Towards an elementary theory of drainage basin evolution: II. A computational evaluation

Smith, Terence R.; Merchant, George E.; Birnir, Björn

doi:10.1016/s0098-3004(97)00067-8

Cited by 31 publications

(76 citation statements)

References 7 publications

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“…For a detailed derivation of these equations we refer to [45] and [8]. When many simulations are performed and an ensemble average over these simulations is taken, a statistically stationary equilibrium water depth emerges.…”

Section: The Model Equationsmentioning

confidence: 99%

“…The second group has produced remarkable simulations of evolving channel networks; see [52,53], [18], [48] and [35]. The third group has lead to an increasing understanding of the physical mechanisms that underlie erosion and channel formation; see [40], [42], [30], [36], [28], [27], [29], [43], [22,23,24,25,21], [44,45,46], [39], [50], [9], [15], [7], [41].…”

Section: Introductionmentioning

confidence: 99%

“…In [44] and [45], a family of two partial differential equations were introduced based on the conservation of water and sediment. These equations describe a transport-limited process [18] in which sediment moves in the same direction in which the surface water flows.…”

Section: Introductionmentioning

confidence: 99%

“…Our focus is the equations developed and studied by the first author and his collaborators in [44], [45], [8], [9]. These equations improve the original model in [42] by including a pressure term (in addition to the gravitational and friction terms) that prevents water from accumulating in an unbounded manner in surface concavities; see [28].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mathematical Models for Erosion and the Optimal Transportation of Sediment

Birnir

Rowlett

2013

International Journal of Nonlinear Sciences and Numerical Simul

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We investigate a mathematical theory for the erosion of sediment which begins with the study of a non-linear, parabolic, weighted 4-Laplace equation on a rectangular domain corresponding to a base segment of an extended landscape. Imposing natural boundary conditions, we show that the equation admits entropy solutions and prove regularity and uniqueness of weak solutions when they exist. We then investigate a particular class of weak solutions studied in previous work of the first author and produce numerical simulations of these solutions. After introducing an optimal transportation problem for the sediment flow, we show that this class of weak solutions implements the optimal transportation of the sediment.

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Section: The Model Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mathematical Models for Erosion and the Optimal Transportation of Sediment

Birnir

Rowlett

2013

International Journal of Nonlinear Sciences and Numerical Simul

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“…WILLGOOSE et al (1991) reverted to an artificial channel indicator variable, and KRAMER and MARDER (1992) used cellular lattice models, a development which has formed the thrust of simulation models since, e.g., those of HOWARD (1994) and TUCKER and SLINGERLAND (1994). There have been efforts to solve the Smith-Bretherton model directly (e.g., SMITH et al, 1997;BIRNIR et al, 2001), although these are problematical, unsurprisingly since the original Smith-Bretherton model is actually ill-posed.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis of a Model of River Channel Formation

Díaz

Fowler

Muñoz

et al. 2008

Pure appl. geophys.

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Abstract-The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin.

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Analytic Theory of Equilibrium Fluvial Landscapes: The Integration of Hillslopes and Channels

Smith

2018

JGR Earth Surface

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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 (qc) 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 qc 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 densities scriptDd=R/2qc. Changes to channel networks induced by environmental fluctuations may occur at any network location because maximum, neutrally stable overland flows qc occur on the CSB. Stable equilibrium surfaces are conjectured to correspond to local minima of the Lagrangian over the space of admissible networks.

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Towards an elementary theory of drainage basin evolution: II. A computational evaluation

Cited by 31 publications

References 7 publications

Mathematical Models for Erosion and the Optimal Transportation of Sediment

Mathematical Models for Erosion and the Optimal Transportation of Sediment

Mathematical Analysis of a Model of River Channel Formation

Analytic Theory of Equilibrium Fluvial Landscapes: The Integration of Hillslopes and Channels

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