Vertical discretizations giving optimal representation of normal modes: general equations of state

Previous work has identified a number of vertical discretizations of the non‐hydrostatic compressible Euler equations that optimally capture the propagation of acoustic, inertio‐gravity, and Rossby waves. Here, that previous work is extend to apply to a general equation of state, making it applicable to a wider range of geophysical fluid systems. It is also shown that several choices of prognostic thermodynamic variables and vertical staggering which were previously thought to be suboptimal can, in fact, give optimal wave propagation when discretized in an appropriate way. The key idea behind constructing these new optimal discretizations is to ensure that their corresponding linear system is equivalent to that of a certain, most fundamental, optimal configuration.