Keplerian Rotation

Spinning Discs in the Lab

Steven A. Balbus
Nature 444, p. 281-2 (2006)

URL: http://www.nature.com/nature/journal/v444/n7117/pdf/444281a.pdf

A relatively simple table-top experiment has shed light on astrophysical processes like accretion.

An accretion disc --- or any other gravitationally bound, rotating fluid disc --- has a velocity that is proportional to the square root of the distance from the center. Apparently this is well known. I can see it from an order of magnitude estimate. The mass inside a disc of radius R is roughly

M \sim \rho\pi R^2 t

where t is the thickness. Equating the centripetal acceleration and the acceleration due to gravity from this mass, one finds

v^2 = G \pi \rho t r.

However, when I try to actually integrate the potential for a uniform disc, I get logarithmic corrections that depend on a cutoff length at the center:

v^2 = G \pi \rho r \, \log [ (R^2 - r^2) / a^2 ]

where R is the radius of the disc, and a is a cutoff to handle the logarithmic divergence in my calculations. The deviation from the order of magnitude estimate is pretty good for r less than about R/2.

Of course, I haven't included any fluid dynamics considerations here --- it's just a continuum model --- no viscosity, no vorticity, no Navier, no Stokes. (It's times like these I wish my undergraduate education included a course on fluid dynamics --- or my graduate education, for that matter.)

O.K. Let's take v^2 \sim r for a Keplerian fluid, and see what the experiments had to say.

Researchers used two independently rotating cylinders (one inside the other) to simulate Keplerian rotation, where the velocity varies inversely with the square root of radius. The shocking result was ... nothing happened. The fluid rotated without turbulence.

According to Balbus, Rayleigh deduced this result for differentially rotating fluids a long time ago. The Rayleigh criterion states that if the specific angular momentum of a fluid increases with radius, then the flow is stable. But the Rayleigh criterion applies only to small, rotationally invariant disturbances. At high enough Reynolds number, it might not hold.

The experiment summarized by Balbus achieved a Reynolds number of 2 000 000, and found stable circulation.

What's interesting about this result is that turbulence in such a rotating fluid was thought to be a major channel for accretion discs to dissipate energy and transport angular momentum. If an accretion disc is rotating with all its particles in stable orbits, it won't accrete!

So what do the theorists do now? The turn from hydrodynamics to magnetohydrodynamics. Charged particles. While a neutral fluid is stable when the Rayleigh criterion is satisfied, "A magnetized gas becomes unstable when the angular velocity decreases as one moves away from the center," as is the case for Keplerian rotation. (This was a little confusing when I first read it. Balbus emphasize that velocity grows as the square root of radius for a neutral disc, but then talks about the angular velocity decreasing with radius for the magnetized gas. The angular velocity of a Keplerian disc decreases with the square root of the radius.)

This article highlights an aspect of astrophysics that amuses me. Astrophysicists have some major problems getting things to work. They can't get accretion discs to accrete without magnetohydrodynamics, supernovae to explode without neutrinos, galaxies to rotate fast enough without dark matter or modified theories of gravity, a universe to expand correctly without dark energy. It's surprising (to me, at least) that the most simple models of these phenomena just don't work. It's an exciting field.