Mathematical Sciences: Multiscale Processes and Stochastic Dynamics in Geosciences

摘要提出了概率方法来分析水文、气候预测和全球变化现象中的三大类问题。所有这些现象都是由具有空间和时间层次的非线性动力学控制的，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致力于污染物在自然介质（如含水层）中的运输问题。控制这种输运的一类福克-普朗克方程涉及多个空间尺度的异质性。主题2涉及湍流和降雨统计理论中出现的一类随机测量的精细尺度结构。这种高度奇异现象所采用的数学理论是随机级联的理论。专题3所涉及的气候变化和全球变化现象的时间范围比专题2所设想的要大得多。这里的目的是利用随机摄动动力系统分析年际气候事件。该提案的主要目标是发展非线性随机动力学理论，用于(1)预测污染物在环境中随时间的扩散，(2)改进短期天气预报，以及(3)理解和最终预测长期气候和全球变化现象。主题1中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。由于在这些系统中，即使水的速度场稍有不同，从长远来看，也会导致污染扩散速度的显著差异，因此需要并建议进行仔细而精确的建模和分析。关于Topic 2，研究雷暴和降雨的时空分布需要不同的技术。预计这些将为通常用于其他预报的数值模式提供更好的输入，从而提高其可预测性。尽管科学家们在这种短期天气预报方面取得了一定的成功，但他们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充分的确定性模型，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。考虑到的一个例子是对厄尔尼诺现象的到来和强度的预测——热带太平洋变暖导致大面积降雨和洪水——以及与之相反的干燥拉尼娜现象。本文寻求实际的非线性随机模型，并对其进行分析。

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.