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，迫切需要研究水资源，特别是地表水和地下水方面的未冻结淡水。地表水和地下水相互作用。一个最好的例子是潜流带，这是河流或小溪河床和河岸的一部分，地表水和地下水在这里混合，以不同的规模交换溶质和水。潜流带对河流廊道的生态至关重要。特别是，在以下方面至关重要：控制河流与地下水之间的水交换通量和位置；为底栖生物和间隙生物提供栖息地；为某些物种提供产卵地和庇护所；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；以及减缓河水温度。这个项目的目的是开发和分析更好的低速带流动模型，以提高我们对与低速流相关的基本物理过程的理解。研究生参与该项目的工作。这位研究人员和他的同事研究了纳维-斯托克斯-达西-热系统，该系统模拟了地表水-地下水相互作用和热效应的耦合。数学模型提出了若干挑战：与河流中的Navier-Stokes流动有关的强烈非线性，包括重要的物理和生物效应引起的复杂性，河流中的流动与周围多孔介质中的流动之间的巨大时空尺度差异(小达西数)，与河床和河岸的几何形状和渗透性有关的不确定性，以及物理领域不同部分的不同物理。虽然这些困难的问题以前已经针对所考虑的模型的一些子系统进行了单独研究，但为了更好地整合与潜流带中的流动和输送相关的物理和生物因素，需要一个更全面的模型。调查员的工作是循序渐进的。首先，对该模型在物理上重要的小达西数区域的渐近行为进行了数学分析。其次，他为模型设计、分析和实现了准确和高效的解耦数值方法，以便将结果与合作者进行的实验进行比较。第三，他使用该模型评估了水资源社区在地下水流研究中使用的各种简化的有效性。研究生参与该项目的工作。

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.