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

Abstract

The spin-up and spin-down of a fluid in a rapidly rotating, fluid-filled, and closed half cone are 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 and Howard [J. Fluid Mech. 17, 385–404 (1963)10.1017/S0022112063001415] since there are no closed geostrophic contours nor a denumerable set of inertial waves (even a modified theory incorporating Rossby waves by Pedlosky and Greenspan [J. Fluid Mech. 27, 291–304 (1967)10.1017/S0022112067000321]—relies on geostrophy to leading order). The linearised spin-up/spin-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/spin-down time scales conventionally with E−1/2, where E is the Ekman number. The numerical coefficient in this time scale 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.