Stieltjes Functions and Spectral Analysis in Sea Ice Physics

极地海冰是地球气候系统的重要组成部分。北极海冰的急剧减少不仅会影响极地海洋环境，还会影响北半球的天气模式、洋流、风暴路径和降水量等。海冰作为一种材料，是纯冰与毫米级盐水包裹体和厘米级多晶微观结构的多尺度复合物。就连浮冰本身也是由海水中的浮冰组成的颗粒状复合物。海冰及其在气候中的作用建模的一个主要挑战是如何利用小尺度结构的信息来发现与过程研究和粗粒度气候模型相关的大尺度上的有效或均质特性。同样令人感兴趣的是从大尺度观测（如遥感）中恢复控制小尺度过程的参数的逆问题。在这个项目中，通过利用有效参数的特殊数学性质来解决这些核心问题，这些参数在海冰建模的几个重要问题领域中是共同的。它们包括通过海冰的流体和电磁传输，盐水对流增强的热通量等扩散过程，以及浮冰中的海洋表面波。海冰与其他自然形成的人造复合材料有很多相似之处。我们关于海冰的数学结果将为其他复合材料提供新的见解和发现，反之亦然。这项研究将有助于推进海冰在气候模型中的表现方式，并改善对地球海冰及其所支持的生态系统命运的预测。该项目还将为本科生和研究生提供支持、培训和研究机会。解析延拓法是均匀化数学理论中一个强有力的方法，它为复合材料的有效参数提供了积分表示，并将其视为参数的Stieltjes函数。复合微几何的复杂性被提取成自伴随算子的谱性质，就像量子物理中的哈密顿算子一样，当系统离散化时，它就变成了随机矩阵。该方法在海冰上的早期应用主要集中在遥感和电磁特性上，海冰被视为两相复合材料。将该方法扩展到多晶介质、平流扩散过程和通过冰堆的表面波，基于最粗略的信息，如积分中光谱测量的质量，即海冰作为两相复合物的盐水体积分数，或表面波被浮冰覆盖的海洋面积分数，只能得出有效参数的基本界限。然而，利用与量子物理的随机矩阵理论描述的深度相似之处，将安德森跃迁概念，如场局域化，迁移率边缘，特征值排斥和带隙带入两相复合材料经典输运的均匀化中。在资助的工作中，我们将探索这种方法在上述问题领域中的新见解，其中有效参数是Stieltjes函数。例如，海冰的有效性质如何取决于晶体的大小和结构，某些海浪能否被冰团的几何形状定位，以及复合几何形状的分形性质如何影响有效输运和光谱性质。这种方法开启了许多数学兴趣问题，同时提供了新的工具来解决在快速变化的极地海洋环境中理解海冰过程至关重要的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Polar sea ice is a critical component of Earth's climate system. Precipitous losses of Arctic sea ice affect not only the polar marine environment, but can impact, for example, weather patterns, ocean currents, storm tracks, and precipitation amounts in the Northern Hemisphere. As a material, sea ice is a multiscale composite of pure ice with millimeter-scale brine inclusions and centimeter-scale polycrystalline microstructure. Even the ice pack itself is a granular composite of ice floes in a sea water host. A principal challenge in modeling sea ice and its role in climate is how to use information on small scale structure to find the effective or homogenized properties on larger scales relevant to process studies and coarse-grained climate models. Also of interest is the inverse problem of recovering parameters controlling small scale processes from large scale observations, such as in remote sensing. These central issues are addressed in this project by exploiting special mathematical properties of effective parameters that are common to several important problem areas in sea ice modeling. They include fluid and electromagnetic transport through sea ice, diffusion processes like heat flux enhanced by brine convection, and ocean surface waves in the ice pack. Sea ice shares close similarities with other naturally occurring and engineered composites. Our mathematical results on sea ice will give new insights and findings about other composite materials, and vice versa. This research will help to advance how sea ice is represented in climate models and improve projections of the fate of Earth's sea ice packs and the ecosystems they support. This project will also provide support, training, and research opportunities for undergraduate and graduate students. A powerful approach in the mathematical theory of homogenization is the analytic continuation method, which provides integral representations for the effective parameters of composite materials, treated as Stieltjes functions of their parameters. The complexities of composite microgeometries are distilled into the spectral properties of self-adjoint operators, like the Hamiltonian in quantum physics, which become random matrices when the system is discretized. Early applications of the method to sea ice focused on remote sensing and electromagnetic properties, with sea ice treated as a two-phase composite. Extensions of the method to polycrystalline media, advection diffusion processes, and surface waves through the ice pack have yielded only elementary bounds on the effective parameters, based on the coarsest information like the mass of a spectral measure in the integral, which is the brine volume fraction for sea ice as a two-phase composite, or the area fraction of ocean covered by ice floes for surface waves. However, exploiting deep parallels with random matrix theory descriptions of quantum physics brings Anderson transition concepts like field localization, mobility edges, eigenvalue repulsion, and band gaps into homogenization for classical transport in two-phase composites. In the funded work we will explore new insights that this approach can give us in the above problem areas, where the effective parameters are Stieltjes functions. For example, how do sea ice effective properties depend on crystal size and structure, can some ocean waves be localized by the ice pack geometry, and how does the fractal nature of composite geometry influence effective transport and spectral properties. This approach opens many questions of mathematical interest, while at the same time provides novel tools to address questions of critical importance in understanding sea ice processes in the rapidly changing polar marine environment.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.