CarpetX which in turn leverages AMReX, an AMR library developed for use by the United States DOE’s Exascale Computing Project. We use the Z4c formalism to evolve the Einstein equations and the Valencia formulation to evolve the equations of GRMHD. GRaM-X supports both analytic as well as tabulated equations of state. We implement TVD and WENO reconstruction methods as well as the HLLE Riemann solver. We test the accuracy of the code using a range of tests on static spacetime, e.g. 1D magnetohydrodynamics shocktubes, the 2D magnetic rotor and a cylindrical explosion, as well as on dynamical spacetimes, i.e. the oscillations of a 3D Tolman-Oppenheimer-Volkhof star. We find excellent agreement with analytic results and results of other codes reported in literature. We also perform scaling tests and find that GRaM-X shows a weak scaling efficiency of ∼40%–50% on 2304 nodes (13824 NVIDIA V100 GPUs) with respect to single-node performance on OLCF’s supercomputer Summit.

Global high-order numerical schemes for the time evolution of the general relativistic radiation magneto-hydrodynamics equations

Modeling correctly the transport of neutrinos is crucial in some astrophysical scenarios such as core-collapse supernovae and binary neutron star mergers. In this paper, we focus on the truncated-moment formalism, considering only the first two moments (M1 scheme) within the grey approximation, which reduces Boltzmann seven-dimensional equation to a system of equations closely resembling the hydrodynamic ones. Solving the M1 scheme is still mathematically challenging, since it is necessary to model the radiation-matter interaction in regimes where the evolution equations become stiff and behave as an advection-diffusion problem. Here, we present different global, high-order time integration schemes based on Implicit-Explicit Runge-Kutta methods designed to overcome the time-step restriction caused by such behavior while allowing us to use the explicit Runge-Kutta commonly employed for the magneto-hydrodynamics and Einstein equations. Finally, we analyze their performance in several numerical tests.

Gravitational versus Magnetohydrodynamic Waves in Curved Spacetime in the Presence of Large-Scale Magnetic Fields

The general-relativistic (GR) magnetohydrodynamic (MHD) equations for a conductive plasma fluid are derived and discussed in the curved spacetime described by Thorne’s metric tensor, i.e., a family of cosmological models with inherent anisotropy due to the existence of an ambient, large-scale magnetic field. In this framework, it is examined whether the magnetized plasma fluid that drives the evolution of such a model can be subsequently excited by a transient, plane-polarized gravitational wave (GW) or not. To do so, we consider the associated set of the perturbed equations of motion and integrate them numerically in order to study the evolution of instabilities triggered by the GW propagation. In particular, we examine to what extend perturbations of the electric and/or the magnetic field can be amplified due to a potential energy transfer from the GW to the electromagnetic (EM) degrees of freedom. The evolution of the perturbed quantities depends on four free parameters, namely, the conductivity of the fluid, σ; the speed of sound square, 13<Csc2≡γ<1, which in this model may serve also as a measure of the inherent anisotropy; the GW frequency, ωg; and the associated angle of propagation with respect to the direction of the magnetic field, θ. We find that GW propagation in the anisotropic magnetized medium under consideration does excite several MHD modes; in other words, there is energy transfer from the gravitational to the EM degrees of freedom that can result in the acceleration of charged particles at the spot and in the subsequent damping of the GW.

Dynamical evolution of the resistive thick accretion Tori around a Schwarzschild black hole

ABSTRACT To study time-dependent phenomena of plasma surrounding a non-rotating black hole with a dipolar magnetic field, we have developed a fully set of 3 + 1 formalism of generalized general relativistic magnetohydrodynamic equations. The general relativistic phenomena, in particular, have been investigated with respect to the Ohm law. Magnetofluid is supposed to flow in three directions and forms a thick disc structure around the central black hole. All physical quantities of the system are functions of three variables: radial distance r, polar angle θ, and time t. The radial, meridional, and time behaviours of all these quantities have been investigated. It has been shown that the electrical conductivity of the fluid is not constant and may be both positive and negative depending on the values of some free parameters. The initially purely rotating non-magnetized plasma in the presence of an external magnetic field gives rise to an azimuthal current density and a charge density measured by the comoving observer. This current generates an electromagnetic field inside the disc which has both poloidal and toroidal components. The dipolar magnetic field lines of the central black hole is able to penetrate the plasma disc, due to the presence of a finite resistivity for the plasma. The accreting plasma pushes them outwards and makes them parallel to the rotation axis of the disc in the meridional plane.

Simulating magnetized neutron stars with discontinuous Galerkin methods

Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to formulate on complex meshes. We perform a detailed comparison of various limiting strategies presented in the literature applied to the equations of general relativistic magnetohydrodynamics. We compare the standard minmod/$\Lambda\Pi^N$ limiter, the hierarchical limiter of Krivodonova, the simple WENO limiter, the HWENO limiter, and a discontinuous Galerkin-finite-difference hybrid method. The ultimate goal is to understand what limiting strategies are able to robustly simulate magnetized TOV stars without any fine-tuning of parameters. Among the limiters explored here, the only limiting strategy we can endorse is a discontinuous Galerkin-finite-difference hybrid method.

Minidisk Accretion onto Spinning Black Hole Binaries: Quasi-periodicities and Outflows

Abstract We perform a full 3D general relativistic magnetohydrodynamical (GRMHD) simulation of an equal-mass, spinning, binary black hole approaching merger, surrounded by a circumbinary disk and with a minidisk around each black hole. For this purpose, we evolve the ideal GRMHD equations on top of an approximated spacetime for the binary that is valid in every position of space, including the black hole horizons, during the inspiral regime. We use relaxed initial data for the circumbinary disk from a previous long-term simulation, where the accretion is dominated by a m = 1 overdensity called the lump. We compare our new spinning simulation with a previous non-spinning run, studying how spin influences the minidisk properties. We analyze the accretion from the inner edge of the lump to the black hole, focusing on the angular momentum budget of the fluid around the minidisks. We find that minidisks in the spinning case have more mass over a cycle than the non-spinning case. However, in both cases we find that most of the mass received by the black holes is delivered by the direct plunging of material from the lump. We also analyze the morphology and variability of the electromagnetic fluxes, and we find they share the same periodicities of the accretion rate. In the spinning case, we find that the outflows are stronger than the non-spinning case. Our results will be useful to understand and produce realistic synthetic light curves and spectra, which can be used in future observations.

Iharm3D: Vectorized General Relativistic Magnetohydrodynamics

Iharm3D is an open-source C code for simulating black hole accretion systems in arbitrary stationary spacetimes using ideal general-relativistic magnetohydrodynamics (GRMHD). It is an implementation of the HARM ("High Accuracy Relativistic Magnetohydrodynamics") algorithm outlined in Gammie et al. (2003) with updates as outlined in McKinney & Gammie (2004) and Noble et al. (2006). The code is most directly derived from Ryan et al. (2015) but with radiative transfer portions removed. HARM is a conservative finite-volume scheme for solving the equations of ideal GRMHD, a hyperbolic system of partial differential equations, on a logically Cartesian mesh in arbitrary coordinates.

Gmunu: paralleled, grid-adaptive, general-relativistic magnetohydrodynamics in curvilinear geometries in dynamical space–times

ABSTRACT We present an update on the General-relativistic multigrid numerical (Gmunu) code, a parallelized, multidimensional curvilinear, general relativistic magnetohydrodynamics code with an efficient non-linear cell-centred multigrid elliptic solver, which is fully coupled with an efficient block-based adaptive mesh refinement module. To date, as described in this paper, Gmunu is able to solve the elliptic metric equations in the conformally flat condition approximation with the multigrid approach and the equations of ideal general-relativistic magnetohydrodynamics by means of high-resolution shock-capturing finite-volume method with reference metric formularised multidimensionally in Cartesian, cylindrical, or spherical geometries. To guarantee the absence of magnetic monopoles during the evolution, we have developed an elliptical divergence cleaning method by using the multigrid solver. In this paper, we present the methodology, full evolution equations and implementation details of Gmunu and its properties and performance in some benchmarking and challenging relativistic magnetohydrodynamics problems.

Kuibit: Analyzing Einstein Toolkit simulations with Python

In the era of gravitational-wave astronomy, general-relativistic simulations of compact objects play a role of paramount importance. These calculations can be performed with the Einstein Toolkit, an open-source and community-supported software for numerical-relativity and relativistic astrophysics. The code comes with multiple solvers for Einstein's equations and for the equations of general-relativistic magneto-hydrodynamics, along with a series of useful diagnostics. However, analyzing the output of the Einstein Toolkit can be a challenging task. Usually, the process involves a series of technical obstacles, like combining data from different restarts or working with HDF5 files. Here, we present kuibit, a Python library that takes care of all these low-level details (and many other more) and that provides high-level, intuitive, representations of the data. Kuibit ships with a wide range of features that include full support for 1-3D ASCII and HDF5 grid data, time and frequency series, gravitational waves, and apparent horizons. With kuibit, users can inspect most of the content of a simulation with just a few lines of code. Importantly, kuibit is designed to be a code for the community: it is user-friendly and does not require any proprietary software to run, it has documentation and examples, and it is openly developed with emphasis on extensibility and maintainability.

A Well Balanced Finite Volume Scheme for General Relativity

In this work we present a novel second order accurate well balanced (WB) finite volume (FV) scheme for the solution of the general relativistic magnetohydrodynamics (GRMHD) equations and the first order CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, as well as the fully coupled FO-CCZ4 + GRMHD system. These systems of first order hyperbolic PDEs allow to study the dynamics of the matter and the dynamics of the space-time according to the theory of general relativity. The new well balanced finite volume scheme presented here exploits the knowledge of an equilibrium solution of interest when integrating the conservative fluxes, the nonconservative products and the algebraic source terms, and also when performing the piecewise linear data reconstruction. This results in a rather simple modification of the underlying second order FV scheme, which, however, being able to cancel numerical errors committed with respect to the equilibrium component of the numerical solution, substantially improves the accuracy and long-time stability of the numerical scheme when simulating small perturbations of stationary equilibria. In particular, the need for well balanced techniques appears to be more and more crucial as the applications increase their complexity. We close the paper with a series of numerical tests of increasing difficulty, where we study the evolution of small perturbations of accretion problems and stable TOV neutron stars. Our results show that the well balancing significantly improves the long-time stability of the finite volume scheme compared to a standard one.

Generalized General Relativistic Magnetohydrodynamic Equations for Plasmas of Active Galactic Nuclei in the Era of the Event Horizon Telescope

Abstract The generalized general relativistic magnetohydrodynamic (generalized GRMHD) equations have been used to study specific relativistic plasma phenomena, such as relativistic magnetic reconnection or wave propagation modified by nonideal MHD effects. However, the Θ term in the generalized Ohm’s law, which expresses the energy exchange between two fluids composing a plasma, has yet to be determined in these equations. In this paper, we determine the Θ term based on the generalized relativistic Ohm’s law itself. This provides closure of the generalized GRMHD equations, yielding a closed system of the equations of relativistic plasma. According to this system of equations, we reveal the characteristic scales of nonideal MHD phenomena and clarify the applicable condition of the ideal GRMHD equations. We evaluate the characteristic scales of the nonideal MHD phenomena in the M87* plasma using the Event Horizon Telescope observational data.

Spritz: a new fully general-relativistic magnetohydrodynamic code

The new era of multimessenger astrophysics requires the capability of studying different aspects of the evolution of compact objects. In particular, the merger of neutron star binaries is a strong source of gravitational waves and electromagnetic radiation, from radio to γ-rays, as demonstrated by the detection of GW170817 and its electromagnetic counterparts. In order to understand the physical mechanisms involved in such systems, it is necessary to employ fully general relativistic magnetohydrodynamic (GRMHD) simulations able to include the effects of a composition and temperature dependent equation of state describing neutron star matter as well as neutrino emission and reabsorption. Here, we present our new code named Spritz that solves the GRMHD equations in 3D Cartesian coordinates and on a dynamical spacetime. The code can support tabulated equations of state, taking into account finite temperature effects and allowing for the inclusion of neutrino radiation. In this first paper, we present the general features of the code and a series of tests performed in special and general relativity to assess the robustness of the basic GRMHD algorithms implemented. Among these tests, we also present the first comparison between a non-staggered and a staggered formulation of the vector potential evolution, which is used to guarantee the divergence-less character of the magnetic field. With respect to other publicly available GRMHD codes, Spritz combines the robust approach of a staggered formulation of the vector potential together with the use of an equation of state driver (EOS_Omni) that can allow the code to use finite temperature equations of state. A next version of the code will fully test the EOS_Omni driver by coupling it with a neutrino leakage scheme.

Numerical relativity in spherical coordinates: A new dynamical spacetime and general relativistic MHD evolution framework for the Einstein Toolkit

We present SphericalNR, a new framework for the publicly available Einstein Toolkit that numerically solves the Einstein field equations coupled to the equations of general relativistic magnetohydrodynamics (GRMHD) in a 3+1 split of spacetime in spherical coordinates without symmetry assumptions. The spacetime evolution is performed using reference-metric versions of either the Baumgarte-Shapiro-Shibata-Nakamura equations or the fully covariant and conformal Z4 system with constraint damping. We have developed a reference-metric version of the Valencia formulation of GRMHD with a vector potential method, guaranteeing the absence of magnetic monopoles during the evolution. In our framework, every dynamical field (both spacetime and matter) is evolved using its components in an orthonormal basis with respect to the spherical reference-metric. Furthermore, all geometric information about the spherical coordinate system is encoded in source terms appearing in the evolution equations. This allows for the straightforward extension of Cartesian high-resolution shock-capturing finite volume codes to use spherical coordinates with our framework. To this end, we have adapted GRHydro, a Cartesian finite volume GRMHD code already available in the Einstein Toolkit, to use spherical coordinates. We present the full evolution equations of the framework, as well as details of its implementation in the Einstein Toolkit. We validate SphericalNR by demonstrating it passes a variety of challenging code tests in static and dynamical spacetimes.

Non-linear evolutions of magnetized thick discs around black holes: dependence on the initial data

ABSTRACT We build equilibrium solutions of magnetized thick discs around a highly spinning Kerr black hole and evolve these initial data up to a final time of about 100 orbital periods. The numerical simulations reported in this paper solve the general relativistic magnetohydrodynamic equations using the bhac code and are performed in axisymmetry. Our study assumes non-self-gravitating, polytropic, constant angular momentum discs endowed with a purely toroidal magnetic field. In order to build the initial data, we consider three approaches, two of which incorporate the magnetic field in a self-consistent way and a third approach in which the magnetic field is included as a perturbation on to an otherwise purely hydrodynamical solution. To test the dependence of the evolution on the initial data, we explore four representative values of the magnetization parameter spanning from almost hydrodynamical discs to very strongly magnetized tori. The initial data are perturbed to allow for mass and angular momentum accretion on to the black hole. Notable differences are found in the long-term evolutions of the initial data. In particular, our study reveals that highly magnetized discs are unstable, and hence prone to be fully accreted and expelled, unless the magnetic field is incorporated into the initial data in a self-consistent way.

Role of active galactic nuclei and flow of relativistic jets

AbstractThe astrophysics of Active Galactic Nuclei (AGN) is one of the long outstanding issues in searches among the scientific communities raised with diverse perspectives like nebula, quasars, etc some decades ago. Currently, this exotic system is at least understood as the center of an active galaxy. Thus, the consensus of this recent theory has opened up a number of research issues for the progress of astrophysical science including how the hosting galaxy evolves with the AGN, how matter and energy flow towards and outwards, etc. Moreover, most of the AGNs possess Supermassive Black Holes (SMBHs) and accrete matter at a very high rate as current observations report. Consequently, both observations of electromagnetic (EM) spectrum and Gravity Waves (GWs) will considered to provide complementary information about the AGNs and the roles in their environments including black holes in their centers, outflow and inflow of matter-energy. Interested with this background rationale, we study the mechanisms of AGN interaction with its environment and flow of relativistic jets where General Relativistic (GR) Magneto-Hydrodynamic (MHD) equations are being considered. The solutions of the field equations are treated with a metric that involves charged systems for the possible relativistic jets including accretions. Then, numerical data is being generated using the latest version Mathematic software. Finally, the theoretical data is being compared with that of observation for validation of the model.

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 general-relativistic 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.

The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project

Abstract Recent developments in compact object astrophysics, especially the discovery of merging neutron stars by LIGO, the imaging of the black hole in M87 by the Event Horizon Telescope, and high- precision astrometry of the Galactic Center at close to the event horizon scale by the GRAVITY experiment motivate the development of numerical source models that solve the equations of general relativistic magnetohydrodynamics (GRMHD). Here we compare GRMHD solutions for the evolution of a magnetized accretion flow where turbulence is promoted by the magnetorotational instability from a set of nine GRMHD codes: Athena++, BHAC, Cosmos++, ECHO, H-AMR, iharm3D, HARM-Noble, IllinoisGRMHD, and KORAL. Agreement among the codes improves as resolution increases, as measured by a consistently applied, specially developed set of code performance metrics. We conclude that the community of GRMHD codes is mature, capable, and consistent on these test problems.

One-fluid Equations of General Relativistic Two-fluid Plasma with the Landau–Lifshitz Radiation Reaction Force in Curved Space

Incorporating the radiation reaction force into two-fluid plasma in curved space, we get a set of one-fluid general relativistic magnetohydrodynamics (GRMHD) equations with the Landau-Lifshitz radiation reaction force. We analyze the importance of the radiation reaction acting on plasma around an astrophysical compact object.

Self-gravitating magnetized tori around black holes in general relativity

We investigate stationary, self-gravitating, magnetised disks (or tori) around black holes. The models are obtained by numerically solving the coupled system of the Einstein equations and the equations of ideal general-relativistic magnetohydrodynamics. The mathematical formulation and numerical aspects of our approach are similar to those reported in previous works modeling stationary self-gravitating perfect-fluid tori, but the inclusion of magnetic fields represents a new ingredient. Following previous studies of purely hydrodynamical configurations, we construct our models assuming Keplerian rotation in the disks and both spinning and spinless black holes. We focus on the case of a toroidal distribution of the magnetic field and build a large set of models corresponding to a wide range of values of the magnetisation parameter, starting with weakly magnetised disks and ending at configurations in which the magnetic pressure dominates over the thermal one. In all our models, the magnetic field affects the equilibrium structure of the torus mainly due to the magnetic pressure. In particular, an increasing contribution of the magnetic field shifts the location of the maximum of the rest-mass density towards inner regions of the disk. The total mass of the system and the angular momentum are affected by the magnetic field in a complex way, that depends on the black hole spin and the location of the inner radius of the disk. The non-linear dynamical stability of the solutions presented in this paper will be reported elsewhere.