End-effects in rapidly rotating cylindrical Taylor-Couette flow

Abstract

We present numerical simulations of the flow in a rapidly rotating cylindrical annulus. We show that at the rotation rates relevant to the magneto-rotational instability, the flow is strongly constrained by the Taylor-Proudman theorem. As a result, it is controlled almost entirely by the end-plates. We then consider two possible options for minimizing these end-effects, namely (i) simply taking a very long cylinder, and (ii) splitting the end-plates into a series of differentially rotating rings. Regarding option (i), we show that the cylinder would have to be hundreds of times as long as it is wide before end-effects become unimportant in the interior. Since this is clearly not feasible, we turn to option (ii), and show that in order to obtain a smooth angular velocity profile, the end-plates would have to be split into around ten rings. If the end-plates are split into fewer rings, perhaps 3-5, the angular velocity profile will not be smooth, but will instead consist of a series of Stewartson layers at the boundaries from one ring to the next. We suggest therefore that the instabilities one obtains in this system will be the familiar Kelvin-Helmholtz instabilities of these Stewartson layers, rather than the magneto-rotational instability. At best, one might hope to obtain the MRI superimposed on these Kelvin-Helmholtz modes. Any subsequent interpretation of results is thus likely to be quite complicated.