The inertial wave activity during spin-down in a rapidly rotating penny shaped cylinder. Part I The quasi-geostrophic trigger

Abstract

In a previous paper, Oruba, Soward & Dormy (J.Fluid Mech., vol.818, 2017, pp.205-240) considered the primary quasi-steady geostrophic (QG) motion of a constant density fluid of viscosity $\nu$ that occurs during linear spin-down in a cylindrical container of radius $L$ and height $H$, rotating rapidly (angular velocity $\Omega$) about its axis of symmetry subject to mixed rigid and stress-free boundary conditions for the case $L=H$. Here, Direct Numerical Simulation (DNS) at large $L= 10 H$ and Ekman number $E=\nu/H^2\Omega=10^{-3}$ reveals structured inertial wave activity on the spin-down time-scale. The analytic study, based on $E\ll 1$, builds on the results of Greenspan & Howard (J.Fluid Mech., vol.17, 1963, pp.385-404) for an infinite plane layer $L\to\infty$. At large but finite distance $r^\dag$ from the symmetry axis, the meridional (QG-)flow, that causes the QG-spin down, is blocked by the lateral boundary $r^\dag=L$, which provides a QG-trigger for inertial waves. The true situation in the unbounded layer is complicated further by the existence of a secondary set of maximum frequency (MF) inertial waves (a manifestation of the transient Ekman layer) identified by Greenspan & Howard. Their blocking at $r^\dag=L$ provides a secondary MF-trigger for yet more inertial waves that we consider in a sequel (Part II). Here, for the QG-trigger, we solve a linear initial value problem by Laplace transform methods. The ensuing complicated inertial wave structure is explained analytically on approximating our cylindrical geometry at large radius by rectangular Cartesian geometry, valid for $L-r^\star=O(H)$ ($L\gg H$). Other than identifying small scale structure near $r^\star=L$, our main finding is that inertial waves radiated away from the outer boundary (but propagating towards it) reach a distance determined by the group velocity.