A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere

TL;DR

This paper extends a mimetic spectral element method to the cubed sphere geometry for rotating shallow water equations, ensuring conservation and spectral convergence on complex geometries.

Contribution

The extension of the mixed mimetic spectral element method to smoothly varying, non-affine cubed sphere geometries with preserved mimetic properties and spectral convergence.

Findings

Conservation of mass, vorticity, and energy achieved.

Spectral convergence maintained on the cubed sphere.

Method effectively handles complex geometries.

Abstract

In a previous article [J. Comp. Phys. $\mathbf{357}$ (2018) 282-304], the mixed mimetic spectral element method was used to solve the rotating shallow water equations in an idealized geometry. Here the method is extended to a smoothly varying, non-affine, cubed sphere geometry. The differential operators are encoded topologically via incidence matrices due to the use of spectral element edge functions to construct tensor product solution spaces in $H(\mathrm{rot})$, $H(\mathrm{div})$ and $L_2$. These incidence matrices commute with respect to the metric terms in order to ensure that the mimetic properties are preserved independent of the geometry. This ensures conservation of mass, vorticity and energy for the rotating shallow water equations using inexact quadrature on the cubed sphere. The spectral convergence of errors are similarly preserved on the cubed sphere, with the generalized Piola transformation used to construct the metric terms for the physical field quantities.