5. Discussion

5.5 Coherent Structures

Another issue of interest in disks, hydrodynamic or magnetohydrodynamic, is the possible formation of coherent structures. For example, Abramowicz et al. (1992) suggested that coherent vortices might produce observable modulations in the luminosity of disks in active galaxies. More recently, vortices have been proposed as sites for planetesimal formation in protoplanetary disks.

Two-dimensional $(R,\phi)$ hydrodynamic simulations of Keplerian disks have been carried out by Nauta & Tóth (1998), Nauta (1999), and Godon & Livio (1999b; 2000). The results to date can be summarized thusly: (1) vortices placed in hydrodynamic Keplerian disks can survive for many orbits, and (2) vortices have not been observed to arise spontaneously in initially Keplerian disks. Point (1) is quite understandable since the counter-rotating epicyclic flow is an equilibrium solution. Point (2) follows from the stability of a hydrodynamic Keplerian flow; there is nothing to drive the transition.

These points are illustrated by an example where coherent hydrodynamic structures were observed to arise in simulations of globally unstable disks. This is the counter-rotating coherent ``planet'' discovered by Hawley (1987) in simulations of the nonlinear saturation of the Papaloizou & Pringle (1984) instability in slender tori. Planets form in thick accretion tori as well (Hawley 1991), although in a less dramatic fashion than in the slender torus limit. These structures were described analytically by Goodman, Narayan, & Goldreich (1987) as fluid undergoing elliptical counter-rotating epicyclic motion. In these planets the Coriolis force is in balance with the fluid pressure, creating an equilibrium structure. These structures can be described as vortices, but the essence of their nature is epicyclic motion. While they are a local equilibrium solution for a differentially rotating fluid, it is unclear when such solutions can arise from an initially axisymmetric disk flow. In the slender torus, the epicyclic planets develop through the action of the Papaloizou-Pringle instability. The instability triggers the transition from one equilibrium to the other. The prospects for such a transition in a stable Keplerian disk seem less promising.

In run HK5 there is no sign of emerging or sustained coherent structures such as counter-rotating epicyclic flows. When looking for coherent structures, however, it is hardly necessary to confine one's attention to hydrodynamic disks. We can also ask whether self-sustained MHD turbulence generates coherent structures. No such structures have been observed in local simulations (e.g., Brandenburg et al. 1995). Similarly, in global MHD simulations, cylindrical or otherwise, there has been no evidence for the formation of coherent structures.

It is worth reiterating that the coherent vortices, i.e., the hydrodynamic planet structures, owe their existence to the presence of stable epicycles, but it is precisely those epicycles that are disrupted by the action of the MRI. Still, analytic MHD equilibrium coherent structures have been found by Balbus & Ricotti (1999) in the local limit which are magnetic analogues to the hydrodynamic planet solution. It has yet to be demonstrated that these solutions are stable, or that they can develop in an MHD disk.

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