Jemma Shipton

Andrea Natale, Jemma Shipton, Colin J. Cotter

Compatible finite elements provide a framework for preserving important structures in equations of geophysical fluid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical fluid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties.

Hiroe Yamazaki, Jemma Shipton, Michael J. P. Cullen et al.

A vertical slice model is developed for the Euler-Boussinesq equations with a constant temperature gradient in the direction normal to the slice (the Eady-Boussinesq model). The model is a solution of the full three-dimensional equations with no variation normal to the slice, which is an idealized problem used to study the formation and subsequent evolution of weather fronts. A compatible finite element method is used to discretise the governing equations. To extend the Charney-Phillips grid staggering in the compatible finite element framework, we use the same node locations for buoyancy as the vertical part of velocity and apply a transport scheme for a partially continuous finite element space. For the time discretisation, we solve the semi-implicit equations together with an explicit strong-stability-preserving Runge-Kutta scheme to all of the advection terms. The model reproduces several quasi-periodic lifecycles of fronts despite the presence of strong discontinuities. An asymptotic limit analysis based on the semi-geostrophic theory shows that the model solutions are converging to a solution in cross-front geostrophic balance. The results are consistent with the previous results using finite difference methods, indicating that the compatible finite element method is performing as well as finite difference methods for this test problem. We observe dissipation of kinetic energy of the cross-front velocity in the model due to the lack of resolution at the fronts, even though the energy loss is not likely to account for the large gap on the strength of the fronts between the model result and the semi-geostrophic limit solution.

Nell Hartney, Thomas M. Bendall, Jemma Shipton

One of the key choices for numerical models of geophysical fluids is how parametrisations of physical processes interact with the numerical methods that handle the resolved flow, known in the atmospheric community as the dynamical core. As both the dynamical core and parametrisations of physics processes continue to evolve and improve, the issue of physics-dynamics coupling - how these two different parts of the model interact - becomes ever more important. In this paper we use two variations of the moist shallow water equations to develop a simplified framework that can be used to investigate some of the questions associated with physics-dynamics coupling. The shallow water equations act as a simplified dynamical core that is computationally cheap but still retains pertinent features of the atmosphere, and the introduction of moisture means the addition to the model of a physical parametrisation. This study uses 'split-physics' moist thermal shallow water equations which couple moisture to the shallow water equations via a parametrisation, and also develops a new 'integrated-physics' formulation of the moist thermal shallow water equations which re-formulates the model so that all the moist processes are captured in the dynamics. This integrated-physics model thus requires no physics-dynamics coupling and acts as a ground-truth to compare coupling strategies in the split-physics formulation against. We use both models to examine the effect of varying the approach to how physics is coupled to the dynamics in the semi-implicit quasi-Newton timestepping scheme. The results demonstrate the usefulness of the integrated-physics moist shallow water equations and provide insights into how best to deal with physics in a model's timestep.

Daniel Witt, Thomas Bendall, Jemma Shipton

The accurate and efficient representation of atmospheric dynamics remains a central challenge in numerical weather prediction. A particular difficulty arises from the strong anisotropy of the atmosphere, in which horizontal and vertical motions occur on very different length scales, motivating numerical discretisations that can reflect this structure. In this study, we introduce a compatible finite element discretisation of the compressible Boussinesq and compressible Euler equations in which the horizontal and vertical polynomial orders are treated independently. The split-order discretisation is constructed using a tensor-product framework that preserves the discrete de Rham complex and associated mimetic properties. Its wave-propagation characteristics are examined through a discrete dispersion analysis that extends previous analyses to configurations with differing horizontal and vertical polynomial orders. The results show that increasing horizontal order improves the representation of gravity waves at low and intermediate wavenumbers, while increasing vertical order can degrade dispersion accuracy near the grid scale and introduce spectral gaps. A series of idealised numerical experiments, including gravity-wave propagation, advective transport, mountain-wave flow, and a global baroclinic-wave test, is used to assess the scheme's accuracy and convergence properties. These experiments demonstrate that increasing the polynomial order in the dominant direction of motion improves convergence, and that increasing the horizontal order yields the greatest gain in accuracy under typical atmospheric conditions. The results indicate that split-order compatible finite element discretisations provide a viable alternative for controlling accuracy and numerical behaviour in atmospheric dynamical cores.

Thomas M. Bendall, Colin J. Cotter, Jemma Shipton

We present a new compatible finite element advection scheme for the compressible Euler equations. Unlike the discretisations described in Cotter and Kuzmin (2016) and Shipton et al (2018), the discretisation uses the lowest-order family of compatible finite element spaces, but still retains second-order numerical accuracy. This scheme obtains this second-order accuracy by first `recovering' the function in higher-order spaces, before using the discontinuous Galerkin advection schemes of Cotter and Kuzmin (2016). As well as describing the scheme, we also present its stability properties and a strategy for ensuring boundedness. We then demonstrate its properties through some numerical tests, before presenting its use within a model solving the compressible Euler equations.