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

数值天气预报（NWP）是一门将大气的计算机模型与观测数据和测量相结合以产生天气预报的科学。任何数值预报模式的一个关键组成部分是“动力核心”，这是一个模型，动态量（风速和风向，温度，压力和密度）如何在大气中随时间演变。所有这些量都存储在网格上（例如，在经纬度网格上，风速存储在每个纬度和经度上），关键问题是网格框需要多小才能准确预测天气。众所周知，数值预报模式的预测能力随着网格框变小而提高（因为该模式更准确地表示小尺度运动，如风暴）。然而，制作更小的盒子意味着需要存储更多的动态量值，因此模型需要更多的计算能力和时间。近年来，人们对使用“自适应网格”越来越感兴趣：在地球仪的不同区域可以具有不同大小的框的网格。这可能意味着在不列颠群岛上有更小的网格框，以便为该地区做出更准确的预测。这也可能意味着在北大西洋风暴周围的区域中具有较小的网格框;风暴正在移动，因此网格框的大小可能需要改变，以便较小的网格框跟随风暴。这种方法被称为自适应网格细化，并允许模型需要更少的计算能力和时间，因为小盒子只在需要的地方使用。当使用自适应网格细化时，通常使用三角形框而不是正方形，因为它们更容易镶嵌，因此可以排列成具有非常复杂结构的网格。使用自适应网格带来了新的挑战，因为多年的研究已经进入了经纬度网格上的NWP模型。特别重要的是，设计一个动力学核心，正确地表示波在大气中的传播以及由于地球旋转而发生的压力和速度之间的相互作用（称为地转）。这可以通过使用“交错网格”来实现，该网格在每个网格框中的不同位置处存储速度、密度和压力。最近有很多关于在三角形上构建交错网格的最佳方法的研究：该项目是关于一种特定的选择（称为P1 dg-P2），与其他方法相比，该方法已被证明具有非常稳定和准确的波传播和地转表示。该项目的目的是在简化的模式背景下对P1 dg-P2进行彻底的调查，这将使业务预报中心（如英国气象局）的科学家能够评估P1 dg-P2是否应该进一步发展为NWP。

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.