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。过去几十年观测到的夏季北极海冰的急剧减少对地球气候系统产生了重大影响，其速度远远超过了大多数全球气候模式的预测。气候科学的基本挑战之一是在气候模型中建立更严格的海冰表示，并将重要的小尺度过程和结构纳入这些大尺度模型中。这项拨款资助的研究将解决这一核心问题。研究人员将使用复合材料的数学和统计物理学来开发严格计算海冰有效或均匀特性的方法，这些方法对于改善对地球冰盖命运的预测以及极地生态系统的反应是必要的。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风暴路径、降水和温度模式等变化的能力，这些变化会影响到大量人口。此外，这项工作将促进我们对复合材料的特性及其在工业、工程和医疗应用中的应用的理解。这项资助的研究课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现出来，而且将推动复合材料数学的边界。边缘冰带（MIZ）是稠密的浮冰向开阔海洋过渡的外围区域，其“宽度”是一个重要的气候长度尺度。最近在客观地确定MIZ宽度和几何形状方面取得的进展导致了气候变暖的惊人趋势的发现，这是基于满足拉普拉斯方程的理想浓度场。在这里，我们将推广这一分析，在多尺度输运方程中包括一个非均匀的有效扩散系数，它可以通过反演方案捕获实际的卫星衍生浓度场。我们将研究这个有效系数是如何与较小尺度的关于浮冰几何形状和结构的信息相关联的。该方法将为包括变形敏感性在内的MIZ动力学分析研究提供基础。在相关工作中，我们将探索冰袋流变学的表示，类似于最近获得的两相复合材料有效弹性的多盘积分公式。这种表征涉及到环面上的光谱测度，其矩与微观结构统计有关，我们将研究光谱测度的结构及其对复合微观几何的依赖。

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.