Introduction

Recent increases in supercomputer performance have significantly improved the ability to evolve the basic equations of accretion disk structure and evolution. These developments, along with continuing progress in understanding the most important physical processes that occur within accretion disks, suggest that predictive disk simulations are a realistic goal. Such disk simulations will be global, fully three dimensional, and incorporate physical processes such as magnetohydrodynamics (MHD) and radiation transport. At present, we are some ways from this goal; global simulations are still rather idealized in terms of disk structure, energetics, and dynamical range. However, because almost any three dimensional disk simulation is relatively novel, there remain many significant questions to be investigated even with such simplified models.

To date there have been several global simulations of three dimensional disks, including Armitage (1998), Matsumoto (1999), Hawley (2000), Machida, Hayashi, & Matsumoto (2000), Hawley & Krolik (2001; hereafter HK), and Armitage, Reynolds & Chiang (2001; hereafter ARC). Much of this work has focused on thick accretion disks. With a pressure scale height H comparable to the disk radius R, the thick disk, or accretion torus, is more easily resolved in a numerical simulation than disks for which $H/R \ll 1$. Matsumoto (1999) followed the evolution of a thick torus embedded in an external vertical field, and found significant outflow collimated along the global vertical field lines. Hawley (2000) considered tori containing toroidal fields and poloidal field loops, and Machida et al. (2000) modeled a thick disk containing a toroidal field. In these studies the initial field was entirely contained within the disk and the resulting outflows were confined to the creation of a magnetized corona. A generic feature of all these thick disk simulations is the presence of large amplitude fluctuations in accretion rate, density, and other variables, in both space and time.

At a minimum these efforts have established that the magnetorotational instability, or MRI, (Balbus & Hawley 1991) is just as efficacious in thick disks as in local simulations to produce MHD turbulence and angular momentum transport. Thick accretion tori with initially non-Keplerian angular momentum distributions are highly unstable. MHD turbulence develops rapidly and is sustained by a self-consistent dynamo process within the disk. The constant or near-constant specific angular momentum distribution of the initial torus rapidly evolves to one that is near Keplerian. The main focus for dynamical studies would therefore seem to be Keplerian disks, both hot (high internal sound speed), and thin and cold (low internal sound speed).

Global simulations have been used to investigate specific physical issues in Keplerian accretion disk models. HK examined the behavior of a thick, nearly Keplerian disk model to study the accretion flow through the radius of the marginally stable orbit (rms) in a pseudo-Newtonian potential. They found significant stress at the location of the marginally stable orbit which creates a continuing decline in the value of the specific angular momentum $\ell $ inside of this point. The equation of state was adiabatic and there was no cooling, and the thickness of the disk remained roughly $H/R \sim 0.15$ throughout.

So far there have been fewer simulations of Keplerian disks with low internal sound speeds corresponding to small H/R. To sidestep the difficulty of resolving both H and R, the cylindrical disk limit has been employed which omits vertical gravity and stratification. Armitage (1998) and Hawley (2000) simulated a few examples of these cylindrically-symmetric Keplerian disks. More recently, ARC modeled several Keplerian cylindrical disks with sound speeds corresponding to H/R of 0.08 and 0.04. They also computed one stratified disk that covered a limited vertical extent. As in the work of HK, they examined the inflow through rms. Although they found a nonzero Maxwell stress inside of rms, it was not large enough to alter $\ell $ to the same extent as seen in the simulation of HK. They emphasized that the differences between their simulations and those of HK were quantitative; the same dynamical effects were present, only at reduced amplitude. In addition to the lower initial sound speed and cylindrical symmetry, the ARC simulations used an angular domain of $\pi/6$ compared with the full $2\pi$ simulation of HK. It is unclear which, if any, of these differences is the most significant in comparing the results of HK and ARC.

Clearly, we still have only a preliminary understanding of the specific processes that establish the precise turbulent stress levels in disks. An attempt to investigate this question through simulation is further complicated by the need to distinguish between influences created purely by the numerical details (e.g. computational domain, numerical resolution, initial conditions), and those determined by physics. In an effort to characterize the array of technical choices in three dimensional simulations, this paper presents a series of disk simulations that begin with a Keplerian angular momentum distribution. These simulations are restricted to the cylindrical limit which omits the gravitational acceleration in the z direction. The cylindrical approximation has the distinct advantage of reducing the number of grid zones required in z: only the wavelengths of the weak-field MRI need be resolved, not several pressure scale heights. Here this advantage is exploited to increase the number of grid zones devoted to the radial extent of the disk, and its angular resolution.

Comparisons of cylindrical disks and vertically stratified disks in Hawley (2000) provided some evidence that the cylindrical disk is a good approximation for studying the evolution of density-averaged properties in a disk. Here we will be exploring the detailed properties and limitations of these cylindrical disks in greater detail. These cylindrical disks will be directly comparable to the simulations of ARC, and are the natural sequel to the local nonstratified shearing box simulations done previously (Hawley, Gammie, & Balbus 1995, hereafter HGB95; 1996; Matsumoto & Tajima 1995). Comparison with shearing boxes provide insight into those phenomena that are truly global versus those that are well-described locally. Cylindrical disk models will also provide a baseline of results that will be compared with future stratified Keplerian disk models.

Some of the questions that this investigation will address include: How do these global results compare with local shearing box simulations? Are there effects that can be attributed to influences from the equation of state, or which arise when compared to simulations of tori with higher internal temperatures? What differences are there in disks computed on grids with angular extent less that $2\pi$? Can a steady state be established in at least part of the disk? What are the amplitudes, and the time- and length-scales for the fluctuations that result from the MRI-induced turbulence?

The plan of this paper is as follows. In §2 the computational setup is described. Section 3 presents the results from a series of MHD cylindrical disk simulations. In §4 a purely hydrodynamic disk is considered for contrast. The results and their consequences are discussed in §5, and the conclusions are summarized in §6.

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