Thermal Rossby waves driven by centrifugal buoyancy in a rotating cylindrical fluid gap become unstable right at the onset of convection when the Prandtl number is small. The Benjamin–Feir–Newell instability leads to modulated thermal Rossby waves which can also be described by a generalized Ginzburg–Landau equation. A resonance instability occurs at a finite distance in Rayleigh number from the neutral curve. It leads to two independent wave patterns propagating past each other and finally gives rise to vacillations of the amplitude of convection. Most of these features can be described to a good approximation by a system of three coupled amplitude equations. Time integrations based on a Galerkin expansion show transitions to chaotic convection at higher Rayleigh numbers.