Geometric foundation of invariant and conservative parameterization schemes

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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. **