On nonlinear Ekman and Stewartson layers in a rotating fluid

Abstract

Boundary and shear layers parallel to the axis of rotation in a rapidly rotating fluid are modified by inertial effects when the Rossby number $R$ is not small enough to be neglected. If $E$ is the Ekman number, inertial modifications to Stewartson $E^{\frac{1}{4}}$ layers become important when the Rossby number is comparable with, or larger than, $E^{\frac{1}{4}}$ and changes in the thickness and the structure of the layer then occur. The nature of the change depends on whether the total component of vorticity parallel to the axis of rotation is increased or diminished by the contribution from the layer. When inertial effects are important in these layers, the Ekman layers are also modified by inertia and a nonlinear Ekman condition must be used. Such a condition is obtained and it is used to discuss the flow between rotating disks when there is a discontinuity in their angular velocity and source-sink flow in an annulus.