On the shallow atmosphere approximation in finite element dynamical cores

TL;DR

This paper introduces a novel method for implementing the shallow atmosphere approximation in 3D finite element models of dynamical cores by embedding the equations in a 4D manifold, simplifying metric calculations.

Contribution

The paper presents a new approach to incorporate the shallow atmosphere approximation using a 4D embedding, making finite element discretizations more straightforward and equivalent to mesh modifications.

Findings

The approach is mathematically equivalent to mesh-based metric modifications.

Demonstrated convergence in a prototypical elliptic problem.

Provides a practical implementation method for atmospheric models.

Abstract

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.