Simulation of Multiphase Flow and Transport in the Partially Molten Mantle

例如，火山喷发会影响当地的供水和全球气候。地球表面的火山活动是由于对流地幔的部分熔融，以及随后熔岩成分的分离。地球的地幔通常低于熔点，因此是固体的。部分熔融一般发生在三个位置，在形成洋壳的大洋中脊，在海洋岛屿和一些大陆遗址(如黄石公园)下方的热点地区，以及洋壳返回地幔和形成陆壳的俯冲带。为了更好地了解火山过程，必须将地表地球化学观测与地幔岩石的组成和分布联系起来，由于地幔对流期间物质的不完全化学混合，地幔岩石是不均匀的。该项目将开发能够解决这些过程的小尺度动力学问题的数值模型。这是一名数学家和一名地球科学家，以及一名本科生和一名研究生的共同努力，他们将在跨学科的环境中接受教育。工业和政府实验室对这样培训的人员需求很大。项目目标包括开发(1)计算模拟地幔流动演变的数学框架，包括不熔化的简并情况，(2)准确逼近地幔流动内温度和化学成分传输的数值方法，(3)实现流动和传输算法的计算机代码，以及处理简单相态的方法。此外，(4)该代码将用于研究地球科学中的重要问题，(5)学生将在跨学科的环境中接受教育和培训。从数学上讲，由于存在没有熔体的区域，地幔中的两相流动受高度退化的方程支配。最近的工作奠定了孔隙率不变时两相流动的数学基础。关键是根据孔隙率，即熔体的体积分数，适当地调整解的变量和方程。这个项目将处理不断演变的情况，目的是为在实际模拟中模拟流动提供一种适当的计算方法。为了模拟温度传输和熔体组分的分离，将为该项目开发适当的WENO和不连续Galerkin方法，该方法将考虑孔隙率可能的简并。

Volcanic eruptions affect, for example, local water supplies and the global climate. Volcanism at the Earth's surface is due to partial melting of the convecting mantle, and subsequent segregation of molten rock components. The Earth's mantle is generally below its melting point and thus solid. Partial melting occurs generally in three locations, at mid-ocean ridges where oceanic crust is formed, hot-spots below ocean islands and some continental sites like Yellowstone, and subduction zones where oceanic crust returns to the mantle and continental crust is formed. To better understand volcanic processes, geochemical surface observations must be linked to the composition and distribution of mantle rocks, which are heterogeneous due to incomplete chemical mixing of material during mantle convection. The project will develop numerical models that can resolve the small-scale dynamics of these processes. It is a joint effort of a mathematician and a geoscientist, as well as an undergraduate and a graduate student, who will be educated in an interdisciplinary setting. People so trained are in high demand by industrial and government labs.Project objectives include the development of (1) a mathematical framework for computational simulation of evolving mantle flow which covers the degenerate case of no melt, (2) a numerical method to accurately approximate the transport of temperature and chemical components within the mantle flow, (3) a computer code to implement the flow and transport algorithms, as well as a method for handling simple phase behavior. Furthermore, (4) the code will be applied to study important problems in the geosciences and (5) students will be educated and trained in an interdisciplinary setting. Mathematically speaking, because there are regions with no melt, two-phase flow in the mantle is governed by highly degenerate equations. Recent work has established the mathematical foundations of two-phase flow when the porosity does not evolve. The key is to scale the solution variables and equations appropriately by the porosity, which is the volume fraction of melt. The evolving case will be treated in this project, with the goal of providing an appropriate computational method for simulating the flow in practical simulations. To model the transport of temperature and the segregation of melt components, appropriate WENO and discontinuous Galerkin methods will be developed for this project, which will respect the possible degeneracies in the porosity.