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Fluid dynamics

Spinning discs in the lab

What causes gas to be drawn in towards black holes, rather than remain in a stable orbit as planets do around the Sun? A laboratory result indicates that something more than just hydrodynamics must be at work.

On page 343 of this issue, Ji et al.1 describe a meticulous experiment in which they confined water between two independently turning cylinders. Through artful experimental design, the authors were able to reduce viscous effects in the resulting 'Couette' flow to a level of one part in two million. They chose the velocities of the cylinders so that they would mimic — and so compel the confined fluid to mimic — so-called keplerian rotation, which is typical of astrophysical disks around black holes. Here, velocity is inversely proportional to the square-root of the distance from the centre of the rotation.

The result was that nothing happened at all: the fluid continued to rotate stably. But why exactly do astrophysicists and fluid dynamicists find this apparently harmless result so surprising?

In the early 1970s, astrophysicists were struggling with the exciting and controversial question of whether black holes — objects whose gravity is so great that nothing, not even light, can escape once captured — were real2. Only slightly earlier, a series of compact sources of X-ray radiation had been discovered. One model held that this radiation originated from gas disks surrounding black holes in binary systems of close stars3 and galactic nuclei4. This gas would dissipate its energy as heat, and ultimately X-ray radiation. Its angular momentum would be transported outward, and the gas would spiral inward towards the hole. By understanding these 'accretion disks', astronomers hoped that they would, in one fell swoop, both explain the mysterious X-ray sources and prove that black holes exist.

The existence of black holes and their accretion disks is now widely accepted by both theorists and observers. But understanding the dynamics of accretion disks, in black holes and in other types of system, has turned out to be an extremely knotty problem. Why do disks accrete at all? Why does gas in motion around a massive centre not remain in a stable, planet-like orbit?

The problem is that energy dissipation and angular-momentum transport are properties of a viscous fluid, and the viscosity of the disk gas is far too small to account for the angular-momentum loss that leads to accretion. This problem might be solved if a keplerian gas were turbulent. A turbulent fluid can have a small particulate viscosity, yet behave in some ways as though it were viscous: correlations between different components of the fluid's velocity fluctuations cause both substantial energy dissipation and enhanced momentum transport.

The importance of viscosity in fluid flow is measured by a dimensionless quantity known as the Reynolds number; a high Reynolds number indicates a small viscosity. Shear flows (those with regions moving at different velocities) of high Reynolds number are generally exquisitely unstable to any small perturbation5. But astrophysical disks both shear and rotate; the rotation introduces Coriolis forces, which can sometimes stabilize an otherwise turbulent fluid.

Starting in the late nineteenth century, the British physicist Lord Rayleigh investigated fluids similar to the gas in an astrophysical disk that rotate differentially (at different speeds according to radial distance) and have a very small viscosity. He found that such fluids would become unstable (and possibly turbulent) only where the specific angular momentum of the flow decreases as one moves to greater radii. In a keplerian disk, however, the specific angular momentum increases with radius. According to the Rayleigh criterion, such a flow is stable.

Because the Rayleigh criterion applies only to infinitesimal disturbances, not those of finite amplitude, and only to perturbations that are symmetrical about the axis of rotation, the formal stability of keplerian disks did not immediately embarrass the theorists. Indeed, although Couette experiments6 persistently failed to reveal instability for keplerian profiles, it was widely believed that differentially rotating flows would ultimately prove to be unstable; it was simply a matter of reaching large enough Reynolds numbers in the laboratory7.

For many, this 'faith-based' approach to turbulent hydrodynamical accretion was less than satisfying. The discovery8 in 1991 that even very weak magnetic fields profoundly alter the stability of rotating gases improved matters. A rotating magnetized gas becomes unstable when its angular velocity decreases as one moves away from the centre, a condition nearly universally satisfied in astrophysical disks. Computer simulations have since shown that this 'magnetorotational instability' leads to precisely the sort of turbulence that disk theorists are seeking9.

Not all disks are ionized throughout to an extent that would allow magnetic fields the necessary influence to seed instability. Only a very small electron component is needed to couple the field and the gas, but accretion disks associated with the formation of stars, known as protostellar disks, seem to contain an extended region that is cold, dense and dusty. In such a region, free electrons are suppressed, and gas and field are decoupled. For this reason and others, many disk theorists continued to believe that a fluid of large Reynolds number undergoing keplerian shear would ultimately prove to be turbulent. In 2001, a laboratory Couette-flow experiment at large Reynolds number seemed to find just that10.

One of the great difficulties in working with Couette flows is that the rotating cylinders have end-caps that rotate uniformly, whereas in the adjacent fluid the rotation is a function of distance from the rotation axis. This mismatch creates a boundary layer near the end-cap and a secondary axial velocity flow known as an Ekman circulation in addition to the purely rotational flow that one wishes to study. This effect can be controlled to some extent by making the cylinders very long, distancing the end-caps from the bulk of the flow.

Ji and colleagues, however, wish to use their apparatus1 in investigations of the magnetorotational instability. These require a very uniform magnetic field that would be difficult to maintain over a long cylinder. So the authors overcame the Ekman circulation problem by splitting the end-caps into two parts that rotate at different speeds. This was sufficient to control the Ekman circulation and to attain an exceptionally high Reynolds number of around 2×106.

The authors' finding that, at such a high Reynolds number, the rotation profile of a keplerian fluid is as stable as the rotation of a solid body should lay to rest the notion that any rotating flow (in general) or an accretion disk (in particular) is nonlinearly unstable simply if its viscosity is sufficiently small. The implication, for which there is now growing astrophysical evidence11, would seem to be that accretion is magnetically, not hydrodynamically, driven.

Ji and colleagues' primary goal for their apparatus was to find evidence for the magnetorotational instability in liquid gallium, and the results of these experiments are yet to come. The researchers might thus be poised for a double coup: tolling the death knell for hydrodynamical shear turbulence in accretion disks, and capturing magnetic instability in a cylindrical flow for the first time.


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