A qualitative argument presented at the end of Sect. 2 shows that the viscosity in a cloudy disk is, in general, much larger than that for a continuous disk. This assertion could be easily confirmed by numerical estimates when we address the circum-nuclear ring (CNR): in a continuous disk with similar global parameters, one would have , where km/s is the sound speed of the gas (Güsten et al. 1987) and is the thickness of the disk. What provides a much larger viscosity than it would be possible in a continuous disk is (although only partly) the clump-clump collisions. As is shown in Sect. 3, the collective mode of instability would dominate the dynamics of a clumpy disk if viscosity in the latter is less than some critical viscosity [Eq. (3.11)] whose value is given by Eq. (3.15) or (3.16). This condition is (marginally) met in the CNR, which results in the turbulent viscosity coefficient . We note in passing that Güsten et al. (1987) suspected the existence of turbulence in the clumpy CNR, although they did not evaluate its viscosity coefficient.
It is worth mentioning that, in spite of the differences in the specific physical parameters of the clumps in the CNR and active galactic nuclei, or AGN (for the latter, see Netzer 1990), the value of evaluated above turns out to be in the range of the values estimated by Fridman & Ozernoy (1992) for cloudy disks in AGN. This might be relevant to the issue whether the CNR could be considered as a prototype for circum-nuclear tori around some types of AGN. (It is interesting that a large value of the viscosity in the CNR implies its rather large scale height, which is a required geometry for the AGN tori). In any case, the angular momentum transport in the CNR seems to be a template while considering similar issues both for quiescent and active galactic nuclei.
The analysis performed above has not accounted for the magnetic field in the CNR. Meanwhile several observational techniques revealed the field strength in the clouds to be mG (for a recent review, see Genzel et al. 1994). A magnetic field as strong as this cannot be ignored in the transport of angular momentum. Conservation of angular momentum transported by both viscous and magnetic stresses can be written for the CNR in which const in the form (Ozernoy & Genzel 1998):
where is the mass inflow rate, is the surface density of the CNR, is the averaged over the z-coordinate, is the Alfvén velocity, is an effective viscosity, and pc is the inner radius of the ring. If we consider the CNR as a magnetized disk for which km/s and is given by Eq. (4.4) the two terms in parentheses in the r.h.s. of Eq. (5.1) turn out to be comparable. This implies that while evaluating the angular momentum transport in the CNR, both magnetic and turbulent viscosity need to be accounted for. One can see that with the parameters listed above and g cm-2 the total mass inflow rate given by Eq. (5.1) at a fiducial distance of pc amounts to .
This result is consistent with a naive, by order-of-magnitude, estimate of , where M is the CNR mass and is the characteristic time for the angular momentum transport: Taking and , i.e. somewhere in between and , the current estimates for the CNR mass (Genzel et al. 1995), one gets yr and . This is consistent as well with the inflow rate toward the Galactic center inferred from the observational data (e.g. Blitz et al. 1993, Genzel et al. 1994).
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998