Semi-implicit integrations of perturbation equations for all-scale atmospheric dynamics

This paper presents a generalised perturbation form of the nonhydrostatic partial differential equations (PDEs) that govern dynamics of all-scale global atmospheric flows. There can be many alternative perturbation forms for any given system of the governing PDEs, depending on the assumed ambient state about which perturbations are taken and on subject preferences in the numerical model design. All such forms are mathematically equivalent, yet they have different implications for the design and accuracy of effective semi-implicit numerical integrators of the governing PDEs. Practical and relevant arguments are presented in favour of perturbation forms that maximise the degree of implicitness of the associated integrators. Two optional forms are implemented in the high-performance finite-volume module (IFS-FVM) for simulating global all-scale atmospheric flows Smolarkiewicz et al. (2016) [32]. Their relative performance is verified with a class of ambient states of reduced complexity. A series of numerical simulations of the planetary baroclinic instability assumes geostrophically balanced zonally uniform ambient flows with significant meridional and vertical shear and exemplifies accuracy gains enabled by the generalised perturbation approach. The newly developed semi-implicit integrators show the potential for numerically accurate separation of a background state from finite-amplitude perturbations of the global atmosphere and provide a base for further development.