On the spin-up by a rotating disk in a rotating stratified fluid

Abstract

We investigate the response of a rotating stratified fluid to the local spin-up by a small rotating disk of radius $R$, with Rossby number $\hbox{\it Ro}\,{=}\,\omega_d/2\Omega$ around unity where $\omega_d$ is the rotating-disk vorticity and $\Omega$ the background rotation frequency. During an initial stage $\tau_{su}\,{=}\,O({E_k}^{-1/2} N^{-1})$ with Ekman number, $E_k\,{=}\,\nu/\Omega R^2$ ($\nu$ the kinematic viscosity and $N$ the buoyancy frequency), fluid ejected by the Ekman boundary layer mixes with ambient fluid, and forms an intermediate-density intrusion the radial spreading of which is arrested by background rotation. This flow resembles a concentric source–sink configuration with the sink represented by the Ekman layer above the disk and the source by the ejected fluid, which, by conservation of potential vorticity, leads to the formation of a cyclonic vortex embedded in an anti-cyclonic ring. In the next stage, the radial and axial diffusion of momentum dominate the flow evolution, and the flow is characterized by a balance between viscous dissipation of momentum and the amount of momentum applied by the rotating disk. Vorticity diffusion dominates the flow and smooths out the flow history when $E_k^{-1/2}(f/N)\,{<}\,3$, whereas the initial stage can be recognized as a separate flow stage when $E_k^{-1/2}(f/N)\,{>}\,3$. The stability of the density front is discussed.