Some of the classical models of tropical cyclone intensification predict tropical cyclones to intensify up to a steady intensity, which depends on surface fluxes only, without any relevant role played by convective motions in the troposphere, typically assumed to have a moist adiabatic lapse rate. Simulations performed using the non-hydrostatic, high-resolution model SAM in idealized settings (rotating radiative-convective equilibrium on a doubly-periodic domain) show early intensification consistent with these theoretical expectations, but different intensity evolution, with the cyclone undergoing an oscillation in wind speed. This oscillation can be linked to feedbacks between the cyclone intensity and air buoyancy: convective heating, radiative heating, and mixing with warm low stratospheric air warm the mid and upper troposphere of the cyclone stabilizing the air column and thus reducing its intensity. After the intensity decay phase, mid and upper tropospheric cooling, mostly through cold advection from the surroundings, cooled by radiation, rebuilds CAPE and triggers a new intensification. These idealized simulations thus highlight the potentially important interactions between a tropical cyclone, its environment and radiation.