Exact spatial and temporal balance of energy exchanges within a horizontally explicit/vertically implicit non-hydrostatic atmosphere

A new horizontally explicit/vertically implicit (HEVI) time splitting scheme for atmospheric modelling is introduced, which the horizontal divergence terms are applied within vertical substep. The HEVI implemented in conjunction with a mixed mimetic spectral element spatial discretisation and semi-implicit stepping that both preserve skew-symmetric structure of non-canonical Hamiltonian form equations motion. Within this context allows exact balance all energetic exchanges space time. However since choice fluxes satisfied not consistent velocity at end level still admits temporal energy conservation error. Linearised eigenvalue analysis shows similar to fully method, neutrally stable buoyancy modes, unlike second order trapezoidal acoustic modes below certain CFL number. validated against standard test cases planetary nonhydrostatic regimes. For scale baroclinic instability case, formulation exhibits secondary oscillation potential kinetic power exchanges, frequency approximately four times exhibited by third order, vertically scheme. upwinding temperature diagnostic equation shown reduce spurious oscillations without altering energetics solution, performed an energetically manner. configured on affine geometry, model run stably any dissipation.