A Semi-Implicit Compressible Model for Atmospheric Flows with Seamless Access to Soundproof and Hydrostatic Dynamics

Abstract

Abstract When written in conservation form for mass, momentum, and density-weighted potential temperature, and with Exner pressure in the momentum equation, the pseudoincompressible model and the hydrostatic model only differ from the full compressible equations by some additive terms. This structural proximity is transferred here to a numerical discretization providing seamless access to all three analytical models. The semi-implicit second-order scheme discretizes the rotating compressible equations by evolving full variables, and, optionally, with two auxiliary fields that facilitate the construction of an implicit pressure equation. Time steps are constrained by the advection speed only as a result. Borrowing ideas on forward-in-time differencing, the algorithm reframes the authors’ previously proposed schemes into a sequence of implicit midpoint step, advection step, and implicit trapezoidal step. Compared with existing approaches, results on benchmarks of nonhydrostatic- and hydrostatic-scale dynamics are competitive. The tests include a new planetary-scale gravity wave test that highlights the scheme’s ability to run with large time steps and to access multiple models. The advancement represents a sizeable step toward generalizing the authors’ acoustics-balanced initialization strategy to also cover the hydrostatic case in the framework of an all-scale blended multimodel solver.