Uncertainty Analysis on a Few Fluid Models

研究人员及其同事研究了使用简化模型的影响，以及噪声对几种流体系统中统计解的不确定性传播的影响。研究的物理问题包括瑞利-贝纳德对流中的大prandtl数或小Ekman数，流体饱和多孔介质中对流中的大prandtl数或小Darcy数，以及随机噪声的随机强迫准地转系统。在参数域的模型约简导致了涉及多个时间和空间尺度的奇异极限问题。这里的目标之一是研究通过奇异极限推导出的简化模型的统计特性可以近似得到什么样的统计特性。在随机偏微分方程模拟的随机噪声系统的情况下，研究者和他的同事研究了可能由噪声引起的统计稳定性，即相对于初始条件和参数的统计特性的稳定性。自然界有许多不确定因素。例如，对飓风路径的预测受到有关大气和海洋表面温度、风和大气喷射流的初始分布的（不完善的）信息以及不完善的模型本身的影响。地球的气候（长期行为）可能受到臭氧层、二氧化碳浓度和其他因素的影响。地下水中污染物的运移受介质孔隙度和渗透率的影响，而介质孔隙度和渗透率是随机的。研究各种不确定性对这里所考虑的地球物理流动的影响有助于我们更好地理解流动的行为。

WangDMS-0606671 The investigator and colleagues study the impact of usingsimplified models as well as the effect of noise on uncertaintypropagation in terms of statistical solutions in several fluidsystems. The physical problems under investigation are largePrandtl number or small Ekman number regimes in Rayleigh-Benardconvection, large Prandtl-Darcy number or small Darcy numberregimes in convection in fluid saturated porous media, andstochastically forced quasi-geostrophic systems with randomnoise. Model reduction in the parameter regimes for theconvection problems leads to singular limit problems involvingmultiple time and spatial scales. One of the goals here is toinvestigate what kind of statistical properties can beapproximated using the statistical properties of the simplifiedmodels derived via singular limits. In the case of systems withrandom noise, which are modeled by stochastic partialdifferential equations, the investigator and his colleaguesinvestigate possible noise-induced statistical stabilities, i.e.,stability of statistical properties with respect to initialcondition and parameters. There are abundant uncertainties in nature. For instance,prediction of the path of a hurricane is influenced by(imperfect) information about the initial distribution ofatmospheric and sea surface temperatures, wind, and theatmospheric jet stream, as well as by the imperfect model itself. The climate (long-time behavior) of the earth may be influencedby the ozone layer, concentrations of carbon dioxide, and otherfactors. Transport of contaminants in ground water is influencedby porosity and permeability of the media, which are random. Studying the influence of various uncertainties on thegeophysical flows considered here helps us better understand thebehavior of the flows.