Bifurcation dynamics of natural drainage networks

Petroff, Alexander P.; Devauchelle, Olivier; Seybold, Hansjörg; Rothman, Daniel H.

doi:10.1098/rsta.2012.0365

Cited by 60 publications

(69 citation statements)

References 39 publications

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“…For a semiinfinite channel on the negative x axis with boundary conditions ϕðθ = ±πÞ = 0, [5] the harmonic field around the tip can be expressed in cylindrical coordinates as (23) ϕðr, θÞ = a 1 r 1=2 cos θ 2 + a 2 r sinðθÞ + O r 3=2 , [6] where, as shown in Fig. 1, r is the distance from the channel head and the channel is located at θ = ±π.…”

Section: Significancementioning

confidence: 99%

Path selection in the growth of rivers

Cohen

Devauchelle

Seybold

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

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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 ...

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Section: Significancementioning

confidence: 99%

Path selection in the growth of rivers

Cohen

Devauchelle

Seybold

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

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“…Petroff et al [3] also consider the dynamics of a network, when analysing the branching of springs and streams. By applying a complex potential method to the seepage of groundwater around a spring, they map this dynamics to the more general problem of growth in a potential field.…”

Section: Contents Of This Issuementioning

confidence: 99%

“…The quantitative investigation of such patterns, however, is now a field of active research. For example, Petroff et al [3] here demonstrate that springs eroding into a slope will generically split, or bifurcate, at an angle of 2π/5, while I present a crack-ordering mechanism that links columnar joints with polygonal terrain and mud-cracks [4]. There are many other situations where regular patterns are generated, from the largest scales of geomorphology, such as the curving subduction arcs of the Earth's crust [5] or the meanders of river networks, to ripples on the beach [6] and periodic chemical precipitation patterns [7].…”

Section: Pattern Formationmentioning

confidence: 99%

Pattern formation in the geosciences

Goehring

2013

Phil. Trans. R. Soc. A.

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Pattern formation is a natural property of nonlinear and non-equilibrium dynamical systems. Geophysical examples of such systems span practically all observable length scales, from rhythmic banding of chemical species within a single mineral crystal, to the morphology of cusps and spits along hundreds of kilometres of coastlines. This article briefly introduces the general principles of pattern formation and argues how they can be applied to open problems in the Earth sciences. Particular examples are then discussed, which summarize the contents of the rest of this Theme Issue.

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“…This equation usually delineates a steady-state diffusive field that can describe a broad array of physical quantities such as chemical concentrations [4], temperature [5], fluid velocity [6,7], groundwater height [8,9] and stress potential [3]. However, in spite of its broad relevance, only a few exact solutions of the Poisson equation exist, in part due to the complexity of the geometry and boundary conditions [3,7,10].…”

Section: Introductionmentioning

confidence: 99%

Exact solution for the Poisson field in a semi-infinite strip

Cohen

Rothman

2017

Proc. R. Soc. A.

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The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.

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Bifurcation dynamics of natural drainage networks

Cited by 60 publications

References 39 publications

Path selection in the growth of rivers

Path selection in the growth of rivers

Pattern formation in the geosciences

Exact solution for the Poisson field in a semi-infinite strip

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