5. Discussion

5.1 Evolution of Keplerian Disks: local versus global

The cylindrical disk is global in radius and azimuthal angle, but essentially local in z; it represents the next step beyond local shearing box simulations. The first cylindrical disk was computed by Armitage (1998), and two cylindrical Keplerian disk models were computed in Hawley (2000), but the limited radial extent of the grid in these models restricted the amount of evolution possible before the outer boundary significantly affected the simulation. ARC computed three cylindrical disks with an initially gaussian density distribution, contained entirely upon the grid. They examine the inflow through the marginally stable orbit in a pseudo-Newtonian potential with an emphasis on the stress there. In this paper we consider the evolution of an MHD turbulent Keplerian cylindrical disks from simple initial conditions, but with a much larger radial extent.

First, how well do local shearing box simulations describe the state of the instability, turbulence, and transport due to the MRI when compared to global models? The answer seems to be that local models do quite well, as long as the questions being asked of them are appropriate to the local approximation. Even in a global disk the MRI is a local instability; the wavelengths of the fastest growing modes are always less than H and R so long as the magnetic field is weak. The great similarity between the $\pi/2$ and $2\pi$ CK7 models illustrates this principle.

One of the major results from the local simulations is the importance of the background field topology. Local simulations (HGB95) find that a net vertical field leads to greater amplification of the initial field compared to a simulation beginning with a purely toroidal field. Stronger field amplification can lead to stronger turbulence and greater stresses. A comparison between models CK5 and CK6, and between as the vertical and toroidal simulations of ARC, support these conclusions in the global context as well. The ``efficiency'' of the Maxwell stress is measured by $\alpha_{mag}$, the ratio of the Maxwell stress to the magnetic pressure. When the toroidal and radial field energies are comparable and the fields fully correlated to produce stress with the correct sign to transport angular momentum outward, $\alpha_{mag} \sim 1$. Simulations show that in both the global and local systems the turbulent state is dominated by toroidal field; this is particularly true when the initial background field is toroidal. With vertical initial fields the toroidal and radial fields are more comparable (although the toroidal field energy remains the largest). This is reflected in the values of $\alpha_{mag}$: for vertical fields $\alpha_{mag} \approx 0.5$, while toroidal fields have a value closer to 0.3.

It should be mentioned that in the local simulations it is straightforward to measure the Reynolds stress, $\rho \delta v_r \delta v_\phi$; there is a well defined background shearing rate which allows an unambiguous definition of $\delta v_\phi$ as well as a limited volume over which to average. It is much more difficult to do this in the global simulations since at any given moment the background flow exhibits substantial deviations from, say, a Keplerian value. In local simulations, however, the Maxwell stress always dominates over the Reynolds stress by a factor of several. While measuring only the Maxwell stress for global simulations provides only a lower limit, $M^{R\phi }$ should nevertheless account for the majority of the stress.

Although the MRI is local and many properties of the resulting MHD turbulence are local as well, the stress is proportional to Pmag, the saturation amplitude of the field, and this might well be determined by global properties such as the scale height H or the ratio H/R. So far this has been difficult to assess. The traditional Shakura-Sunyaev $\alpha$ parameter is set by the relation $\alpha = \alpha_{mag}/\beta$. Thus $\alpha = 0.1$-0.01 requires $\beta \sim 3$-50. In local simulations vertical fields tend to saturate near the lower end of this $\beta$ range, and toroidal fields at the upper end, unless the toroidal field began with $\beta \sim $1-10. If one considers the current global simulation results, both from this paper and from previous cited works, the impression is that global simulations produce lower $\beta$ at saturation and larger $\alpha$ values than the local models. However, one should be cautious drawing a general conclusion at this stage. The global simulations have initial vertical fields (or poloidal field loops), or initially strong toroidal fields. For such fields the resulting saturation levels near $\alpha \sim 0.1$ are fully consistent with the local simulations. Local simulations saturate at higher $\beta$ values when the initial field consists of weak random field, or weak toroidal field, and these cases have not yet been investigated globally, in part due to the higher resolution required.

One aspect that emerges from these and other global simulations is the difficulty of characterizing a ``steady state'' disk. In the global simulations all quantities vary strongly both in time and in space. Although one can average over space and over many orbits and obtain relatively smooth spatial distributions, significant fluctuations are always present.

Absent from the local models, but present in global disks are effects such as a net accretion and spiral wave propagation. The MRI is inherently time unsteady and produces fluctuations in the disk at frequencies close to the local orbital frequency, generating magnetoacoustic waves. It not surprising that strong MHD turbulence should generate such magnetoacoustic waves. Blaes and Balbus (1994) show that the presence of toroidal fields couples the compressible and incompressible modes of the MRI, and when $\beta$ approaches order unity, significant acoustic modes are expected. In These spiral waves are generated at small radii and propagate out through the full radial extent of the disk. This is one way that the turbulence at the inner part of the disk exerts a global effect on the disk.

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