Discussion

In this paper, we have carried out global MHD simulations of accretion tori. We have also made use of several different limits or approximations: two dimensional axisymmetry, and three dimensional cylindrical gravitational potential, in addition to the full three dimensional global model. In addition to the intrinsic interest in the results, these simulations begin to map out what is currently possible with the present hardware and algorithms.

First, how useful are the two-dimensional and cylindrical approximations? While neither should be relied on exclusively, they both have appropriate applications, and they greatly reduce the computational difficulty of the simulations. Two-dimensional axisymmetric torus simulations demonstrate effects from the generation of toroidal field due to shear, and from the development of the poloidal field MRI. The latter leads to turbulence, rapid angular momentum transport and evolution toward a nearly Keplerian angular momentum distribution. A significant limitation of the axisymmetric system, however, is embodied in Cowling's antidynamo theorem. The Aphi component of the vector potential is conserved, except for losses due to dissipative processes. Poloidal field can grow from axisymmetric stretching and folding, but this is ultimately limited. In the simulations, after the nonlinear saturation of the MRI, the poloidal magnetic field energies decline and the turbulence begins to die out.

Despite this, considerable similarity is seen between the two- and three-dimensional simulations. This is due to the dominance in the initial stages of the torus evolution by what is largely axisymmetric dynamics: increase in toroidal field pressure due to shear, the development of the ``channel solution,'' and the rapid redistribution of angular momentum by large stresses. During phases of a disk's evolution when such effects are most important, two-dimensional simulations are a good approximation. Their utility must be limited, however, since genuine nonaxisymmetric effects, including the development of the toroidal field MRI, dynamo amplification and maintenance of poloidal fields, and nonaxisymmetric spiral waves, are generally of dominant importance over the long term.

Cylindrical disks are a natural extension of the shearing box model. The cylindrical disk, like the shearing box simulations of HGB and HGB2, does not include the effects of vertical gravity. Cylindrical Keplerian disk simulations, initialized with vertical fields or toroidal fields show rapid development of the MRI consistent with the shearing box results. Field amplification and stress levels are comparable between the two types of simulation as well. With the cylindrical disk, however, we are able to observe the direct consequences of the stress: redistribution of angular momentum toward Keplerian, and mass accretion. Cylindrical disk simulations can provide detailed information about the radial dependence of the growth and saturation of the MRI, the resulting MHD turbulence, the transport of angular momentum, and the net accretion flow.

Simulations of cylindrical constant angular momentum tori with vertical or toroidal initial fields illustrate many aspects of a full global evolution. Regardless of the initial field configuration, the initially constant angular momentum tori evolve toward radial angular momentum distributions that are nearly, but not quite, Keplerian. The final outcome is largely the same for either initial vertical or toroidal fields, although the early growth of the MRI is more rapid for vertical fields, and dominated by small scale structure for toroidal fields. This is entirely consistent with the local linear analysis. Of course, we want to simulate disks fully globally and with as few approximations or geometric constraints as possible. Indeed, full three dimensionality is essential. Cylindrical simulations can provide little, if any, information about the vertical structure of a disk, energy transport, or the possible formation of winds or jets.

As with the cylindrical disk simulations, the overall evolution of fully global tori are consistent with the intuitions developed from shearing box simulations. The MRI grows rapidly, and produces MHD turbulence that transports angular momentum. In all cases, toroidal field dominates, followed by radial and then vertical field. One difference is that the shearing box calculations with zero net field (e.g., HGB2) typically have a total stress value of alpha ~ 0.01, whereas here alpha ~ 0.1 in the heart of the turbulent disks. This is more a matter of the magnetic pressure that is sustained in the torus versus the shearing box, rather than some qualitative difference in the behavior of the MRI. In both the global and local simulations, the stress is directly proportional to B2/8 pi: alpham ~ 0.4-0.5. All of the present global simulations began with relatively strong field, either in the form of poloidal field loops which immediately generated strong toroidal field through shear, or from the presence of an initially strong toroidal field. This field strength is by and large sustained, and thus the observed alphavalues are consistent with beta < 10. It is suggestive that Matsumoto (1999) also obtained beta ~ 10 for his toroidal field simulations, even one that began with beta=1000. Global (rather than local) length scales make it natural to have larger magnetic field strengths in the saturated state.

Computational problems associated with the inner boundary are greatly reduced through the use of the pseudo-Newtonian potential. This provides a physical inner radius for the disks, and ensures that gas and field will flow smoothly off the inner radial grid. The accretion flow rapidly accelerates inward near rms, with the radial speed quickly exceeding the sound speed. Interestingly, the angular momentum continues to drop inside of rms, indicating the continuing presence of Maxwell stress. The absence of a stress-free inner disk boundary is one of the ways in which these disks differ from standard analytic models.

One immediate conclusion from these simulations is that the constant, or nearly-constant angular momentum torus is a remarkably unstable structure in the presence of a weak magnetic field. The MHD turbulence resulting from the MRI simply transports angular momentum too efficiently. Within a few orbits the angular momentum distribution changes from q=2 to q ~ 1.6. The tori considered here did not remain maintain their initial constant angular momentum distribution long enough to develop the coherent structures associated with the nonlinear outcome of the Papaloizou-Pringle instability. This does not, however, necessarily imply that accretion disks must be thin. Moderately thick configurations, as seen in the grayscale plots, are still possible even for nearly Keplerian angular momentum distributions, if there is significant internal pressure.

Averaged properties of the torus such as density, pressure or angular momentum distribution fail to convey the impression of disorder seen in animations of the evolution. Low density, high magnetic field filaments entwine themselves throughout the torus. Regions of strong field develop and rise through the torus into the surrounding low-density atmosphere. Low density material flows outward around the bulk of the torus. Since angular momentum transport is by MHD turbulence, it follows that the disk should be highly dynamic. The effect of this on observed properties of disks must be considered in subsequent, more sophisticated simulations. It is not premature, however, to question the relevance of the traditional image of a quiescent steady-state disk.

There remain several limitations to the present simulations to be addressed in subsequent work. First, greater grid resolution is always welcome, particularly where the accretion inflow is squeezed into a narrow equatorial flow, or for following the growth of the MRI from substantially weaker initial field strengths. Because the disks develop oscillations across the equatorial plane, particularly in the simulations that begin with poloidal field loops, the full (R,z) plane must be simulated in global models, without applying explicit (e.g., reflecting) equatorial boundary conditions. This doubles the number of grid zones required, but appears to be necessary. Higher resolution simulations are feasible with the addition of more processors to the parallel computation. The simulations described here used a maximum of 64. Indeed, a great many issues can be profitably investigated using simulations with more grid zones. For example, the global simulations presented here featured tori that began close to the marginally stable orbit. Little evolution is required to accrete through rms and into the central hole. Simulating a greater dynamic range in disk radii, for longer times and with larger numbers of radial grid zones, is an immediate next goal.

The present simulations focus only on the dynamical properties of MHD in tori. As the thermodynamic properties of the disk are of obvious importance in establishing an overall global disk structure, further algorithm development is desirable. In the present simulations the simplified equation of state and lack of explicit resistivity mean that the only sources of gas heating are artificial shock viscosity and adiabatic compression. Some amount of energy is necessarily lost through the numerical reconnection of magnetic field. Further, neither radiative transport nor simple radiative losses were included. These and other issues of global disk evolution must be deferred to subsequent work.

This work is supported in part by NASA grants NAG5-3058 and NAG5-7500. Three dimensional simulations were run on the Cray T90 and T3E systems operated by the San Diego Supercomputer Center with resources provided through a National Resource Allocation grant from the National Science Foundation. The three-dimensional torus simulation GT4 was run by Greg Lindahl on a cluster of 64 alpha processors, part of the Centurion system developed by the Legion Project at the University of Virginia. A portion of this work was completed while attending the program on Black Hole Astrophysics at the Institute for Theoretical Physics, supported in part by the NSF under Grant No. PHY94-07194. I thank Steve Balbus, Charles Gammie, James Stone, and Wayne Winters, and the members of the ITP Black Hole Astrophysics program for helpful comments and discussions.

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