The firmest result of our analysis of the Couette-Taylor experiment is that the criterion for finite amplitude instability may be expressed in terms of the gradient Reynolds number , where the critical Reynolds number is independent of the width of the gap between cylinders, for wide enough gap. The turbulent transport of angular momentum then seems also to be independent of gap width; it proceeds always down the angular velocity gradient, as confirmed by the behavior of the initially "neutral" flows examined by Wendt.
Though the experimental evidence is somewhat less compelling, we have established empirically an expression which links the turbulent viscosity to the local shear. The value of , and that of in Eq. (7), have been derived from Wendt's experiment with the inner cylinder at rest, and it is not obvious that these parameters would be the same for different ratios of cylinder speeds. Also, the linear scaling may be valid only for those moderate gradient Reynolds numbers which could be reached in the laboratory
Nevertheless, it is tempting to apply this expression (7) to accretion disks, as an alternate for the commonly used prescription , where H is the scale height of the disk and the local sound speed (Shakura & Sunyaev 1973). Some caution is required because this viscosity has been derived from experiments performed with an incompressible fluid. Moreover (7) implies that the eddies which dominate in the transport of angular momentum have a size of order , independent of the strength of the local shear, and that their velocity is of order . When applying this prescription to a compressible flow, one has to make sure that this velocity is smaller than the sound speed and, in the case of an accretion disk, that the size of the eddies, which are three-dimensional, does not largely exceed the scale height H. The behavior of "neutral" flows demonstrates that the shear instability always transports angular momentum down the angular velocity gradient, which means outward for accretion disks.
Note that in a keplerian disk our expression is equivalent to
Such a prescription has been suggested originally by Lynden-Bell and Pringle (1974), and recently it was used again by Duschl et al. (1998). As a test, it is being applied to the modelling of accretion discs in active galactic nuclei (Huré & Richard 1999).
The reader may wonder why we have only used experimental results dating from the thirties, namely those of Wendt (1933) and Taylor (1936). The reason is that no one, since them, has studied in such extent the regime of outward increasing angular momentum. 1 We suspect that it is because the flow becomes then turbulent at once, without undergoing a series of bifurcations associated with enticing patterns. But we hope that experimentalists will turn again to this classical problem, which is of such great interest for geophysical and astrophysical fluid dynamics, and that they will explore the rotation regimes for which the data are so incomplete. In the meanwhile, the quest will continue to detect the finite amplitude instability in computer simulations.
© European Southern Observatory (ESO) 1999
Online publication: June 30, 1999