Instability of oscillatory Stokes-Stewartson layers in a rotating fluid

Abstract

A cylinder rotating about its vertical axis is filled with homogeneous liquid and subjected to oscillatory mechanical forcing. Depending on the ratio, τ, of the forcing period to the spin-down time, the flow adjacent to the sidewall resembles either a classic Stokes layer (small τ), or a modulated Stewartson layer (large τ). Laboratory experiments show that the flow becomes unstable to columnar disturbances that are aligned with the axis of rotation. This azimuthally wavy instability can lead to the formation of strong vertical vortices which penetrate into the interior. Quasigeostrophic depth-invariant linear instability theory is compared with the experiments. The theory is much too stable and grossly overestimates the experimental critical points. An inertial adjustment model, which in a crude way takes account of observed small-scale (ageostrophic) instability and turbulence in the near-wall region, is in much better agreement with the laboratory measurements of the onset of the columnar vortices. Thus, the origin of the vertically coherent structures appears to be crucially related to alterations of the laminar Stokes-Stewartson profiles by fine structure in the boundary layer.