  We study the destabilization of a shear layer, produced by differential
rotation of a rotating axisymmetric container. For small forcing, this produces
a shear layer, which has been studied by Stewartson and is almost invariant
along the rotation axis. When the forcing increases, instabilities develop. To
study the asymptotic regime (very low Ekman number $E$), we develop a
quasi-geostrophic two-dimensional model, whose main original feature is to
handle the mass conservation correctly, resulting in a divergent
two-dimensional flow, and valid for any container provided that the top and
bottom have finite slopes. We use it to derive scalings and asymptotic laws by
a simple linear theory, extending the previous analyses to large slopes (as in
a sphere), for which we find different scaling laws. For a flat container, the
critical Rossby number for the onset of instability evolves as $E^{3/4}$ and
may be understood as a Kelvin-Helmoltz shear instability. For a sloping
container, the instability is a Rossby wave with a critical Rossby number
proportional to $\beta E^{1/2}$, where $\beta$ is related to the slope. We also
investigate the asymmetry between positive and negative differential rotation
and propose corrections for finite Ekman and Rossby numbers. Implemented in a
numerical code, our model allows us to study the onset over a broad range of
parameters, determining the threshold but also other features such as the
spatial structure. We also present a few experimental results, validating our
model and showing its limits.
