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 ν𝜈 ) rotating rapidly (angular velocity Ω𝛺 ) inside an axisymmetric cylindrical container (radius LL , height HH ) with rigid boundaries, which follows the instantaneous small change in the boundary angular velocity at small Ekman number E=ν/H2Ω≪1E=𝜈/H2𝛺≪1 , was provided by Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385–404). For that problem E1/2E1/2 Ekman layers form quickly, triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the O(E−1/2Ω−1)O(E−1/2𝛺−1) time scale. On the longer lateral viscous diffusion time scale O(L2/ν)O(L2/𝜈) , the QG flow responds to the E1/3E1/3 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, ω⋆𝜔⋆ , evolution on the O(L2/ν)O(L2/𝜈) time scale: spatially, ω⋆𝜔⋆ increases (decreases) outwards from the axis for a slippery (rigid) sidewall; temporally, the long-time (≫L2/ν)(≫L2/𝜈) 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 E1/2×E1/2E1/2×E1/2 corner region that forms at the sidewall intersection with the rigid base is responsible for a lnEln⁡E singularity within the E1/3E1/3 layer, causing our asymptotics to apply only at values of EE 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 E1/3E1/3 boundary layer equations for given EE numerically. Our hybrid asymptotic–numerical approach yields results in excellent agreement with our DNS.