Simulation of Multiphase Flow and Transport in the Partially Molten Mantle

例如，火山爆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，以及随后熔融岩石成分的分离。地幔通常低于熔点，因此是固态的。部分融化通常发生在三个地方：形成海洋地壳的洋中脊，海洋岛屿和一些大陆遗址（如黄石公园）下面的热点，以及海洋地壳返回地幔和大陆地壳形成的俯冲带。为了更好地理解火山过程，地表地球化学观测必须与地幔岩石的组成和分布联系起来，地幔岩石在地幔对流过程中由于物质的化学混合不完全而具有非均质性。该项目将开发能够解决这些过程的小规模动力学的数值模型。这是一位数学家和一位地球科学家，以及一位本科生和一位研究生的共同努力，他们将在跨学科的环境中接受教育。工业和政府实验室对受过此类培训的人员需求量很大。项目目标包括开发(1)计算模拟演化地幔流的数学框架，该框架涵盖无熔体的简并情况；(2)精确近似地幔流中温度和化学成分输运的数值方法；(3)实现流和输运算法的计算机代码，以及处理简单相行为的方法。此外，(4)代码将应用于研究地球科学中的重要问题，(5)学生将在跨学科的环境中接受教育和培训。从数学上讲，由于存在没有熔融的区域，地幔中的两相流动由高度简并的方程控制。最近的工作已经建立了孔隙度不变化时两相流的数学基础。关键是用孔隙率（即熔体的体积分数）来适当地缩放溶液变量和方程。在这个项目中，我们将处理不断变化的情况，目的是为实际模拟中模拟流动提供一种适当的计算方法。为了模拟温度的传递和熔体成分的偏析，本项目将开发适当的WENO和不连续Galerkin方法，这些方法将考虑孔隙度可能的简并。

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.