Homogenization for Sea Ice

海冰是地球气候系统的重要组成部分，也是气候变化的主要指标。过去几十年观察到的夏季北极海冰急剧减少对地球气候系统产生了重大影响，远远超过了大多数全球气候模型的预测。气候科学的基本挑战之一是在气候模式中开发更严格的海冰表示法，并将重要的小尺度过程和结构纳入这些大尺度模式中。由这笔资金资助的研究将解决这一核心问题。研究人员将利用复合材料的数学和统计物理来开发严格计算海冰的有效或同质化属性的方法，这对于改进对地球冰包命运的预测以及极地生态系统可能做出的反应是必要的。全球变暖的影响是深远的，更好地了解海冰及其在气候中的作用将提高我们预测影响到大量人口的风暴路径、降水和温度模式等变化的能力。此外，这项工作将促进我们对复合材料的性能以及它们在工业、工程和医疗应用中的应用的理解。这笔赠款中的研究主题包括一系列关键问题，这些问题不仅将推动海冰如何在气候模型中表示，而且将推动复合材料数学的界限。边缘冰带(MIZ)是冰盖向开阔海洋过渡的外部区域，其宽度是一个重要的气候长度尺度。最近在客观识别Miz宽度和几何形状方面的进展导致了随着气候变暖而发现的显著趋势，这些趋势基于满足拉普拉斯方程的理想化浓度场。在这里，我们将把这一分析推广到多尺度传输方程中包括一个不均匀的、有效的扩散系数，它可以通过反演方案捕捉到实际的卫星导出的浓度场。然后，我们将研究该有效系数如何与有关浮选几何形状和构型的较小尺度信息相关联。这种方法将为分析研究MIZ动力学，包括其对变形的敏感性提供基础。在相关工作中，我们将探索类似于最近获得的两相复合材料有效弹性的多圆盘积分公式的冰袋流变学的表示式。这种表示涉及环面上的谱测量，其矩与微结构统计有关，我们将研究谱测量的结构及其对复合微几何的依赖。

Sea ice is a critical component of Earth's climate system, and a leading indicator of climate change. The precipitous losses of summer Arctic sea ice observed in the past few decades have a significant impact on Earth's climate system, and have far outpaced the projections of most global climate models. One of the fundamental challenges of climate science is to develop more rigorous representations of sea ice in climate models, and incorporate important small scale processes and structures into these large scale models. The research funded by this grant will address this central issue. The investigators will use the mathematics of composite materials and statistical physics to develop methods of rigorously calculating the effective or homogenized properties of the sea ice pack which are necessary for improving projections of the fate of Earth's ice packs, and how polar ecosystems may respond. The effects of planetary warming are far reaching, and a better understanding of sea ice and its role in climate will improve our ability to predict changes in storm tracks, precipitation and temperature patterns, etc., affecting large populations. Moreover, this work will advance our understanding of the properties of composite materials and their use in industrial, engineering, and medical applications. The research topics in this grant encompass a range of key problems which will not only advance how sea ice can be represented in climate models, but will the push the boundaries of the mathematics of composite materials. The marginal ice zone (MIZ) is the outer region of the ice pack where dense pack ice transitions to open ocean, and its "width" is an important climatic length scale. Recent advances in objectively identifying MIZ width and geometry have led to the discovery of striking trends as the climate has warmed, which are based on an idealized concentration field satisfying Laplace's equation. Here we will generalize this analysis to include an inhomogeneous, effective diffusivity in the multiscale transport equation, which can, through inversion schemes, capture the actual satellite-derived concentration field. We will investigate how this effective coefficient is then related to smaller scale information about floe geometry and configurations. This approach will provide a basis for analytical investigation of MIZ dynamics including its susceptibility to deformation. In related work, we will explore representations for ice pack rheology similar to a recently obtained polydisc integral formula for the effective elasticity of two phase composites. Such representations involve spectral measures on the torus whose moments are related to microstructural statistics, and we will investigate the structure of the spectral measures and their dependence on composite microgeometry.