Choice of function spaces for thermodynamic variables in mixed finite-element methods

We study the dispersion properties of three choices for the buoyancy space in a mixed finite-element discretization of geophysical fluid flow equations. The problem is analogous to that of the staggering of the buoyancy variable in finite-difference discretizations. Discrete dispersion relations of the two-dimensional linear gravity wave equations are computed. By comparison with the analytical result, the best choice for the buoyancy space basis functions is found to be the horizontally discontinuous, vertically continuous option. This is also the space used for the vertical component of the velocity. At lowest polynomial order, this arrangement mirrors the Charney–Phillips vertical staggering known to have good dispersion properties in finite-difference models. A fully discontinuous space for the buoyancy corresponding to the Lorenz finite-difference staggering at lowest order gives zero phase velocity for high vertical wavenumber modes. A fully continuous space, the natural choice for scalar variables in a mixed finite-element framework, with degrees of freedom of buoyancy and vertical velocity horizontally staggered at lowest order, is found to entail zero phase velocity modes at the large horizontal wavenumber end of the spectrum. Corroborating the theoretical insights, numerical results obtained on gravity wave propagation with fully continuous buoyancy highlight the presence of a computational mode in the poorly resolved part of the spectrum that fails to propagate horizontally. The spurious signal is not removed in test runs with higher-order polynomial basis functions. Runs at higher order also highlight additional oscillations, an issue that is shown to be mitigated by partial mass-lumping. In light of the findings and with a view to coupling the dynamical core to physical parametrizations that often force near the horizontal grid scale, the use of the fully continuous space should be avoided in favour of the horizontally discontinuous, vertically continuous space.