Mathematical Sciences: Multiscale Processes and Stochastic Dynamics in Geosciences

9504557巴塔查亚摘要 概率方法提出了三大类问题的分析水文，气候预测和全球变化现象。 所有这些现象都受到具有空间和时间层次的非线性动力学的支配，这使得它们的分析和预测变得复杂和具有挑战性。 在这里考虑的三个广泛的主题中，主题1致力于研究污染物在含水层等自然介质中的迁移问题。 支配这种传输的福克-普朗克方程类涉及多个空间尺度的异质性。题目2涉及湍流和降雨统计理论中出现的一类随机测度的精细尺度结构。 对这种高度奇异现象所采用的数学理论是随机级联理论。专题3涉及的气候变化和全球变化现象的时间尺度比专题2所设想的要大得多。 这里的目的是分析年际气候事件使用随机扰动动力系统。 该提案的主要目标是发展非线性随机动力学理论，用于：（1）预测环境中污染物随时间的扩散;（2）改进短期天气预报;（3）了解并最终预测长期气候和全球变化现象。 专题1下的一个重要例子是化学和其他污染物在天然地下水系统中的扩散问题。 由于这些系统中的水的速度场即使略有不同，从长远来看也会导致污染物扩散速率的显著不同，因此需要进行仔细和精确的建模和分析。 关于专题2，研究雷暴和降雨的时空分布需要不同的技术。 预期这些资料可为一般用于天气预测的数值模式提供更佳的输入数据，从而改善预测的准确性。虽然科学家在短期天气预报方面取得了一定的成功，但在理解和预测长期气候现象方面却收效甚微。 由于不可能为后者提供适当的确定性模型，主题3的主要目的是使用随机扰动模型分析这些长期现象。 所考虑的一个例子是对厄尔尼诺现象的出现和强度的预测--热带太平洋变暖导致大范围降雨和洪水--而它与干燥的拉尼娜现象相反。 现实的非线性随机模型寻求在这里，并提出了他们的分析。

9504557 Bhattacharya Abstract Probabilistic methods are proposed for the analysis of three broad classes of problems in hydrology, climatic prediction and global change phenomena. All these phenomena are governed by nonlinear dynamics with a hierarchy of scales in space and time, which make their analysis and prediction complex and challenging. Of the three broad topics considered here, Topic 1 is devoted to the problem of contaminant transport in natural media such as aquifers. The class of Fokker-Planck equations which govern this transport involve multiple spatial scales of heterogeneity. Topic 2 concerns the fine scale structure of a class of random measures which arise in the statistical theories of turbulence and rainfall. The mathematical theory to be employed for such highly singular phenomena is that of random cascades. Topic 3 is concerned with climatic changes and global change phenomena over much larger scales of time than envisaged under Topic 2. Here the aim is to analyze interannual climatic events using randomly perturbed dynamical systems. The main objective of the proposal is the development of nonlinear stochastic dynamical theories for (1) the prediction of the spread of pollutants in the environment with time, (2) making improvements in short-term weather forecasting, and (3) the understanding and eventual prediction of long-term climatic and global change phenomena. An important example under Topic 1 is the problem of the spread of chemical and other pollutants in natural underground water systems. Since even slightly different velocity fields of water in these systems often lead to dramatically different rates of spread of the pollution in the long run, careful and precise modeling and analysis are required and proposed here. With regard to Topic 2, different techniques are needed for the study of the space-time distribution of thunderstorms and rainfall. These are expected to provide better inputs into the numerical models which are generally used for we ather forecasts, thus improving their predictability. Although scientists have been somewhat successful in such short-term weather forecasting, they have had little success in the understanding and prediction of long-term climatic phenomena. Since it is impossible to provide an adequate deterministic model for the latter, the main aim under Topic 3 is the analysis of these long-term phenomena using models with random perturbations. An example considered is the prediction of the advent and intensity of the El Nino--a warming of the Tropical Pacific causing extensive rainfall and flooding--and its opposite the dry La Nina. Realistic nonlinear stochastic models are sought here and their analysis is proposed.