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

Highlights: • Implicit integrator for vertical dynamics exactly conserves energy in space and time. • Physics based preconditioner results in modified Helmholtz equation for inner linear system. • Robust convergence at planetary and non-hydrostatic scales. • Coupled to horizontal dynamics via trapazoidal splitting scheme. 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.