Ramification of stream networks

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

doi:10.1073/pnas.1215218109

Cited by 123 publications

(155 citation statements)

References 41 publications

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“…For instance, if a channel grows along the stream line that intersects its tip (Fig. 3A), it should bifurcate at 2π=5 = 72° (24). This value accords with field measurements collected in a small river network near Bristol, Florida.…”

Section: Evaluation Of the Principle Of Local Symmetrysupporting

confidence: 68%

“…First, we justify its application to channel networks based on simple physical reasoning. We then show that the PLS is mathematically equivalent to assuming that a channel grows along the groundwater flow lines, an assumption that has been shown to be consistent with field observations (24). Finally, we describe a numerical procedure to grow a network according to this principle.…”

Section: Evaluation Of the Principle Of Local Symmetrymentioning

confidence: 99%

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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: Evaluation Of the Principle Of Local Symmetrysupporting

confidence: 68%

Section: Evaluation Of the Principle Of Local Symmetrymentioning

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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“…It also indicates the typical distance for interaction between channels-a screening length-that may be related to the regular spacing of channels in ramified networks [13]. Figure 8 suggests how this length scale manifests itself in a network incised by groundwater seepage located near Bristol, FL, USA [11,13]. In this network, the field U represents the groundwater height squared.…”

Section: Screening Lengthsmentioning

confidence: 99%

“…That is not the case when the source term is not zero, and the field becomes non-harmonic. Recent work shows how stream networks can be considered as absorbing slits in a diffusive groundwater field [11,12]. Motivated by this problem, we refer to these slits as channels and find an exact solution for the Poisson field around a rudimentary channel network.…”

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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“…Such structures are observed e.g., in the electrochemical deposition experiments [2,9], wormhole formation in dissolving rocks [10], smoldering combustion [11][12][13], side-branches growth in crystallization [14,15], or the evolution of seepage channel networks [16,17]. For systems of this kind, we show that the dynamics can be followed using a higher-level description, in which the microscopic details are neglected and the system is treated as a collection of emergent structures -thin lines interacting with each other through a continuous field.…”

mentioning

confidence: 99%

Effective description of the interaction between anisotropic viscous fingers

Budek¹,

Szymczak²

2014

EPL

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We study patterns formed by viscous fingering in a rectangular network of microfluidic channels. Due to the strong anisotropy of such a system, the emerging patterns have a form of thin needle-like fingers, which interact with each other, competing for an available flow. We develop an upscaled description of this system in which only the fingers are tracked and the effective interactions between them are introduced, mediated through the evolving pressure field. Due to the quasi-2d geometry of the system, this is conveniently accomplished using conformal mapping techniques. A complex two-phase flow problem is thus reduced to a much simpler task of tracking evolving shapes in a 2d complex plane. This description, although simplified, turns out to capture all the key features of the system's dynamics and allows for the effective prediction of the resulting growth patterns.

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Ramification of stream networks

Cited by 123 publications

References 41 publications

Path selection in the growth of rivers

Path selection in the growth of rivers

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

Effective description of the interaction between anisotropic viscous fingers

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