CMG: Analytical and Computational Studies of Magma Dynamics

研究人员致力于加深对地幔中部分熔岩(岩浆)动力学行为的理解。这一问题对大尺度地幔对流和板块构造的动力学以及地球的地球化学演化具有重要的意义。岩浆运移也是流体/固体耦合力学中一个非常有趣的问题，因为它需要了解低粘度反应流体在强变形和可渗透基质中的非线性相互作用。在过去的20年里，地幔中岩浆迁移的主要理论是一套数学模型(偏微分方程或PDE)，这些模型描述了宏观参数的演化，例如给定位置和时间的熔岩的孔隙度或比例。通过简化的模型问题和更复杂的地学具体应用，这些方程被证明对探索岩浆系统的行为是有用的。然而，最近的实验工作表明，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，随着孔隙率变小，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，以衡量这些模型的实用性，并改进对大尺度地幔动力学中部分熔融区的数学描述(例如，通过引入物理上合理的正则化)。这一建议的目的是将非线性偏微分方程分析方面的专业知识与岩浆动力学的物理和计算相结合，以开发更好的洞察力和复杂的流体/固体耦合系统的模型。这项工作是迈克尔·I·温斯坦(哥伦比亚应用数学)和马克·斯皮格尔曼(哥伦比亚大学地球和环境的联合任命)共同努力的结果。科学和应用数学)，将是吉迪恩·辛普森的主要博士研究，他由温斯坦和斯皮格尔曼共同指导。这个项目将解决两个独立的问题，这两个问题在部分熔融地区出现的规模不同。在大尺度上，孔隙度的变化可以传播为具有鲜为人知的非线性色散项的色散非线性波。在较小的尺度上，实验室实验表明，固体基质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的交叉点工作的主要资金来源。这项工作还应该在科学和工程中广泛应用于涉及可渗透变形固体中的流体流动的问题，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.