9418674 Duffy This research will test the hypothesis that the rainfall-runoff process can be represented as a low-dimensional dynamical system, forced by topographic, soil, geologic and climatic variability. By "low-dimensional " we mean the minimum number of state variables required to approximate the processes as a system of nonlinear ordinary differential equations. We examine the case where rainfall-runoff is controlled by porous soils and shallow groundwater circulation, and where the ability of the catchment to store water and yield runoff depends on the nature of the storage-flux relationships of the system. The role of evapotranspiration as a parametic function of soil-moisture storage will also be examined. The "low-dimensional" model will serve as a physically-based alternative to the nonlinear partial differential equations (Richard's equation) representing the local processes. In constructing the dynamical model, the essential problem is to separate or distinguish among spatial and temporal components of watershed dynamics, such that the important mechanics of the processes involved are elucidated, without loss of critical nonlinear structure. The research has the following elements: (1) Comprehensive numerical experiments based on finite element solutions to the partial differential equations for saturated-unsaturated flow will be performed To establish terrain-integrated constitutive relations (e.g. storage-flux relations). the proposed research will build on a series of steady-state numerical experiments (Lee, 1993; Duffy, 1994) for two dimensional hillslope geometry with uniform soil properties. This previous work found that recharge to the water table and subsurface flow to the stream were nonlinear functions of at least two state variables: the integrated soil moisture and integrated saturated storage. The present objective is to extend these experiments to the case of fully three dimensional and time varying flow, and test the role of soil stratification and variability on nonlinear storage-flux relations and runoff response. (2) Develop and test procedures for scaling and spatial integration of the state variables and fluxes for the Shale Hills watershed, 8 hectare, forested, catchment in central PA (J. Lynch et al, 1976). Shale Hills was the site of a unique experiment in the 1970's to evaluate the effects of antecedent soil moisture on stormflow volume and timing. Rainfall was artificially applied for 8 events with initial moisture ranging from dry to very wet. a comprehensive accounting of soil moisture and saturated storage at multiple depths was made over the entire watershed. Although if may seem to be a straight forward problem, spatial integration of scattered field observations requires an appropriate weighting function. Duffy (1994) has proposed weighting function derived from the hypsometric distribution, and a local rescaling of hillslope trajectories for each hillslope or zero-order basin. This scaling and averaging method will be carried out for the Shale Hills data base. Field-estimated storage-flux relations will be compared with the numerical experiments in (1). (3) An independent method known as proper orthogonal decomposition (POD), allows the essential spatial structure of the dynamics to be reconstructed directly from random field data or from the governing pde's. The method is widely applied to detecting coherent structures in hydrodynamics turbulence (Lumley, 1967), the evolution of climatic fields (North et al, 1982), and nonlinear vibration (Cusumano, and Bai, 1993, Cusumano et al, 1993, Lin and Cusumano, 1993, Cusumano et al, 1994). The POD method will be applied to the field data of the Shale Hills experiment and the Richard's equation to provide a measure of the dimensionality, or number of state variables required to model the system.
