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