The onset of convection driven by centrifugal buoyancy in the annular region between two coaxial corotating cylinders is investigated when the height of the annular region not only depends on the distance from the axis as in most previous works on the problem but on the azimuthal coordinate as well. The onset of convection in the form of stationary flows or in the form of thermal Rossby waves is investigated in dependence on the parameters of the problem. The evolution of the various modes with increasing amplitude of motion is followed through integrations in time of the fully nonlinear basic equations. Various forms of periodic, quasiperiodic and aperiodic time dependence are found. ©1998 American Institute of Physics.