Energy-conserving finite-difference schemes for quasi-hydrostatic equations

The pressure-based hydrostatic primitive equations model LMD-Z is extended to solve the quasi-hydrostatic deep-atmosphere as well as the non-traditional shallow-atmosphere equations (with a complete Coriolis force representation). The continuous equations are first derived in their curl form using Eulerian horizontal and non-Eulerian vertical coordinates. The equations are then interpreted as a Hamiltonian system, as they are expressed in terms of functional derivatives of the Hamiltonian. Using a finite-difference scheme on a longitude/latitude grid and based either on a Lagrangian or mass-based vertical coordinate, the discrete scheme is obtained by imitating the Hamiltonian formulation at the discrete level. It is shown how this form leads straightforwardly to the conservation of discrete total energy. The relation between the discrete equations and the discrete antisymmetry property of the Poisson bracket is discussed. The computing infrastructure of the dynamical core is kept essentially unchanged but the modification of the hydrostatic balance requires a mass-based vertical coordinate. Also, absolute angular momentum is used as a prognostic variable instead of relative velocity, which allows time-dependent metric terms and the non-traditional Coriolis force to be absorbed into it. The prototype implementation is applied to idealized circulations of an Earth-like small planet and validates the stability and accuracy of the new dynamical core.