5. Discussion

5.2 Viscous Instability

A prominent feature of model CK6 at late time is the presence of a dense ring in the inner region and low density gaps that have formed within the disk. These gaps arise due to a type of ``viscous'' instability, specifically an inverse relationship between the Maxwell stress and the density. Although these rings and gaps are most prominent in CK6, there is evidence for a similar tendency in CK5, the toroidal field simulation.

The dense rings that form in CK6 are likely to be enhanced by the cylindrical approximation. The rings per se might not form in simulations with vertical stratification since the strong magnetic field located between the dense rings would be buoyant. Cylindrical symmetry eliminates the possibility of buoyancy as well as any development of magnetized outflows. The tendency for greater stress in regions of lower density is not due to the cylindrical limit, however, but due to the increased Alfvén speeds in such regions. The probable outcome of this tendency in stratified disks will be larger stresses in the lower density region above the equatorial plane (see, for example, in the simulations of Miller & Stone 2000), and the creation of local regions of strong toroidal field that will be ejected into a corona. In fact, strong toroidal fields rising out of the disk are a common feature of the global thick torus simulations of Hawley (2000), HK, and Machida et al. (2000). Another consequence is that the accretion rate through the disk will be inherently unsteady, and spatially inhomogeneous. The two-dimensional axisymmetric global MHD simulations of Stone & Pringle (2000), for example, provide evidence of just such strong radial inhomogeneities in accretion rate and density.

How generic is this tendency to develop a viscous instability? Interestingly there is no evidence for the formation of dense rings in CK7, in contrast to CK6. The most obvious systematic differences between CK6 and CK7 are the domain size ($\pi/2$ versus $2\pi$), the equation of state (isothermal versus adiabatic), the absence or presence of net accretion into the central hole, and total evolution time. The influence of the $\phi $ domain seems too small to account for the difference. Although an isothermal equation of state can create greater density fluctuations, a direct comparison between adiabatic and isothermal models CK7a and CK7b does not find any systematic effect due to the choice of equation of state. Further, ARC used an isothermal equation of state and did not observe any tendency toward the formation of dense rings.

This leaves the influence of the inflow through the marginally stable orbit and the duration of the simulation. In models that do not accrete through the marginally stable orbit, mass tends to pile up at the inner edge of the disk. The accretion through rms prevents this, and creates an inward pointing pressure gradient out through the disk to radii that are several times rms. This appears to inhibit the development of the viscous instability, at least in the inner disk over the time scales simulated. The viscous instability might yet manifest itself in the disk at larger radii, but CK7 was not run far enough in time for this to occur. Further simulations and analysis would be required to test this conjecture further.

Title Title Page   |   Method 5.1 Evolution of Keplerian disks: local versus global   |   Discussion 5.3 Stress at the marginally stable orbit