Towards a consistent numerical compressible non‐hydrostatic model using generalized Hamiltonian tools

A set of compressible non‐hydrostatic equations for a turbulence‐averaged model atmosphere comprising dry air and water in three phases plus precipitating fluxes is presented, in which common approximations are introduced in such a way that no inconsistencies occur in the associated budget equations for energy, mass and Ertel's potential vorticity. These conservation properties are a prerequisite for any climate simulation or NWP model. It is shown that a Poisson bracket form for the ideal fluid part of the full‐physics equation set can be found, while turbulent friction and diabatic heating are added as separate ‘dissipative’ terms. This Poisson bracket is represented as a sum of a two‐fold antisymmetric triple bracket (a Nambu bracket represented as helicity bracket) plus two antisymmetric brackets (so‐called mass and thermodynamic brackets of the Poisson type). The advantage of this approach is that the given conservation properties and the structure of the brackets provide a good strategy for the construction of their discrete analogues. It is shown how discrete brackets are constructed to retain their antisymmetric properties throughout the spatial discretisation process, and a method is demonstrated how the time scheme can also be incorporated in this philosophy.