How barotropic and stable are differential-rotation cylindrical flows?

In rotating cylindrical containers it is possible to generate a highly depth-independent vertical shear layer, akin to the layers studied by Stewartson theoretically in 1957, by driving the inner radial part of the fluid at a different speed through rotation of disks embedded in the top and base of a fluid-filled enclosure. This configuration finds laboratory application in the study of shear layers in rotating flows motivated by geophysical flows such as planetary polar vortices and terrestrial hurricanes. We combine high-order axisymmetric computations with a linear stability analysis and three-dimensional simulation to characterize regimes of depth-independent (``barotropic'') flow, and the modes by which both axial (depth-dependent) and azimuthal symmetry are broken in the system. Azimuthal instability produces striking symmetrical polygonal patterns closely resembling patterns seen in atmospheric polar vortices.