# Moving meshes for global atmospheric modelling

Lead Research Organisation:
Imperial College London

Department Name: Dept of Mathematics

### Abstract

This project is about using moving meshes - r-adaptivity - to improve the predictive power of atmospheric flow simulations, which are used in the fields of numerical weather prediction and climate modelling.

When the atmosphere is simulated on a computer, this is done by dividing the sphere into cells which are arranged in a mesh. There is a conflict between the need for accuracy, which requires smaller (and hence more) cells, and computational efficiency, which increases with the number of cells. A reasonable question to ask is: for a given amount of accuracy, what size of cells do I need? The answer can be provided mathematically, but it depends on what is actually happening in the atmosphere simulation. Much smaller cells are required in the regions of smaller scale features such as atmospheric fronts, cyclones, jets, convective cells etc. It then seems like a waste to choose the same cell size all over the globe even in regions where these features are absent.

An attractive idea is to try to stretch, deform and move the mesh around so that smaller cells are used in the regions of small scale features, and larger cells are used elsewhere. This would mean that a better compromise can be made between accuracy and computational efficiency, thus improving predictive power for the same resource. This idea has been used successfully in many engineering applications, and the goal of this project is to transmit this technology to atmosphere simulation, where it can be used by meteorologists and climate scientists to take their science forward.

There are, however, a number of challenging aspects. Efficient mesh movement algorithms have not previously been developed for the sphere geometry which is needed for global atmosphere simulations. There is the question of how to detect where the mesh should be moved to. It is also the case that it is very challenging to design stable and accurate numerical algorithms for simulating the atmosphere, and these must be adapted to remain stable and accurate under mesh movement. All of these questions and issues will be addressed in this project.

When the atmosphere is simulated on a computer, this is done by dividing the sphere into cells which are arranged in a mesh. There is a conflict between the need for accuracy, which requires smaller (and hence more) cells, and computational efficiency, which increases with the number of cells. A reasonable question to ask is: for a given amount of accuracy, what size of cells do I need? The answer can be provided mathematically, but it depends on what is actually happening in the atmosphere simulation. Much smaller cells are required in the regions of smaller scale features such as atmospheric fronts, cyclones, jets, convective cells etc. It then seems like a waste to choose the same cell size all over the globe even in regions where these features are absent.

An attractive idea is to try to stretch, deform and move the mesh around so that smaller cells are used in the regions of small scale features, and larger cells are used elsewhere. This would mean that a better compromise can be made between accuracy and computational efficiency, thus improving predictive power for the same resource. This idea has been used successfully in many engineering applications, and the goal of this project is to transmit this technology to atmosphere simulation, where it can be used by meteorologists and climate scientists to take their science forward.

There are, however, a number of challenging aspects. Efficient mesh movement algorithms have not previously been developed for the sphere geometry which is needed for global atmosphere simulations. There is the question of how to detect where the mesh should be moved to. It is also the case that it is very challenging to design stable and accurate numerical algorithms for simulating the atmosphere, and these must be adapted to remain stable and accurate under mesh movement. All of these questions and issues will be addressed in this project.

### Planned Impact

The UK economy relies on accurate forecasts in a number of sectors, for example insurance, energy, agriculture, food retail and many leisure activities. The UK and the world also urgently need predictions of the regional impacts of climate change. For example it is currently not known if the UK will become wetter or dryer as a result of climate change. Predictions of the regional impacts of climate change might be improved by using adaptive meshes - having more resolution in the region of interest, within a global model. We cannot promise improved weather and climate forecasts within the lifetime of this project. But the numerical methods described in this proposal are aimed at improving forecast skill for computational cost and improving performance on parallel computers. We expect this project to have its biggest impact by influencing model development at the Met Office, ECMWF and other operational forecasting centres and consequently improving weather and climate predictions. This will be achieved through a number of routes:

* Collaboration with groups in the Met Office:

- Developing the next weather forecasting model suitable for massively parallel computers.

- Data assimilation (the process of initialising a model from observations and satellite data).

- Dispersion of atmospheric pollution.

* Collaboration with the team at ECMWF who are designing and building their next forecasting model suitable for massively parallel computers. ECMWF are currently exploring the use of moving meshes and this project will compliment their efforts.

* Holding two workshops, one near the beginning and one near the end of the project and inviting scientists from international forecasting and research centres such as the National Centre for Atmospheric Science in the US and Los-Alamos National Laboratory.

We will also engage with the public and with school children in a variety of outreach activities aimed at explaining the role of mathematics and computation in weather and climate forecasting. This will include:

* Presentations to the general public on the topics of Forecasting Weather and Climate change at:

- The British Science Festival

- The Cheltenham Science Festival

- The Big Bang Fair

- Irish Maths Week

* Similar presentations to sixth formers at:

- The Royal Institution

- The Maths Inspiration programme

* We will write popular articles based on the material in the proposal project. These will be made available to the general public through the PLUS Maths website, an Internet Maths Magasine which has a large international readership.

* We will engage with the recently launched Climate-Pi project, a collaboration between CliMathNet and the Met Office which aims to develop weather and climate software for

schools which can be implemented and run on a Raspberry Pi platform.

* Collaboration with groups in the Met Office:

- Developing the next weather forecasting model suitable for massively parallel computers.

- Data assimilation (the process of initialising a model from observations and satellite data).

- Dispersion of atmospheric pollution.

* Collaboration with the team at ECMWF who are designing and building their next forecasting model suitable for massively parallel computers. ECMWF are currently exploring the use of moving meshes and this project will compliment their efforts.

* Holding two workshops, one near the beginning and one near the end of the project and inviting scientists from international forecasting and research centres such as the National Centre for Atmospheric Science in the US and Los-Alamos National Laboratory.

We will also engage with the public and with school children in a variety of outreach activities aimed at explaining the role of mathematics and computation in weather and climate forecasting. This will include:

* Presentations to the general public on the topics of Forecasting Weather and Climate change at:

- The British Science Festival

- The Cheltenham Science Festival

- The Big Bang Fair

- Irish Maths Week

* Similar presentations to sixth formers at:

- The Royal Institution

- The Maths Inspiration programme

* We will write popular articles based on the material in the proposal project. These will be made available to the general public through the PLUS Maths website, an Internet Maths Magasine which has a large international readership.

* We will engage with the recently launched Climate-Pi project, a collaboration between CliMathNet and the Met Office which aims to develop weather and climate software for

schools which can be implemented and run on a Raspberry Pi platform.

### Publications

Wimmer G
(2020)

*Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations*in Journal of Computational Physics
Van Sebille E
(2018)

*Lagrangian ocean analysis: Fundamentals and practices*in Ocean Modelling
Shipton J
(2018)

*Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere*in Journal of Computational Physics
Natale A
(2018)

*Corrigendum to: A variational $\boldsymbol{H}({\rm div})$ finite-element discretization approach for perfect incompressible fluids*in IMA Journal of Numerical Analysis
Natale A
(2016)

*Compatible finite element spaces for geophysical fluid dynamics*in Dynamics and Statistics of the Climate System
Natale A
(2018)

*A variational $\boldsymbol{H}({\rm div})$ finite-element discretization approach for perfect incompressible fluids*in IMA Journal of Numerical Analysis
Melvin T
(2018)

*Choice of function spaces for thermodynamic variables in mixed finite-element methods*in Quarterly Journal of the Royal Meteorological Society
McRae A
(2018)

*Optimal-Transport--Based Mesh Adaptivity on the Plane and Sphere Using Finite Elements*in SIAM Journal on Scientific Computing
Cotter CJ
(2016)

*Mixed finite elements for global tide models.*in Numerische mathematik
Cotter C
(2016)

*Embedded discontinuous Galerkin transport schemes with localised limiters*in Journal of Computational Physics
Budd C
(2018)

*The scaling and skewness of optimally transported meshes on the sphere*in Journal of Computational Physics
Bendall T
(2020)

*A compatible finite-element discretisation for the moist compressible Euler equations*in Quarterly Journal of the Royal Meteorological SocietyDescription | We have developed a robust numerical algorithm for adapting meshes for weather forecasting by evolving the mesh in time so that more gridpoints are concentrated in regions where more accuracy is needed. We have also demonstrated that weather can be simulated on these meshes in a stable and accurate manner, and that more accuracy can be obtained for the same number of grid points if this approach is used. |

Exploitation Route | This technology could be used by atmosphere/ocean scientists to produce meshes for models that make more efficient use of computational resources. They could also be used by operational weather forecasting centres in their data assimilation algorithms. |

Sectors | Environment |

Title | Gusto |

Description | A Python library for compatible finite element dynamical cores |

Type Of Technology | Software |

Year Produced | 2016 |

Open Source License? | Yes |

Impact | This software is providing a testbed for the development of the Gung Ho dynamical core for the Met Office forecast model. |

URL | http://firedrakeproject.org/gusto/ |

Title | dcore |

Description | Software implementing a numerical model for a 3D dynamical core on the sphere using compatible finite element methods. |

Type Of Technology | Software |

Year Produced | 2016 |

Open Source License? | Yes |

Impact | This software is being used to benchmark numerical algorithms being developed for the UK Met Office forecast model. |

URL | https://github.com/firedrakeproject/dcore |