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Analytic Theory of Equilibrium Fluvial Landscapes: The Integration of Hillslopes and Channels
Author(s) -
Smith Terence R.
Publication year - 2018
Publication title -
journal of geophysical research: earth surface
Language(s) - English
Resource type - Journals
eISSN - 2169-9011
pISSN - 2169-9003
DOI - 10.1002/2016jf004073
Subject(s) - surface runoff , fluvial , scaling , geology , channel (broadcasting) , open channel flow , boundary (topology) , flow (mathematics) , hydrology (agriculture) , sediment transport , geometry , mechanics , sediment , geomorphology , physics , mathematics , geotechnical engineering , structural basin , mathematical analysis , ecology , biology , engineering , electrical engineering
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 ( q c ) 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 q c 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 densitiesD d = R / 2 q c . Changes to channel networks induced by environmental fluctuations may occur at any network location because maximum, neutrally stable overland flows q c occur on the CSB. Stable equilibrium surfaces are conjectured to correspond to local minima of the Lagrangian over the space of admissible networks.