On the shallow atmosphere approximation in finite element dynamical cores
This provides a principled way to incorporate the shallow atmosphere approximation into finite element dynamical cores, which is relevant for atmospheric modeling but is an incremental methodological contribution.
The paper presents a method to implement the shallow atmosphere approximation in 3D finite element dynamical cores by embedding the equations in a 4D manifold, showing equivalence to using a modified 3D mesh. A convergence test on an elliptic problem demonstrates the approach.
We provide an approach to implementing the shallow atmosphere approximation in three dimensional finite element discretisations for dynamical cores. The approach makes use of the fact that the shallow atmosphere approximation metric can be obtained by writing equations on a three-dimensional manifold embedded in $\mathbb{R}^4$ with a restriction of the Euclidean metric. We show that finite element discretisations constructed this way are equivalent to the use of a modified three dimensional mesh for the construction of metric terms. We demonstrate our approach via a convergence test for a prototypical elliptic problem.