## 4. ResultsIt is convenient to characterize the properties of the considered
acoustic oscillations in the disk by a dimensionless frequency
and a dimensionless wave number
## 4.1. Fundamental S-mode in the 2D- and 3D-modelsOur analysis confirms the good agreement between the thin disk
model and the results of the 3D-problem solution in the case of the
dissipative acoustic instability. The eigenfrequencies of oscillations
obtained from Eq. (29) at using
Eq. (30) and the solution of Eqs. (24), (25) practically coincide in
the range (see Fig. 1a,b).
Appreciable differences occur when .
We emphasize that in this region the formal condition of applicability
of the MTD () is obviously broken.
In Fig. 1e,f, the dependences of the group velocity
and
on the wave number
This approximation is sufficiently exact. Our results shows that the MTD adequately describes the dynamics of perturbations with characteristic spatial scales in the disk plane. The differential rotation () and the dependence of dynamic viscosity on the thermodynamic parameters (for example, the dependence on density and temperature) are responsible for the instability. A simple model can demonstrate this. The last term in Eq. (3) (i.e., containing ) is a source of instability of the acoustic modes. The expression ( is the surface density, is the kinematic viscosity) is used in the 2D-model. For simplicity we assume here that is constant. Then a perturbation of the surface density generates a perturbation of the dynamic viscosity. Now let us consider how a viscous force, which is caused by , leads to an amplification of the amplitude of a sound wave in the plane of the disk. The evolution equation for the surface density, without including Coriolis's force, is , where . This equation is linear and the appropriate dispersion relation is , which means that acoustic waves are unstable with a growth rate and entropy oscillations damping (). These estimates can only be used in the short-wavelength limit, because we have neglected the epicyclic oscillations. ## 4.2. Fundamental AS-modeWe now consider the low-frequency bending oscillations for which Eq. (27) is fulfilled. In the long-wavelength limit both symmetric and antisymmetric perturbations have the same frequency in a Keplerian disk. This result can easily be obtained from Eq. (29) when . In the adiabatic model two other branches of oscillations () always damp at and we do not examine them. The dispersion behaviour of S- and AS-modes is very similar (Fig. 1a-d, solid lines). For the fundamental bending mode it is possible to set in Eq. (29). The dispersion curves on Fig. 1c,d are shown as dotted lines for this case. The exact solution in the 3D-model exhibits differences from the result derived from Eq. (29). In the range these differences are caused by the following factors: 1) the 3D-model includes radial inhomogeneities of the equilibrium quantities; 2) the Eq. (29) does not take into account the vertical inhomogeneity of the disk. The result obtained is very unexpected and significant, since we used Eq. (29) beyond the limit of the 2D-model. The physical reason for the AS-mode instability is similar to that for the S-oscillations. ## 4.3. High-frequency S- and AS-modesBesides the fundamental S-mode, unstable harmonics can be
generated. These harmonics differ from each other by the number of
nodes of the perturbed pressure across the disk plane. The following
estimate of the effective wave number in the where The dependences of the eigen-frequency
on the radial wave number
The vertical structure of the disk in the 3D-model is determined
primarily by the parameters and
The dispersion properties of the acoustic perturbations do not depend on the values of the parameters and in the range . The reason for this is that the radial gradients of equilibrium pressure and density give only a small contribution to the equilibrium balance in the case of a thin disk. © European Southern Observatory (ESO) 1999 Online publication: April 12, 1999 |