# A Mixed Mimetic Spectral Element Model of the 3D Compressible Euler Equations on the Cubed Sphere

## TL;DR

This paper introduces a novel 3D compressible Euler equations model on the cubed sphere using mixed mimetic spectral elements, enabling energy conservation and efficient vertical-horizontal decoupling for atmospheric simulations.

## Contribution

It presents a new discretization approach combining mimetic spectral elements with dimensional splitting, improving energy conservation and computational efficiency in atmospheric modeling.

## Key findings

- Successfully validated with baroclinic instability tests
- Accurately modeled non-hydrostatic gravity waves
- Achieved energy exchanges consistent with physical laws

## Abstract

A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is presented. The model uses a mixed mimetic spectral element discretization which allows for the exact exchanges of kinetic, internal and potential energy via the compatibility properties of the chosen function spaces. A Strang carryover dimensional splitting procedure is used, with the horizontal dynamics solved explicitly and the vertical dynamics solved implicitly so as to avoid the CFL restriction of the vertical sound waves. The function spaces used to represent the horizontal dynamics are discontinuous across vertical element boundaries, such that each horizontal layer is solved independently so as to avoid the need to invert a global 3D mass matrix, while the function spaces used to represent the vertical dynamics are similarly discontinuous across horizontal element boundaries, allowing for the serial solution of the vertical dynamics independently for each horizontal element. The model is validated against standard test cases for baroclinic instability within an otherwise hydrostatically and geostrophically balanced atmosphere, and a non-hydrostatic gravity wave as driven by a temperature perturbation.