The structure of black hole magnetospheres I. Schwarzschild black holes

We introduce a multipolar scheme for describing the structure of stationary, axisymmetric, force-free black hole magnetospheres in the '3+1' formalism. We focus here on Schwarzschild spacetime, giving a complete classification of the separable solutions of the stream equation. We show a transparent term-by-term analogy of our solutions with the familiar multipoles of flat-space electrodynamics. We discuss electrodynamic processes around disc-fed black holes in which our solutions find natural applications: (i) 'interior' solutions in studies of the BlandfordZnajek process of extracting the rotational energy of holes, and of the formation of relativistic jets in active galactic nuclei and 'microquasars'; (ii) 'exterior' solutions in studies of accretion disc dynamos, disc-driven winds and jets. On the strength of existing numerical studies, we argue that the poloidal field structures found here are also expected to hold with good accuracy for rotating black holes, except for the cases of the maximum possible rotation rates. We show that the closed-loop exterior solutions found here are not in contradiction with the MacdonaldThorne theorem, as these solutions, which diverge logarithmically on the horizon of the hole , only apply to those regions that exclude .     

Steady shocks around black holes produced by sub-Keplerian flows with negative energy

We discuss a special case of formation of axisymmetric shocks in the accretion flow of ideal gas on to a Schwarzschild black hole: when the total energy of the flow is negative. The result of our analysis enlarges the parameter space for which these steady shocks are exhibited in the accretion of gas rotating around relativistic stellar objects. Since Keplerian discs have negative total energy, we guess that, in this energy range, the production of the shock phenomenon might be easier than in the case of positive energy. So our outcome reinforces the view that sub-Keplerian flows of matter may significantly affect the physics of the high energy radiation emission from black hole candidates. We give a simple procedure to obtain analytically the position of the shocks. The comparison of the analytical results with the data of one-dimensional (1D) and two-dimensional (2D) axisymmetric numerical simulations confirms that the shocks form and are stable.     

Non-axisymmetric relativistic Bondi–Hoyle accretion on to a Schwarzschild black hole

We present the results of an exhaustive numerical study of fully relativistic non-axisymmetric Bondi–Hoyle accretion on to a moving Schwarzschild black hole. We have solved the equations of general relativistic hydrodynamics with a high-resolution shock-capturing numerical scheme based on a linearized Riemann solver. The numerical code was previously used to study axisymmetric flow configurations past a Schwarzschild black hole. We have analysed and discussed the flow morphology for a sample of asymptotically high Mach number models. The results of this work reveal that initially asymptotic uniform flows always accrete on to the hole in a stationary way, which closely resembles the previous axisymmetric patterns. This is in contrast with some Newtonian numerical studies where violent flip-flop instabilities were found. As discussed in the text, the reason can be found in the initial conditions used in the relativistic regime, as they cannot exactly duplicate the previous Newtonian setups where the instability appeared. The dependence of the final solution on the inner boundary condition as well as on the grid resolution has also been studied. Finally, we have computed the accretion rates of mass and linear and angular momentum.     

Subcritical dynamos in shear flows

Identifying generic physical mechanisms responsible for the generation of magnetic fields and turbulence in differentially rotating flows is fundamental to understand the dynamics of astrophysical objects such as accretion disks and stars. In this paper, we discuss the concept of subcritical dynamo action and its hydrodynamic analogue exemplified by the process of nonlinear transition to turbulence in non‐rotating wall‐bounded shear flows. To illustrate this idea, we describe some recent results on nonlinear hydrodynamic transition to turbulence and nonlinear dynamo action in rotating shear flows pertaining to the problem of turbulent angular momentum transport in accretion disks. We argue that this concept is very generic and should be applicable to many astrophysical problems involving a shear flow and non‐axisymmetric instabilities of shearinduced axisymmetric toroidal velocity or magnetic fields, such as Kelvin‐Helmholtz, magnetorotational, Tayler or global magnetoshear instabilities. In the light of several recent numerical results, we finally suggest that, similarly to a standard linear instability, subcritical MHD dynamo processes in high‐Reynolds number shear flows could act as a large‐scale driving mechanism of turbulent flows that would in turn generate an independent small‐scale dynamo. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Magnetized tori around Kerr black holes: analytic solutions with a toroidal magnetic field

The dynamics of accretion discs around galactic and extragalactic black holes may be influenced by their magnetic field. In this paper, we generalize the fully relativistic theory of stationary axisymmetric tori in Kerr metric of Abramowicz, Jaroszynski & Sikora by including strong toroidal magnetic field and construct analytic solutions for barotropic tori with constant angular momentum. This development is particularly important for the general relativistic computational magnetohydrodynamics that suffers from the lack of exact analytic solutions that are needed to test computer codes.     

The excitation, propagation and dissipation of waves in accretion discs: the non-linear axisymmetric case

We analyse the non-linear propagation and dissipation of axisymmetric waves in accretion discs using the ZEUS-2D hydrodynamics code. The waves are numerically resolved in the vertical and radial directions. Both vertically isothermal and thermally stratified accretion discs are considered. The waves are generated by means of resonant forcing, and several forms of forcing are considered. Compressional motions are taken to be locally adiabatic  ( γ =5/3)  . Prior to non-linear dissipation, the numerical results are in excellent agreement with the linear theory of wave channelling in predicting the types of modes that are excited, the energy flux by carried by each mode, and the vertical wave energy distribution as a function of radius. In all cases, waves are excited that propagate on both sides of the resonance (inwards and outwards). For vertically isothermal discs, non-linear dissipation occurs primarily through shocks that result from the classical steepening of acoustic waves. For discs that are substantially thermally stratified, wave channelling is the primary mechanism for shock generation. Wave channelling boosts the Mach number of the wave by vertically confining the wave to a small cool region at the base of the disc atmosphere. In general, outwardly propagating waves with Mach numbers near resonance  ℳ_r≳0.01  undergo shocks within a distance of order the resonance radius.     

Dissipative accretion flows around a rotating black hole

We study the dynamical structure of a cooling dominated rotating accretion flow around a spinning black hole. We show that non-linear phenomena such as shock waves can be studied in terms of only three flow parameters, namely the specific energy     , the specific angular momentum (λ) and the accretion rate     of the flow. We present all possible accretion solutions. We find that a significant region of the parameter space in the     plane allows global accretion shock solutions. The effective area of the parameter space for which the Rankine–Hugoniot shocks are possible is maximum when the flow is dissipation-free. It decreases with the increase of cooling effects and finally disappears when the cooling is high enough. We show that shock forms further away when the black hole is rotating compared to the solution around a Schwarzschild black hole with identical flow parameters at a large distance. However, in a normalized sense, the flow parameters for which the shocks form around the rotating black holes are produced shocks closer to the black hole. The location of the shock is also dictated by the cooling efficiency in that higher the accretion rate     , the closer is the shock location. We believe that some of the high-frequency quasi-periodic oscillations may be due to the flows with higher accretion rate around the rotating black holes.     

On the angular momentum transfer on to compact stars in binary systems

Results of three-dimensional numerical simulations of the gas transfer in close binary systems show that, in addition to the formation of a tidally induced spiral shock wave, it is also possible for accretion streams to be produced, having low specific angular momentum in a region close to the accreting star. These streams are mainly placed above the orbital disc but are also unevenly present in the equatorial plane. The relevance of such flows is related to formation of hot coronae or bulges in regions very close to the accretor centre. The eventual formation of such bulges and shock-heated flows is interesting in the context of advection-dominated solutions and for the explanation of spectral properties of the black hole candidates in binary systems.     

Self-similar solution of hot accretion flows with ordered magnetic field and outflow

Observations and numerical magnetohydrodynamic (MHD) simulations indicate the existence of outflows and ordered large-scale magnetic fields in the inner region of hot accretion flows. In this paper, we present the self-similar solutions for advection-dominated accretion flows (ADAFs) with outflows and ordered magnetic fields. Stimulated by numerical simulations, we assume that the magnetic field has a strong toroidal component and a vertical component in addition to a stochastic component. We obtain the self-similar solutions to the equations describing the magnetized ADAFs, taking into account the dynamical effects of the outflow. We compare the results with the canonical ADAFs and find that the dynamical properties of ADAFs such as radial velocity, angular velocity and temperature can be significantly changed in the presence of ordered magnetic fields and outflows. The stronger the magnetic field is, the lower the temperature of the accretion flow will be and the faster the flow rotates. The relevance to observations is briefly discussed.     

Global structure of self-excited magnetic fields arising from the magnetic shear instability

The global structure of a self-excited magnetic field arising from the magnetic shear instability has been simulated in spherical geometry by a 3D fully non-linear approach. In order to model the structure of an accretion disc we prescribe a rotation profile of the Brandt type which is Keplerian in the outer regions but yields rigid rotation at the inner core. We performed a whole series of runs at different dynamo numbers with an increasing number of modes in spectral space, thereby checking the influence of the numerical resolution in our simulations. Starting from arbitrary small perturbations, the magnetic and kinetic energies grow by several orders of magnitude as soon as a certain azimuthal resolution of at least m =15 was used at a dynamo number of order C =10⁵. Several phases of the transition to turbulence are realized and interpretations are given for the respective effects occurring at each stage. The resulting magnetic field is highly non-axisymmetric and possesses a pronounced inhomogeneous vortex structure of twisted flux tubes. The flow is almost axisymmetric but shows a Kolmogorov-like behaviour for small scales. The outer surface of the shell is penetrated by magnetic field lines in spot-like regions, which are located mainly in the equatorial plane. For very high dynamo numbers we find a cyclic behaviour of the averaged magnetic field amplitude. The problem of angular momentum transport is discussed in terms of the ShakuraSunyaev viscosity alpha , which depends exponentially on the radial distance and adopts values in the range 10³10⁵.     

Hydrodynamical non-radiative accretion flows in two dimensions

Two-dimensional (axially symmetric) numerical hydrodynamical calculations of accretion flows that cannot cool through emission of radiation are presented. The calculations begin from an equilibrium configuration consisting of a thick torus with constant specific angular momentum. Accretion is induced by the addition of a small anomalous azimuthal shear stress which is characterized by a function ν . We study the flows generated as the amplitude and form of ν are varied. A spherical polar grid which spans more than two orders of magnitude in radius is used to resolve the flow over a wide range of spatial scales. We find that convection in the inner regions produces significant outward mass motions that carry away both the energy liberated by and a large fraction of the mass participating in the accretion flow. Although the instantaneous structure of the flow is complex and dominated by convective eddies, long-time averages of the dynamical variables show remarkable correspondence to certain steady-state solutions. The two-dimensional structure of the time-averaged flow is marginally stable to the Høiland criterion, indicating that convection is efficient. Near the equatorial plane, the radial profiles of the time-averaged variables are power laws with an index that depends on the radial scaling of the shear stress. A stress in which ν ∝ r ^1/2 recovers the widely studied self-similar solution corresponding to an ' α -disc'. We find that, regardless of the adiabatic index of the gas, or the form or magnitude of the shear stress, the mass inflow rate is a strongly increasing function of radius, and is everywhere nearly exactly balanced by mass outflow. The net mass accretion rate through the disc is only a fraction of the rate at which mass is supplied to the inflow at large radii, and is given by the local, viscous accretion rate associated with the flow properties near the central object.     

Luminous hot accretion discs

We find a new two-temperature hot branch of equilibrium solutions for stationary accretion discs around black holes. In units of Eddington accretion rate defined as 10 L _Edd c ², the accretion rates to which these solutions correspond are within the range ̇ ₁≲ ̇ ≲1, where ̇ ₁ is the critical rate of advection-dominated accretion flow (ADAF). In these solutions, the energy loss rate of the ions by Coulomb energy transfer between the ions and electrons is larger than the viscously heating rate and it is the advective heating together with the viscous dissipation that balances the Coulomb cooling of ions. When ̇ ₁≲ ̇ ≲ ̇ ₂, where ̇ ₂∼5 ̇ ₁<1, the accretion flow remains hot throughout the disc. When ̇ ₂≲ ̇ ≲1, Coulomb interaction will cool the inner region of the disc within a certain radius ( r _tr∼several tens of Schwarzschild radii or larger depending on the accretion rate and the outer boundary condition) and the disc will collapse on to the equatorial plane and form an optically thick cold annulus. Compared with ADAF, these hot solutions are much more luminous because of the high accretion rate and efficiency; therefore, we call them luminous hot accretion discs.     

Non-linear bending waves in Keplerian accretion discs

The non-linear dynamics of a warped accretion disc is investigated in the important case of a thin Keplerian disc with negligible viscosity and self-gravity. A one-dimensional evolutionary equation is formally derived that describes the primary non-linear and dispersive effects on propagating bending waves other than parametric instabilities. It has the form of a derivative non-linear Schrödinger (DNLS) equation with coefficients that are obtained explicitly for a particular model of a disc. The properties of this equation are analysed in some detail and illustrative numerical solutions are presented. The non-linear and dispersive effects both depend on the compressibility of the gas through its adiabatic index Γ. In the physically realistic case Γ < 3, non-linearity does not lead to the steepening of bending waves but instead enhances their linear dispersion. In the opposite case Γ > 3, non-linearity leads to wave steepening and solitary waves are supported. The effects of a small effective viscosity, which may suppress parametric instabilities, are also considered. This analysis may provide a useful point of comparison between theory and numerical simulations of warped accretion discs.     

The effects of thermal conduction on the ADAF with a toroidal magnetic field

The observation of the hot gas surrounding Sgr A * and a few other nearby galactic nuclei imply that electron and proton mean free paths are comparable to the gas capture radius. So, the hot accretion flows are likely to proceed under week-collision conditions. Hence, thermal conduction has been suggested as a possible mechanism by which the sufficient extra heating is provided in hot advection-dominated accretion flow (ADAF) accretion discs. We consider the effects of thermal conduction in the presence of a toroidal magnetic field in an ADAF around a compact object. For a steady-state structure of such accretion flows, a set of self-similar solutions are presented. We find two types of solutions which represent high and slow accretion rate. They have different behaviours with saturated thermal conduction parameter, φ.     

Two- and three-dimensional numerical simulations of accretion discs in a close binary system

We perform 2D and 3D numerical simulations of an accretion disc in a close binary system using the simplified flux vector splitting (SFS) finite volume method. In our calculations, the gas is assumed to be ideal with γ =1.01, 1.05, 1.1 and 1.2 . The mass ratio of the mass-losing star to the mass-accreting star is unity. Our results show that spiral shocks are formed on the accretion disc in all cases. In 2D calculations we find that the smaller γ is, the more tightly the spiral winds. We observe this trend in 3D calculations as well in a somewhat weaker sense. Mach numbers in our discs are less than 10. These values are lower than the values in observed accretion discs in close binary systems.
Recently, Steeghs, Harlaftis & Horne found the first convincing evidence for spiral structure in the accretion disc of the eclipsing dwarf nova binary IP Pegasi, using the technique known as Doppler tomography. Although the Mach numbers in present calculations are rather low, we may claim that the spiral structure that we discovered in earlier numerical simulations is now found observationally.     

A steady-state solution for warped accretion discs

We consider a thin accretion disc warped due to the Bardeen–Petterson effect, presenting both analytical and numerical solutions for the situation in which the two viscosity coefficients vary with radius as a power law, with the two power-law indices not necessarily equal. The analytical solutions are compared with numerical ones, showing that our new analytical solution is more accurate than the previous one, which overestimated the inclination change in the outer disc. Our new analytical solution is appropriate for moderately warped discs, while for extremely misaligned discs only a numerical solution is appropriate.     

Dynamics of oscillating relativistic tori around Kerr black holes

We present a comprehensive numerical study of the dynamics of relativistic axisymmetric accretion tori with a power-law distribution of specific angular momentum orbiting in the background space–time of a Kerr black hole. By combining general relativistic hydrodynamics simulations with a linear perturbative approach we investigate the main dynamical properties of these objects over a large parameter space. The astrophysical implications of our results extend and improve two interesting results that have been recently reported in the literature. First, the induced quasi-periodic variation of the mass quadrupole moment makes relativistic tori of nuclear matter densities, as those formed during the last stages of binary neutron star mergers, promising sources of gravitational radiation, potentially detectable by interferometric instruments. Secondly, p-mode oscillations in relativistic tori of low rest-mass densities could be used to explain high-frequency quasi-periodic oscillations observed in X-ray binaries containing a black hole candidate under conditions more generic than those considered so far.