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

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1. Introduction

Accretion disks (AD) are important in many observable astrophysical objects (close binary systems, quasars, active galactic nuclei, young stars, protoplanetary disks). It is usually assumed that the disk is geometrically thin, and has turbulent viscosity (Shakura & Sunyaev 1973; Lightman & Eardley 1974). The physical mechanism of turbulent viscosity can be connected with various instabilities.

The basis for various viscous models of AD is the assumption that the dynamic viscosity [FORMULA] is caused by turbulence of the medium and that [FORMULA] ([FORMULA] is the characteristic amplitude of the most large-scale turbulent velocity, [FORMULA] is the spatial scale, [FORMULA] is the density) (Shakura & Sunyaev 1973). In this connection emphasis is given to the search of multimode instabilities, which result in a complex perturbations structure. Different spatial and temporal scales may lead to the development of turbulence in the disk.

The thin disk model (MTD) is the widely used for studies of the accretion disk dynamics. The MTD involves averaging over the z coordinate of the three-dimensional hydrodynamic equations, assuming a number of additional conditions to be fulfilled (Shakura & Sunyaev 1973; Gor'kavyi & Fridman 1994; Khoperskov & Khrapov 1995). In the context of two-dimensional models without magnetic field, four unstable modes of oscillations are present: two acoustic modes (Wallinder 1991; Wu & Yang 1994; Khoperskov & Khrapov 1995; Wu et al. 1995), a thermal mode and a viscous one (Lightman & Eardley 1974; Shakura & Sunyaev 1976; Szuszkiewicz 1990; Wallinder 1991; Wu & Yang 1994). The instability growth rate of acoustic waves increases with smaller wavelengths [FORMULA]. However, the thin disk model imposes a restriction on wavelength, [FORMULA], and therefore it is necessary to consider AD z-structure for a correct treatment when [FORMULA]. The thin disk model applies only to pinching oscillations. Then the perturbed pressure is a symmetric function, and the displacement of gas does not change the mass centre in a disk with respect to the plane of symmetry ([FORMULA]). The MTD cannot account for bending oscillations (AS-mode) because then the perturbed pressure is antisymmetric. Also, the high-frequency (reflective) harmonics with characteristic spatial scales in the z-direction [FORMULA] cannot be investigated in the MTD.

In this paper we investigate the dynamics of acoustic perturbations taking into account the non-homogeneous z-structure of a viscous disk. Apart from the special problem of the limits of MTD applicability, the main question is the existence of short-wave instabilities and, in addition, the stability of the high-frequency harmonics. In Sect. 2 we define the AD model and we choose the viscosity law. In Sect. 3 we consider the dynamics of linear acoustic perturbations, and formulate the mathematical problem of eigenfrequency determination for various unstable modes in the disk. Lastly we discuss in Sects. 4 and 5 results of the numerical solution of the boundary problem and summarize the main conclusions.

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© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999