The importance of the catchment area–length relationship in governing non-steady state hydrology, optimal junction angles and drainage network pattern

Abstract

Analytical and numerical models of landscape evolution in general either assume steady state hydrology or use empirically based functions in the form of a drainage area power law to model runoff. A method to compute non-steady state runoff on the basis of the area–length relationship of river basins is proposed, for application in landscape evolution models. This shape-sensitive runoff function is derived analytically for environments with short storm duration (storm duration < concentration time) and it is supported numerically for environments with small storms (storm cell size < basin size). The effect of these area–length dependent relationships on drainage network development is analysed. The methodology is optimization through stream power minimization both on the level of individual junctions and on the level of the entire network. It is demonstrated that the way the area–length dependent runoff function influences the stream power of the network yields a systematic downstream increase of the network's optimal junction angles. An example for this prediction is given by a DEM-derived network from the Colorado High Plains. The inverse dependence of the runoff rate on the basin's flow path length implies that minimum total stream power is achieved through maximizing the flow path length of the system. This effect is responsible for the lateral instability of junction positions when runoff conditions shift from steady towards non-steady state. We offer this phenomenon as an explanation for the lateral instability and pronounced planation activity of semi-arid channels, especially on gently sloping piedmont surfaces. Finally throughout the paper much attention is paid to the runoff character in terms of advectivity vs. diffusivity, defined as the ratio between instream flow and lateral inflow. It is shown how optimal network pattern also relates to the advectivity/diffusivity of the runoff conditions.