Unstructured mesh dynamical core for atmospheric modelling using geophysically-optimal finite elements

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Numerical Weather Prediction (NWP) is the science of combining computer models of the atmosphere with observational data and measurements to produce a forecast of the weather. A key ingredient of any NWP model is the 'dynamical core' which is a model of how the dynamical quantities (wind speed and direction, temperature, pressure and density) evolve in the atmosphere over time. All of these quantities are stored on a grid (for example on a latitude-longitude grid with wind speed being stored at each degree of latitude and longitude), and a crucial question is how small the grid boxes need to be for accurate weather prediction. It is known that the predictive skill of an NWP model increases as the grid boxes get smaller (because the model represents small scale motion such as storms more accurately). However, making smaller boxes means that more values of the dynamical quantities need to be stored, and hence the model requires more computational power and time. In recent years, there has been increased interest in using 'adaptive grids': grids which can have different sizes of boxes in different regions of the globe. This could mean having smaller grid boxes over the British Isles so that a more accurate forecast is made for that region. It could also mean having smaller grid boxes in the region surrounding a storm in the North Atlantic; the storm is moving and hence the size of grid boxes might need to change so that the smaller grid boxes follow the storm. This approach is called adaptive mesh refinement, and allows models which require less computational power and time because small boxes are only used where they are needed. When adaptive mesh refinement is used, triangular boxes are often used instead of squares, because they tessellate more easily and so can be arranged into grids with a very complex structure. Using adaptive meshes creates new challenges, since many years of research have gone into making NWP models on latitude-longitude grids. In particular, it is important to design a dynamical core which correctly represents the propagation of waves through the atmosphere and the interaction between the pressure and the velocity (called geostrophy) that takes place due to the rotation of the Earth. This can be achieved by using a 'staggered grid' that stores velocity, density and pressure at different locations in each grid box. There has been much recent research on the best way to construct a staggered grid on triangles: this project is about a particular choice (called P1dg-P2) which has been shown to have very stable and accurate wave propagation and representation of geostrophy compared to other methods. The aim of this project is to make a thorough investigation of P1dg-P2 in a simplified model context which will allow scientists at operational forecast centres (such as the UK Met Office) to assess whether P1dg-P2 should be developed further for NWP.