River networks evolve as migrating drainage divides reshape river basins and change network topology by capture of river channels. We demonstrate that a characteristic metric of river network geometry gauges the horizontal motion of drainage divides. Assessing this metric throughout a landscape maps the dynamic states of entire river networks, revealing diverse conditions: Drainage divides in the Loess Plateau of China appear stationary; the young topography of Taiwan has migrating divides driving adjustment of major basins; and rivers draining the ancient landscape of the southeastern United States are reorganizing in response to escarpment retreat and coastal advance. The ability to measure the dynamic reorganization of river basins presents opportunities to examine landscape-scale interactions among tectonics, erosion, and ecology.

Bedrock river profiles are often interpreted with the aid of slope-area analysis, but noisy 11 topographic data make such interpretations challenging. We present an alternative 12 approach based on an integration of the steady-state form of the stream power equation.

[1] 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 Péclet 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.

Bedrock fracture systems facilitate weathering, allowing fresh mineral surfaces to interact with corrosive waters and biota from Earth's surface, while simultaneously promoting drainage of chemically equilibrated fluids. We show that topographic perturbations to regional stress fields explain bedrock fracture distributions, as revealed by seismic velocity and electrical resistivity surveys from three landscapes. The base of the fracture-rich zone mirrors surface topography where the ratio of horizontal compressive tectonic stresses to near-surface gravitational stresses is relatively large, and it parallels the surface topography where the ratio is relatively small. Three-dimensional stress calculations predict these results, suggesting that tectonic stresses interact with topography to influence bedrock disaggregation, groundwater flow, chemical weathering, and the depth of the "critical zone" in which many biogeochemical processes occur.

One of the most striking examples of self-organization in landscapes is the emergence of evenly spaced ridges and valleys 1-6. Despite the prevalence of uniform valley spacing, no theory has been shown to predict this fundamental topographic wavelength. Models of long-term landscape evolution can produce landforms that look realistic 7-9 , but few metrics exist to assess the similarity between models and natural landscapes. Here we show that the ridge-valley wavelength can be predicted from erosional mechanics. From equations of mass conservation and sediment transport, we derive a characteristic length scale at which the timescales for erosion by diffusive soil creep and advective stream incision are equal. This length scale is directly proportional to the valley spacing that emerges in a numerical model of landform evolution, and to the measured valley spacing at five field sites. Our results provide a quantitative explanation for one of the most widely observed characteristics of landscapes. They also imply that valley spacing is a fundamental topographic signature that records how material properties and climate regulate erosional processes. The spacing between adjacent ridges and valleys is a fundamental dimension of hilly topography 1-6. Even a casual observer can see from an airplane window that ridges

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[1] Erosion by bedrock river channels is commonly modeled with the stream power equation. We present a two-part approach to solving this nonlinear equation analytically and explore the implications for evolving river profiles. First, a method for non-dimensionalizing the stream power equation transforms river profiles in steady state with respect to uniform uplift into a straight line in dimensionless distance-elevation space. Second, a method that tracks the upstream migration of slope patches, which are mathematical entities that carry information about downstream river states, provides a basis for constructing analytical solutions. Slope patch analysis explains why the transient morphology of dimensionless river profiles differs fundamentally if the exponent on channel slope, n, is less than or greater than one and why only concave-up migrating knickpoints persist when n < 1, whereas only concave-down migrating knickpoints persist when n > 1. At migrating knickpoints, slope patches and the information they carry are lost, a phenomenon that fundamentally limits the potential for reconstructing tectonic histories from bedrock river profiles. Stationary knickpoints, which can arise from spatially varying uplift rates, differ from migrating knickpoints in that slope patches and the information they carry are not lost. Counterparts to migrating knickpoints, called "stretch zones," are created when closely spaced slope patches spread to form smooth curves in distance-elevation space. These theoretical results are illustrated with examples from the California King Range and the Central Apennines.Citation: Royden, L., and J. Taylor Perron (2013), Solutions of the stream power equation and application to the evolution of river longitudinal profiles,

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