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.
