"Foundations of Black Hole Accretion Disk Theory"
Marek A. Abramowicz and P. Chris Fragile 
1 Introduction
2 Three Destinations in Kerr’s Strong Gravity
2.1 The event horizon
2.2 The ergosphere
2.3 ISCO: the orbit of marginal stability
2.4 The Paczyński–Wiita potential
2.5 Summary: characteristic radii and frequencies
3 Matter Description: General Principles
3.1 The fluid part
3.2 The stress part
3.3 The Maxwell part
3.4 The radiation part
4 Thick Disks, Polish Doughnuts, & Magnetized Tori
4.1 Polish doughnuts
4.2 Magnetized Tori
5 Thin Disks
5.1 Equations in the Kerr geometry
5.2 The eigenvalue problem
5.3 Solutions: Shakura–Sunyaev & Novikov–Thorne
6 Slim Disks
7 Advection-Dominated Accretion Flows (ADAFs)
8 Stability
8.1 Hydrodynamic stability
8.2 Magneto-rotational instability (MRI)
8.3 Thermal and viscous instability
9 Oscillations
9.1 Dynamical oscillations of thick disks
9.2 Diskoseismology: oscillations of thin disks
10 Relativistic Jets
11 Numerical Simulations
11.1 Numerical techniques
11.2 Matter description in simulations
11.3 Polish doughnuts (thick) disks in simulations
11.4 Novikov–Thorne (thin) disks in simulations
11.5 ADAFs in simulations
11.6 Oscillations in simulations
11.7 Jets in simulations
11.8 Highly magnetized accretion in simulations
12 Selected Astrophysical Applications
12.1 Measurements of black-hole mass and spin
12.2 Black hole vs. neutron star accretion disks
12.3 Black-hole accretion disk spectral states
12.4 Quasi-Periodic Oscillations (QPOs)
12.5 The case of Sgr A*
13 Concluding Remarks

9 Oscillations

Even when analytic disk solutions are stable against finite perturbations, it is often the case that these perturbations will, nevertheless, excite oscillatory behavior. Oscillations are a common dynamical response in many fluid (and solid) bodies. Here we briefly explore the nature of oscillations in accretion disks. This topic is particularly relevant to understanding the physical mechanisms that may be behind quasi-periodic oscillations (or QPOs, which are discussed in Section 12.4).

There are a number of local restoring forces available in accretion disks to drive oscillations. Local pressure gradients can drive oscillations via sound waves. Buoyancy forces can act through gravity waves. The Coriolis force can operate through inertial waves. Surface waves can also exist, with the restoring force given by the local effective gravity.

Of particular interest are families of low order modes that may exist in various accretion geometries. Such modes will tend to have the largest amplitudes and produce more easily observed changes than their higher-order counterparts. Here we briefly review a couple relatively simple examples for the purpose of illustration. More details can be found in the references given.

9.1 Dynamical oscillations of thick disks

A complete analysis of the spectrum of modes in thick disks has not yet been done. Some progress has been made by considering the limiting case of a slender torus, where slender here means that the thickness of the torus is small compared to its radial separation from the central mass (i.e., the torus has a small cross-sectional area). In this limit, the complete set of modes have been determined for the case of constant specific angular momentum in a Newtonian gravitational potential [43]. A more general analysis of slender torus modes is given in [45Jump To The Next Citation Point].

Any finite, hydrodynamic flow orbiting a black hole, such as the Polish doughnuts described in Section 4, is susceptible to axisymmetric, incompressible modes corresponding to global oscillations at the radial (σ = ωr) and vertical (σ = ω 𝜃) epicyclic frequencies. Other accessible modes are found by solving the relativistic Papaloizou–Pringle equation [4]

1 { [ ]} ( ) ----1∕2- ∂i (− g)1∕2gijfn∂jW − m2g ϕϕ − 2m ωgtϕ + ω2gtt fnW (− g) (ut)2(ω − m Ω )2 n−1 = − --------2-------f W , (108 ) cs0
where the function f is defined by
P P0 1 [ ℰ2 ] --= ---f(r,𝜃), f (r,𝜃 ) = 1 − --2- -0(𝒰eff − 𝒰eff,0) + Φ , (109 ) ρ ρ0 ncs0 2
m is the azimuthal wave number, n is the polytropic index (assuming an equation of state of the form P = K ρ1+1∕n), c s0 is the sound speed at the pressure maximum r 0, and g is the determinant of the metric. The necessary boundary condition comes from requiring that the Lagrangian pressure perturbation vanish at the unperturbed surface (f = 0):
ΔP = (δP + ξα∇ P ) = 0, (110 ) α
where ξα is the Lagrangian displacement vector. Eqs. (108View Equation) and (110View Equation) describe a global eigenvalue problem for the modes of Polish doughnuts, with three characteristic frequencies: the radial epicyclic frequency ωr, the vertical epicyclic frequency ω𝜃, and the characteristic frequency of inertial modes κ.

In the slender-torus limit, one can write down analytic expressions for a few of the lowest order modes [45Jump To The Next Citation Point] besides just the radial and vertical epicyclic ones. In terms of local coordinates measured from the equilibrium point,

( r − r ) ( π∕2 − 𝜃 ) x ≡ g1rr∕20 ----0- and y ≡ g1𝜃∕𝜃20 -------- , (111 ) r0 r0
an eigenfunction of the form W = Axy, for some constant A, yields two modes with eigenfrequencies [45Jump To The Next Citation Point]
2 1-{ 2 2 [ 2 2 2 2 2]1∕2} ¯σ0 = 2 ω r + ω 𝜃 ± (ω r + ω𝜃) + 4 κ0ω𝜃 , (112 )
where ¯σ0 = σ0∕Ω0. The positive square root gives a surface gravity mode that has the appearance of a cross(×)-mode. The negative square root gives a purely incompressible inertial (c-) mode, whose poloidal velocity field represents a circulation around the pressure maximum.
View Image

Figure 13: Poloidal velocity fields (δux, δuy) of the lowest order, non-trivial thick disk modes. Image reproduced by permission from [45Jump To The Next Citation Point].

An eigenfunction of the form 2 2 W = A + Bx + Cy, with arbitrary constants A, B, C, also has two modes with eigenfrequencies [45Jump To The Next Citation Point]

{ 2 1 2 2 2 ¯σ 0 = --- (2n + 1)(ωr + ω 𝜃) − (n + 1)κ 0 (113 ) 2n } [ 2 2 2 2 2 2 2 2]1∕2 ± ((2n + 1)(ω 𝜃 − ωr) + (n + 1)κ 0) + 4(ω r − κ 0)ω 𝜃 .
The upper sign results in an acoustic mode with the velocity field of a breathing mode. This mode is comparable to the acoustic mode in the incompressible Newtonian limit for ℓ = const., while in the Keplerian limit, the mode frequency becomes that of a vertical acoustic wave. The lower sign corresponds to a gravity wave, with a velocity field reminiscent of a plus(+)-mode. The poloidal velocity fields of all these lowest order modes are illustrated in Figure 13View Image.

9.2 Diskoseismology: oscillations of thin disks

To analyze oscillations of thin disks, one can express the Eulerian perturbations of all physical quantities through a single function δW ∝ δP ∕ρ. Since the accretion disk is considered to be stationary and axisymmetric, the angular and time dependence are factored out as i(mϕ− σt) δW = W (r,z)e, where the eigenfrequency σ (r,z ) = ω − m Ω. Here it is useful to assume that the variation of the modes in the radial direction is much stronger than in the vertical direction, z = r cos𝜃. The resulting two (separated) differential equations for the functional amplitude W (r,z) = Wr (r)Wy (r,y) are given by [244Jump To The Next Citation Point, 311, 312]

[ ] ( ) d2Wr 1 d ( 2 2) dWr (ut)2grr ( 2 2) Ψ ----2-− --2----2-- --- ω − ω r -----+ -------- ω − ωr 1 − -2- Wr = 0, (114 ) dr (ω − ωr) dr dr cs &tidle;ω 2 d2Wy-- dWy-- [ 2 2 2 ] (1 − y ) dy2 − 2gy dy + 2g &tidle;ω y + Ψ (1 − y ) Wy = 0. (115 )
The radial eigenfunction, Wr, varies rapidly with r, while the vertical eigenfunction, Wy, varies slowly. Here y = (z∕H )[γg∕(γg − 1)]1∕2 is the re-scaled vertical coordinate, γg is the adiabatic index, &tidle;ω(r) = ωr ∕ω𝜃 is the ratio of the epicyclic frequencies from Section 2.3, Ψ is the eigenvalue of the (WKB) separation function, and g = (P ∕P )∕(ρ∕ρ ) c c, where P c and ρ c are the midplane values. The radial boundary conditions depend on the type of mode and its capture zone (see below). Oscillations in thin accretion disks, then, are described in terms of Ψ (r,σ), along with the angular, vertical, and radial mode numbers (number of nodes in the corresponding eigenfunction) m, j, and n, respectively. Modes oscillate in the radial range where
( ) 2 2 Ψ-- (ω − ωr) 1 − &tidle;ω2 > 0 (116 )
outside the inner radius, r > ri.

p-modes are inertial acoustic modes defined by Ψ < &tidle;ω2 and are trapped where ω2 > ω2r, which occurs in two zones. The inner p-modes are trapped between the inner disk edge and the inner “Lindblad” radius, i.e., ri < r < r−, where gas is accreted rapidly. The outer p-modes occur between the outer Lindblad radius and the outer edge of the disk, i.e., r+ < r < ro. The Lindblad radii, r− and r+, occur where ω = ωr. The outer p-modes are thought to be more consequential as they produce stronger luminosity modulations [233]. In the corotating frame these modes appear at frequencies slightly higher than the radial epicyclic frequency. Pressure is the main restoring force of p-modes.

g-modes are inertial gravity modes defined by Ψ > &tidle;ω2. They are trapped where ω2 < ω2r in the zone r− < r < r+ given by the radial dependence of ωr, i.e., g-modes are gravitationally captured in the cavity of the radial epicyclic frequency and are thus the most robust among the thin-disk modes. Since this is the region where the temperature of the disk peaks, g-modes are also expected to be most important observationally [244]. In the corotating frame these modes appear at low frequencies. Gravity is their main restoring force.

c-modes are corrugation modes defined by 2 Ψ = ω&tidle;. They are non-radial (m = 1) and vertically incompressible modes that appear near the inner disk edge and precess slowly around the rotational axis. These modes are controlled by the radial dependence of the vertical epicyclic frequency. In the corotating frame they appear at the highest frequencies.

All modes have frequencies ∝ 1∕M. Upon the introduction of a small viscosity (ν ∝ α, α ≪ 1), most of the modes grow on a dynamical timescale tdyn, such that the disk should become unstable. However, evidence for these modes has so far mostly been lacking in MRI turbulent simulations (see Section 11.6). This leaves their relevance in some doubt.

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