TY - JOUR

T1 - Spin-up and spin-down in a half cone: A pathological situation or not?

AU - Li, Ligang

AU - Patterson, Michael D

AU - Zhang, Keke

AU - Kerswell, Richard

PY - 2012/11/6

Y1 - 2012/11/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84872418488&partnerID=8YFLogxK

UR - http://pof.aip.org/

UR - http://dx.doi.org/10.1063/1.4765333

U2 - 10.1063/1.4765333

DO - 10.1063/1.4765333

M3 - Article

VL - 24

JO - Physics of Fluids

JF - Physics of Fluids

IS - 11

M1 - 116601

ER -