Abstract A climatology of the three-dimensional life cycle of the gravest waves in the tropical lower and middle stratosphere is presented. It shows that at periods around 10 days the gravest waves correspond to Kelvin and Rossby–gravity wave packets that substantially affect specific regions in the lower stratosphere. It also shows that the planetary-scale Kelvin waves with zonal wavenumber s = 1 and periods between 10 and 20 days produce a substantial signal. Still at the planetary scales, the climatology also shows that the global planetary Rossby waves with s = 1 and periods around 5 and 16 days have a substantial equatorial signature. This climatology is for all the dynamical fields (horizontal wind, temperature, and geopotential height) and relates the equatorial waves to the equatorial zonal mean flow evolution associated with the quasi-biennial oscillation. The method used to extract the climatology is a composite analysis of the dynamical fields keyed on simple indexes measuring when the waves enter in the stratosphere. For the Kelvin waves, the Rossby–gravity waves, and the 5- and 16-day Rossby planetary waves, these indexes are related to the latitudinal means over the equatorial band of the temperature, the meridional wind, the geopotential height, and the zonal wind respectively. The method is applied first to ERA-40 and then to a simulation done with the LMDz GCM. When compared to the results from ERA-40, this reveals that the LMDz GCM underestimates the Rossby–gravity wave packets and a fraction of the Kelvin wave packets. This deficit is attributed to the fact that the model has a too coarse vertical resolution and an insufficient tropospheric forcing for the horizontal wavenumbers s > 3.