Transient attractors: towards a theory of the graded stream for alluvial and bedrock channels

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

doi:10.1016/s0098-3004(99)00128-4

Cited by 30 publications

(41 citation statements)

References 37 publications

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“…(11) would be a migrating wave of erosion travelling either up or down the catchment (Braun et al, 2015). This wave could also potentially take the form of a shock wave, in which due to the change in gradient the lower reaches of the migrating wave could travel faster than the upper reaches, creating a breaking wave (Smith et al, 2000;Pritchard et al, 2009). The time evolution of Eq.…”

Section: Erosion Within a Single Dimension Systemmentioning

confidence: 99%

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Numerical modelling of landscape and sediment flux response to precipitation rate change

et al. 2018

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Abstract. Laboratory-scale experiments of erosion have demonstrated that landscapes have a natural (or intrinsic) response time to a change in precipitation rate. In the last few decades there has been growth in the development of numerical models that attempt to capture landscape evolution over long timescales. However, there is still an uncertainty regarding the validity of the basic assumptions of mass transport that are made in deriving these models. In this contribution we therefore return to a principal assumption of sediment transport within the mass balance for surface processes; we explore the sensitivity of the classic end-member landscape evolution models and the sediment fluxes they produce to a change in precipitation rates. One end-member model takes the mathematical form of a kinetic wave equation and is known as the stream power model, in which sediment is assumed to be transported immediately out of the model domain. The second end-member model is the transport model and it takes the form of a diffusion equation, assuming that the sediment flux is a function of the water flux and slope. We find that both of these end-member models have a response time that has a proportionality to the precipitation rate that follows a negative power law. However, for the stream power model the exponent on the water flux term must be less than one, and for the transport model the exponent must be greater than one, in order to match the observed concavity of natural systems. This difference in exponent means that the transport model generally responds more rapidly to an increase in precipitation rates, on the order of 10 5 years for post-perturbation sediment fluxes to return to within 50 % of their initial values, for theoretical landscapes with a scale of 100 × 100 km. Additionally from the same starting conditions, the amplitude of the sediment flux perturbation in the transport model is greater, with much larger sensitivity to catchment size. An important finding is that both models respond more quickly to a wetting event than a drying event, and we argue that this asymmetry in response time has significant implications for depositional stratigraphies. Finally, we evaluate the extent to which these constraints on response times and sediment fluxes from simple models help us understand the geological record of landscape response to rapid environmental changes in the past, such as the Paleocene-Eocene thermal maximum (PETM). In the Spanish Pyrenees, for instance, a relatively rapid (10 to 50 kyr) duration of the deposition of gravel is observed for a climatic shift that is thought to be towards increased precipitation rates. We suggest that the rapid response observed is more easily explained through a diffusive transport model because (1) the model has a faster response time, which is consistent with the documented stratigraphic data, (2) there is a high-amplitude spike in sediment flux, and (3) the assumption of instantaneous transport is difficult to justify for the transport of large grain sizes as an alluvi...

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Section: Erosion Within a Single Dimension Systemmentioning

confidence: 99%

“…Kirkby, 1971;Smith and Bretherton, 1972;Smith et al, 2000;Whipple and Tucker, 2002;Crosby et al, 2007). From solving both Eqs.…”

Section: Generating Similar Landscapesmentioning

confidence: 99%

Numerical modelling of landscape and sediment flux response to precipitation rate change

et al. 2018

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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 following separable solutions were found by the first author and studied in great detail in [46] Remark 2. When a and b have opposite signs, and the sign changes across x À x 0 ¼ y À y 0 , the functions h, H o , and H defined in the preceding lemma are called mountain ridges (see Fig.…”

Section: Model Solutions: Mountains and Ridgesmentioning

confidence: 96%

“…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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Transient attractors: towards a theory of the graded stream for alluvial and bedrock channels

Cited by 30 publications

References 37 publications

Numerical modelling of landscape and sediment flux response to precipitation rate change

Numerical modelling of landscape and sediment flux response to precipitation rate change

Turbulent rivers

Mathematical Models for Erosion and the Optimal Transportation of Sediment

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