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. Copyright (C) 2008 Royal Meteorological Society.