Validity of anelastic and other equation sets as inferred from normal‐mode analysis

Various simplifications of the fully compressible inviscid (Euler) equations have been made over the years to filter out rapidly‐propagating acoustic oscillations, resulting in various anelastic, hydrostatic and pseudo‐incompressible equation sets. The principal tool used to develop approximate equation sets, and to assess their validity as a function of flow regime, has been scale analysis which has proven quite subtle to apply. Here it is shown that normal‐mode analysis provides a useful complementary tool for assessing the validity of the abovementioned approximate equation sets for both small‐ and large‐scale flows, and leads to the following conclusions. Whilst of key importance for small‐scale theoretical studies and process modelling, the anelastic equations are not recommended for either operational numerical weather prediction or climate simulation at any scale. The pseudo‐incompressible set appears to be viable for numerical weather prediction, but only at short horizontal scales. For global non‐hydrostatic modelling, only the fully compressible equations are suitable. Advances in numerical techniques in the past decade allow these to be integrated in a computationally efficient manner.