Convective plumes emanating from fixed buoyant sources such as volcanoes, hot springs and oil spills are common in the atmosphere and the ocean. Most of what we know about their dynamics comes from scaling laws, laboratory experiments and numerical simulations. A plume grows laterally during its ascent mainly due to the process of turbulent entrainment of fluid from the environment into the plume. In an unstratified system, nothing hampers the vertical motion of the plume. By contrast, in a stratified system, as the plume rises, it reaches and overshoots the neutral buoyancy height – due to the non-zero momentum at that height. This rising fluid is then dense relative to the environment and slows down, ceases to rise and falls back to the height of the intrusion. For buoyant plumes occurring in the ocean or atmosphere, the rotation of the Earth adds an additional constraint via the conservation of angular momentum. In fact, the effect of rotation is still not well understood, and we addressed this issue in the study reported here. We looked for the steady states of an axisymmetric model in both the rotating and non-rotating cases. At the non-rotating limit, we isolated two regimes of convection depending on the buoyancy flux/momentum flux ratio at the base of the plume, in agreement with scaling laws. However, the inclusion of rotation in the model strongly affects these classical convection patterns: the lateral extension of the plume is confined at the intrusion level by the establishment of a geostrophic balance, and non-trivial swirl speed develops in and around the plume.