Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We illustrate our discretisation with some standard rotating sphere test problems.Comment: accepted versio

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

This is the final version. Available from Elsevier via the DOI in this record.We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor–Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We demonstrate that our discretisation achieves the expected second order convergence and provide results from some standard rotating sphere test problems.Natural Environment Research Council (NERC)Natural Environment Research Council (NERC)Engineering and Physical Sciences Research Council (EPSRC)Engineering and Physical Sciences Research Council (EPSRC

A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes

Copyright © 2015 Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics Vol. 290 (2015), DOI: 10.1016/j.jcp.2015.02.045A new numerical method is presented for solving the shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes. The second is the use of dual finite element spaces on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.Natural Environment Research Council under the “GungHo” projec

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

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

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

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this recordTo meet the challenges posed by future generations of massively parallel supercomputers a reformulation of the dynamical core for the Met Office’s weather and climate model is presented. This new dynamical core uses explicit finite-volume type discretisations for the transport of scalar fields coupled with an iterated-implicit, mixed finite-element discretisation for all other terms. The target model aims to maintain the accuracy, stability and mimetic properties of the existing Met Office model independent of the chosen mesh while improving the conservation properties of the model. This paper details that proposed formulation and, as a first step towards complete testing, demonstrates its performance for a number of test cases in (the context of) a Cartesian domain. The new model is shown to produce similar results to both the existing semi-implicit semi-Lagrangian model used at the Met Office and other models in the literature on a range of bubble tests and orographically forced flows in two and three dimensions.Natural Environment Research Council (NERC

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

This is the author accepted manuscriptThe 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 do- mains. 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 diver- gence. The resulting model is then applied to a standard set of dry dynamical core tests and compared to the exist- ing semi-implicit semi-Lagrangian dynamical core currently used in the Met Office’s operational model.Natural Environment Research Council (NERC)Natural Environment Research Council (NERC)Engineering and Physical Sciences Research Council (EPSRC)Engineering and Physical Sciences Research Council (EPSRC