Spin-down in a rapidly rotating cylinder container with mixed rigid and stress-free boundary conditions
Journal of Fluid Mechanics
Cambridge University Press (CUP)
Reason for embargo
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 lnElnE 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.
This project was partially supported by the French National Center for Scientific Research (CNRS) under the interdisciplinary grant Inphyniti/MI/CNRS. The numerical simulations were performed using HPC resources from GENCI-IDRIS (Grant 2015- 100584 and 2016-100610). L.O. and E.D. visited the School of Mathematics and Statistics, Newcastle University (25–30 November 2015) and wish to thank the university for its hospitality and support. We also thank the anonymous referees for their many helpful comments.
This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.
Vol. 818, pp. 205 - 240
- Mathematics