Controls on Branching in Valley Networks

Branching valley networks are a widespread planetary feature, yet significant questions remain about one of their most visually striking properties: how does the interaction between hillslopes and channels determine whether or not a valley branches, and what is the nature of the transition from the unbranched to the branched state? I address these questions by examining a simple case in which erosion is dominated by detachment-limited stream incision and slope-dependent creep, such that the long-term evolution of the topography can be modeled with a nonlinear advection-diffusion equation. Basic topographic dimensions of the equilibrium model solutions, such as valley spacing and relief, are functions of the ratio of the characteristic timescales for diffusion and advection, which can be expressed as a quantity analogous to a Péclet number, Pe. In a landscape consisting of first-order valleys, valley spacing narrows linearly as the rate of stream incision quickens relative to the rate of hillslope transport (Pe increases). This scaling regime is bounded by two critical values of Pe. The lower of these is the critical value for the formation of valleys, below which the solutions are unchanneled hillslopes. The upper value marks the onset of branching: valleys develop tributaries, and the spacing among the second-order valleys begins to widen. Two mechanisms contribute to this abrupt change in behavior. First, as Pe increases, there is a transition from a regime in which the equilibrium spacing of first-order valleys is stable with respect to perturbations in valley width or depth, to a regime in which the predicted spacing is unstable. The range of Pe over which this transition occurs corresponds to the critical Pe for branching. Second, tributaries form when Pe for the valley side slopes reaches the critical value for valley formation. The formation of tributaries accelerates the evolution of the topography away from the unstable equilibrium and toward a state consisting entirely of branched valleys. In nature, irregularities in the initial topography of an evolving landscape will drive the drainage network toward the branched state rather than the unstable state of narrowly-spaced, unbranched valleys. I compare these results with observations of experimental and natural valley networks, and explore the implications for valley network morphology at scales much larger than a hillslope length.