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

In a recently published article [1], Thuburn and Woollings present a detailed analysis of the effect of the vertical discretization on the numerical representation of wave motion in the atmosphere. Among the schemes they analyzed, the ones that had the most accurate linear normal mode frequencies were not suited for preservation of discrete analogs of conservation properties. For example, the most straightforward way to achieve mass conservation is to predict density using the flux-form of the mass continuity equation, yet some of their optimal schemes did not involve such a direct prediction of mass. Here we augment their analysis by starting from a continuous form of the vertical momentum equation that involves the vertical derivative of the Exner function instead of pressure. This improves the accuracy of the normal mode frequencies for the discretizations that may best preserve discrete analogs of conservation properties. Thuburn and Woollings (hereafter TW) begin their article by discussing the continuous dry, non-hydrostatic, compressible Eulerian equations, in which, besides the three components of velocity, a minimum of two thermodynamic variables must be predicted. They present the possible pairs of prognostic thermodynamic variables and their corresponding prognostic equations within the framework of three different vertical coordinate systems: geometric height, isentropic and a terrain-following mass-based coordinate. Then they derive the analytical normal modes for wave motion in an atmosphere with an isothermal, hydrostatic basic state.