*Astron. Astrophys. 345, 307-314 (1999)*
## 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 is caused by
turbulence of the medium and that
( is the characteristic amplitude of
the most large-scale turbulent velocity,
is the spatial scale,
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 .
However, the thin disk model imposes a restriction on wavelength,
, and therefore it is necessary to
consider AD *z*-structure for a correct treatment when
. 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
(). 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
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.
© European Southern Observatory (ESO) 1999
Online publication: April 12, 1999
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