Some Mathematical Problems Associated with Hyporheic Flow

淡水供应面临压力的原因有很多，包括自然变化、人口增长、商业活动扩张、快速城市化、气候变化、含水层枯竭和污染。因此，迫切需要对水资源，特别是地表水和地下水方面的未冻淡水进行研究。地表水和地下水相互作用。一个典型的例子是暗流带，这是河流或溪流的河床和河岸的一部分，地表水和地下水在这里混合，在不同的尺度上交换溶质和水。潜流带是河流廊道生态的重要组成部分。特别是，它在控制河流与地下水交换的通量和位置方面具有重要意义；为底栖生物和间质生物提供栖息地；为某些鱼类提供产卵地和避难所的；为水生植物提供生根区；作为碳、能量和养分循环的重要区域；通过生物降解、吸附和混合为某些污染物提供自然衰减区；缓和河水温度。该项目的目的是开发和分析更好的下隐带流动模型，以提高我们对与下隐带流动相关的基本物理过程的理解。研究生参与项目工作。研究者和同事们研究了纳维-斯托克斯-达西-热系统，该系统模拟了地表水-地下水相互作用与热效应的耦合。数学模型提出了几个挑战：河流中Navier-Stokes流的强非线性、包括物理和生物上重要的热效应所引起的复杂性、河流和周围多孔介质（小达西数）的流动在空间和时间尺度上的巨大差异、与河床和河岸的几何形式和渗透率相关的不确定性，以及物理域不同部分的不同物理。虽然这些困难的问题之前已经对本文所考虑的模型的某些子系统分别进行了研究，但为了更好地整合与潜流带的流动和运输有关的物理和生物因素，需要一个更全面的模型。调查者的工作是分步骤进行的。首先对模型在重要的小达西数域的渐近行为进行了数学分析。其次，他为模型设计、分析并实现了准确高效的解耦数值方法，使结果可以与合作者进行的实验进行比较。第三，他使用该模型来评估水资源界在潜流研究中使用的各种简化方法的有效性。研究生参与项目工作。

1715504Wang The supply of fresh water is under pressure for many reasons, including natural variability, population growth, expansion of business activities, rapid urbanization, climate change, depletion of aquifers, and pollution. Therefore, there is an urgent need to study water resources, especially unfrozen fresh water in terms of surface water and groundwater. Surface water and groundwater interact with each other. A prime example is the hyporheic zone, a portion of the bed and bank of a river or stream where surface water and groundwater mix, exchanging solutes and water at various scales. The hyporheic zone is critical to the ecology of river corridors. In particular, it is important in controlling the flux and location of water exchange between stream and subsurface; providing a habitat for benthic and interstitial organisms; providing a spawning ground and refuge for certain species of fish; providing a rooting zone for aquatic plants; serving as an important zone for cycling of carbon, energy, and nutrients; providing a natural attenuation zone for certain pollutants by biodegradation, sorption, and mixing; and moderating river water temperature. The purpose of this project is to develop and analyze better models of flows in the hyporheic zone in order to improve our understanding of the fundamental physical processes associated with hyporheic flows. Graduate students participate in the work of the project. The investigator and colleagues study the Navier-Stokes-Darcy-heat system that models coupled surface water-groundwater interaction together with thermal effects. The mathematical model presents several challenges: the strong nonlinearity associated with the Navier-Stokes flow in the river, the complications arising from including physically and biologically important thermal effects, the substantial disparity of spatial and temporal scales between flows in the river and in surrounding porous media (small Darcy number), the uncertainty associated with the geometric form and the permeability of the riverbed and riverbank, and the different physics in different parts of the physical domain. Although these difficult issues have been studied separately before for some subsystems of the models under consideration here, the need for a better integration of the physical and biological factors relevant to flow and transport in the hyporheic zone requires a more comprehensive model. The work of the investigator proceeds in steps. First is mathematical analysis of the model in terms of asymptotic behavior in the physically important small Darcy number regime. Second, he designs, analyzes, and implements accurate and efficient decoupled numerical methods for the model so that the results can be compared to experiments performed by collaborators. Third, he uses the model to assess the validity of various simplifications utilized by the water resources community in hyporheic flow studies. Graduate students participate in the work of the project.