A physical basis for drainage density
Drainage density, a basic length scale in the landscape, is recognized to be the transition point between scales where unstable channel-forming processes yield to stable diffusive processes. This notion is examined in terms of equations for the evolution of landscapes that include the minimum necessary mathematical complexity. The equations, a version of the equations studied by Smith and Bretherton (1972), consist of convervation of sediment, an assumption that sediment movement is in the steepest downslope direction, and a constitutive relationship which gives the sediment transport rate as a function of slope and upslope area. The difference between processes is embedded in the constitutive relation. Instability to a small perturbation can be determined according to a criteria given by Smith and Bretherton and results when the sediment transport rate is strongly dependent on upslope area, whereas stability occurs if the main dependence is on slope. Where multiple processes are present, the transition from stability to instability occurs at a particular scale. Based on the idea that instability leads to channelization, the transition scale gives the drainage density. This scale can be determined as a maximum, or turn over point in a slope-area scaling function, and can be used practically to determine drainage density from digital elevation data. Fundamentally different scaling behavior, an example of which is the slope-area scaling, is to be expected in the stable and unstable regimes below and above the basic scale. This could explain the scale-dependent fractal dimension measurements that have been reported by others.