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

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5. Discussion and conclusions

Acoustic waves in a differentially rotating gaseous disk can play an important role for understanding the turbulent viscosity in accretion disks. The amplitude of small-scale (in r and in z) waves grows most rapidly. Such instabilities do not destroy the initial flow at a nonlinear stage, but can effectively make the disk matter turbulent, and in turn, the turbulent viscosity generates unstable sound modes. A self-consistent regime with turbulent viscosity arises. Beside the dissipation mechanism analysed above, the development of global resonant Papaloizou-Pringle modes (Papaloizou & Pringle 1987; Savonije & Heemskerk 1990), the resonant amplification of acoustic oscillations in the regime of double-flow accretion (Mustsevoj & Khoperskov 1991) and in the case of disk accretion onto a magnetized compact object (Hoperskov et al. 1993) may be important for understanding the turbulent viscosity. It is remarkable that in all cases the characteristic time scale does not exceed the dynamical time scale ([FORMULA]), and that all values are of the same order.

We have demonstrated the possibility of unstable high-frequency acoustic waves in a differentially rotating gaseous disk. They exist in the system for a limited period, and the presence of a positive growth rate does not mean that the perturbations reach the nonlinear stage. The perturbations leave the disk with a speed [FORMULA] and the characteristic life-time in the disk is [FORMULA] disk rotation periods. In view of the derived estimation [FORMULA] for [FORMULA] we have obtained the growth rate of wave amplitude ([FORMULA]). Nonaxisymmetric perturbations may stay in the system for a longer time and reach a non-linear stage.

The main conclusions of this study are:

  1. The linear wave dynamics of a thin disk model is compared to an exact solution, which takes into account the vertical structure. A small quantitative difference in the dispersion properties occur for a wavelength [FORMULA], and even if [FORMULA] the dispersion properties remain qualitatively similar. At [FORMULA] the eigenfrequencies differ less than on 5%.

  2. In addition to the fundamental dissipational unstable sound mode in MTD, an arbitrary number of high-frequency unstable harmonics of the pinch-oscillations is found. The different harmonics have a different vertical structure. The waves with small scales along the z-coordinate have a maximum growth rate at short wavelengths in the r-direction. The differential rotation and the variable dynamic viscosity cause instability of all oscillation branches, i.e., a dissipational mechanism lead to the growth of perturbations.

  3. Taking into account the vertical structure of the disk we studied new bending modes, which cannot be investigated in the context of MTD. The bending oscillations, as well as the pinch-wave, are unstable. The physical mechanism of instability and the dispersion properties are similar to those of the pinch-oscillations. The significant feature of all these unstable acoustic modes is the fact that the spatial scales of the perturbations differ from each other, but that their characteristic growth rates have all the same order of magnitude.

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© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999