Reflections and focusing of inertial waves in a tilted librating cube

Abstract

A fluid-filled cube rotating about an axis passing through the midpoints of opposite edges is subjected to small librations (i.e. modulation of the mean rotation). Low viscosity regimes, with Ekman number as small as $10^{-8}$ and equally small relative forcing amplitude, are explored numerically. The full inertial range of forcing frequencies, from 0 to twice the mean rotation rate are considered. The response flows are dominated by inertial wavebeams emitted from edges and/or vertices, depending on the forcing frequency. How these reflect on the cube's walls and focus onto edges and vertices lead to intricate patterns. Most of the results can be reconciled using linear inviscid ray-tracing theory with careful attention to wavebeam emissions and reflections. However, even at the low Ekman number and relative forcing amplitude considered, other effects are discernible which are not captured by ray tracing. These include a symmetry breaking due to viscous effects and a progressive wave, retrograde to the mean rotation and localized in the boundary layers of the cube, due to nonlinear effects and the librational forcing.