Numerical weather- and climate-prediction models have traditionally applied the hydrostatic approximation and also, in particular, the shallow-atmosphere approximation. In addition, and probably as a result, studies of the normal modes of the atmosphere too have made the shallow-atmosphere approximation. The approximation appears to be based on simple scaling arguments. Here, the forms of the unforced, linear normal modes for the deep atmosphere on a sphere are considered and compared with those of the shallow atmosphere. Also, the impact of ignoring the vertical variation of gravity is investigated. For terrestrial parameters, it is found that relaxing either or both of these approximations has very little impact on the spatial form of the energetically significant components of most normal modes. The frequencies too are only slightly changed. However, relaxing the shallow-atmosphere approximation does lead to significant changes in the tropical structure of internal acoustic modes. Relaxing the shallow-atmosphere approximation also leads to non-zero vertical-velocity and potential-temperature fields for external acoustic and Rossby modes; these fields are identically zero when the shallow-atmosphere approximation is made.
For a finite-difference numerical model to be able to represent well the behaviour of the free atmosphere it must be able to capture accurately the structures of the normal modes. Therefore, the structures of normal modes can have implications for the choice of prognostic variables and grid staggering. In particular, the vertical structure of normal modes suggests that density and temperature should be analytically eliminated in favour of pressure and potential temperature as the prognostic thermodynamic variables, and that potential temperature and vertical velocity should be staggered in the vertical with respect to the other dynamic prognostic variables, the so-called Charney– Phillips grid. © Royal Meteorological Society, 2002. N. Wood's and A. Staniforth's contributions are Crown copyright.