Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers

Detached shear layers in a rotating container, known as Stewartson layers, become unstable for a critical shear measured by the Rossby number Ro. To study rapidly rotating flows at asymptotically small Rossby and Ekman numbers E, a quasigeostrophic (QG) model is developed, whose main original feature is to enforce mass conservation properly, and valid for any container provided that the top and bottom have finite slopes. Asymptotic scalings of the Stewartson instability are deduced from the linear QG model, extending the previous analyses to large slopes (as in a sphere). For a flat container, the critical Rossby number evolves as E3∕4 and the instability can be understood as a shear instability. For a depth-varying container, the instability is quite different: it is a Rossby wave with a critical Rossby number proportional to βE1∕2, where β is related to the slope. A numerical implementation of the QG model is used to determine the critical parameters for different Ekman numbers (as low as 10−10) and to ...