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