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Astron. Astrophys. 337, 105-112 (1998)

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Appendix A: shear viscosity in a clumpy disk at elastic/inelastic encounters

The shear viscosity in a differentially rotating clumpy disk, with accounting for gravitational interactions between the clumps of mass m, is given by Eq. (60) in Stewart & Kaula (1980):

[EQUATION]

where [FORMULA], and [FORMULA] is frequency of clump-clump collisions given by

[EQUATION]

Eq. (A2) corresponds to Eq. (57) of SK when one puts [FORMULA] because in a flat disk the long-range gravitational interactions could be neglected compared to the short-range ones. The value of [FORMULA] is related to the characteristic time of gravitational encounters, [FORMULA], given by Eq. (2.3) simply by [FORMULA]. By substituting Eq. (A2) into Eq. (A1), the latter could be written in the form:

[EQUATION]

where [FORMULA] for a Keplerian disk, [FORMULA] for a solid-body rotating disk, and [FORMULA].

The viscosity coefficient for clump-clump collisions, when the clump gravity is negligible [Goldreich & Tremaine 1978, Eq. (46)], can be written in the form analogous to Eq. (A3):

[EQUATION]

Eq. (2.1) unifies the representation of viscosity both in the case when interactions are inelastic, while gravitation is negligible [Eq. (A4)] and in the case when gravitational encounters are dominating [Eq. (A3)].

Appendix B: character of perturbations in a differentially rotating vs. a solid-body rotating disk

Let us consider an incompressible, differentially rotating, liquid cylinder. The linearized equations of motion take the form:

[EQUATION]

Substituting a perturbation in the form of the wave packet f [FORMULA] [FORMULA] results in the following dispersion relation ([FORMULA]):

[EQUATION]

As always [FORMULA] [FORMULA] [FORMULA] and therefore the condition of instability is [FORMULA] [FORMULA] 0, which corresponds to the growth of the leading spirals. Under this condition the instability is monotonous without any oscillations with epicyclic frequency. If the rotation is solid-body, one obtains [FORMULA] = 0, i.e. the perturbations are stationary.

The above solution has a drawback since the r-component of the group velocity for perturbations under consideration is zero. A more detailed analysis should be performed to find out what is the role of r-dependence for [FORMULA]. This difficulty can be overcome if one considers the system in a corotating frame of reference. In this case, an exact time-dependent solution can be derived (see Fridman, 1989) which demonstrates a qualitatively the same behaviour as is obtained above. Specifically, in any differentially rotating disk whose angular velocity decreases with radius, the leading spirals grow monotonously. As for a solid-body rotating disk, all perturbations are stationary.

Note that if one would consider absolutely compressible case, i.e. [FORMULA] in the above equations, the continuity equation takes the form:

[EQUATION]

and we immediately obtain

[EQUATION]

The contrast to other work is illuminating. Julian & Toomre (1966) considered perturbations in a very thin collisionless self-gravitating stellar disk. In fact, for such perturbations a collisionless stellar disk is absolutely compressible, in contrast to incompressible character of perturbations with wavelengths much smaller than the disk's width for a gaseous disk considered here. Goldreich & Lynden-Bell (1965) studied dynamics of perturbations in a gaseous disk, but only at high frequencies. Really, perturbations considered in their paper demonstrate (asymptoticaly) oscillations with sound frequency kc, where k is the wave number and c is the sound speed. As the scale of perturbations is smaller than the disk width, i.e. [FORMULA], their frequency is higher than the epicyclic one. But as the equations of motions contain three derivatives with respect to time, they describe three modes of perturbations. Besides the two sonic high frequency modes, one mode of low frequency perturbations exists as well. What is examined in our paper is the role of low frequency mode in dynamics of the circumnuclear disk. As was shown in Fridman (1989), this mode does not demonstrate any oscillations and is stationary in a solid-body rotating disk.

Appendix C: Velocity perturbations in a plane shear layer

We start from linearized equations for short-wave, low-frequency perturbations in a barotropic gaseous disk in the corotating frame of reference (for derivation, see Fridman 1989):

[EQUATION]

Here [FORMULA] = const is the angular velocity of the corotating frame of reference; A [FORMULA] [FORMULA] characterizes the value of the shear; [FORMULA] [FORMULA] [FORMULA] + [FORMULA]; [FORMULA] [FORMULA] [FORMULA] + [FORMULA].

This system has an exact solution in the form (Fridman, 1989):

[EQUATION]

For small t the exponential term is negligent and we come up with vorticity conservation:

[EQUATION]

For a plane shear layer, a similar set of equations can be derived. The simpliest way to do this would be to put in the above system [FORMULA] [FORMULA] 0, r [FORMULA] [FORMULA], [FORMULA] [FORMULA] [FORMULA]. As a result, we obtain the following system:

[EQUATION]

Here B [FORMULA] [FORMULA] characterizes the value of the shear; [FORMULA] [FORMULA] [FORMULA] - [FORMULA]; [FORMULA] [FORMULA] [FORMULA] + [FORMULA]. One can easily see that this system is almost similar to the previous one and it has a similar solution:

[EQUATION]

Obviously, in a plane layer there are no any Coriolis forces at all. Nevertheless, the solution (C3) has the same form as (C1). What is relevant to the real physical meaning of our solution is conservation of vorticity (Eq. C2), which has been emphasized here.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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