Physical interpretation of spiralling-columnar convection in a rapidly rotating annulus with radial propagation properties of Rossby waves

To aid the physical understanding of spiralling-columnar convection emerging in rapidly rotating spheres and spherical shells, two-dimensional thermal convection in a rapidly rotating annulus is investigated through the radial propagation properties of topographic Rossby waves. Two kinds of the boundaries containing the fluid in the axial direction are considered: a convex type modelling a spherical geometry and a concave type for comparison. The linear stability of a basic state with no motion and uniformly unstable stratification is examined and spirally elongated Structures of critical convection are obtained for small Prandtl numbers. An analysis of the energy budget shows that a part of the kinetic energy generated in the region with slightly inclined boundaries is dynamically transferred and dissipates through viscosity in the region with strongly inclined boundaries. This indicates that the Rossby waves propagate from the region with slightly inclined boundaries to the region with strongly inclined boundaries. It is presented that the appearance of a spiral Structure corresponds to an Increase of the local radial wavenumber of the Rossby waves propagating in the radial direction. The flow patterns obtained using the dispersion relation of the Rossby waves coincide with those of the tailing part of the spiral structure obtained numerically. As the Prandtl number increases, the Rossby waves barely propagate because of strong Viscous dissipation, and the flow pattern is localized in the region with slightly inclined boundaries. For convex boundaries with unstable stratification concentrating near the Outer boundary and concave boundaries with unstable stratification confined near the inner boundary, the flow patterns tilt in the direction inverse to the case of uniform unstable stratification. The tilting direction of the flow pattern is not determined by the Curvature of the boundaries considered but instead by the radial propagation direction of the Rossby waves excited by thermal convection.