Alluvial and transport-limited bedrock rivers constitute the majority of fluvial systems on Earth. Their long profiles hold clues to their present state and past evolution. We currently possess first-principles-based governing equations for flow, sediment transport, and channel morphodynamics in these systems, which we lack for detachment-limited bedrock rivers. Here we formally couple these equations for transport-limited gravel-bed river long-profile evolution. The result is a new predictive relationship whose functional form and parameters are grounded in theory and defined through experimental data. From this, we produce a power-law analytical solution and a finite-difference numerical solution to long-profile evolution. Steady-state channel concavity and steepness are diagnostic of external drivers: concavity decreases with increasing uplift rate, and steepness increases with an increasing sediment-to-water supply ratio. Constraining free parameters explains common observations of river form: to match observed channel concavities, gravel-sized sediments must weather and fine - typically rapidly - and valleys typically should widen gradually. To match the empirical square-root width-discharge scaling in equilibrium-width gravel-bed rivers, downstream fining must occur. The ability to assign a cause to such observations is the direct result of a deductive approach to developing equations for landscape evolution.
Bibliographical noteFunding Information:
Acknowledgements. This study was funded in large part by the Emmy-Noether-Programme of the Deutsche Forschungs-gemeinschaft (DFG) grant no. SCHI 1241/1-1 awarded to Taylor F. Schildgen. Field and lab observations and discussions with Stefanie Tofelde, who revised an early version of the paper, helped to motivate the work. Conversations on channel concavity with Kelin Whipple and Greg Tucker stimulated initial thoughts on Sect. 5 of this paper, Chris Paola commented on an early form of the derivation, and Sam Holo pointed out an error in our treatment of sinuosity that appeared in the Discussions paper. Comments from Rebecca Hodge and one anonymous reviewer helped us to improve the paper.