9421445 Waymire One of the classic problems in fluvial geomorphology and river basin hydrology is prediction of flows from ungauged river basins. This is a very difficult problem because not only does it involve notions of dynamics via the conservation equations, but also that of geometry via branching of river networks, and of probability due to the presence mf fluctuations in river flows and river networks. Since these notions recur across a broad range of spatial scales, the focus of our proposed research is on developing and empirically testing scaling invariance ideas and theories. For the past several years, both the PIs have been extensively involved in exploring a scaling framework to study spatial rainfall, peak river flows viewed as channel forming discharges, and the 3-D structure of river networks. Several unexpected and significant developments in this line of research within the last three years show that the scaling invariance ideas provide a very rich framework to tackle this problem. In this proposal our objectives are: (1) to develop a geometric-statistical scaling theory of river networks, (2) to develop a statistical-dynamic scaling theory of river runoff, which are viewed as channel-forming discharges, (3) to develop a geometric-statistical-dynamic scaling theory to connect the 3-D geometry of river networks and flows and (4) to carry out tests of the theories in Objectives 1,2, and 3 via spatial data analysis and simulations. This proposed research addresses how different notions of scaling, namely, geometric, dynamic, and statistical, can be woven together into a single, coherent theoretical framework to understand the physics of this problem and interpret space-time hydrologic data. It has applications to a variety of hydrologic problems spanning a broad range of space and time scales.
