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
SN - 1089-7666
VL - 24
JO - Physics of Fluids
JF - Physics of Fluids
IS - 11
M1 - 116601
ER -