Towards a new generation of adaptive climate and weather models

小波多分辨率分析是近30年来数学领域最重要的突破之一。 小波方法在纯数学和应用数学中的影响得到了认可，Yves Meyer因“他在小波数学理论发展中的关键作用”而获得2017年阿贝尔奖。 小波方法是分析、统计和逼近理论中常用的方法。 该方案应用小波分析的多尺度形式自适应求解球面上的非线性偏微分方程。 特别是，我将使用我开发的数学工具来建立一个完整的气候模型。这一创新的气候模型将用于研究大气和海洋物理学中的开放性问题。该研究计划将影响数值分析（球形拓扑结构中PDE的自适应数值方法）和气候建模。 天气和气候模型使用静态计算网格，无法适应快速发展的风暴或在精度要求较低的地方进行粗化。这意味着计算资源的使用效率低下，并且所产生的误差在空间和时间上不是均匀的。在需要更高精度的情况下，例如在人口稠密的地区，也难以提高分辨率。 第一个目标是将我们的自适应二维浅水模式扩展到三维多层静力模式。在这一点上，它将形成一个所谓的“动力核心”，可以在其上添加物理效应（例如辐射、云、地形的影响、植被），以建立一个真正的气候模型。 这将是第一个大气和海洋的动态适应动力核心。 第二个目标是开发一个尺度感知物理模型，以考虑次网格尺度的物理效应，如云的形成和太阳辐射传输。这些过程发生的尺度从一万公里到亚毫米，从几百年到毫秒不等。天气模型使用数据同化，以最佳方式将温度、风等观测结果纳入数值模拟。 第三个目标是开发新的技术，利用我们的数值模式的动态适应的数据同化。 我将培养11名高素质的人才（1名博士后研究员，2名博士生，3名硕士生和5名理学士生）在这个研究计划的五年跨度。 该提案有助于国际社会（例如加拿大环境部、ECMWF、NCAR、法国气象局）开发下一代天气和气候模型的努力。加拿大在数值天气预报和气候建模方面一直处于世界领先地位，在这一研究项目中获得的知识将使加拿大在开发新模型以应对新出现的挑战方面保持领先地位。

Wavelet multiresolution analysis is one of the most significant mathematical breakthroughs of the last 30 years. The impact of wavelet methods in both pure and applied mathematics was recognized when Yves Meyer was awarded the 2017 Abel Prize for “his pivotal role in the development of the mathematical theory of wavelets.” Wavelet methods are now common in analysis, statistics and approximation theory. This proposal applies the multiscale formalism of wavelet analysis to adaptively solve nonlinear partial differential equations (PDEs) on the sphere. In particular, I will use the mathematical tools I develop to build a full climate model. This innovative climate model will be applied to investigate open problems in the physics of atmospheres and oceans. The research program will impact both numerical analysis (adaptive numerical methods for PDEs in spherical topology) and climate modelling. Weather and climate models use static computational grids which are unable to adapt to resolve rapidly developing storms or coarsen where less accuracy is needed. This means that computational resources are used inefficiently and that the resulting errors are not uniformly bounded in space and time. It is also difficult to increase resolution where more accuracy is needed, for example over densely populated areas. The first objective extends our adaptive 2D shallow water model to a 3D multilayer hydrostatic model. At this point it will form a so-called "dynamical core" on which physical effects (e.g. radiation, clouds, the effect of topography, vegetation) can be added to build a true climate model. This will be the first dynamically adaptive dynamical core for both atmospheres and oceans. The second objective develops a scale aware physics model to account for subgrid scale physical effects, such as cloud formation and solar radiation transfer. These processes occur on scales varying from ten thousand kilometres to submillimeter and from centuries to milliseconds. Weather models use data assimilation to optimally incorporate observations of temperature, wind etc. into the numerical simulation. The third objective is to develop new techniques in data assimilation that take advantage of the dynamic adaptive of our numerical model. I will train 11 Highly Qualified Personnel (1 post-doctoral fellow, 2 PhD students, 3 MSc students and 5 BSc students) over the five-year span of this research program. This proposal contributes to an international effort (e.g. by Environment Canada, ECMWF, NCAR, Météo France) to develop the next generation of weather and climate models. Canada has been a world leader in numerical weather prediction and climate modelling, and the knowledge gained in this research project will keep Canada at the forefront as new models are developed to deal with emerging challenges.