A graphical method to interpret how incision thresholds influence topographic and scaling properties of landscapes

Theoretical analysis of the governing equations of numerical models can reveal relationships between topographic properties, such as drainage area, slope, and curvature, in simulated landscapes. These relationships are testable predictions; they can diagnose whether real-world landscapes could potentially arise from similar mechanisms. For example, the stream-power incision model is consistent with drainage area and slope data that plot as straight lines on logarithmic axes.

Here we graph theoretical relationships between topographic curvature and the steepness index, which depends on drainage area and slope. These relationships plot as straight lines for steady-state landscapes that have evolved according to a model that combines stream-power incision, linear diffusion, and uplift. Further, they link topography (drainage area, slope, and curvature) to characteristic length scales of the landscape, which depend on the competition between the processes of incision, diffusion, and uplift.

Adding an incision threshold to the model changes the relationship between the steepness index and topographic curvature. We examine these changes graphically and we show that they shed light on how incision thresholds influence topographic and scaling properties of landscapes. Specifically, we present a graphical method that consists of plotting steepness index–curvature lines and of tracing their intersections with each other and with the coordinate axes. This simple method reveals both how topography and process competition are influenced by the incision threshold, and how these influences vary within a given landscape and across different landscapes.