Moving meshes for Global Atmospheric Modelling

Lead Research Organisation: University of Reading
Department Name: Meteorology


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.

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. Moving meshes may enable simulations that are currently impractical - resolving high impact weather in the UK in a global climate model. 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.


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Description R-adaptivity, or moving meshes, means mesh re-distribution in order to vary the local resolution. It is an attractive form of adaptivity for global atmospheric modelling since it does not involve altering the mesh connectivity, does not create load balancing problems on parallel computers, does not require mapping solutions between different meshes, does not lead to sudden changes in resolution and can be retro-fitted into existing models. R-adaptivity requires the numerical solution of equations to move the mesh so that it does not tangle and this can be achieved by using methods based on optimal transport. Whilst these methods have been developed successfully in two and three dimensional Euclidean space, r-adaptivity using optimal transport has not yet been implemented on the surface of a sphere which would be necessary for global atmospheric modelling. Here we show that the optimal transport problem on the surface of a sphere will lead to meshes that do not tangle and we develop a numerical technique to solve an equation of Monge-Ampère type on the surface of the sphere in order to solve the optimal transport problems and generate meshes that are equidistributed with respect to various prescribed monitor functions. This optimal transport problem or the Monge-Ampere equation have not before been solved on the surface of a sphere. This is necessary for creating r-adaptive models of the global atmosphere with locally high resolution, which could enable otherwise impossible simulations, with locally high resolution where required within a global model.

Having created the first numerical solution of the optimal transport problem on the surface of the sphere, we next attacked the problem of finding an efficient numerical method. Some existing numerical methods for solving the optimal transport problem on a plane were compared and some new, more efficient ones developed, aiming for sufficiently cheap computational cost so that the optimal transport problem can be incorporated at every time-step of a weather forecasting model.

A recurring problem with using adaptivity over a complex terrain is the changing volume of the domain as terrain is resolved. We have developed a novel solution to this problem, solving a transport equation for the local size of the domain. This enables us to conserve mass without squashing the mass into a different domain size.
Exploitation Route Further collaboration with the Met Office and ECMWF
Sectors Aerospace, Defence and Marine,Environment

Title AtmosFOAM 
Description A suite of numerical models and methods for atmospheric modelling using emerging numerical techniques 
Type Of Material Computer model/algorithm 
Year Produced 2014 
Provided To Others? Yes  
Impact Papers and collaborative code development with students 
Title AMMM - Adaptive Moving Mesh Methods for Atmospheric Modelling 
Description Code for numerical solution of the equations governing mesh movements and numerical methods for solution of equations of atmospheric motion on moving meshes. 
Type Of Technology Software 
Year Produced 2016 
Open Source License? Yes  
Impact Scientific publications 
Title AtmosFOAM 
Description A set of library routines and applications for simulating the atmosphere using arbitrary meshes 
Type Of Technology Software 
Year Produced 2015 
Open Source License? Yes  
Impact This is a research tool and has enabled the research for all of my publications. 
Description Dynamical Core Model Intercomparison Project (DCMIP) 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Series of lectures at the Dynamical Core Model Intercomparison Project workshop in order to train post-graduates and weather forecast model developers in the latest modelling techniques.
Year(s) Of Engagement Activity 2016
Description Moving mesh workshop 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Workshop to disseminate the results of the project to operational weather forecasting centres. The UK Met Office are considering the methods developed for their next model
Year(s) Of Engagement Activity 2018