6. Conclusion

Cylindrical disk simulations are a useful tool for investigating global evolution of disks evolving due to magnetically driven turbulence. Such simulations demonstrate that the conclusions developed in the local shearing box model hold in the global context as well. As in the local model the MRI grows rapidly and produces MHD turbulence with a significant Maxwell stress. The turbulence is more vigorous and more efficient in producing stress for a given total magnetic pressure when driven by an initial field that is vertical rather than toroidal. Hydrodynamics alone seems no more effective at creating or sustaining turbulence in a global model than it is in a local one.

In addition to reaffirming local properties of the MRI, cylindrical disk simulations illuminate disk characteristics that are truly global. A net accretion rate is one such property, but there are also important nonlocal structural features. Tightly-wrapped low-m spiral waves are prominent. The final accretion through the marginally stable orbit provides an example of a highly nonaxisymmetric spiral flow. Particularly interesting in the present simulations is the tendency for radial variations in Maxwell stress to concentrate gas into rings, creating substantial spatial inhomogeneities.

A perennial question is the degree to which these simulations resemble traditional steady state $\alpha$-disk models. They do in so far as they accrete in direct response to internal stress, specifically due to MHD turbulence. Beyond that, however, there are significant differences. The simulations are characterized by large scale variability in space and time in all variables. The stress is proportional to the magnetic pressure which is itself only indirectly related to other disk parameters. In the simulations it is possible to approximate a quasi-steady state only with broad-stroke averages. In part this is due to the initial conditions (e.g., isolated tori or constant density slabs) which are far from a possible accreting steady state solution. To address this issue it will be useful to attempt simulations that begin with more realistic initial states. Results to date indicate, however, that analytic disk models are likely to prove woefully inadequate in describing detailed spatial and temporal disk properties so long as they are based upon a strict $\alpha$ formulation with $\alpha$ a constant in space and/or in time.

Although cylindrical disk simulations provide a valuable point of reference for future work, the lack of vertical stratification is clearly a major limitation for investigating many important physical processes. This is obviously true for the development of a magnetized corona, or the launching of winds or jets. It appears also to be an important factor in measuring the stress in the disk at the marginally stable orbit. Stratified global thin disk simulations are the next logical step to contrast with existing global thick disk models. Stratified thin simulations will require far more vertical grid zones centered around the equator than are used in cylindrical disks. However, the tests presented here suggest that a reduction in the $\phi $ domain is a acceptable problem simplification, as long as the potential for some small quantitative reduction in energy levels is kept in mind.

I thank Steve Balbus, Julian Krolik, Jim Stone, and Wayne Winters for useful discussions related to this work. Wayne Winters supplied data from his unpublished simulations NK1 and NK1a for the analysis in \S3.4. This work was supported by NSF grant AST-0070979, and NASA grants NAG5-9266 and NAG5-7500. Simulations were carried out on the Cray T3E and T90 systems of the San Diego Supercomputer Center of the National Partnership for Advanced Computational Infrastructure, funded by the NSF.

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