We study a semi-analytical model of convection in a rapidly-rotating, differentially-heated annulus with sloping top and bottom lids. Rapid rotation leads to a preservation of relatively simple, two-dimensional (2-D) structure in the experimentally-observed flow, while temporal complexity increases with the Rayleigh number. The model is, therefore, two-dimensional; it exhibits a sequence of bifurcations from steadily-drifting, azimuthally-periodic convection columns, also called thermal Rossby waves, through vacillation and a period-doubling cascade, to aperiodic, weakly-turbulent solutions.