CMG: Analytical and Computational Studies of Magma Dynamics

研究人员专注于对地幔中部分熔融岩石（岩浆）的动力学行为有更深入的了解。这一问题对大尺度地幔对流动力学和板块构造动力学以及地球地球化学演化具有重要意义。岩浆运移在流体/固体耦合力学中也是一个非常有趣的问题，因为它需要理解低粘度反应性流体在强变形和渗透性基质中的非线性相互作用。在过去的20年里，地幔岩浆迁移的主要理论是一个数学模型系统（偏微分方程或PDEs），它描述了宏观数量的演化，例如在给定位置和时间的孔隙度或熔融岩石的比例。这些方程已被证明对通过简化模型问题和更复杂的地球科学特定应用来探索岩浆系统的行为是有用的。然而，最近的实验工作表明，在这些模型的实验和预测之间存在一些重要的定量差异。特别是，当孔隙度变小时，这些模型变得奇异。这些和其他结果表明，有必要对当前方程的行为进行建模和数学理解，以衡量这些模型的效用，并为大尺度地幔动力学中的部分熔融区域开发改进的数学描述（例如，通过引入物理上合理的正则化）。本提案的目的是将非线性偏微分方程的分析专业知识与岩浆动力学的物理和计算相结合，以更好地了解复杂的流体/固体耦合系统并建立更好的模型。这项工作是Michael I. Weinstein（哥伦比亚大学应用数学）和Marc Spiegelman（哥伦比亚大学地球与环境联合任命）的合作成果。科学与应用数学)，并将成为Gideon Simpson的主要博士研究，他由Weinstein和Spiegelman共同指导。该项目将解决在部分熔融区域以不同规模出现的两个独立问题。在大尺度上，孔隙度的变化可以作为色散的非线性波传播，其中有一个鲜为人知的非线性色散项。在更小的尺度上，实验室实验表明，固体基体的剪切变形可以驱动富熔体带的局部化。我们将考虑一系列的解析和数值模型问题，以更好地理解当前的模型，并探索改进公式的方法。这个项目将成为一个有前途的研究生（吉迪恩·辛普森）在应用数学和地球科学交叉领域工作的主要资金来源。这项工作还应在科学和工程中具有普遍的应用，以解决涉及可渗透可变形固体中流体流动的问题，例如石油工程、水文学和核/有毒废物限制中出现的问题。

The investigators are focused on developing a deeper understanding of the behavior of the dynamics of partially molten rock (magma) in the Earth's mantle. This problem has important implications for the dynamics of large scale mantle convection and plate tectonics as well as the geochemical evolution of the planet. Magma migration is also an intrinsically interesting problem in coupled fluid/solid mechanics as it requires understanding the non-linear interactions of low viscosity reactive fluids in a strongly deformable and permeable matrix. For the past 20 years, the predominant theory for magma migration in the mantle has been a system of mathematical models (partial differential equations or PDEs), which describe the evolution of macroscopic quantities, e.g. the porosity or proportion of molten rock at a given position and time. These equations have proved useful for exploring the behavior of magmatic systems through both simplified model problems and more complex geoscience specific applications. However, more recent experimental work suggests that there exist some important quantitative discrepancies between the experiments and predictions of these models. In particular, these models become singular as the porosity becomes small. These and other results suggest that a modeling and mathematical understanding of the behavior of the current equations is necessary to gauge the utility of these models and to develop improved mathematical descriptions (e.g. by introducing physically reasonable regularizations) for partially molten regions in large scale mantle dynamics. The purpose of this proposal is to combine expertise in the analysis of non-linear PDE's with the physics and computation of magma dynamics to develop better insight and better models for complex coupled fluid/solid systems. This work is collaborative effort between Michael I. Weinstein (Columbia Applied Mathematics) and Marc Spiegelman (Columbia Joint appointment between Earth and Env. Sciences and Applied Math) and will be the primary Ph. D. research of Gideon Simpson who is jointly supervised by Weinstein and Spiegelman. This project will attack two separate problems that arise at different scales in partially molten regions. At large scales, variations in porosity can propagate as dispersive non-linear waves with a poorly understood non-linear dispersion term. At smaller scales, laboratory experiments demonstrate that shear deformation of the solid matrix can drive localization of melt-rich bands We will consider a series of analytic and numerical model problems to develop a better understanding of the current models as well as exploring ways to improve the formulation. This project will form the primary source of funding for a promising graduate student (Gideon Simpson) to work at the intersection of Applied Mathematics and Earth Science. This work should also have general applications in science and engineering to problems involving the flow of fluids in permeable deformable solids such as those arising in petroleum engineering, hydrology and nuclear/toxic waste confinement.