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

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 be 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. 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.