Astron. Astrophys. 345, 307-314 (1999)

4. Results

It is convenient to characterize the properties of the considered acoustic oscillations in the disk by a dimensionless frequency and a dimensionless wave number kh. We define the basic model as follows: , , , , , (), . If not specified otherwise, the parameters take these values.

4.1. Fundamental S-mode in the 2D- and 3D-models

Our 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 k for the 3D-model and the 2D-model at and are shown. In the case of small values of n, a difference occur at large kh as Fig. 1a,b shows. The reason is that the characteristic scale of inhomogeneity in the vertical direction increases at smaller n. The growth rate and the phase velocity of perturbations in the framework of the 3D-model are smaller than for the thin disk model (2D-model) in the short-wavelength region. This effect is caused by the inhomogeneous distribution of the equilibrium quantities in z-direction and by the transverse gravitational force. When the parameter increases the growth rates grow linearly. The value of the wave number k, at which the differences between the models appear, does not depend on . The considered low-frequency mode exists in both the 2D-model and in the 3D-model, and it depends weakly on the z-structure. We therefore call it the fundamental mode . When , the expression for frequency is easily obtained from Eq. (29) as:

Fig. 1a-f. The dependence of the dimensionless eigenfrequency W on the dimensionless wave number kh for different values of n (1 - , 2 - ) (a-d ). The dependence of on kh for the S-mode (e,f ). The solid lines correspond to the 3D-model, dotted lines to the 2D-model at , long-dashed lines to the 2D-model at .

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-mode

We 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-modes

Besides 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 z-direction can be made:

where j is the number of the harmonic. The fundamental () and reflective () harmonics exist for both symmetric (S-) and antisymmetric (AS-) modes.

The dependences of the eigen-frequency on the radial wave number k for the fundamental mode and the first four reflective harmonics are displayed in Fig. 2a-d. The pinch-oscillations and the bending modes are both unstable (). The imaginary part of the frequency grows with k and reaches a maximum at some value. The maximum of moves to the region of shorter wavelengths when the harmonic number j grows, and the value increases with reduction of the characteristic scale of perturbations in z-direction. As indicated by Fig. 2a-d, the growth rate increases with , while the perturbed phase velocity does not depend on . It should be noted that for very short-wave perturbations (), the damping of oscillations due to the presence of a perturbed velocity gradient in the viscous stress tensor can be of decisive importance (Khoperskov & Khrapov 1995). This factor can be essential for small-scale waves (as on r- and on z-coordinate) and it can result in complete stabilization.

Fig. 2a-d. The dependence of W on kh for S- and AS-modes for various values of . The solid line corresponds to , the dotted line to . The harmonic numbers j are indicated.

The vertical structure of the disk in the 3D-model is determined primarily by the parameters and n. When , the entropy does not vary along the z-coordinate, i.e. the disk is stable against convective z-motions. When we have and the conditions for convective instability are fulfilled; in the opposite case , , so convection cannot appear. In Fig. 3a-d, the dispersion curves for various values of n (, and ) are shown for the fundamental modes () and for the third reflective harmonic (). The phase velocity of the fundamental S- and AS-modes weakly depends on the parameter n (see Fig. 3b,d). The adiabatic sound speed in the disk, grows with n (see Eq. (14)), and since for the high-frequency reflective harmonics , the phase velocity of the perturbations with also increases (see Fig. 3b,d).

Fig. 3a-d. The dependence of W on kh for the fundamental mode () and for the high-frequency one with for various values of n. Solid line: , dotted line: , dashed line: . The harmonic numbers j are specified.

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
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