A Variational Principle for the Integrated Channels and Slopes of Stable Equilibrium Landscapes

• Published in: Journal of geophysical research. Earth surface 2021-09, Vol.126 (9), p.n/a
• Main Author: Smith, Terence R.
• Format: Article
• Language: English
• Subjects: Channels; Constraint modelling; Discharge; Equilibrium; equilibrium fluvial landscapes; Hills; integration of channels and slopes; Kinetic energy; Landforms; least action; minimum energy; Optimization; Rain; Rainfall; River valleys; Rivers; Runoff; Slope; Slope runoff; slope stability; Slopes; Stability; Symmetry; Valleys; variational principles; Water flow
• ISSN: 2169-9003; 2169-9011
• DOI: 10.1029/2020JF006014

The equations and constraints of a general class of models of equilibrium fluvial landscapes are represented as a variational principle (VP). The VP minimizes a functional defined as the sum of (a) the difference between the total kinetic and potential energies of flows over the unchanneled slopes and (b) the sum of the kinetic and potential energies of the channelized flows. When the flows over the slopes are subject to a stability constraint on their magnitude, landscapes that minimize the functional are characterized by: (a) the occurrence on channel boundaries of the minimally stable value q=qc of slope runoff; (b) the symmetry of inflows into the channels; (c) the minimal total channel length L=RA/2qc for rainfall R and area A; (d) the minimization of the action of the slopes if and only if they are stable and their slope‐discharge relation q=q(S) is monotone‐increasing in slope; and (e) the minimization of channel energy if and only if the slope‐discharge relation for channels Q(S) is monotone‐decreasing in slope. Applications of the variational theory to first order channels on a linear ridge lead easily to solutions with equal‐length, uniformly‐spaced channels in symmetrical valleys staggered at the ridge‐crests. These solutions minimize the functional of the VP when the equilibrium landscape is in a minimally‐stable state and correspond well to numerical solutions to analogous time‐dependent models driven to equilbrium. The theory offers a promising approach for investigating the structure of equilibrium channel networks that are stably integrated into equilibrium slopes. Plain Language Summary The usual approach to understanding the forms of rivers, valleys, and hills is to study equations that describe how water flows over a landscape, creating such landforms by erosion. This paper takes a simpler, and perhaps more illuminating, approach by showing that a measure of the kinetic energy of the water moving over the surface minus the energy that it possesses as a result of its height above sea‐level, is minimized by the shapes into which it erodes the hills and valleys. The paper shows that the usual equations for describing landscapes have this apparent “laziness” of nature encoded into them. It also shows that the forms that we usually take for granted in landscapes, such as the distance apart of rivers and the symmetry of river valleys, are reflections of this laziness. Key Points Minimizing a simple functional of slope and channel energy implie