Nonlinear Dynamics in Geosciences

pp 121-151

Towards a Nonlinear Geophysical Theory of Floods in River Networks: An Overview of 20 Years of Progress

  • Vijay K. GuptaAffiliated withUniversity of Colorado, Dept. of Civil, Environmental and Architectural Engineering, Cooperative Institute for Research in Environmental Sciences
  • , Brent M. TroutmanAffiliated withU. S. Geological Survey
  • , David R. DawdyAffiliated withHudrologic Consultant


Key results in the last 20 years have established the theoretical and observational foundations for developing a new nonlinear geophysical theory of floods in river basins. This theory, henceforth called the scaling theory, has the explicit goal to link the physics of runoff generating processes with spatial power-law statistical relations between floods and drainage areas across multiple scales of space and time. Published results have shown that the spatial power law statistical relations emerge asymptotically from conservation equations and physical processes as drainage area goes to infinity. These results have led to a key hypothesis that the physical basis of power laws in floods has its origin in the self-similarity (self-affinity) of channel networks. Research within the last 20 years has also shown that self-similarity is the basis for the widely observed fractal structure and Horton relations in river networks. Observed power laws in floods span a broad range of spatial scales and multiple time scales that range from hours of individual flood events to annual time scale of flood frequencies. They serve as the foundation for developing a new diagnostic framework to test different assumptions governing spatial variability in physical processes that arise in predicting power law statistical relations between floods and drainage areas. The structure of the diagnostic framework is illustrated using two examples. Contemporary relevance and future applications of the scaling theory come from predicting floods in the context of a significant warming of the Earth’s climate, which is altering local, regional, continental and global balances of water and energy. This anthropogenic perturbation to the planetary hydro-climate precludes making flood predictions at gauged and ungauged sites based on regional statistics from historical stream flow data, which is a well-established practice in hydrologic engineering.