Title Page
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2.
Numerical Method
Title Page
|
2.
Numerical Method
Many black hole accretion sources are remarkably underluminous, and
pose a stern challenge for accretion theory. Generally, these sources
emit a significant fraction of their radiated energy in the form of
X-rays (Elvis et al. 1994), and standard, optically thick Keplerian
disk theory cannot account for this type of spectrum. In a classical
Keplerian accretion disk (Shakura & Sunyaev 1973), dissipated
mechanical energy is lost as local blackbody radiation, and there is a
simple relationship between the accretion rate 
and the
luminosity L. There is little freedom to adapt the model to
more complex spectra.
Another class of models is characterized by inefficient cooling and
high temperatures (e.g., Shapiro, Lightman & Eardley 1976; Ichimaru
1977). One particular model that of late has enjoyed considerable
attention is known as an Advection Dominated Accretion Flow, or ADAF
for short. The epithet ADAF originated with the influential paper of
Narayan & Yi (1994; hereafter NY94), which presented a
one-dimensional, time-stationary, self-similar solution (fixed
power-law dependencies in r for dynamical variables). All velocities
--radial, nonradial and thermal--are assumed to have the same
Keplerian r dependence, r-1/2. Angular momentum transport is
effected by a Navier-Stokes viscosity along the lines of the Shakura &
Sunyaev (1973) and Lynden-Bell & Pringle (1974) 
formalism.
The gas is hot, with the sound speed comparable to the orbital speed.
The internal energy is advected along with the mass into the central
black hole; very little is radiated. An ADAF is more like a Bondi flow
than a Keplerian disk. The quasi-spherical nature of the flow is often
cited as a defining characteristic feature. ADAFs are sufficiently
distinct from a Keplerian disk that a transition radius rt is
invoked to define the location separating the (outer) Keplerian flow
from the (inner) ADAF.
There are two dynamical features of the NY94 self-similar ADAF solution that have attracted special attention. The first is that the specific energy flux of the gas (the Bernoulli parameter) is everywhere positive, the second is that entropy increases inwards, rendering the accretion convectively unstable by the Schwarzschild criterion. We review these in turn.
In a nonradiative accretion flow a positive Bernoulli parameter is a consequence of the fact that the stress transports energy, as well as angular momentum, outwards. In a standard thin disk solution, this transported energy is radiated on its outward journey, so that the local radiated flux (outside of the inner boundary layer) is always greater by a factor of 3 than the energy released locally. By contrast, in a nonradiative accretion flow the transported energy is retained, locally raising the specific energy. If the flow originates at large radius as a marginally bound fluid, the Bernoulli parameter will accordingly be everywhere positive. NY94 noted that a positive Bernoulli parameter might make outflows possible, but since an outflow is very distinct from their ADAF solution, the significance of this observation is unclear. Blandford & Begelman (1999) argued that this property is likely to change fundamentally the dynamical flow properties of the nonradiative solution, and proposed that not only were outflows possible, but that the bulk of the gas and energy would be carried off by a wind. They refer to this concept as an Adiabatic Inflow-Outflow Solution, or ADIOS.
More recently, attention has focused upon the increasing-inward entropy
profile that is a straightforward consequence of viscous dissipation,
and is a property of any nonadiabatically heated, nonradiative
inflow. Narayan & Yi (1995) argued that radial convection would
inevitably arise, and serve as a source of outward angular
momentum transport--either as the self-consistent source of ,
or at a minimum, as a supplement to it. Narayan, Igumenshchev, &
Abramowicz (2000) later explored the consequences of convection moving
angular momentum either outward or inward. In one scheme, inward
angular momentum transport by convection is envisioned to be capable of
balancing the outward
transport, creating zero (net) mass and
angular momentum flux. Quataert & Gruzinov (2000) argued that
convection would produce a state of near marginal convective
stability rather like stellar convective zones, described by the
classical Høiland criteria. Quataert & Gruzinov referred to this new
variation as a Convective Dominated Accretion Flow (CDAF), though
formally there is no mass accretion at all!
We regard the defining property of a specific flow to be the fate of the accreting mass and energy. In the Shakura-Sunyaev thin disk model, liberated mechanical energy is both transported outwards and radiated locally. In an ADAF, this liberated energy is advected with the flow through the event horizon of the black hole. In a CDAF, the accretion flow is effectively stifled with no significant mass inflow or outflow. What little energy is extracted from the inner region of the flow is simply transported outward by convection to large radius. In an ADIOS the net accretion is very small because systematic outflows remove the bulk of the mass and energy. The existence of what seem to be under-luminous X ray sources lends credence to the existence of nonradiative flows. The question is, are they best described as ADAFs, CDAFs, ADIOS, or something else?
While flow energetic properties are likely to be complex and (for the moment) highly model-dependent, gross dynamical features--inflow versus outflow, rotational versus pressure support--should be more robust. In this work, we report the results of three-dimensional magnetohydrodynamic (MHD) simulations that follow the accretion of rotating, nonradiating, magnetized gas onto a black hole. We find that the bulk of the mass in nonradiative flows is not slowly rotating and quasispherical like an ADAF, but is much closer to Keplerian: rotationally supported and disk-like. Most of the liberated energy is carried outward by streaming backflow and turbulent transport. The nature of the flow is entirely determined by MHD turbulence, triggered by the magnetorotational instability (MRI; Balbus & Hawley 1991).
An outline of the paper is as follows. In §2 we discuss the numerical model for the simulations. In §3 we present two such simulations, one with resistive dissipation and one without. In §4 we summarize the main features of our model, and discuss the implications of the simulation for the accreting black hole in Sgr A*. Our conclusions are given in §5.