A blended semi-implicit numerical model for weakly compressible atmospheric dynamics

Abstract

Physical processes in the atmosphere develop on a wide range of spatial and temporal scales. Meteorologically relevant phenomena move at speeds much lower than that of sound waves. The latter, despite their unimportance in weather and climate studies, enforce the use of very small time steps in explicit discretizations of the fully compressible equations. Traditionally, the problem has been analytically tackled using reduced formulations – anelastic and pseudo-incompressible models on the small scales, hydrostatic models on large scales – that lack the terms that generate acoustics. Alternatively, fully compressible equations have been solved with split-explicit or semi-implicit numerical methods free of acoustic-dependent stability constraints. However, most existing approaches in this context resort to various forms of numerical filtering to achieve stability at the expense of accuracy. The present study discusses a semi-implicit fully compressible numerical model for the simulation of low-speed flows in the atmosphere. The second-order accurate finite volume scheme extends a projection method for the pseudo-incompressible model and agrees with it by construction in the small-scale limit. Unlike most numerical approaches in meteorology, equations are solved in non-perturbational form and in terms of the thermodynamic pressure variable. Quantities are advanced in time in an explicit advection step limited by a stability threshold independent of sound speed. Compressibility is handled implicitly in a correction step that solves two elliptic problems for the pressure increments. Well-balancing techniques are used to discretize buoyancy without reference to a hydrostatically balanced background state. Convergence properties are evaluated on the advection of a smooth vortex and compressibility effects are assessed on the case of a simple acoustic wave. Then, we test the ability of the scheme to accurately simulate gravity-driven flows with large time steps on thermal benchmarks in neutrally and stably stratified atmospheres. Obtained numerical solutions are found to be in line with published work. Equations are then cast in a blended soundproof-compressible multimodel formulation allowing for controlled introduction of compressibility in the scheme. In a unified and uniformly accurate numerical framework, the blending feature is employed to filter acoustic perturbations in the initial stages of thermal simulations. The technique can find application in a data assimilation context, enabling on-the-fly incorporation of unbalanced data in the numerical model. The proposed extension to an implicit treatment of buoyancy envisages the use of the scheme as a flexible tool for the simulation of multiscale atmospheric flows.