A Numerical Study of the Factors Controlling Valley Spacing in Landscapes

Landscapes often exhibit a characteristic spatial scale of dissection, which is most apparent as the spacing between adjacent valleys (or the intervening ridges). Simple physical and geometrical arguments imply that valley spacing could be determined by erosion thresholds and/or by the relative rates of advective (e.g., stream incision) and diffusive (e.g., soil creep) sediment transport processes. We solve a simplified, nonlinear advection-diffusion equation numerically to investigate the factors controlling valley spacing. Using dimensional analysis, we infer the form of a relationship between valley spacing, the rates of the advective and diffusive processes, and spatial and temporal boundary conditions. Numerical solutions of the advection-diffusion equation confirm the general form of this relationship. We show that a similar relationship should hold for more complicated expressions that are more realistic representations of sediment transport processes in nature. Our results also provide insight into the factors controlling local relief, the causes of transitions from rill-like drainage networks to branching networks, and the upper and lower bounds on the length-to-width ratios of first-order drainage basins.