Uncertainty Analysis on a Few Fluid Models

王DMS-0606671 研究者和同事们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。 研究的物理问题是Rayleigh-Benard对流中的大Prandtl数或小Ekman数区域、流体饱和多孔介质中对流中的大Prandtl-Darcy数或小Darcy数区域以及具有随机噪声的随机强迫准地转系统。 对流问题的参数域模型降阶导致涉及多个时间和空间尺度的奇异极限问题。 这里的目标之一是调查什么样的统计特性可以近似使用的统计特性的离散模型通过奇异极限。 在具有随机噪声的系统的情况下，由随机偏微分方程建模，研究者和他的同事研究了可能的噪声引起的统计稳定性，即，关于初始条件和参数统计性质的稳定性。 自然界中存在着大量的不确定性。 例如，对飓风路径的预测受到有关大气和海洋表面温度、风和大气急流的初始分布的（不完美的）信息的影响，以及不完美的模型本身。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。 污染物在地下水中的运移受介质孔隙度和渗透率的影响，而孔隙度和渗透率是随机的。研究各种不确定性对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.