Controls on the spacing of first-order valleys

Many landscapes are composed of ridges and valleys that are uniformly spaced, even where valley locations are not controlled by bedrock structure. Models of long-term landscape evolution have reproduced this phenomenon, yet the process by which uniformly spaced valleys develop is not well understood, and there is no quantitative framework for predicting valley spacing. Here we use a numerical landscape evolution model to investigate the development of uniform valley spacing. We find that evenly spaced valleys arise from a competition between adjacent drainage basins for drainage area ( a proxy for water flux) and that the spacing becomes more uniform as the landscape approaches a topographic equilibrium. Valley spacing is most sensitive to the relative rates of advective erosion processes ( such as stream incision) and diffusion-like mass transport ( such as soil creep) and less sensitive to the magnitude of a threshold that limits the spatial extent of stream incision. Analysis of a large number of numerical solutions reveals that valley spacing scales with a ratio of characteristic diffusion and advection timescales that is analogous to a Peclet number. We use this result to derive expressions for equilibrium valley spacing and drainage basin relief as a function of the rates of advective and diffusive processes and the spatial extent of the landscape. The observed scaling relationships also provide insight into the cause of transitions from rill-like drainage networks to branching networks, the spatial scale of first-order drainage basins, the contributing area at which hillslopes transition into valleys, and the narrow range of width-to-length ratios of first-order basins.