Geometric foundation of invariant and conservative parameterization schemes

我的研究计划的首要目标是推动几何参数化领域的发展。我的研究领域是应用数学、数值分析、流体力学和气象学。*数值模式是可靠的天气和气候预报的主要来源。随着计算能力的增加，模型变得更加准确，能够比以往任何时候都能分辨更广泛的空间和时间尺度。尽管如此，即使是最先进的地球数值模型也不能解决对准确的天气和气候预测至关重要的所有过程。由于这些未解决的过程不能被省略，它们必须以近似的方式并入模型中。这被称为参数化法。参数化是建立大气-海洋系统数值模式的重要手段，也是现代气象学和海洋学的主要研究方向之一。关键的范例是利用水热动力学原始控制方程的几何性质来制定在近似方程中保留这些性质的参数化方案。这将构成数值模型的基础。这些性质将包括对称性和守恒定律，这是任何物理模型最重要的特征之一。几何保持的参数化方案的构建将主要依赖于微分方程组的群分析方法。*虽然本研究计划的主要重点将是发展可用于寻找保持几何的参数化方案的数学技术，但我们也将通过为包括湍流、海洋中的涡流和大气对流在内的重要物理过程构建几个闭合模型来说明这些技术。*本研究计划将为传统上必须在数值模式中参数化的各种过程提供必要的工具来构建物理和数学上一致的参数化方案。这将为未来天气和气候模式的改进提供基础。由于对称性和守恒定律在物理学、工程学和数学科学中具有核心意义，预期的进展也将推动这些领域的发展。这项研究计划还将在应用数学和计算数学的交叉点上培养几名本科生和研究生，从而为他们提供机会学习各种同样重要的数学和技术技能，这些技能在学术界和行业中都是出类拔萃的。**

The overarching goal of my research program is to advance the field of geometric parameterization. My research area lies at the intersection of applied mathematics, numerical analysis, fluid mechanics and meteorology.***Numerical models are the prime source of reliable weather and climate predictions. With increase in computational power, models become more accurate and are capable of resolving a wider range of spatial and temporal scales than ever before. Despite this, even the most advanced numerical Earth models cannot resolve all processes that are important for precise weather and climate predictions. Since these unresolved processes cannot be omitted, they have to be incorporated into the model in an approximate manner. This is referred to as parameterization. Parameterization is important to close a numerical model for the atmosphere-ocean system and thus one of the main research directions in modern meteorology and oceanography.***I propose a novel framework for developing physical parameterization schemes. The key paradigm is to use the geometric properties of the original governing equations of hydro-thermodynamics to formulate parameterization schemes that preserve these properties in the approximated equations. This will then form the basis for numerical models. These properties will include both symmetries and conservation laws, which are among the most important features of any physical model. The construction of geometry-preserving parameterization schemes will mainly rely on methods from the group analysis of differential equations.***While the main emphasis of this research program will be the development of mathematical techniques that can be used for finding geometry-preserving parameterization schemes, we will also illustrate these techniques by constructing several closure models for important physical processes, including turbulence, eddies in the ocean, and atmospheric convection.***This research program will provide the necessary tools for constructing physically and mathematically consistent parameterization schemes for a variety of processes that traditionally have to be parameterized in numerical models. This will provide a basis for the improvement of future weather and climate models. As symmetries and conservation laws are of central relevance in physics, engineering and the mathematical sciences, progress anticipated will advance these fields as well. This research program will also train several undergraduate and graduate students at the intersection of applied and computational mathematics, thus providing them with the opportunity to learn a wide variety of mathematical and technical skills equally important to excel in both academia and industry. **