Stability

8 Stability

Having reviewed some of the main analytic models of accretion disks, it is important now to discuss the issue of stability. Since all analytic models presume steady-state solutions, such models are only useful if the resulting solutions are stable. One reason to suspect accretion disks may not be stable is that the systematic differential rotation that is a signature feature of accretion is a potential source of energy, and therefore, of instability. Another is that some level of instability may be essential in accretion disks as it can provide a pathway to the kind of sustained turbulence anticipated by Shakura and Sunyaev (see Section 3.2.1).

8.1 Hydrodynamic stability

Within ideal hydrodynamics, local linear stability of an axisymmetric rotating flow is guaranteed if the Høiland criterion is satisfied [302 ]:

2 -1-∂-ℓ-− -1--∇p ⋅ ∇S >0, (102a ) R3 ∂R Cpρ ( 2 2 ) ∂p- ∂-ℓ-∂S-− ∂ℓ--∂S- <0, (102b ) ∂z ∂R ∂z ∂z ∂R

where $Cp$ is the specific heat at constant pressure and $R$ is the cylindrical radius (see [275 ] for the criterion for relativistic stars). This criterion can be easily understood in two limits: For non-rotating equilibria (e.g., a non-rotating star), the criterion reduces to the Schwarzschild criterion ( $∂S ∕∂r > 0$ ) that the entropy must not increase toward the interior (for stability against convection). Provided this is true, local fluid elements will simply oscillate under stable buoyancy forces. To see the effects of rotation, we can consider an equilibrium that has constant entropy everywhere. Then the Høiland criterion reduces to the Rayleigh criterion ( $dℓ2∕dR > 0$ ): the specific angular momentum must not decrease outward. Physically, if one perturbs a fluid element radially outward, it conserves its own specific angular momentum. If the ambient specific angular momentum decreases outward, then the fluid element will be rotating too fast to stay in its new position, and centrifugal forces will push it further outward. Stability would be a fluid element that oscillates at the local epicyclic frequency.

As it turns out, the Høiland criterion is a huge disappointment for understanding why turbulence might exist in accretion disks. This is because it indicates that accretion disks with rotation profiles that do not differ too much from Keplerian should be strongly stable!

8.1.1 Papaloizou–Pringle Instability (PPI)

The Høiland criterion is only a local stability criterion. Flows can be locally stable, yet have global instabilities. An example of this occurs in the Polish doughnut solution (Section 4). Papaloizou and Pringle [238 ] showed that this solution is marginally stable with respect to local axisymmetric perturbations yet unstable to low-order nonaxisymmetric modes. As with all global instabilities, the existence of the Papaloizou–Pringle instability (PPI) is sensitive to the assumed boundary conditions [44 ]. In cases where the disk overflows its potential barrier (Roche lobe) and accretes through pressure-gradient forces across the cusp, the PPI is generally suppressed [117 ].

8.1.2 Runaway instability

Another instability associated with the Polish doughnut is the runaway instability [5 ]. If matter is overflowing its Roche lobe and accreting onto the black hole, then one of two evolutionary tracks are possible: (i) As the disk loses material it contracts inside its Roche lobe, slowing the mass transfer and resulting in a stable situation, or (ii) as the black hole mass grows, the cusp moves deeper inside the disk, causing the mass transfer to speed up, leading to the runaway instability. Recent numerical simulations show that, while this instability grows very fast, on timescales of a few orbital periods, over a wide range of disk-to-black hole mass ratios when $ℓ = const.$ , i.e., a constant specific angular momentum profile [98 ], it is strongly suppressed whenever the specific angular momentum of the disk increases with the radial distance as a power law, $ℓ ∝ rp$ [63 ]. Even values of $p$ much smaller than the Keplerian limit ( $p = 1∕2$ ) suffice to suppress this particular instability. [This is equivalent to angular velocity profiles, $Ω ∝ r−q$ , with $q > 3∕2$ .]

8.2 Magneto-rotational instability (MRI)

Although it had long been suspected that some sort of MHD instability might provide the necessary turbulent stresses to make accretion work, the nature of this instability remained a mystery until the rediscovery of the magneto-rotational instability by Balbus and Hawley [26 , 118 , 27 ]. Originally discovered by Velikhov [309 ], and generalized by Chandrasekhar [58 ], in the context of vertically magnetized Couette flow between differentially rotating cylinders, the application of this instability to accretion disks was originally missed.

The instability itself can be understood through a simple mechanical model. Consider two particles of gas connected by a magnetic field line. Arrange the particles such that they are initially located at the same cylindrical distance from the black hole but with some vertical separation. Give one of the particles (say the upper one) a small amount of extra angular momentum, while simultaneously taking away a small amount of angular momentum from the lower one. The upper particle now has too much angular momentum to stay where it is and moves outward to a new radius. The lower particle experiences the opposite behavior and moves to a smaller radius. In the usual case where the angular velocity of the flow drops off with radius, the upper particle will now be orbiting slower than the lower one. Since these two particles are connected by a magnetic field line, the differing orbital speeds mean the field line will get stretched. The additional tension coming from the stretching of the field line provides a torque, which transfers angular momentum from the lower particle to the upper one. This just reinforces the initial perturbation, so the separation grows and angular momentum transfer is enhanced. This is the fundamental nature of the instability.

In more concrete terms, consider a disk threaded with a vertical magnetic field $Bz$ and having an Alfvén speed $2 2 vA = B z∕(4π ρ)$ . The dispersion relation for perturbations of a fluid quantity $δX ∼ exp [i(kz − ωt )]$ is [27 ]

This equation has an unstable solution ( $ω2 < 0$ ), if and only if, $kvA + rdΩ2∕dr < 0$ . Since

in accretion disks, the instability criteria can generally be met for weakly magnetized disks. More specifically, the MRI exists for intermediate magnetic field strengths. In terms of the natural length scale of the instability ( $∼ vA∕Ω$ ), the field strength is constrained at the upper limit by the requirement that the unstable MRI wavelength fit inside the vertical thickness of the disk ( $vA∕Ω ≲ H$ ). This corresponds to field energy densities that are less than the thermal pressure, i.e., $2 b < Pgas$ . At the lower end, dissipative processes set a floor on the relevant length scales, and hence, field strengths.

If the conditions for the instability are met, the fastest-growing mode, which dominates the early evolution, has the form of a “channel flow” involving alternating layers of inward- and outward-moving fluid. The amplitude of this solution grows exponentially until it becomes unstable to three-dimensional “parasitic modes” that feed off the gradients of velocity and magnetic field provided by the channel flow. The flow rapidly reaches a state of magnetohydrodynamic turbulence [118 , 119 ]. This instability can be self-sustaining through a nonlinear dynamo process [52 ] – nonlinear because the motion that sustains or amplifies the magnetic field is driven by the field itself through the MRI. A more complete description of the linear and non-linear evolution of the MRI is provided in the review article by Balbus and Hawley [27 ]. A general relativistic linear analysis is presented in [20 ].

8.3 Thermal and viscous instability

It was realized by Shakura and Sunyaev themselves [280 ], as well as other authors [176 , 287 ], that the Shakura–Sunyaev solution (Section 5.3) should be thermally and viscously unstable for disks in which radiation pressure dominates (when the opacity is governed by electron scattering). The most general and elegant arguments are presented in the classic paper by Piran [246 ]. This discovery started a long debate, which continues unresolved to this day. A recent update is provided in [60 ].

To understand the thermal instability better, we consider a disk cooling through radiative diffusion. The local emergent flux at radius $r$ is given by

where $a$ is the radiation density constant, $T$ is a measure of the disk interior temperature, and $τ$ is half the total vertical optical depth. If the opacity is dominated by electron scattering, then $τ ∼ κTΣ ∕2$ , where $Σ$ is the surface density of the disk, which is constant on the time scales of the instabilities (very much less than the radial flow time scale). The Thomson opacity $κ T$ is also constant, being independent of temperature, provided there is already sufficient ionization. Therefore, the optical depth is independent of temperature and the cooling rate per unit area is $F − ∝ T 4$ .

The dissipation rate per unit area is

Vertical hydrostatic equilibrium implies that the disk half thickness $H ∼ 2P ∕(Ω2K Σ )$ , so that Eq. (106

) becomes

In the radiation pressure dominated inner region, $P ≃ aT4∕3$ , so that Eq. (107

) plus the Shakura–Sunyaev assumption $𝒯rϕ = − αP$ imply that $F + ∝ T 8$ ! Hence a perturbative increase in temperature increases both the local cooling and heating rates, but the heating rate increases much faster, leading to a thermal runaway.

Note, though, that this argument only applies when the viscous stress is proportional to the total pressure $PTot$ ( $α$ being the proportionality constant). For some time it seemed that a plausible way to avoid this instability was to argue that the stress is proportional instead to the gas pressure $P gas$ . Recent numerical simulations, though, of the magneto-rotational instability in radiation-pressure dominated disks have shown that the stress is, in fact, proportional to the total pressure [128 ]. Interestingly, these simulations exhibit no sign of the predicted thermal instability.

Most observations also argue against the existence of this instability. In the case of accretion onto black holes, the instability is supposed to set in for luminosities in excess of $L > 0.01L Edd$ . However, during outbursts, many stellar-mass black hole sources cross this limit both during their rise to peak luminosity and on their decline to quiescence, showing no dramatic symptoms (although they do undergo state changes, as described in Section 12.3). On the contrary, observations suggest that disks in black hole X-ray binaries are stable up to at least $L ≈ 0.5LEdd$ [80 ]. Certainly there is no evidence for the sensational behavior anticipated by some models [172 , 300 ].

	Abstract
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
	Acknowledgements
	References
	Footnotes
	Figures
	Tables