Splitting horizontal and vertical polynomial order in a compatible finite element discretisation for numerical weather prediction

Abstract

The accurate and efficient representation of atmospheric dynamics remains a central challenge in numerical weather prediction. A particular difficulty arises from the strong anisotropy of the atmosphere, in which horizontal and vertical motions occur on very different length scales, motivating numerical discretisations that can reflect this structure. In this study, we introduce a compatible finite element discretisation of the compressible Boussinesq and compressible Euler equations in which the horizontal and vertical polynomial orders are treated independently. The split-order discretisation is constructed using a tensor-product framework that preserves the discrete de Rham complex and associated mimetic properties. Its wave-propagation characteristics are examined through a discrete dispersion analysis that extends previous analyses to configurations with differing horizontal and vertical polynomial orders. The results show that increasing horizontal order improves the representation of gravity waves at low and intermediate wavenumbers, while increasing vertical order can degrade dispersion accuracy near the grid scale and introduce spectral gaps. A series of idealised numerical experiments, including gravity-wave propagation, advective transport, mountain-wave flow, and a global baroclinic-wave test, is used to assess the scheme's accuracy and convergence properties. These experiments demonstrate that increasing the polynomial order in the dominant direction of motion improves convergence, and that increasing the horizontal order yields the greatest gain in accuracy under typical atmospheric conditions. The results indicate that split-order compatible finite element discretisations provide a viable alternative for controlling accuracy and numerical behaviour in atmospheric dynamical cores.