Towards a new generation of adaptive climate and weather models

小波多分辨率分析是近30年来最重要的数学突破之一。当Yves Meyer因“他在小波数学理论发展中的关键作用”而被授予2017年阿贝尔奖时，小波方法在纯数学和应用数学中的影响得到了认可。******小波方法现在在分析、统计和近似理论中很常见。本文将小波分析的多尺度形式应用于球面上非线性偏微分方程的自适应求解。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。这个创新的气候模型将被应用于研究大气和海洋物理学中的开放性问题。该研究项目将影响数值分析（球面拓扑中偏微分方程的自适应数值方法）和气候模拟。******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需要较低精度的粗糙天气。这意味着计算资源的使用效率低下，并且产生的错误在空间和时间上没有统一的界限。在需要更高精度的地方，例如人口密集地区，也很难提高分辨率。******第一个目标是将我们的自适应2D浅水模型扩展到3D多层流体静力模型。在这一点上，它将形成一个所谓的“动力核心”，物理效应（如辐射，云，地形的影响，植被）可以加入其中，以建立一个真正的气候模型。这将是大气和海洋的第一个动态自适应动态核心。******第二个目标是开发一个尺度感知物理模型，以解释亚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，从几个世纪到几毫秒不等。天气模式利用数据同化将温度、风等观测数据最优地纳入数值模拟。第三个目标是开发新的数据同化技术，利用我们的数值模型的动态适应性。******我将在这个研究项目的五年时间里培养11名高素质人才（1名博士后，2名博士，3名硕士，5名理学士）。******这一建议有助于国际社会（例如加拿大环境部、ECMWF、NCAR、法国气象组织）开发下一代天气和气候模式。加拿大在数值天气预报和气候模拟方面一直处于世界领先地位，在该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挑战时保持领先地位。**************

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.**************