2. Viscosity due to cloud-cloud interactions
In a recent version of the unified model for fueling AGN, Begelman et al. (1989) proposed that the inflow of matter (driven by global axisymmetric gravitational instabilities on large scales) proceeds on small scales in the form of a `disk' composed of randomly moving clouds, which are embedded in a low-density medium with a small filling factor. There is an essential uncertainty in our knowledge about the overall configuration, dynamics, and confinement mechanisms for thermal gas clouds observed in AGN, but as for our Galaxy, such a disk is known to exist as the circum-nuclear disk, or ring that is rather clumpy, indeed (Jackson et al. 1993). Begelman et al. (1989) assumed that the viscosity in the disk is provided by collisions between clouds. Below, this collisional mechanism of viscosity is compared with some others.
We consider the following simple model of a cloudy disk: The clouds are orbiting in an external gravitational field and have some random peculiar velocities. It is assumed that the clouds have a small filling factor and are embedded in a low-density medium that provides a confinement of the clouds. In the differentially-rotating cloudy disk, the angular momentum is transported due to cloud-cloud interactions (which include, and are not just restricted to, collisions). Let us address the shear viscosity associated with these interactions.
The most elaborated models to calculate the shear viscosity have been considered by Goldreich & Tremaine (1978) and Stewart & Kaula (1980), hereinafter referred to as GT and SK, correspondingly. GT considered contact inelastic collisions of non-gravitating spherical particles obeying an anisotropic distribution function, whereas SK considered gravitational (elastic) encounters of particles obeying a Maxwellian distribution. In the both cases, as shown in Appendix A, the viscosity coefficient can be represented in the form
Here is the one-dimensional velocity dispersion, is the orbital angular velocity in the disk, is the `optical depth' to cloud-cloud interactions (see below), is the free path time (index ; c stands for collisional, or contact, interactions, G stands for gravitational ones), and are the constant coefficients defined below.
Collisions between the clouds result in diminishing , whereas gravitational encounters tend to increase it. For cloud-cloud collisions,
where a and n are the typical size of a cloud and the spatial number density of the clouds, respectively. For gravitational encounters between the clouds
(Braginskii 1965), where m is the typical mass of a cloud. Evidently, can be considered as an effective size of the domain for gravitational influence of the cloud.
Coefficients and in Eq. (2.1) take the following values:
The optical depth of the disk is a convenient parameter describing how effective are the interactions between the clouds. By order of magnitude, it is nothing but the mean number of interactions suffered by a cloud in passing through the disk. More accurately,
assuming a to be the largest of geometrical and gravitational influence sizes. Here h is the thickness of the disk given by
Since , where is the surface density of the disk, Eq. (2.2) can be rewritten as
which implies one more interpretation for : it is the covering factor, C, or the fraction of disk area covered by clouds when they are placed as a monolayer.
The filling factor of the system of clouds, i.e. the fractional volume filled by the clouds is
Eqs. (2.5), (2.6), and (2.8) yield one more expression for containing F:
Evidently, for any cloudy disk with unless .
Here we emphasize that cloud-cloud collisisons cannot be discussed without noting that the magnetic fields permeate clouds, and are likely to make such collisions more efficient by increasing the effective cross section: When two clouds collide, it is unlikely that they just slide along a given straight flux tube. First of all, if this imagined flux tube were not exactly on a circle, then by angular momentum conservation the clouds would not go in a straight line, and second, by virtue of the general distribution of velocities it is very unlikely that two clouds would just match in proper velocities to be able to slide along a flux tube, and, third, the energy density in flux tubes is unlikely to sufficiently overpower the kinetic energy of clouds to do this. It follows that it is indeed likely that the flux tube will be twisted, thus strengthened in magnetic field, and therefore the clouds may interact even at some distance. This means that cloud-cloud collisions may involve a larger effective cross section than just the geometry would imply.
Furthermore, with magnetic loops and reconnection in the region above the disk, the effective scale height may well be larger than the scale height of the visible cloud distribution.
It follows that the estimate above may be an underestimate just as well as an overestimate; the observational fit and interpretation given to the data by von Linden et al. suggests that the viscosity derived above for cloud-cloud collisions is an underestimate.
It is instructive to compare the viscosity coefficient in a cloudy disk [Eq. (2.1)] with that in a typical thin, but continuous disk: , where and l are the sound velocity and the mean free path length, correspondingly. Qualitatively, in a continuous disk l is anticipated to be much smaller than l in a cloudy disk. If , the viscosity in a cloudy disk exceeds that in a continuous disk, bearing in mind some reasonable assumptions about the disk parameters. Before specifying them, we would like to discuss one more mechanism for viscosity in a cloudy disk, this time of a collective origin, proposed recently by Fridman & Ozernoy (1992).
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998