  A comprehensive study of the classical linear spin-down of a constant density
viscous fluid (kinematic viscosity \nu) rotating rapidly (angular velocity
\Omega) inside an axisymmetric cylindrical container (radius L, height H) with
rigid boundaries, that follows the instantaneous small change in the boundary
angular velocity at small Ekman number $E=\nu/H^2\Omega \ll 1$, was provided by
Greenspan & Howard (1963). $E^{1/2}$-Ekman layers form quickly triggering
inertial waves together with the dominant spin-down of the quasi-geostrophic
(QG) interior flow on the $O(E^{-1/2}\Omega^{-1})$ time-scale. On the longer
lateral viscous diffusion time-scale $O(L^2/\nu)$, the QG-flow responds to the
$E^{1/3}$-side-wall shear-layers. In our variant the side-wall and top
boundaries are stress-free; a setup motivated by the study of isolated
atmospheric structures, such as tropical cyclones, or tornadoes. Relative to
the unbounded plane layer case, spin-down is reduced (enhanced) by the presence
of a slippery (rigid) side-wall. This is evinced by the QG-angular velocity,
\omega*, evolution on the O(L^2/\nu) time-scale: Spatially, \omega* increases
(decreases) outwards from the axis for a slippery (rigid) side-wall;
temporally, the long-time ($\gg L^2/\nu)$ behaviour is dominated by an
eigensolution with a decay rate slightly slower (faster) than that for an
unbounded layer. In our slippery side-wall case, the $E^{1/2} \times E^{1/2}$
corner region that forms at the side-wall intersection with the rigid base is
responsible for a $\ln E$ singularity within the $E^{1/3}$-layer causing our
asymptotics to apply only at values of E far smaller than can be reached by our
Direct Numerical Simulation (DNS) of the entire spin-down process. Instead, we
solve the $E^{1/3}$-boundary-layer equations for given E numerically. Our
hybrid asymptotic-numerical approach yields results in excellent agreement with
our DNS.
