Spin-down in a rapidly rotating cylinder container with mixed rigid and stress-free boundary conditions

TL;DR

This paper investigates the classical spin-down of a viscous fluid in a cylindrical container with mixed boundary conditions, revealing how slip and rigid walls influence the decay rates and flow structures, using asymptotic analysis and numerical simulations.

Contribution

It introduces a hybrid asymptotic-numerical method to analyze spin-down with mixed boundary conditions, extending classical results to more realistic boundary scenarios.

Findings

Slip boundary reduces spin-down rate; rigid boundary enhances it.

Long-term flow behavior dominated by eigenmodes with modified decay rates.

Asymptotic results agree well with direct numerical simulations.

Abstract

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.