arXiv subjects

Thomas M. Bendall

Publications and source records attributed to Thomas M. Bendall.

9 recordsLinked to original sources

arXiv2026

Exploring physics-dynamics coupling using moist shallow water equations

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.

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arXiv2026

A conservative, discontinuous Galerkin, tracer transport scheme using compatible finite elements

This paper outlines a conservative transport scheme for scalar tracers within a compatible finite element model for geophysical fluid equations. Instead of using the advective transport equation for a mixing ratio, a conservative transport equation is solved for the tracer density of the mixing ratio multiplied by the dry density. This ensures mass conservation in the continuous equations, which can be preserved in the discrete equations with a discontinuous Galerkin transport scheme. Our method is designed to work for two placements of the mixing ratio in a Charney-Phillips vertical staggering: either co-located with the dry density or vertically staggered from it. The new scheme is designed to conserve the tracer density and ensure consistency by maintaining a constant mixing ratio. Additionally, a mass-conserving limiter is developed to ensure non-negativity in the co-located configuration. Tests with terminator toy chemistry and a moist rising bubble show the use of the new transport scheme with physics terms and its ability to accurately model mass conservation of moisture species in a dynamical core setup.

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arXiv2025

Exploring forms of the moist shallow water equations using a new compatible finite element discretisation

The moist shallow water equations offer a promising route for advancing understanding of the coupling of physical parametrisations and dynamics in numerical atmospheric models, an issue known as 'physics-dynamics coupling'. Without moist physics, the traditional shallow water equations are a simplified form of the atmospheric equations of motion and so are computationally cheap, but retain many relevant dynamical features of the atmosphere. Introducing physics into the shallow water model in the form of moisture provides a tool to experiment with numerical techniques for physics-dynamics coupling in a simple dynamical model. In this paper, we compare some of the different moist shallow water models by writing them in a general formulation. The general formulation encompasses three existing forms of the moist shallow water equations and also a fourth, previously unexplored formulation. The equations are coupled to a three-state moist physics scheme that interacts with the resolved flow through source terms and produces two-way physics-dynamics feedback. We present a new compatible finite element discretisation of the equations and apply it to the different formulations of the moist shallow water equations in three test cases. The results show that the models capture generation of cloud and rain and physics-dynamics interactions, and demonstrate some differences between moist shallow water formulations and the implications of these different modelling choices.

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arXiv2024

SWIFT: A Monotonic, Flux-Form Semi-Lagrangian Tracer Transport Scheme for Flow with Large Courant Numbers

Local conservation of mass and entropy are becoming increasingly desirable properties for modern numerical weather and climate models. This work presents a Flux-Form Semi-Lagrangian (FFSL) transport scheme, called SWIFT, that facilitates this conservation for tracer variables, whilst maintaining other vital properties such as preservation of a constant, monotonicity and positivity. Importantly, these properties all hold for large Courant numbers and multi-dimensional flow, making the scheme appropriate for use within a dynamical core which takes large time steps. The SWIFT scheme presented here can be seen as an evolution of the FFSL methods of Leonard et al and Lin and Rood. Two-dimensional and three-dimensional schemes consist of a splitting into a sequence of one-dimensional calculations. The new SWIFT splitting presented here allows monotonic and positivity properties from the one-dimensional calculations to be inherited by the multi-dimensional scheme. These one-dimensional calculations involve separating the mass flux into terms that correspond to integer and fractional parts of the Courant number. Key to achieving conservation is coupling the transport of tracers to the transport of the fluid density, through re-use of the discrete mass flux that was calculated from the fluid density in the transport of the tracers. This work also describes how these properties can still be attained when the tracer is vertically-staggered from the density in a Charney-Phillips grid.

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arXiv2023

Physics-Dynamics-Chemistry Coupling Across Different Meshes in LFRic-Atmosphere: Formulation and Idealised Tests

The main components of an atmospheric model for numerical weather prediction are the dynamical core, which describes the resolved flow, and the physical parametrisations, which capture the effects of unresolved processes. Additionally, models used for air quality or climate applications may include a component that represents the evolution of chemicals and aerosols within the atmosphere. While traditionally all these components use the same mesh with the same resolution, we present a formulation for the different components to use a series of nested meshes, with different horizontal resolutions. This gives the model greater flexibility in the allocation of computational resources, so that resolution can be targeted to those parts which provide the greatest benefits in accuracy. The formulation presented here concerns the methods for mapping fields between meshes, and is designed for the compatible finite element discretisation used by LFRic-Atmosphere, the Met Office's next-generation atmosphere model. Key properties of the formulation include the consistent and conservative transport of tracers on a mesh that is coarser than the dynamical core, and the handling of moisture to ensure mass conservation without generation of unphysical negative values. Having presented the formulation, it is then demonstrated through a series of idealised test cases which show the feasibility of this approach.

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arXiv2021

Perspectives on the Formation of Peakons in the Stochastic Camassa-Holm Equation

A famous feature of the Camassa-Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite element discretisation for it, which we use to explore the formation of peakons. Our simulations using this discretisation reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

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arXiv2019

A Compatible Finite Element Discretisation for the Moist Compressible Euler Equations

We present new discretisation of the moist compressible Euler equations, using the compatible finite element framework identified in Cotter and Shipton (2012). The discretisation strategy is described and details of the parametrisations of moist processes are presented. A procedure for establishing hydrostatic balance for moist atmospheres is introduced, and the model's performance is demonstrated through several test cases, two of which are new.

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arXiv2019

The Recovered Space Advection Scheme for Lowest-Order Compatible Finite Element Methods

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.

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arXiv2018

Statistical properties of an enstrophy conserving discretisation for the stochastic quasi-geostrophic equation

A framework of variational principles for stochastic fluid dynamics was presented by Holm (2015), and these stochastic equations were also derived by Cotter et al. (2017). We present a conforming finite element discretisation for the stochastic quasi-geostrophic equation that was derived from this framework. The discretisation preserves the first two moments of potential vorticity, i.e. the mean potential vorticity and the enstrophy. Following the work of Dubinkina and Frank (2007), who investigated the statistical mechanics of discretisations of the deterministic quasi-geostrophic equation, we investigate the statistical mechanics of our discretisation of the stochastic quasi-geostrophic equation. We compare the statistical properties of our discretisation with the Gibbs distribution under assumption of these conserved quantities, finding that there is agreement between the statistics under a wide range of set-ups.

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