Instabilities in the spin-up of a rotating, stratified fluid

Theoretical analyses and laboratory experiments have been performed on the stability of a flow generated by the differential cyclonic corotation of a flat, rigid disk in a uniformly rotating, linearly stratified fluid contained within a cylindrical tank. The undisturbed fluid is stably stratified with salt (Schmidt number $σ≈670$⁠) and the (vertical) axes of rotation of the disk and the fluid container are coincident. The theoretical analysis shows that when the interior flow satisfies gradient wind balance (or, alternatively, thermal wind balance), it is destabilized by the action of viscosity. In the experiments, the manifestation of the viscous overturning instability is seen to be the formation of steplike internal microstructures in the density field, observed as regularly spaced, curved ring-shaped sheets with associated localized sharp, vertical density gradients. A stability analysis of the flow shows that the instability criterion is dependent on local values of the vertical and radial gradients of zonal velocity and the background density field. These quantities are measured in the experiments using a combination of horizontal-plane particle image velocimetry and an array of traversing microconductivity probes. The stability criterion based on this linear analysis predicts that the interior of the fluid is unstable. Using the $σ⪢1$ condition, simple asymptotic expressions for the maximum growth rate and associated wave number have been derived from the cubic dispersion relation. The theoretically predicted length scales and e-folding times associated with the fastest growing modes are found to give excellent agreement with the corresponding values obtained from the laboratory experimental data.