# The inertial wave activity during spin-down in a rapidly penny shaped cylinder. Part II The inertial wave of maximum frequency trigger

## TL;DR

This paper analytically investigates inertial wave activity triggered by maximum frequency waves during spin-down in a rapidly rotating cylindrical fluid, comparing results with numerical simulations to understand boundary effects on wave dynamics.

## Contribution

It provides a combined analytical model for inertial wave activity triggered by both quasi-geostrophic and maximum frequency mechanisms, aligning well with numerical simulations.

## Key findings

- Analytic results match DNS data closely.
- Boundary blocking significantly influences inertial wave activity.
- Combined triggers better explain observed wave patterns.

## Abstract

In an earlier 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 $L=H$. Direct Numerical Simulation (DNS) at large $L= 10 H$ and Ekman number $E=\nu/H^2\Omega=10^{-3}$ by Oruba, Soward & Dormy (J.Fluid Mech., sub judice and referred to as Part I) reveals significant inertial wave activity on the spin-down time-scale. The analytic study in Part I, 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 the primary QG-trigger for the inertial waves studied in Part I. For the laterally unbounded layer, Greenspan & Howard also identified inertial waves of maximum frequency (MF), which are a manifestation of the transient Ekman layer. The blocking of the MF-waves by the lateral boundary provides a secondary MF-trigger for yet more inertial waves. Here we obtain analytic results for the wave activity caused by the combined-trigger (QG+MF) that faithfully captures the character of the laterally unbounded base flow including its transients. The results are compared with the inertial wave part of the DNS (the so called "filtered DNS" or simply "FNS"), for which the agreement is excellent and accounts for minor discrepancies evident in the Part I results for the QG-trigger.