An energetically balanced, quasi-Newton integrator for non-hydrostatic vertical atmospheric dynamics

An energetically balanced, implicit integrator for non-hydrostatic vertical atmospheric dynamics on the sphere is presented. The allows exact balance of energy exchanges in space and time motions by preserving skew-symmetry non-canonical Hamiltonian formulation compressible Euler equations. performance accelerated a preconditioning strategy that reduces dimensionality inner linear system. Here we reduce four component velocity, density, density weighted potential temperature Exner pressure system into single equation via repeated Schur complement decomposition careful selection coupling terms. As currently formulated, based horizontal-vertical spatial splitting does not permit bottom topography. validated standard test cases baroclinic instability gravity wave rising bubble high-resolution planar geometry, shows robust convergence across all these regimes.