Oscillations of Disks
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The main part of this subsection is a duplicate of Kato et al. Since the oscillations are nearly horizontal, we neglect both the vertical component of the perturbed motions and the vertical derivation of the perturbed quantities, i. It should be noticed that the relativistic disks are strongly unstable in the innermost region.
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The oscillations are assumed to occur isothermally in isothermal disks. The case of the Newtonian potential is also shown for comparison. Before showing this, we derive a general instability criterion in the next section. Viscous excitation of disk oscillations was first shown by Kato and Blumenthal et al. To derive perspectively a general form of the stability criterion, however, the adoption of Lagrangian variables is better.
In this section we derive a general criterion of an oscillatory instability overstability of geometrically thin disks by using the Lagrangian variables. The basic assumptions involved in this formulation are that the unperturbed disks are steady isolated systems. The isolation implies that the disks are bounded with surfaces with no pressure.
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This does not represent real situations of accretion flows, but would bring about slight differences in the final results. The unperturbed flows are assumed to be adiabatic and inviscid. On such disks, oscillations are superposed, which are assumed to occur quasi-adiabatically and quasi-inviscidly. The necessary tools used to derive the criterion have been prepared by Lynden-Bell and Ostriker Applying their formalism, Cox and Everson derived a criterion of the oscillatory instability of disks.
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In the following, a different form of the criterion is presented by a method somewhat different from that of Cox and Everson Kato ; Kato et al. The final expression of our present criterion is already given by Kato without any derivation. Kato et al. Inside the radius the wave energy is negative, while it is positive outside. This is well known in the density wave theory of galactic dynamics and also in the disk instability theory in relation to the Papaloizou-Pringle instability.
The above formulation also shows that the perturbation method adopted here is invalid when the perturbations have a non-negligible amplitude around the corotation radius, since there is a singularity there. In problems treating one-armed oscillations, however, the corotation radius usually occurs outside the oscillating region.
In this sense, the criterion can be used, in practice, in studying the stability criterion. Equation clearly demonstrates that viscosity has two effects on excitation of oscillations.
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The other one is a dynamical process process  , and is represented as the first integral on the right-hand side of equation This dynamical process of viscosity is characteristic of the accretion disk oscillations, as mentioned before. In subsection 6. The second term in the brackets is related to rotation and is the term of our concern. The results of numerical simulations that inertial—acoustic oscillations are generated in the innermost region of relativistic disks see subsection 5.
One of the possible processes would be the dynamical actions of turbulence Goldreich, Keeley , which is known as the excitation process of solar oscillations, and is suggested to also be a possible source of excitation of disk oscillations Nowak, Wagoner See Ortega-Rodriguez and Wagoner for careful studies in the case of Newtonian disks.
This kind of anisotropy of the turbulent stress tensor is theoretically expected in accretion-disk turbulence e. The importance of anisotropic turbulence on the excitation of the oscillations has already been suggested by Nowak and Wagoner The latter warps are known to be excited by a reaction due to the re-emission of irradiation from the central object self-irradiated destabilization Pringle and subsequent papers. Possible excitation mechanisms of various oscillation modes discussed here are briefly summarized in table 4 see section 9.
Before closing the main parts of this article, we discuss some issues which have not been considered so far. In addition to these kinds of trapped oscillations, we can expect another types of trapped oscillations, which are realized even in the Newtonian disks. Two possible examples are presented here:. This model says that in the stage of outbursts the dwarf nova disks, which are non-relativistic Keplerian, undergo limit-cycle oscillations because of the thermal instability.
This instability brings about a transition front separating hotter and colder disks. Eigenfunctions of this type of trapped oscillations were studied by Yamasaki et al.
Oscillations of Disks
Schematic picture showing a trapped oscillation on dwarf-novae accretion disks in the outburst stage. Because of these situations, the wave is trapped in the region hatched in the upper panel. After Yamasaki et al. The advection-dominated accretion flows around compact objects have been extensively studied in recent years, since such flows can explain well the spectra of the hard state of X-ray stars and of the low-luminosity galactic nuclei see a review by Narayan et al. The accretion disks generally cannot be ADAFs from their outermost region, unless they start from the beginning with a hot virial temperature.
There is, however, no widely recognized model of the transition region. One of the possible models is that the transition occurs in a radially narrow region Honma ; Manmoto, Kato In this transition model, the pressure decreases sharply inwards in a narrow transition region. This sharp pressure decrease is accompanied by a much sharper inward decrease in density, although the temperature increases inward.
One of important characteristics of this transition is that a super-Keplerian rotation is realized in the transition region Honma ; Abramowicz et al. In the ADAF region the disk rotation is sub-Keplerian, but it becomes super-Keplerian in the transition region and sharply decreases in the outer boundary of the transition region to join the Keplerian rotation in the SSD region. Since a super-Keplerian rotation changes to a Keplerian one in a narrow region, the specific angular momentum also sharply decreases outward in this narrow region.
An outward decrease of specific angular momentum leads to the Rayleigh instability of disks if the stabilizing effects due to the inhomogeneity of the region is negligible. In the present problem, however, the physical quantities, such as density, are sharply change in this region, which acts against the onset of the Rayleigh instability.
If this latter action against the Rayleigh instability is stronger, perturbations in the region are oscillations rather than growing motions. Furthermore, the frequency of the oscillations would be low if the above two actions acting in the opposite directions are close to each other and are almost cancelled out. An important point is that such low frequency oscillations will be trapped in the narrow transition region by the following reasons.
In the surrounding regions of the transition region i. This means that the waves are trapped in the narrow boundary region of the transition region. Hence, one may suppose that such trapped oscillations cannot cause luminosity variations of the observable amount. This is, however, not the case, since the transition region emits a comparable amount of radiation with the whole luminosity emitted in the SSD region Honma ; Kato, Nakamura ; Manmoto, Kato Finally, it is noted that the presence of trapped oscillations in the transition layer between ADAF and SSD is based on the assumption that the transition occurs in a sharp narrow region.
For this to be realized, a large turbulent conductivity is necessary. It is not yet clear whether such a large turbulent conductivity is expected in real disks cf.
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Abramowicz et al. The form of viscosity has crucial effects on excitation of oscillations. Roughly speaking, an increase decrease of the viscosity in the compressed expanded phase acts in favor of oscillation amplification. A diffusion-type viscous force, however, does not always represent real situations. This is because a basic assumption involved in the diffusion approximation is that the turbulence responds instantly to a change in the environment. In other words, any information is transported with an infinite speed. This unphysicalness becomes prominent as an violation of causality when the transonic flows beyond the inner edge of accretion flows are treated Pringle To resolve this problem, many attempts to improve the expression for the viscous force have been made Popham, Narayan ; Narayan ; Narayan et al.
When we consider the effects of viscous forces on oscillations, the drawback of the diffusion-type viscous force again becomes noticeable, although we adopted its expression so far.
In the opposite case, where the frequency of oscillations is higher than that of turbulent motions, the turbulence cannot respond instantly to a time change of the medium. In other words, we must consider a time lag of turbulence in response to the oscillations.
To study the effects of the time lag of the response of the turbulent viscous stress force on oscillatory motions, an evaluation of the turbulent viscous stress tensor based on a second-order closure modeling Kato a is appropriate, since the time lag in the response of the turbulence is taken into account automatically in the modeling.
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Kato b see also Yamasaki, Kato applied this seconder-order closure modeling of turbulence to stydying excitation of disk oscillations. The results actually show, as expected, that the time lag of the response of turbulence acts in a direction so as to stabilize the oscillations. As is well known, there is yet no established theory of turbulence describing inhomogeneous time-dependent turbulent flows.
Hence, our understanding on the excitation of oscillations on disks by turbulent viscosity is still far from a satisfactory stage.