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 twodimensional
linear gravity ...
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 twodimensional
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 CharneyPhillips
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.
