Exploring mathematical formulations for a next-generation compatible finite element dynamical core

Compatible finite element methods are attractive for modelling geophysical fluids because they can replicate many of the desirable properties of the Arakawa C-grid, such as good wave dispersion. Compatible finite elements also facilitate alternative grid structures which avoid the clustering of grid points at the poles without the associated downsides these grids have with a finite difference scheme. For this reason compatible finite element methods are used in the Met Office's next-generation dynamical core, GungHo. The mathematical formulation used in GungHo is designed to be similar to that used in the Met Office's previous dynamical core, ENDGame. For instance, GungHo uses an advective form of the momentum equation and the lowest-order finite element spaces  

 

We present an investigation into formulation options, carried out in Gusto, a geophysical fluid toolkit built upon Firedrake, an automated code generation framework for solving PDEs via finite element methods. Gusto shares the same fundamental compatible finite element formulation as GungHo but provides greater flexibility to investigate other choices

 

Specifically, we investigate the effects of increasing the finite element order on the model. Due to the flexibility of Gusto, we can consider a generally higher order model as well as separately altering the horizontal and vertical orders. Additionally we investigate the effects brought about by writing the advective term in the momentum equation in its vector invariant form. We evaluate the impacts of these choices by conducting several test cases considering the short-term fluid dynamics and long-term statistical climate properties. 

 

Understanding how these options impact the dynamical core will inform future research directions and improvements for GungHo.