Comment on the article “Vertical discretizations for compressible Euler equation atmospheric models giving optimal representation of normal modes” by Thuburn and Woollings

The accuracy of the numerical representation of atmospheric wave motion as a function of the vertical discretization was analyzed in a recent study. The schemes which provided the most accuracy were not well suited for satisfying conservation properties. In this comment, the analysis is extended to include alternative forms of the vertical momentum equation that use the Exner function, instead of pressure, in the pressure gradient term. It is found that these forms lead to more accurate representations of the normal mode frequencies for the schemes having good conservation properties.