Dynamos in Accretion Disks:

A general review and some moderately biased comments

Chicago - October 2003

What is an Accretion Disk?

• Flattened, rotionally supported, gas accreting onto a central object over the course of many rotation periods.

• A good fluid, i.e. a high collision rate and a short mean free path.

• Negligible gravity due to disk. (ignoring galaxies, outer parts of AGN disks)

• Not necessarily ionized, but here we will discuss only good conductors. (ignoring intermediate radii in protostellar disks)

Original theoretical description by Shakura and Sunyaev

A Theoretical Cartoon:

Disk Plasma - collisional, rotationally supported - resistive MHD

Central Object - source of all gravity

Corona/Outflow -collisionless, hot, nonthermal emitter

Jet - very fast, very low density

An Observational Perspective:

• Broad spectral energy distribution.

• Power law emission at high frequencies (X-rays).

• Radio emission seen in some cases, associated with jets.

• Conspicuous emission lines in conjunction with optically thick continuum.

Accretion Disks are Ubiquitous

• Protostars - accretion from environment- winds & jets.

• Cataclysmic variables (white dwarf accretors) - accretion from binary companion - winds.

• Active Galactic Nuclei (supermassive black holes) - accretion from environment - winds & jets.

• Galactic black holes/neutron stars - accretion from companion - winds & jets.

Viscosity and Other Fictions

• Accretion disks rotate differentially, with the angular frequency decreasing outwards.

• Consequently, friction between annuli will automatically transfer angular momentum outward.

• Real viscosity is orders of magnitude too small in real accretion disks. (We can estimate from cataclysmic variable systems.)

• Shakura and Sunyaev (1973) proposed a useful parameterization, inspired by expectations of turbulence.

Magnetic field tension?

Time Scales

Dynamic rates (for vertical pressure balance etc.)

Thermal rates (for vertical heat transfer)

Infall time

The Physics of Angular Momentum Transport• Originally (1973) it was assumed that accretion disks

would be violently, hydrodynamically unstable, giving rise to an of order unity. After 30 years it has become apparent that any hydrodynamical instability is weak, and may not be present at all.

• The evidence for a substantially larger in ionized disks led to a widespread belief that in such disks the transport of angular momentum is driven by magnetic fields, which are created in the disk by some dynamo process.

• In the last decade this belief has become the community consensus, based on the existence of magnetic field instability which acts to transport angular momentum outward (Balbus and Hawley 1991 and many papers thereafter).

The Disk Dynamo: Generating B

• An accretion disk is a standard example of an “-” dynamo in which the large scale radial field is stretched to produce an azimuthal field

• The radial field must be produced from eddy-scale motions acting on the azimuthal field, typically written as

The usual treatment involves keeping only vertical gradients and concentrating on

The Magneto-Rotational Instability

• Radial ripples in an azimuthal or vertical magnetic field embedded in an accretion disk will be stretched by the differential rotation of the disk. This stretching will have the effect of adding angular momentum to segments that are displaced outward and substracting it from line segments that are displaced inward. (Like the tethered satellite experiment, only it works.)

• This torque will reinforce the outward (or inward motion) of the perturbed field line segments, leading to an instability with a growth rate

Velikhov (1959), Chandrasekhar (1961), Balbus and Hawley (1991)

Simulations of the MRI

• This MRI has been simulated in `shearing box’ and `global’ 3D simulations by a variety of groups (Hawley, Stone, Matsumoto, Brandenberg)

• In three dimensions the process saturates in a turbulent state with and somewhat larger than

• The angular momentum transport is mostly mediated by the magnetic field

• The local value of is typically 0.01 or less near the disk mid-plane (or everywhere in simulations with no vertical gravity). However, the average magnetic stress is almost uniform over a few density scale heights, so that the vertically averaged is probably close to 0.1. The mechanism for saturation is not well understood.

• The magnetization of the corona has not been properly simulated.

• The simulations all produce “large scale” magnetic fields in addition to turbulent components, which are more or less axisymmetric.

• The simulations are all “toy models” with very simple models for the disk plasma. The hope is that if can understand the dynamo process and the MRI instability in detail, we won’t need to run some very large number of incredibly detailed numerical simulations. (This isn’t a realistic goal anyhow.)

There may be a few problems…..

• The (very) large literature on this topic assumes various kinds of turbulent motions in the disk, but not the MRI, which is the only instability we are sure exists in disks.

• The kinematic dynamo, in which the evolution of the large scale magnetic field proceeds with some given form for the kinetic helicity tensor, is known to be subject to very large corrections, even when the large scale magnetic field is still small (Cattaneo and Vainshtein; Hughes; Gruzinov and Diamond).

• The dynamo equations assume, in effect, that reconnection is efficient. In an accretion disk (or in a star) we are dealing with a collisionally dominated fluid, where resistive MHD ought to be a good description of the magnetic field dynamics.

To which people have a proposed a few solutions…

• There are no accretion disk dynamos. (Of course, then there are no stellar dynamos either.)

• Reconnection is efficient in disks (and stars) or else unnecessary.

• The objections to conventional mean-field dynamo theory are mistaken. (Of course, the objections are supported by computational simulations. Conventional mean-field dynamo theory is not.)

• Mean-field dynamo theory can be recast in a form which survives the criticisms.

• Mean-field dynamo theory is unnecessary.

What’s wrong with the - dynamo?

In order for the azimuthal field component to produce a radial field component we need three things:

1. The turbulence has to deform the azimuthal field lines into spirals with same helicity (locally).

2. The must be a vertical gradient in the strength of the spirals.

3. The spirals must reconnect, so that the gradient in their amplitude produces a net radial field component.

As far as I know, no one has a problem with the second item.

Making Spirals?• Taking a straight field line and deforming it into a spiral

creates a magnetic helicity on the field line:

• However, in the context of ideal MHD, magnetic helicity is strictly conserved. As long as reconnection takes place only on 2D surfaces and the turbulent magnetic field power spectrum falls off at large wavenumber, it remains a good conserved quantity.

These last two conditions are equivalent. The coulomb gauge turns out to be uniquely useful here.

The Inverse Cascade of Magnetic Helicity

• If we average over eddy scales, then the magnetic helicity becomes

• The evolution of H is given by

• In the absence of any magnetic helicity current, the dynamo can only work by creating equal and opposite amounts of magnetic helicity on large and small scales, limiting the large scale magnetic energy to the ratio of the eddy scale to the large scale, times the small scale magnetic field energy.

• Even this is unrealistic. The magnetic helicity h will interact coherently with the large scale field, inducing motions (through the current helicity ) which will transform h into H.

This is the step that makes the Coulomb gauge the natural choice, since it ties the current and magnetic helicities together.

A successful dynamo requires a systematic magnetic helicity current, driving local accumulations of “h”, which then drives the dynamo through the nonlinear inverse cascade.

The Eddy-Scale Magnetic Helicity Current

The eddy scale magnetic helicity current can be calculated explicitly. It is

If we make the approximation that the inverse cascade is faster than anything else, we have

Here sigma is the symmetrized large scale shear tensor. This current will be zero in perfectly symmetric turbulence. However, if we have symmetry breaking in the radial and azimuthal directions (due to differential rotation) then it will be non-zero, despite the vertical symmetry.

or

Bhattacharjee and Hameiri 1986; Kleeorin, Moss, Rogachevskii and Sokoloff,2000; Vishniac and Cho 2001

A Toy Model of the Accretion Disk Dynamo

• In order to see what this does to the accretion disk dynamo, we need to plug in the correlations expected from the MRI. If we use quasilinear theory as a guide, then we find that the overall sense of the magnetic helicity current is that it is aligned with the rotation axis has a sign given by

• It points up for an accretion disk and down for the Sun.

• If we treat the accretion disk as a periodic shearing box, then this gives us a dynamo growth rate of

• The existence of a dynamo depends on the sign of the magnetic helicity current. If it ran the other way the dynamo would be suppressed (as it is in magnetic Kelvin-Helmholtz simulations).

• The implication is that the dynamo field always grows on a time scale of a few eddy turn over times, and is always dominated by scales which are a few eddy scales in size (vertically). This is what is seen in the MRI simulations.

• The sliding scale of the turbulence ensures that a nonlinear dynamo mechanism is always viable and there is no kinematic dynamo regime. This is not true for stars where the scale of the turbulence is set by convective instability.

Towards More Realistic Models

• Real disks are vertically stratified, which will affect the properties of the MRI.

• We need to look at the interplay between radiative and convective transport of energy in an accretion disk and the behavior and structure of the dynamo field.

• The dynamo field will probably vary in time, and the effects of this on disk structure are largely unknown. The time scale for variations, the dynamo time, is ~ the disk thermal time.

• We expect, based on the simulations done to date, that the time averaged Shakura-Sunyaev “” will not be a constant but will vary with the distance from the midplane. (The simulations suggest that “P” is close to constant.)

The Environment of Accretion Disks: Jets and Coronae

• In real disks, as opposed to periodic simulations, the magnetic helicity current will emerge from the disk photosphere.

• Since the scale of the helicity current will change from <h at the photosphere to ~r at the edge of the corona/wind, it will shed most of its energy, amounting to about of the disk energy budget, in the corona. This is similar to a popular explanation for the solar corona, although the disk can also inject additional material from runaway heating. It may also run an additional dynamo in the corona. (The MRI persists in collisionless plasmas.)

• Finally, will the corona build up a large scale organized field? (Simulations are not helpful because they do not conserve magnetic helicity in this part of the grid.)

• How does all of this lead to jets? Why aren’t they universal?

Beyond Mean-Field Dynamo Theory

This is two scale dynamo theory, but the scale separation in accretion disks is only by a factor of several. We need to incorporate these considerations into a more general treatment, that allows for a continuous range of magnetic field scales from large scales down to eddy sizes.

Computational Projects and Problems

• The theory that driven magnetic helicity currents are responsible for fast dynamo activity needs to be tested. The simplest way to do this is to revisit the Kelvin-Helmholtz problem, but to add additional driving which is designed to bias the magnetic helicity current so that it has the correct sign.

• The single greatest problem with the current generation of numerical simulations is that only spectral codes do a good job of conserving magnetic helicity throughout the grid. Grids with variable spacing tend to lose magnetic helicity.

• The ejection of magnetic helicity (and flux) from a disk seems to happen in simulations, but it’s not clear if this process is being modelled adequately.

Summary

• Accretion disk dynamos are unique in that simulations naturally yield large scale magnetic fields. This is because the crucial physical ingredients are a strong shear and a particular magnetic field instability.

• While kinematic dynamo theory does not lead to a useful understanding of the simulations, they are consistent with a modified version of mean-field dynamo theory, which depends on a systematic, eddy-scale, magnetic helicity current.