INSTABILITY AND TURBULENCE IN ACCRETION DISKS
JOHN F. HAWLEY AND STEVEN A. BALBUS
Dept. of Astronomy, University of Virginia
Charlottesville, VA 22903, USA
1.
Introduction
Accretion disks are believed to be the power sources for such diverse astronomical systems as X-ray binaries, quasars, active galaxies, and cataclysmic
variables. In each case, the luminosity emitted from the system comes from
the gravitational energy released as gas in the disk spirals down toward
the central compact object. The trajectory of an individual volume of gas
as it orbits is determined by its specic angular momentum, which will be
a conserved quantity of the motion in the absence of torques. Hence inspiral requires a torque. The most fundamental physical process governing
the structure and evolution of accretion disks, therefore, is the extraction
of angular momentum from the gas and its transfer outward through the
disk.
Recently, we have made substantial progress in understanding this angular momentum transport process. The key conceptual point is the recognition that a subthermal magnetic eld rapidly generates magnetohydrodynamic (MHD) turbulence via a simple linear instability. This turbulence
transports angular momentum outward through the disk, allowing accretion
to proceed. Although turbulence seems like a natural and straightforward
transport mechanism, it turns out that the magnetic elds are essential.
Not all forms of turbulence work. Purely hydrodynamic turbulence, in particular, fails to arise spontaneously in disks, is not self-sustaining (even if
initiated \by hand"), and does not produce sustained outward transport of
angular momentum.
In this paper we will briey review these results. In particular, we focus
on the role that numerical simulations have played in obtaining our conclusions. For a more complete review of this work see Balbus & Hawley
(1998).
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2.
JOHN F. HAWLEY AND STEVEN A. BALBUS
Hydrodynamic Turbulence
Although the angular momentum in a disk would be transported outward
by ordinary (molecular) viscosity, the viscosity in accretion disks is extremely small. The importance of viscosity is measured by the Reynolds
number, a dimensionless quantity given by the system's characteristic length
times velocity divided by the viscosity. Accretion disks are characterized by
a huge Reynolds number. Far from being regarded as an obstacle to angular
momentum transport, however, a disk's high Reynolds number has traditionally been seen as evidence in support of the presence of turbulence.
Turbulence is, after all, a characteristic of high Reynolds number terrestrial uids. This view goes back at least as far as the rst detailed disk
model developed by Crawford and Kraft (1956) who wrote that because
\the Reynolds number [of the gas] is extremely high, it is quite certain
that the gas is turbulent." The more cautious have long recognized that
even with a high Reynolds number, it is far from obvious that turbulence
develops in Keplerian disks.
Turbulence, even in a high Reynolds number ow, requires a source,
that is, an instability. It is not sucient that the viscosity be low enough to
allow for small-scale, disorganized velocity uctuations to persist. There has
to be something producing those velocity uctuations. The problem is that
the dierentially rotating ows in accretion disks are dynamically stable to
small perturbations according to the Rayleigh criterion (angular momentum must increase outward). This means that the forces that are present
in the disk (gravitational, centrifugal, and Coriolis) do not amplify small
(linear) velocity uctuations. Despite this, people clung to the possibility
that nonlinear (large amplitude) instabilities might develop and produce
self-sustaining turbulence. It is known from laboratory experiments, for
example, that simple shear ows break down into turbulence without the
presence of a linear instability. Many felt that accretion disks should behave
in the same way.
This would seem like an ideal question to put to a numerical test: do
accretion disk ows spontaneously break down into turbulence due to nonlinear instabilities? There are several important numerical issues to bear in
mind before proceeding. First, the question is inherently three-dimensional
(3D), since the known nonlinear shear ow instabilities are 3D. Second,
because the Reynolds number in disks is so high, there is a huge disparity
between the macroscopic lengths in the disk (e.g. the scale height) and the
(physical) viscous dissipation length. If this full range of scales must be
numerically resolved, one would need an enormous number of grid zones.
As it turns out, although the disk stability problem is 3D, it is not
necessary to evolve the full global, large-scale accretion disk. The dynamics
189
Instability and Turbulence in Accretion Disks
φ
R
Accretion Disk
Z
Central Mass
Z
Shearing
Box
Y
X
Figure 1.
The local shearing box model. A small region of the full global accretion disk
is mapped to a small volume that is locally Cartesian in geometry but not in dynamics:
the gravitational tidal eld and Coriolis forces are retained. The center of the box is
considered to orbit with frequency . Vertical gravity can be included to model a stratied
local disk section.
of a dierentially rotating uid can more easily be studied in a local shearing
box system (see Figure 1 and the description in Hawley, Gammie & Balbus
1995). The local model preserves the orbital dynamics while using a small,
well-resolved, locally geometrically Cartesian computational domain. Longterm evolution is made possible by the use of shearing-periodic boundary
conditions in the radial direction. With these boundary conditions, the
radial direction is strictly periodic at t = 0, but a relative shear is applied
as time advances. As a uid element exits from the outer radial boundary
(for example), it reappears at the inner boundary at the appropriate sheared
location with its angular velocity adjusted to account for the mean shear
across the box.
Even with a local model, just how much resolution is required to search
for nonlinear instabilities? Since these instabilities, if they exist, are dynamically, not viscously, driven it is not necessary to resolve all lengths
down to the viscous damping scale. Instead, one evolves the inviscid Euler equations with enough resolution to capture a representative range of
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JOHN F. HAWLEY AND STEVEN A. BALBUS
dynamical lengths. By perturbing the computational domain with large
amplitude perturbations over the full range of length scales, one should
be able to observe the growth of any reasonably rapidly growing unstable
modes, linear or nonlinear.
Balbus, Hawley & Stone (1996) carried out just such a series of local hydrodynamic simulations. The simulation results were understood through
a moment analysis of the dynamical equations. The analysis, in turn, suggested further numerical experiments and additional ow quantities to measure. Simulation and analysis work hand in hand.
Consider an initial uniform Keplerian ow, perturbed with large amplitude velocity uctuations and evolved forward in time. Stability is determined by whether these uctuations grow or decay with time. Balbus et al.
found that for all Keplerian simulations the initial perturbations rapidly
die out and the system returns to a simple smooth dierential rotation.
As an important control on the results, Balbus et al. also carried out simulations of simple shear ows. These ows become unstable and develop
turbulence, in stark contrast to the Keplerian results. Since the Keplerian
system diers from the simple shearing system only by the addition of orbital dynamical forces, the absence of instabilities in Keplerian ows cannot
be due to numerical eects associated with the use of a nite number of
grid zones. Indeed, there is nothing to set a distinct physical scale to differentiate the two systems. If the shear ow develops turbulence then the
Keplerian system should as well, if turbulence were the correct physical
outcome.
This conclusion is supported by another experiment which retains the
orbital dynamics but assumes that the background shear corresponds to a
constant angular momentum distribution. Such a disk is marginally stable
to the linear Rayleigh criterion: the epicyclic frequency is zero. By symmetry the constant angular momentum system should be equivalent to the
simple shear ow; in the local analysis their moment equations are formally equivalent. This implies that the constant angular momentum system should also develop the same type nonlinear instabilities that aict
simple shear ows. Simulations conrm this expectation, and provide further evidence that the stability of Keplerian ow is due to dynamical, not
numerical, eects.
Another way to check the validity of a numerical simulation (and the
conclusions that follow from it) is by increasing the grid resolution. We have
recently carried out a resolution study using grids ranging from a coarse
163 zones up to 2563 zones. The evolution of the energy in the angular
velocity perturbations for the dierent resolutions is displayed in Figure
2. Interestingly, the greater the resolution, the more rapid the decline in
energy. While turbulence is present, the higher resolution simulations have
Instability and Turbulence in Accretion Disks
191
Figure 2.
The time evolution of the kinetic energy in angular velocity uctuations for
hydrodynamic turbulence in Keplerian disk, in units of (L
)2 where L is the size of the
computational domain. The curves correspond to dierent resolutions for the same initial
conditions. They are labeled by the number of grid zones used. Instead of promoting
greater turbulence, higher resolution results in more rapid decline in the vy uctuation
energy.
larger positive values of the Reynolds stress. The Reynolds stress transports
angular momentum outward, but it also acts as a sink for energy in the
angular velocity uctuations (see Balbus et al. 1996). This appears to hasten
the damping of the turbulence compared to less well-resolved simulations.
From the resolution experiments, from the direct comparison with the
simple shear ow, and from the behavior of the constant angular momentum
system, we conclude that Keplerian ows are hydrodynamically stable to
local perturbations of any reasonable amplitude.
3.
Magnetohydrodynamical Turbulence
If the source of the angular momentum transport in Keplerian disks does
not arise from simple hydrodynamical turbulence, we must look elsewhere.
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JOHN F. HAWLEY AND STEVEN A. BALBUS
Because accretion disks are generally suciently hot to be fully ionized,
they are highly conducting. Such hot, ionized plasmas are quite capable
of supporting the currents that create magnetic elds. Therefore, magnetic
forces are a good candidate to transport angular momentum. The idea is a
venerable one: Lynden-Bell's pioneering (1969) quasar accretion disk model
favored transport by magnetic stresses. What prevented a wide consensus
in favor of magnetic transport was the perception that the eld could not
be important unless raised to an energy comparable to the disk's thermal
energy. Again, it was assumed, it would be necessary to demonstrate that
magnetic elds are amplied to signicant strengths.
As it turns out, the presence of even a weak magnetic eld fundamentally alters the stability of orbiting ow. Balbus & Hawley (1991) demonstrated through a linear analysis that a Keplerian ow is linearly unstable in
the presence of a weak magnetic eld. (The earliest version of this instability
was uncovered by Velikhov [1959] in a study of magnetized Couette ows.)
The disk stability criterion is quite simple: weakly magnetized disks are dynamically unstable if the angular velocity decreases outwards, a condition
that always holds for Keplerian orbits. This magnetorotational instability
means that disk turbulence arises naturally, just as it does in other circumstances, such as the convective turbulence that develops in the outer layers
of low mass stars. In both the star and the disk, turbulence develops when
a simple local linear stability criterion is violated.
Since the elucidation of the linear instability, numerical simulations have
played an essential role in illuminating the instability's physical nature and
nonlinear consequences. Again the local shearing box model has proven
invaluable for these investigations. Hawley, Gammie & Balbus (1995) and
Matsumoto & Tajima (1995) examined the evolution of uniform initial vertical and toroidal elds. Hawley, Gammie & Balbus (1996) simulated models
with random initial elds. The general outcome of all such simulations is
MHD turbulence and vigorous transport of angular momentum.
What limits the growth of the instability and sets the level of the turbulence (and hence angular momentum transport)? In a disk there are (at
least) two principal eects which limit the eld strength: local dissipation
(i.e. through reconnection and resistivity), and loss of ux from the disk
through buoyancy.
In the local models without vertical stratication, dissipation and the
nite size of the computational domain are the limiting mechanisms. In a
numerical simulation, the primary dissipation process is numerical: elds reconnect at the grid scale. To some extent this numerical eect must mimic
the physical one; reconnection can occur when oppositely directed elds
are brought close enough together. But nite resolution certainly exaggerates its ecacy. To gauge such numerical inuences we have run many
Instability and Turbulence in Accretion Disks
193
Maxwell Stress (as a fraction of the thermal energy) for a series of identical
models at dierent grid resolutions. The curves are labeled by the number of grid zones
in the x direction.
Figure 3.
tests including resolution studies. Here we present a recent example. Consider a shearing box simulation initialized with a vertical eld that varies
sinusoidally with radius; there is no net eld piercing the computational
domain. This initial conguration is given a long-wavelength spectrum of
velocity perturbations (identical for all resolutions) and evolved forward in
time. The time evolution of the Maxwell stress in the dierent simulations
is shown in Figure 3. Although the peak at the end of the linear growth
phase shows a signicant increase with resolution, the dierences are much
smaller once turbulence sets in. This is because the magnetic stresses are
dominated by the largest scales in the box. These scales are reasonably well
resolved by all but the lowest resolution model.
The issue of eld loss due to buoyancy can be investigated with models
that include the vertical gravity. Local models with vertical stratication
have been done by Brandenburg et al. (1995) and Stone et al. (1996). An
example of a vertically stratied disk simulation is included as an MPEG
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JOHN F. HAWLEY AND STEVEN A. BALBUS
movie in the CD accompanying this volume. In these simulations we nd
that while magnetic ux is buoyantly lifted out of the disk, the rate at
which this occurs is limited to about the vertical sound crossing time. The
magnetorotational instability operates on shorter time scales and eld is
regenerated faster than it is lost due to buoyancy.
4.
Conclusions
It now seems clear from the work done to date that accretion disks are
fundamentally MHD systems. Further insights into the consequences of the
magnetorotational instability and the nature of the resulting turbulence will
require increasingly detailed numerical work. There are many potential avenues of investigation to pursue. It is important to develop full-scale global
simulations in three dimensions. Although such numerical simulations are
technically quite challenging, the diculties are slowly being overcome. An
example of such a global model is presented by R. Matsumoto elsewhere in
these proceedings. We also need to increase the physical complexity used
in disk simulations, rst in the local disk domains, and eventually in the
global models. Examples include radiation transport, partial ionization, resistivity and reconnection, and improved equations of state. Of course these
goals (increased resolution, increased physical content) apply across the full
range of astrophysical problems that are being addressed with numerical
simulations. The proceedings of this workshop provide several examples
of progress in all these areas, both for disks and for other astrophysical
systems.
Acknowledgements
This work is supported in part by NASA grants NAG-53058, NAG5-4600,
and by NSF grant AST-9423187. The computations were carried out on
the Cray C90 and T3E systems of the Pittsburgh Supercomputing Center.
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