Libration-driven topographic Rossby waves in a tilted cube

The flow of an incompressible fluid in a rapidly rotating cubic cavity librating at a low frequency around an axis through the midpoints of opposite edges features synchronous waves with a foliation pattern that is quasi-invariant in the axial direction. These waves are emitted from the equatorial edges (the edges furthest away from the axis) and travel into the interior in a retrograde fashion about the eastern equatorial vertices. These waves are interpreted as topographic Rossby waves, consistent with the lack of closed geostrophic contours for the rotating container. They are analysed in detail at small Ekman numbers, both in the linear regime, corresponding to the limit of zero libration amplitude (Rossby number $ Ro \to 0$ ), and in the weakly nonlinear regime with small but finite $ Ro$ . The waves subsist in the linear regime and coexist with a network of shear layers that are predicted by linear inviscid analysis to focus towards the equatorial edges. However, viscous effects stop the focusing at a distance from the edges that scales with $E^{1/2}$ . The large inclination of the oblique walls with the rotation axis, together with the vanishing depth at the equatorial edges, provide the conditions for singular behaviour in the Rossby waves as $E\to 0$ . Within a distance of the eastern equatorial vertices also scaling with $E^{1/2}$ , the nonlinear contributions have a self-similar structure whose enstrophy density scales as $E^{-16/3} Ro ^2$ . This means that $ Ro$ must be reduced considerably faster than $E$ for nonlinear contributions to be negligible as $E\to 0$ .