Abstract
We previously reported experimental and theoretical results on the linear spin-up of a linearly stratified, rotating fluid in a uniform-depth square cylinder [M. R. Foster and R. J. Munro, “The linear spin-up of a stratified, rotating fluid in a square cylinder,” J. Fluid Mech. 712, 7–40 (2012)]. Here we extend that analysis to a “sliced” square cylinder, which has a base-plane inclined at a shallow angle α. Asymptotic results are derived that show the spin-up phase is achieved by a combination of the Ekman-layer eruptions (from the perimeter region of the cylinder's lid and base) and cross-slope-propagating stratified Rossby waves. The final, steady state limit for this spin-up phase is identical to that found previously for the uniform depth cylinder, but is reached somewhat more rapidly on a time scale of order E−1/2Ω−1/log (α/E1/2) (compared to E−1/2Ω−1 for the uniform-depth cylinder), where Ω is the rotation rate and E the Ekman number. Experiments were performed for Burger numbers, S, between 0.4 and 16, and showed that for \documentclass[12pt]{minimal}\begin{document}$S\gtrsim \mathcal {O}\left(1\right)$\end{document}S≳O1, the Rossby modes are severely damped, and it is only at small S, and during the early stages, that the presence of these wave modes was evident. These observations are supported by the theory, which shows the damping factors increase with S and are numerically large for \documentclass[12pt]{minimal}\begin{document}$S\gtrsim \mathcal {O}\left(1\right)$\end{document}S≳O1.
Highlights
A quite thorough understanding of the way in which rapidly rotating fluids, in axisymmetric containers, adjust to small changes in rotation rates – either “spin-up” or “spin-down” – has developed over the past 50 or so years
We know that an increase in rotation rate of the container results in the formation of thin, quasi-steady viscous Ekman layers at the horizontal lid and base, within which layers the spun-up fluid is transported radially outward, where it erupts into the vertical, sidewall Stewartson layers.[1,4]
We found a number of interesting features not noted in stratified spin-up in axisymmetric containers: (a) Fluid in the horizontal-wall Ekman layers collides with the vertical walls in the corners, and erupts into the interior, even on the spin-up time scale. (b) The vertical-wall boundary layers are not buoyancy layers, but rather conventional Prandtl boundary layers on the horizontal velocity, and are fully nonlinear even for very small Rossby numbers. (c) Due to finite-time boundary-layer singularity formation, cyclonic eddies form in the vertical corner regions of the container, and that formation occurs for times that scale with the inverse Rossby number
Summary
A quite thorough understanding of the way in which rapidly rotating fluids, in axisymmetric containers, adjust to small changes in rotation rates – either “spin-up” or “spin-down” – has developed over the past 50 or so years. The authors of this paper reported experimental and asymptotic results for the spin-up of a linearly stratified fluid in a closed, uniform-depth cylinder of square cross-section.[14] We found a number of interesting features not noted in stratified spin-up in axisymmetric containers: (a) Fluid in the horizontal-wall Ekman layers collides with the vertical walls in the corners, and erupts into the interior, even on the spin-up time scale. (c) Due to finite-time boundary-layer singularity formation, cyclonic eddies form in the vertical corner regions of the container, and that formation occurs for times that scale with the inverse Rossby number These results stand in contrast with what is known about the spin-up of a linearly stratified fluid in a axisymmetric container: (a) Unless there is long-time differential rotation between the horizontal and vertical walls,[15] there is no Ekman layer eruption on the spin-up time scale. V makes possible quantitative comparisons of experimental and theoretical velocity-profiles, which are presented at the end of that section
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