Path selection in the growth of rivers

Abstract

River networks exhibit a complex ramified structure that has inspired decades of studies. However, an understanding of the propagation of a single stream remains elusive. Here we invoke a criterion for path selection from fracture mechanics and apply it to the growth of streams in a diffusion field. We show that, as it cuts through the landscape, a stream maintains a symmetric groundwater flow around its tip. The local flow conditions therefore determine the growth of the drainage network. We use this principle to reconstruct the history of a network and to find a growth law associated with it. Our results show that the deterministic growth of a single channel based on its local environment can be used to characterize the structure of river networks.river channels | principle of local symmetry | harmonic growth | Loewner equation | fracture mechanics A s water flows, it erodes the land and produces a network of streams and tributaries (1-3). Each stream continues to grow with the removal of more material, and evolves in a direction that corresponds to the water flux entering its head. The prediction of the trajectory of a growing channel and the speed of its growth are important for understanding the evolution of complex patterns of channel networks. Several models address their evolution and ramified structure. One, the Optimal Channel Networks model (4), is based on the concept of energy minimization and suggests a fractal network. The landscape evolution method and many diffusion-based models (5-7) have also proven useful for modeling erosion and sediment transport. These models distinguish between two regimes: one is a diffusion-dominated regime where topographic perturbations are diminished, which leads to a smoother landscape and uniform symmetric drainage basins. In this case, the shape of a channel cannot deviate from a straight line. In the second regime, advection dominates, and channel incisions are amplified. The channel effectively continues to the next point that attracts the largest drainage basin, which corresponds to the direction where it receives the maximum water flux. These models nicely predict the formation of ridges and valleys and provide insight into the interaction between advective and diffusive processes (8, 9). However, they do not address directly the evolution of a single channel and do not explicitly address the nature of a growing stream based on its local environment.Here we address two basic questions in the evolution and the dynamics of a growing channel: where it grows and at what velocity. We propose that, when streams are fed by groundwater, the direction of the growth of a stream is defined by the groundwater flow in the vicinity of the channel head. This theory is widely used in the framework of continuum fracture mechanics and accurately predicts crack patterns in different fracture modes, for both harmonic and biharmonic fields and for different stress singularities (10-12). The theory, known as the principle of local symmetry, states that a crack propagates along the ...

A boundary-integral framework to simulate viscous erosion of a porous medium

“…Flow-induced erosion deteriorates and reshapes solid material over a range of scales found in nature, from massive land formations [1,16,23,30,37,[49][50][51], to centimeter-scale features and patterns [9,20,24,47]. Though less visible, these same forces are working at the very smallest scales, slowly deteriorating the individual constituents of porous media (e.g.…”

Laplacian networks: Growth, local symmetry, and shape optimization

“…In fact, these two propositions are equivalent. For instance, in a wording inspired from fracture mechanics, we can say that rivers formed by seepage erosion grow according to the principle of local symmetry (PLS) [28,29]. This reformulation explicitly constrains the geometry of a network during its growth, and can therefore be used to identify the velocity and bifurcation rules based on its shape, at least in principle.…”

Congruent Bifurcation Angles in River Delta and Tributary Channel Networks

“…Fluvial tributary networks have been modeled for over a century as the subset of a landscape where there is sufficient shear stress imparted by a flow to transport sediment via fluvial processes (Gilbert & Murphy, ; Horton, ; Izumi & Parker, , ). Recent studies have advanced the theory to show that tributary networks are partially (Perron et al, , , ) or completely (Abrams et al, ; Cohen et al, ; Devauchelle et al, ; Petroff et al, , ) controlled by diffusive transport of fluid and/or sediment outside of the channel network. Distributary channel networks, on the other hand, have been understood as the product of sedimentation from turbulent jets that form at the mouths of rivers entering basins (Bates, ; Edmonds & Slingerland, ; Fagherazzi et al, ; Jerolmack & Swenson, ; Wright, ).…”

“…Fluvial tributary networks have been modeled for over a century as the subset of a landscape where there is sufficient shear stress imparted by a flow to transport sediment via fluvial processes (Gilbert & Murphy, 1914;Horton, 1945;Izumi & Parker, 1995, 2000. Recent studies have advanced the theory to show that tributary networks are partially (Perron et al, 2008(Perron et al, , 2009(Perron et al, , 2012 or completely (Abrams et al, 2009;Cohen et al, 2015;Devauchelle et al, 2012;Petroff et al, 2011Petroff et al, , 2013 controlled by diffusive transport of fluid and/or sediment outside of the channel network. Distributary channel networks, on the other hand, have been understood as the product of sedimentation from turbulent jets that form at the mouths of rivers entering basins (Bates, 1953;Edmonds & Slingerland, 2007;Fagherazzi et al, 2015;Jerolmack & Swenson, 2007;Wright, 1977 Table S1 • Table S2 • Table S3 Correspondence to: T. S. Coffey, tscoffey@email.uark.edu 10.1002/2017GL074873 to the distance from a channel mouth to the locus of mouth bar aggradation, which in turn is linked to aspects of channelized flow at a river mouth, including inflow buoyancy and inertia, bed friction, the width-to-depth ratio of the channel, and sediment characteristics (Axelsson, 1967;Edmonds & Slingerland, 2007;Edmonds, Shaw, et al, 2011;Jerolmack & Swenson, 2007;Wright, 1977.…”