The magneto-rotational instability and MHD turbulence

Until the early 1990s, there was no clear candidate for the actual angular momentum transport mechanism in black hole (or, indeed, neutron star, white dwarf or protostellar) accretion disks -- normal atomic/molecular viscosity turns out to be orders of magnitude too small to drive the accretion that was presumed to cause the prodigious radiative powers observed from such sources. Detailed theoretical studies of accretion disks were plagued by our lack of understanding of the basic angular momentum transport mechanism. However, in 1991, Steve Balbus and John Hawley [118] showed that the presence of a (arbitrarily) weak magnetic field in the accretion disk material will render it susceptible to a powerful local MHD shear instability. Assuming that the disk is in the MHD limit, they showed that the only criterion for linear instability is that the angular velocity of the disk decreases outwards, $d\Omega^2/dr<0$; this condition is easily satisfied for any accretion disk around a compact object. It is now widely accepted that this instability, known as the magneto-rotational instability (MRI), lies at the heart of angular momentum transport in the inner regions of accretion disks around compact objects (for a detailed review of accretion disk MHD, see [108]).

A full exploration of the non-linear evolution of the MRI requires numerical MHD simulations. The initial era of simulations [119,120,121] examined the dynamics in a local patch of the disk, with an extent much smaller than the radius of the disk. These simulations confirmed the linear MRI analysis, and allows the evolution to be followed into the non-linear regime. It is found that the MRI enters the non-linear regime (some aspects of which have been studied analytically by Goodman & Xu [122]) and MHD turbulence results after a small number of dynamical time scales. The process becomes a self-sustaining dynamo, and the turbulence does not decay away with time. Of course, Cowling's antidynamo theorem dictates that non-axisymmetric simulations are required in order to produce self-sustained dynamo action. In recent years, simulators have been able to leap beyond these local simulations and tackle complete accretion disks (so-called ``global'' simulations, [123,124,125]). As in the local simulations, the global models also produce sustained MHD turbulence.

The importance to accretion disks comes from the fact that the MRI has the properties of an exchange instability in which high angular momentum material is exchanged with low angular momentum material, thereby facilitating angular momentum transport. From the simulations, one can extract effective values of the Shakura-Sunyaev $\alpha$-parameter. For local simulations, one obtains $\alpha=10^{-3}-10^{-2}$ if there is no net vertical magnetic flux threading the shearing box, and $\alpha=10^{-2}-10^{-1}$ when there is a significant (but still dynamically weak) net vertical flux [120]. The global simulations (which, to date, have only been performed for the case of zero net vertical flux) show $\alpha=10^{-2}-10^{-1}$. There is no inconsistency between these global and local results; large-scale field structures generated within the global models can effectively impose a net magnetic flux on the local level, thereby enhancing the local angular momentum transport.