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

Abstract

A comprehensive study of the classical linear spin-down of a constant-density viscous fluid (kinematic viscosity $\unicode[STIX]{x1D708}$) rotating rapidly (angular velocity $\unicode[STIX]{x1D6FA}$) inside an axisymmetric cylindrical container (radius $L$, height $H$) with rigid boundaries, which follows the instantaneous small change in the boundary angular velocity at small Ekman number $E=\unicode[STIX]{x1D708}/H^{2}\unicode[STIX]{x1D6FA}\ll 1$, was provided by Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385–404). For that problem $E^{1/2}$ Ekman layers form quickly, triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the $O(E^{-1/2}\unicode[STIX]{x1D6FA}^{-1})$ time scale. On the longer lateral viscous diffusion time scale $O(L^{2}/\unicode[STIX]{x1D708})$, the QG flow responds to the $E^{1/3}$ sidewall shear layers. In our variant, the sidewall and top boundaries are stress-free, a set-up 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) sidewall. This is evidenced by the QG angular velocity, $\unicode[STIX]{x1D714}^{\star }$, evolution on the $O(L^{2}/\unicode[STIX]{x1D708})$ time scale: spatially, $\unicode[STIX]{x1D714}^{\star }$ increases (decreases) outwards from the axis for a slippery (rigid) sidewall; temporally, the long-time $(\gg L^{2}/\unicode[STIX]{x1D708})$ behaviour is dominated by an eigensolution with a decay rate slightly slower (faster) than that for an unbounded layer. In our slippery sidewall case, the $E^{1/2}\times E^{1/2}$ corner region that forms at the sidewall 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 linear equations governing 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.