Three-dimensional quasi-geostrophic convection in the rotating cylindrical annulus with steeply sloping endwalls

The rotating cylindrical annulus geometry was first developed by Busse (J. Fluid Mech., vol. 44, 1970, pp. 441-460) as a simplified analogue for studying convection in rapidly rotating spherical geometries. Although it has provided a more tractable two-dimensional model than the sphere, it is formally limited to asymptotically small slopes and thus weak velocities in the direction parallel to the rotation axis. We present an asymptotically reduced three-dimensional equation set to model quasi-geostrophic convection in the annulus geometry where order-one slopes are permissible; this model provides a closer analogue to quasi-geostrophic convection in spheres and spherical shells where steeply sloping boundaries are present. A linear stability analysis of the reduced equations shows that a new class of three-dimensional, convectively driven Rossby waves is present in this system. The gravest modes exhibit strong axial variations as the slope of the boundaries becomes large. In addition, higher-order eigenmodes showing increasingly complex axial dependence are found that possess critical Rayleigh numbers close to that of the gravest mode.