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^{Design Aspects of the�Met Office Dynamical Core: LFRic��}^{DCMIP, 2025}

Thomas Bendall

Dynamics Research, Met Office

www.metoffice.gov.uk

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First, some context …

The Met Office’s current model, the Unified Model, has been used for 35 years
It is used for climate modelling, global and regional NWP
The dynamical core is called ENDGame, which uses a longitude-latitude grid for global simulations

We are replacing the Unified Model with a new atmospheric model, called LFRic-Atmosphere
Named after Lewis Fry Richardson
The new dynamical core GungHo, is designed for the next-generation of supercomputers
The whole modelling framework is called Momentum^®
So how have we chosen to design LFRic?

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Outline

Underlying Principles

Model Design

Transport

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Why build a new model?

The longitude-latitude mesh has points clustered at the poles

At 10km global resolution, horizontal spacing at the pole is 12m. For 50m/s winds and timesteps of 4 minutes, this gives Courant numbers of 1000!

Not only is this unphysical, it requires enormous parallel data communication costs

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Scalability

This creates a bottleneck in parallel scalability -- eventually we require exponentially more processors to increase resolution

Back in 2011, it was thought this would need solving to exploit the next-generation of supercomputers (which have now arrived!)

(17km)

Courtesy of Nigel Wood

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A New Grid

This led to the GungHo Project from 2011-2016, to provide the research underpinning the new dynamical core

So how have we chosen to design a dynamical core on a quasi-uniform grid?

Principle I: Quasi-Uniform

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ABC of Building a Dynamical Core

Stability: it can’t crash!
Efficiency: we have a 1 hour forecast window
Accuracy

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A Seamless Approach

Seamlessness has been at the heart of the Met Office’s approach since 1990

Builds trust in both weather and climate configurations

An efficient use of human resources

Principle II: Seamlessnses

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GungHo by name but not by nature?

The UM has been a very successful model

Changing the grid and modelling system has a big impact on many users and partners

Try to keep as much the same as we can

Principle III: Change as little as possible!

Lower is better!

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How to choose a grid and discretisation?

A very important paper was that of Staniforth and Thuburn (2012), which laid out desirable properties of a dynamical core:

Mass conservation
Accurate representation of balanced flow and adjustment
Absence or control of computational modes
Geopotential gradient and pressure gradient should produce no unphysical source of vorticity
Terms involving pressure should be energy conserving
Coriolis terms should be energy conserving
No spurious fast propagation of Rossby modes; geostrophic balance should not break down
Axial angular momentum should be conserved
Accuracy approaching second-order
Minimal grid imprinting

Staniforth, A. and Thuburn, J., 2012. Horizontal grids for global weather and climate prediction models: a review. Quarterly Journal of the Royal Meteorological Society, 138(662), pp.1-26.

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How to choose a grid and discretisation?

A very important paper was that of Staniforth and Thuburn (2012), which laid out desirable properties of a dynamical core:

Mass conservation
Accurate representation of balanced flow and adjustment
Absence or control of computational modes
Geopotential gradient and pressure gradient should produce no unphysical source of vorticity
Terms involving pressure should be energy conserving
Coriolis terms should be energy conserving
No spurious fast propagation of Rossby modes; geostrophic balance should not break down
Axial angular momentum should be conserved
Accuracy approaching second-order
Minimal grid imprinting

ENDGame had a lot of these properties, and we didn’t want to lose them!

Principle IV: Good Wave Properties

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How to choose a grid and discretisation?

A very important paper was that of Staniforth and Thuburn (2012), which laid out desirable properties of a dynamical core:

Mass conservation
Accurate representation of balanced flow and adjustment
Absence or control of computational modes
Geopotential gradient and pressure gradient should produce no unphysical source of vorticity
Terms involving pressure should be energy conserving
Coriolis terms should be energy conserving
No spurious fast propagation of Rossby modes; geostrophic balance should not break down
Axial angular momentum should be conserved
Accuracy approaching second-order
Minimal grid imprinting

Principle V: Local Mass Conservation

When surveyed, the top request from UM users was for LFRic to add local mass conservation

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Our design principles

Seamlessness
Quasi-Uniform
Change as little as possible!
Good wave dispersion properties
Local mass conservation

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How did these principles lead to the design of LFRic?

Let’s start with Principles 1 and 3: how do we get good wave properties on a quasi-uniform grid?

Slide courtesy of Tom Melvin

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How did these principles lead to the design of LFRic?

ENDGame’s good wave dispersion properties are associated with the C-grid and Charney-Phillips variables staggerings

Quad A: 2:1

Quad C:2:1

Tri C: 3:2

Hex C: 3:1

Slide courtesy of Tom Melvin

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Compatible Finite Element Methods

Cotter, C.J. and Shipton, J., 2012. Mixed finite elements for numerical weather prediction. Journal of Computational Physics, 231(21), pp.7076-7091.

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What will we keep the same?

Physics schemes are remaining (largely) the same

Equation set is still non-hydrostatic, but doesn’t (yet!) include deep gravity or correctly account for the temperature-dependence of latent heats^†

Most importantly, time-stepping scheme is still Semi-Implicit Semi-Lagrangian…

^†If you’re interested, please ask me about this later!

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Semi-Implicit Timestep

Semi-Implicit discretisation allows large time steps, but this can mean large Courant numbers!

At 20 km resolution with time steps of 10 mins, horizontal Courant numbers can be as big as 10

Transport

Forcings

“Slow” Physics

“Fast” Physics

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Semi-Implicit Timestep

Transport

Fast physics

Implicit Forcing

Cloud/�Chemistry/Aerosol

Slow physics

x2

Courtesy of Ben Shipway

Explicit Forcing

Slow Physics
Explicit Forcing
Outer loop:

Transport
Fast Physics
Inner loop:

Implicit Forcing
Linear Solve

Cloud/Chemistry/Aerosol

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Dissipation Mechanisms

Like ENDGame, GungHo does not have any explicit dissipation schemes. However, damping is present through:

Physics schemes which representing turbulent or mixing processes

Off-centering in the semi-implicit scheme

A sponge / damping layer at the model top

Upwinding and monotonicity from transport…

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Desirable properties of a transport scheme

Stability and efficiency at large Courant numbers
Local mass conservation
Consistency (preservation of constant mixing ratios)
Positivity/Monotonicity
Suitable for use on the Charney-Phillips grid

Fluxes are computed at faces of each cell

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How does a Flux-Form Semi-Lagrangian scheme work?

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Flux-Form Semi-Lagrangian Scheme

State-of-the-art FFSL scheme is Lin-Rood/COSMIC: accurate and efficient…

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Flux-Form Semi-Lagrangian Scheme

State-of-the-art FFSL scheme is Lin-Rood/COSMIC: accurate and efficient… but…

It is not formally monotonic in 2D when 1D steps have monotone limiter applied

Can’t (cheaply) obtain both conservation and monotonicity

Issues particularly at large Courant number

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SWIFT (Splitting With Improved FFSL for Tracers)

A new formulation of the FFSL scheme, developed by myself and James Kent

Ensures both conservation and monotonicity (when desired)

This is achieved by using a different 2D splitting, and careful definition of departure points

Designed for tracers, but has since been expanded to all variables

Bendall, T.M. and Kent, J., 2025. SWIFT: A Monotonic, Flux-Form Semi-Lagrangian Tracer Transport Scheme for Flow with Large Courant Numbers. Monthly Weather Review, 153(4), pp.565-587.

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Our design principles

Seamlessness
Quasi-Uniform
Change as little as possible!
Good wave dispersion properties
Local mass conservation

This has led to our dynamical core:

Cubed-sphere global mesh
Compatible finite element spatial discretisation
Flux-Form Semi-Lagrangian transport
Semi-Implicit Semi-Lagrangian timestepping

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Some C896 (10km) output …

Melvin, T., Shipway, B., Wood, N., Benacchio, T., Bendall, T., Boutle, I., Brown, A., Johnson, C., Kent, J., Pring, S. and Smith, C., 2024. A mixed finite‐element, finite‐volume, semi‐implicit discretisation for atmospheric dynamics: Spherical geometry. Quarterly Journal of the Royal Meteorological Society, 150(764), pp.4252-4269.

Johnson, C., Shipway, B., Melvin, T., Bendall, T., Kent, J., Boutle, I., Brown, A., Zerroukat, M., Buchenau, B. and Wood, N., 2025. A regional implementation of a mixed finite-element, semi-implicit dynamical core. arXiv preprint arXiv:2503.11528.

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UM vs LFRic

Property	ENDGame	GungHo
Global Grid	Longitude-Latitude	Equiangular Cubed-Sphere
Spatial Discretisation (wave terms and forces)	Finite difference	Lowest-order compatible finite element
Horizontal Staggering	C-grid	Finite element C-grid equivalent
Vertical Staggering	Charney-Phillips	Finite element CP equivalent
Equation Set	Non-hydrostatic compressible Euler, deep atmosphere	Non-hydrostatic compressible Euler, deep atmosphere
Transport	Semi-Lagrangian	Flux-Form Semi-Lagrangian (Conservative)
Timestepping	Semi-Implicit	Semi-Implicit

Physics parametrisations (almost) unchanged

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vertical direction

Cell tops/bottoms on the shifted mesh lie halfway between those on the primary mesh

Bendall, T.M., Wood, N., Thuburn, J. and Cotter, C.J., 2023. A solution to the trilemma of the moist Charney–Phillips staggering. Quarterly Journal of the Royal Meteorological Society, 149(750), pp.262-276.

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(Approximate) Entropy Conservation


-1.52 W/m²	0.25 W/m²

Size of energy correction in climate model improved by conservative potential temperature transport

Reduced frequency of unphysical updrafts in climate model

Thuburn, J., 2022. Numerical entropy conservation without sacrificing Charney–Phillips grid optimal wave propagation. Quarterly Journal of the Royal Meteorological Society, 148(747), pp.2755-2768.

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Other developments

Higher-order elements: investigating use of higher-order elements for wave-like terms. Joshua Dendy has implemented capability to separate horizontal and vertical degree

Physics/chemistry at different resolutions: have the infrastructure to run chemistry and idealised physics on a coarser mesh. See Brown et al (2024)

Mixed precision: much of the model can now run at single precision

Improved thermodynamics: temperature-dependent latent heats, etc.

Deep gravity: current obstacle is initialisation from DA / shallow models

Brown, A., Bendall, T.M., Boutle, I., Melvin, T. and Shipway, B., 2024. Physics–dynamics–chemistry coupling across different meshes in LFRic‐Atmosphere: Formulation and idealised tests. Quarterly Journal of the Royal Meteorological Society, 150(764), pp.4650-4670.

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Global Performance

Overall scientific performance is now close to the UM, but still more to do:

Ancils files need to be computed on cubed-sphere mesh (currently they are regridded from lon-lat mesh)

DA still running on lon-lat mesh

Stability looks at least as good as UM!

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Regional Model

Snapshot from “UKV” domain

One of our main objectives this year is to have scientific performance of regional models matching the UM

Early indications are that conservation properties have improved representation of convection

Domain set up courtesy of Christine Johnson

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