A mixed finite-element, finite-volume, semi-implicit discretisation for atmospheric dynamics: Spherical geometry

TL;DR

This work extends a semi-implicit mixed finite-element dynamical core to spherical geometries using a cubed-sphere mesh, integrating an explicit finite-volume transport scheme that accommodates non-uniform, non-orthogonal grids. The transporter employs a Strang-split, advective-then-flux formulation to ensure conservation and stability, while the implicit wave dynamics are advanced with an iterated semi-implicit timestepping scheme and a Krylov solver with multigrid preconditioning. The discretisation leverages compatible FE spaces on a spherical shell, with a Piola-based mapping to the sphere and a semi-analytic Jacobian, enabling accurate geopotential gradients and reduced grid imprinting. Computational experiments across resting and dynamic tests (including a Held-Suarez climate run) show good agreement with established semi-implicit semi-Lagrangian cores and demonstrate balanced behavior over terrain and minimal imprinting on cubed-sphere grids, supporting the approach’s viability for weather and climate prediction on spherical domains.

Abstract

The reformulation of the Met Office's dynamical core for weather and climate prediction previously described by the authors is extended to spherical domains using a cubed-sphere mesh. This paper updates the semi-implicit mixed finite-element formulation to be suitable for spherical domains. In particular the finite-volume transport scheme is extended to take account of non-uniform, non-orthogonal meshes and uses an advective-then-flux formulation so that increment from the transport scheme is linear in the divergence. The resulting model is then applied to a standard set of dry dynamical core tests and compared to the existing semi-implicit semi-Lagrangian dynamical core currently used in the Met Office's operational model.

Figures (8)

• Figure 1: $C12$ cubed-sphere horizontal mesh with $6x12x12$ cells.
• Figure 2: Cell $j$ with points staggered half a grid point located on cell faces $\Delta_i/2,$$i=1,2,3$. For a field stored at $j$ the spatial reconstruction is computed at $j\pm\Delta_i/2$.
• Figure 3: Zonal (left panels) and vertical (right panels) wind fields for the resting atmosphere case after 6 days on a C96L30 mesh with $\Delta t = 600$ s. The top row shows results from this paper and the bottom row shows results from the semi-implicit semi-Lagrangian ENDGame model with a 1 degree resolution and $\Delta t = 600$ s, (wood14).
• Figure 4: $700 hPa$ geopotential height (left column) and temperature (right column) at days 5, 10 and 15 for the flow over a Gaussian hill test at C96L40 resolution with $\Delta t = 900 s$. The location of the cubed-sphere panel boundaries are overlaid on the $T700$ plots.
• Figure 5: Surface pressure (top row) and 850 hPa temperature (bottom row) for the Baroclinic wave test on a C96L30 mesh with $\Delta t = 900$ s. Left panels: after 8 days simulation and Right panels: after 10 days simulation.