Simulation of Multiphase Flow and Transport in the Partially Molten Mantle

例如，火山爆发会影响当地的水供应和全球气候。 地球表面的火山作用是由于对流地幔的部分熔融，以及随后熔融岩石成分的分离。 地球的地幔通常低于其熔点，因此是固体。 部分熔融通常发生在三个地方，即洋中脊，洋壳形成的地方;洋岛和黄石公园等大陆遗址下方的热点;以及俯冲带，洋壳返回地幔，形成大陆壳。为了更好地了解火山过程，地球化学表面观测必须与地幔岩石的成分和分布联系起来，地幔岩石由于地幔对流期间物质的不完全化学混合而具有异质性。 该项目将开发能够解决这些过程的小规模动态的数值模型。 这是一个数学家和地球科学家的共同努力，以及本科生和研究生，谁将在一个跨学科的环境中接受教育。 工业和政府实验室非常需要受过这种训练的人。项目的目标包括：（1）发展一个计算模拟地幔流演变的数学框架，其中包括没有熔体的退化情况;（2）一种精确近似地幔流内温度和化学成分输运的数值方法;（3）一种实现流动和输运算法的计算机代码;以及用于处理简单相行为的方法。 此外，（4）该代码将应用于研究地球科学中的重要问题，（5）学生将在跨学科环境中接受教育和培训。 从数学上讲，因为有没有熔体的区域，地幔中的两相流是由高度退化的方程。 最近的工作建立了两相流的数学基础时，孔隙度不演变。 关键是通过孔隙率（熔体的体积分数）适当地缩放解变量和方程。 在这个项目中，将处理不断变化的情况下，提供一个适当的计算方法，在实际模拟的流动模拟的目标。 为了模拟温度的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，这将考虑孔隙度中可能的退化。

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.