Analytic Theory of Equilibrium Fluvial Landscapes: The Integration of Hillslopes and Channels

• Published in: Journal of geophysical research. Earth surface 2018-03, Vol.123 (3), p.557-615
• Main Author: Smith, Terence R.
• Format: Article
• Language: English
• Subjects: analytic theory; bifurcation of solutions; Bifurcations; Boundary conditions; Channel flow; Climate change; Drainage channels; Drainage density; Energy conservation; Equilibrium; Erosion; Flow stability; fluvial landscape; hillslopes; Lagrangian function; Laminar flow; Landscape; Mathematical models; Networks; Power law; Rain; Rainfall; Rainfall rate; river channels; Scaling; Sediment; Sediment transport; Segments; Slopes; Solutions; Stability analysis; Stable channels; stable equilibrium; Transport; Water flow
• ISSN: 2169-9003; 2169-9011
• DOI: 10.1002/2016JF004073

A physically based theory for equilibrium fluvial landscapes undergoing transport‐limited or detachment‐limited erosion is derived from equations for water flow, erosion, and hillslope stability. A scaling analysis of the equations identifies submodels for overland and channelized flows. The channel‐slope boundary (CSB) defines the interface between the two flow environments, marking the bifurcation of stable overland flows into unstable sheet flows and stable channel flows. The characteristics of CSBs are derived from an equilibrium condition of the hillslope submodel and stability conditions derived from the time‐dependent model. Equilibrium hillslopes are characterized by proportional flows of water and sediment, uniformly constant and maximum values of water flow (qc) and slope occurring on the CSB, and the minimization of a Lagrangian function of energy. Equilibrium solutions are readily derived for laminar overland flows. Power law sediment transport functions for channel flow lead to stable self‐similar solutions for channel segments characterized by realistic hydraulic geometries. Uniform equilibrium inflows qc result in straight segments of channel that are combinable into networks satisfying admissibility constraints. The aggregate length of CSB upstream of any network point determines total channel discharge and hence channel slope and elevation, providing boundary conditions that determine hillslope elevations. Large equilibrium landmasses with rainfall rate R have relatively uniform drainage densities Dd=R/2qc. Changes to channel networks induced by environmental fluctuations may occur at any network location because maximum, neutrally stable overland flows qc occur on the CSB. Stable equilibrium surfaces are conjectured to correspond to local minima of the Lagrangian over the space of admissible networks. Key Points A theory of fluvial equilibria describes how solutions to submodels for hillslopes and channels are integrated into a continuous surface Stability and equilibrium conditions determine the slope-discharge relation for hillslopes and the location of uniform inflows into channels Channel solutions have realistic, stable geometries, determine elevations of channel networks, and provide boundary conditions for hillslopes