Towards an elementary theory of drainage basin evolution: I. The theoretical basis

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

doi:10.1016/s0098-3004(97)00068-x

Cited by 36 publications

(75 citation statements)

References 22 publications

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“…The classical hydro-geomorphic model for channel head locations is based on a stability analysis for geomorphic hillslope evolution (Smith and Bretherton, 1972;Tarboton et al, 1992;Smith et al, 1997): infinitesimally small hollows on a hillslope will act as a focal point for water and sediment transported from upslope, due to flowline convergence just upflow of the hollow. If the (increased) amount of sediment delivered to the hollow is larger than the (increased) sediment transport capacity from the hollow downwards, the hollow will tend to fill up and disappear.…”

Section: Hillslope Lengthmentioning

confidence: 99%

Curvature distribution within hillslopes and catchments and its effect on the hydrological response

Bogaart

Troch

2006

Hydrol. Earth Syst. Sci.

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Abstract. Topographic convergence and divergence are first order controls on the hillslope and catchment hydrological response, as evidenced by similarity parameter analyses. Hydrological models often do not take convergence as measured by contour curvature directly into account; instead they use comparable measures like the topographic index, or the hillslope width function. This paper focuses on the question how hillslope width functions and contour curvature are related within the Plynlimon catchments, Wales. It is shown that the total width function of all hillslopes combined suggest that the catchments are divergent in overall shape, which is in contrast to the perception that catchments should be overall convergent. This so-called convergence paradox is explained by the effect of skewed curvature distributions and extreme curvatures near the channel network. The hillslopestorage Bossiness (hsB) model is used to asses the effect of within-hillslope convergence variability on the hydrological response. It is concluded that this effect is small, even when the soil saturation threshold is exceeded. Also described in this paper is a novel algorithm to compute flow path lengths on hillslopes towards the drainage network, using the multidirectional flow redistribution method.

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Section: Hillslope Lengthmentioning

confidence: 99%

Curvature distribution within hillslopes and catchments and its effect on the hydrological response

Bogaart

Troch

2006

Hydrol. Earth Syst. Sci.

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“…Surprisingly it is still not completely understood even in one or two-dimensional approximations of the full three-dimensional flow. Erosion by water seems to determine the features of the surface of the earth, up to very large scales where the influence of earthquakes and tectonics is felt; see [33,34,32,6,4,36]. Thus water flow and the subsequent erosion give rise to the various scaling laws known for river networks and river basins; see [12,8,9,10,11].…”

Section: Introductionmentioning

confidence: 99%

Turbulent rivers

Birnir

2008

Quart. Appl. Math.

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Abstract. The existence of solutions describing the turbulent flow in rivers is proven. The existence of an associated invariant measure describing the statistical properties of this one-dimensional turbulence is established. The turbulent solutions are not smooth but Hölder continuous with exponent 3/4. The scaling of the solutions' second structure (or width) function gives rise to Hack's law (1957), stating that the length of the main river, in mature river basins, scales with the area of the basin l ∼ A h , h = 0.568 being Hack's exponent. Introduction.The flow of water in streams and rivers is a fascinating problem with many applications that has intrigued scientists and laymen for many centuries; see Levi [21]. Surprisingly it is still not completely understood even in one or two-dimensional approximations of the full three-dimensional flow. Erosion by water seems to determine the features of the surface of the earth, up to very large scales where the influence of earthquakes and tectonics is felt; see [33,34,32,6,4,36]. Thus water flow and the subsequent erosion give rise to the various scaling laws known for river networks and river basins; see [12,8,9,10,11].One of the best known scaling laws of river basins is Hack's law [16], which states that the area of the basin scales with the length of the main river to an exponent known as Hack's exponent. Careful studies of Hack's exponent, see [11], show that it actually has three ranges, depending on the age and size of the basin, apart from very small and very large scales where it is close to one. The first range corresponds to a spatial roughness coefficient of one half for small channelizing (very young) landsurfaces. This has been explained, see [4] and [13], as Brownian motion of water and sediment over the channelizing surface. The second range with a roughness coefficient of 2/3 corresponds to the evolution of a young surface forming a convex (geomorphically concave) surface, with young rivers that evolve by shock formation in the water flow. These shocks are called bores (in front) and hydraulic jumps (in rear); see Welsh, Birnir and Bertozzi [36]. Between them sediment is deposited. Finally there is a third range with a roughness

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“…The initial surface is unstable, but in spite of this it is possible to solve the two equations (2.1) and (2.2) numerically with modern numerical methods. The first author and his collaborators did this in [44], [45], [8], [9] and [50]. Thus they gained considerable insight into the properties of the solutions, and the main purpose of this paper is to develop the full nonlinear analysis based on these insights.…”

Section: Solutions Of 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%

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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Towards an elementary theory of drainage basin evolution: I. The theoretical basis

Cited by 36 publications

References 22 publications

Curvature distribution within hillslopes and catchments and its effect on the hydrological response

Curvature distribution within hillslopes and catchments and its effect on the hydrological response

Turbulent rivers

Mathematical Models for Erosion and the Optimal Transportation of Sediment

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