On the inertial wave activity during spin-down in a rapidly rotating penny shaped cylinder: a reduced model

In a previous paper, Oruba et al. (J. Fluid Mech., vol. 818, 2017, pp. 205-240) considered the 'primary' quasi-steady geostrophic (QG) motion of a constant density fluid of viscosity ⌫ that occurs during linear spin-down in a cylindrical container of radius r † = L and height z † = H, rotating rapidly (angular velocity ⌦) about its axis of symmetry subject to mixed rigid and stress-free boundary conditions for the case L = H. Here, Direct numerical simulation at large L = 10H and Ekman numbers E = ⌫/H 2 ⌦ in the range = 10 3-10 7 reveals inertial wave activity on the spin-down time scale E 1/2 ⌦ 1. Our analytic study, based on E ⌧ 1, builds on the results of Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385-404) for an infinite plane layer L ! 1. In addition to QG spin-down, they identify a 'secondary' set of quasi-maximum frequency ! † ! 2⌦ (MF) inertial waves, which is a manifestation of the transient Ekman layer, decaying algebraically / 1/ p t †. Here, we acknowledge that the blocking of the meridional parts of both the primary-QG and the secondary-MF spin-down flows by the lateral boundary r † = L provides a trigger for other inertial waves. As we only investigate the response to the primary QG-trigger, we call the model 'reduced' and for that only inertial waves with frequencies ! † < 2⌦ are triggered. We explain the ensuing organised inertial wave structure via an analytic study of the thin disc limit L H restricted to the region L r † = O(H) far from the axis, where we make a Cartesian approximation of the cylindrical geometry. Other than identifying a small scale fan structure emanating from the corner [r † , z † ] = [L, 0], we show that inertial waves, on the gap length scale H, radiated (wave energy flux) away from the outer boundary r † = L (but propagating with a phase velocity towards it) reach a distance determined by the mode with the fastest group velocity.