Earth's atmosphere is thin relative to its horizontal extent, and weather phenomena reflect this asymmetry: gravity waves propagate differently in the vertical than horizontal winds spread laterally. Witt, Bendall, and Shipton introduce a compatible finite element discretization for atmospheric simulation that treats horizontal and vertical polynomial orders independently, maintaining the discrete de Rham complex and its mimetic properties throughout.
Their dispersion analysis reveals an asymmetry in the returns from increasing polynomial order. Raising horizontal order improves gravity wave representation at lower wavenumbers -- the scales that matter for large-scale weather dynamics. Raising vertical order, by contrast, can degrade accuracy near the grid scale, introducing spurious oscillations where the vertical mesh cannot resolve the higher-order basis functions. This is not a failure of the method but an expression of the physical asymmetry: the atmosphere's strong stratification means that vertical structure is already well-captured at lower order, while horizontal flow features benefit from the additional resolution. Testing across gravity waves, mountain waves, advective transport, and baroclinic instabilities confirms that increasing horizontal order yields the greatest accuracy gain under typical atmospheric conditions.
The broader lesson is that mimetic numerical methods -- which preserve topological relationships from the continuous equations -- can accommodate anisotropic resolution without sacrificing their structural guarantees. The de Rham complex does not require isotropy; it requires compatibility between function spaces. Decoupling horizontal and vertical orders exploits the physical geometry of the atmosphere while preserving the mathematical geometry of the discretization.
(arXiv:2603.16571)