Stratified spin-up in a sliced, square cylinder

Munro, Richard J.; Foster, M. R.

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/E 1/2) (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 S≳O(1)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 S≳O(1)S≳O1 .

Citation

Munro, R. J., & Foster, M. R. (in press). Stratified spin-up in a sliced, square cylinder. Physics of Fluids, 26(2), https://doi.org/10.1063/1.4864266

• Journal Article Type: Article
• Acceptance Date: Jan 21, 2014
• Online Publication Date: Feb 18, 2014
• Deposit Date: Jul 9, 2016
• Publicly Available Date: Jul 9, 2016
• Journal: Physics of Fluids
• Print ISSN: 1070-6631
• Electronic ISSN: 1089-7666
• Publisher: American Institute of Physics
• Peer Reviewed: Peer Reviewed
• Volume: 26
• Issue: 2
• DOI: https://doi.org/10.1063/1.4864266
• Public URL: https://nottingham-repository.worktribe.com/output/723081
• Publisher URL: http://dx.doi.org/10.1063/1.4864266
• Contract Date: Jul 9, 2016