Instabilities of the Stewartson layer Part 1. The dependence on the sign of $Ro$

Abstract

We consider the fluid flow in a spherical shell in rapid overall rotation, with additionally a differential rotation imposed on the inner sphere. The basic state consists of the axisymmetric Stewartson shear layer situated on the tangent cylinder, the cylinder parallel to the axis of rotation and just touching the inner sphere. In this work we consider the non-axisymmetric instabilities that arise when the differential rotation becomes sufficiently large. We find that the sign of the differential rotation, that is, whether the inner sphere is rotating slightly faster or slightly slower than the outer sphere, is crucial, with positive differential rotations yielding a progression to higher wavenumbers $m$ as the overall rotation rate increases, but negative differential rotations yielding $m\,{=}\,1$ over almost the entire range of rotation rates. This difference is particularly intriguing, as it has been seen before in one closely related experimental study, but not in another. A prior asymptotic analysis also suggested there should be no difference. We therefore try to understand what subtle features of the flow structures and/or geometries should cause this difference in results. We show that the geometry is the critical feature, with the height along the axis of rotation changing abruptly across the tangent cylinder. We are not able to identify why this should make such a difference, and why only for negative differential rotations. We suggest instead additional experiments and asymptotics to further clarify this point.