A fully-compressible nonhydrostatic cell-integrated semi-Lagrangian solver (CSLAM-NH) with consistent and conservative transport

Semi-Lagrangian semi-implicit (SLSI) schemes are popular in the dynamical cores of global and regional numerical weather prediction models, primarily for their stability and computational efficiency with larger time steps. While traditional SLSI schemes are not inherently mass-conserving due to their use of grid-point interpolation, several recently developed cell-integrated semi-Lagrangian (CISL) transport schemes evaluate the mass continuity equation in an inherently mass-conserving manner. However, semi-implicit fluid-flow solvers using these CISL schemes have relied on an implicit correction in the continuity equation that is based on the linearization around a time-independent mean reference state. This dependence makes mass conservation more difficult and forgoes numerical consistency between mass and constituent mass transport, leading to possible spurious generation or removal of constituent mass.

A new semi-implicit fully-compressible nonhydrostatic CISL solver proposed here uses the Conservative Semi-Lagrangian Multi-tracer scheme (CSLAM; a CISL approach) as the transport scheme and uses a new flux-form formulation of the discrete semi-implicit continuity equation that ensures numerically consistent transport. The formulation is based on a recent successful approach used in the shallow-water equations solver, CSLAM-SW. The algorithm is constructed to be similar to typical conservative SLSI schemes, requiring at each time step a single linear Helmholtz equation solution and a single application of CSLAM. The new semi-implicit CISL solver is extended for the fully-compressible nonhydrostatic equations and tested for the desirable properties of local (and thus global) mass conservation, consistency, and shape-preservation of moisture variables and tracers. The formulation of the nonhydrostatic solver and results from several idealized dry and moist test cases will be presented.