Discharge is the independent variable primarily responsible for shaping the hydraulic geometry and longitudinal profile of rivers. The assumption is frequently made that discharge and drainage area scale linearly or nearly linearly, i.e., Q = kAc, where k is the theoretical discharge for a unit area watershed (A = 1), Q is river discharge (m3/s), A is drainage area (m2), and c is the scaling power dependency. Watershed and longitudinal profile modeling enjoy simplified assumptions if discharge grows linearly with drainage area, and this assumption is widely applied. This paper investigates the scaling relationship between discharge and drainage area for five large rivers, with an emphasis on exploring the linearity of the discharge-area relationship and suggesting causes for significant departure from linearity. The five large main-stem rivers explored (John Day, Salmon, Wabash, Greenbrier, and Yellowstone) all have a minimum of 60 years of continuous discharge records and have been selected to represent a wide geographic area spanning different land uses, climate, and topography. Peak annual flow and mean annual flow are compiled from the U.S. Geological Survey national surface-water database, and a linear regression analysis was completed for each year for the discharges. The five rivers were selected to minimize, but not eliminate, the impacts of dams and diversions such as for irrigation on river discharges.
The scaling factor (c) exhibits both secular and nonsecular trends over the length of record for these five rivers. The results show that the studied watersheds can be grouped into two broad categories based on their respective c values: (1) those rivers where c is 1 or nearly 1, and (2) those rivers where c is significantly <1. The John Day, Salmon, Wabash, and Greenbrier rivers scale at values of ∼0.8 with natural variables including slope, elevation, and evapotranspiration potentially accounting for c values slightly <1. The second category is c values of ∼0.5, as exhibited by the Yellowstone River. The Yellowstone watershed is unique for our study because of its secular trend, as well as its overall lower average c values. Climatic trends that control the timing of winter snowpack melting, increased frequency and intensity of forest fires, and increased human consumptive water use in downstream areas may all contribute to the observed behavior in c for this watershed. The results from this set of rivers have broad implications for studies ranging from the modeling of fluvial erosion in numeric landscape evolution models to allocations of water resources for human and environmental purposes.