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

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 ν that occurs during linear spin-down in a cylindrical container of radius L and height H, rotating rapidly (angular velocity Ω) 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 = 10H and Ekman number E = ν/H 2 Ω = 10 −3 reveals structured inertial wave activity on the spin-down time-scale. The 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 → ∞. At large but finite distance r † from the symmetry axis, the meridional (QG-)flow, that causes the QG-spin down, is blocked by the lateral boundary r † = 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 † = 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 † = O(H) (L H). Other than identifying small scale structure near r † = 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. † For our unit of relative velocity v † , we adopt the velocity increment Ro LΩ of the initial flow at the outer boundary r † = L. So, relative to cylindrical components, we set(1.3d,e) and refer to [u, v] and w as the horizontal and axial components of velocity, respectively. Relevant to our previous = 1 study, but of even greater importance to our present 1 case, are the aspects of the seminal work of Greenspan & Howard (1963) that pertain to the unbounded limit → ∞. That study is complicated and many of the key concepts, as they relate to our study are not easily identified. Indeed the most important ideas stem from the even simpler problem of the transient Ekman layer above a flat plate in an otherwise unbounded fluid, which we outline in the following subsection.

Inertial wave activity during spin-down in a rapidly rotating penny shaped cylinder

“…On the one hand, we refrain from repeating some of the references cited in Oruba et al. (2020), which apply equally here. On the other, as we need to refer extensively to equations (say ()), sections (say §) and figures (say figure ) from Oruba et al.…”

“…Here, and in our previous work (Oruba et al. 2020), we are not concerned with that QG-development, but rather focus attention on the evolution of inertial waves triggered simply by the impermeable boundary condition at on the shorter time scales . For that, whether the lateral boundary is no slip or stress free is irrelevant and has no influence on the inertial wave activity ‘triggered’ by the blocking of the radial flow found in the unbounded domain .…”

“…In Oruba, Soward & Dormy (2020), we investigated the inertial wave response during spin-down in a shallow cylinder height , of large radius , equivalently by consideration of a ‘reduced model’. Here, we consider the complete problem.…”

“…On the other, as we need to refer extensively to equations (say ()), sections (say §) and figures (say figure ) from Oruba et al. (2020) (say ), we use the notation ‘(: )’, ‘§:’ and ‘figure :’ respectively to identify them.…”

“…Oruba et al. (2020) ignored the early time behaviour. Although this is adequate to capture the main features of the triggered wave solution, it is not asymptotically correct.…”