Abstract
For disk oscillations to be observed, excitation processes are necessary. Otherwise, they will be damped with time by dissipative processes which usually act so as to dampen oscillations. It is well-known, however, that in thermally non-equilibrium open systems dissipative processes can often excite oscillations. The excitation of stellar pulsation by non-adiabatic processes is one of these typical examples in classical astrophysics. Three typical excitation processes are known in stellar pulsation, which are κ-, ε-, and δ-mechanisms (see Unno et al. (Nonradial oscillations of stars. University of Tokyo Press, Tokyo, 1989) for details of stellar pulsation theory). The above excitation mechanisms of oscillations can also operate in accretion disks, if favorable situations are realized. In case of accretion disks, however, another important mechanism exists, which does not work in stellar pulsation. This is a process due to angular momentum transport in disks. In accretion disks angular momentum is transported outward by viscous processes. If this outward angular momentum flow is modulated by oscillations, it can work so as to excite the oscillations when the modulation occurs in a phase relevant to excitation of oscillations.
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Notes
- 1.
It is noted that
$$\displaystyle\begin{array}{rcl} & & \delta \biggr (\frac{1} {\rho } \nabla p\biggr ) = - \frac{\delta \rho } {\rho _{0}^{2}}\nabla p_{0} + \frac{1} {\rho _{0}}[\nabla p_{1} + (\boldsymbol{\xi }\cdot \nabla )\nabla p_{0}] {}\\ & & = - \frac{\delta \rho } {\rho _{0}^{2}}\nabla p_{0} + \frac{1} {\rho _{0}}[\nabla (\delta p) -\nabla (\boldsymbol{\xi }\cdot \nabla )p_{0} + (\boldsymbol{\xi }\cdot \nabla )\nabla p_{0}]. {}\\ \end{array}$$If δ p in the above equation is written as δ p = Γ 1(p 0∕ρ 0)δ + (δ p)na, we have ρ 0 δ[(1∕ρ 0)∇p] = ∇(δ p)na+ parts of \(\mathcal{L}(\boldsymbol{\xi })\).
- 2.
Non-adiabatic and viscous processes introduce not only an imaginary part of frequencies but also a slight change of the real part. By neglecting the latter change, however, we regard ω 0 as the frequency of non-adiabatic and inviscid oscillations.
- 3.
The expression for wave energy is now slightly modified by the presence of accretion flows, but it is negligible.
- 4.
The formula
$$\displaystyle{\mathfrak{R}(A)\mathfrak{R}(B) = \frac{1} {2}\mathfrak{R}[AB + AB^{{\ast}}] = \frac{1} {2}\mathfrak{R}[AB + A^{{\ast}}B]}$$is used, where A and B are complex variables and B ∗ is the complex conjugate of B.
- 5.
The growth rate of oscillations comes from both non-adiabatic and viscous processes. Hereafter, however, −ω i is used to represent the growth rate due to viscous process, in order to avoid many subscripts.
- 6.
It is noted that in the case where the force acting on perturbations in the azimuthal direction is neglected, \(\breve{u}_{\varphi }\) and \(\breve{u}_{r}\) are related by
$$\displaystyle{i(\omega -m\varOmega )\breve{u}_{\varphi } + \frac{\kappa ^{2}} {2\varOmega }\breve{u}_{r} = 0,}$$but \(\breve{\xi }_{\varphi }\) and \(\breve{\xi }_{r}\) are related by (see equation (3.13))
$$\displaystyle{i(\omega -m\varOmega )\breve{\xi }_{\varphi } + 2\varOmega \breve{\xi }_{r} = 0.}$$ - 7.
In the expression for (−ω i)shear given by equation (9.45), η 1 does not appear explicitly, because η 1 is written as η 1 = η 0 ρ 1∕ρ 0.
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Kato, S. (2016). Overstability of Oscillations by Viscosity. In: Oscillations of Disks. Astrophysics and Space Science Library, vol 437. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56208-5_9
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