Spin-up and spin-down in a half cone: A pathological situation or not?

Ligang Li, Michael D Patterson, Keke Zhang, Richard Kerswell

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Abstract

The spin-up and spin-down of a fluid in a rapidly-rotating, fluid-filled, closed half cone is studied both numerically and experimentally. This unusual set up is of interest because it represents a pathological case for the classical linear theory of Greenspan & Howard (J. Fluid Mech., 17, 385, 1963) since there are no closed geostrophic contours nor a denumerable set of inertial waves (even a modified theory incorporating Rossby waves by Pedlosky & Greenspan - J. Fluid Mech., 27, 291, 1967 - relies on geostrophy to leading order). The linearised spin-up/-down dynamics in a half cone is found to dominated by topographical effects which force an ageostrophic leading balance and cause the large-scale starting vorticity to coherently move into the 'westward' corner of the half cone for both spin-up and spin-down. Once there, viscous boundary layer effects take over as the dominant process ensuring that the spin-up/-down time scales conventionally with E^{−1/2} , where E is the Ekman number. The numerical coefficient in this timescale is approximately a quarter of that for a full cone when the semi-angle is 30^o . Nonlinear spin up from rest is also studied as well as an impulsive 50% reduction in the rotation rate which shows boundary layer separation and small scales. We conclude that spin-up in a rapidly-rotating half cone is not pathological because the fluid dynamics is fundamentally the same as that in a container with small topography: in both topography-forced vortex stretching is to the fore.
Original languageEnglish
Article number116601
JournalPhysics of Fluids
Volume24
Issue number11
DOIs
Publication statusPublished - 6 Nov 2012

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half cones
fluids
topography
boundary layer separation
rotating fluids
downtime
fluid dynamics
containers
vorticity
boundary layers
cones
vortices

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Li, L., Patterson, M. D., Zhang, K., & Kerswell, R. (2012). Spin-up and spin-down in a half cone: A pathological situation or not? Physics of Fluids, 24(11), [116601]. https://doi.org/10.1063/1.4765333

Spin-up and spin-down in a half cone: A pathological situation or not? / Li, Ligang; Patterson, Michael D; Zhang, Keke; Kerswell, Richard.

In: Physics of Fluids, Vol. 24, No. 11, 116601, 06.11.2012.

Research output: Contribution to journalArticle

Li, L, Patterson, MD, Zhang, K & Kerswell, R 2012, 'Spin-up and spin-down in a half cone: A pathological situation or not?', Physics of Fluids, vol. 24, no. 11, 116601. https://doi.org/10.1063/1.4765333
Li, Ligang ; Patterson, Michael D ; Zhang, Keke ; Kerswell, Richard. / Spin-up and spin-down in a half cone: A pathological situation or not?. In: Physics of Fluids. 2012 ; Vol. 24, No. 11.
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