Reply [to “Comment on ‘Chaos in rainfall’ by I. Rodriguez‐Iturbe et al.”]

Flossmann, Andrea I.; Power, B. Febres de; Sharifi, Mohammad Bagher; Georgakakos, Konstantine P.

doi:10.1029/wr026i008p01841

Cited by 11 publications

(9 citation statements)

References 10 publications

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“…1. The straight line on this graph indicates that the drainage area has a power-law distribution, and the gradient agrees well with empirical data[18].…”

supporting

confidence: 67%

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Mechanism for Global Optimization of River Networks from Local Erosion Rules

Sinclair

Ball

1996

Phys. Rev. Lett.

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We show that landscapes and their drainage networks evolve under erosion rules to obey a variational principle. Starting from hydrodynamics we first find by separation of variables a general relationship between erosion and a modified slope-area law. This law encompasses previous work and observations, and we confirm it by simulation. Secondly we show that this corresponds to a solution of a variational minimization model, generalized from Rodriguez-Iturbe et al. Thus a mechanism by which the local process of erosion acts to globally optimize a river network is produced. [S0031-9007 (96)00098-1] PACS numbers: 64.60.Ak, 92.40.Fb Empirical observations on natural rivers by Leopold and Maddock [1] and Leopold [2] revealed power-law relationships between the slope (s), width (w), depth (d), velocity (y), and discharge (Q) of a channel, measured at the same time for different locations amongst river networks in the United States: s~Q 20.5 , w~Q 0.5 , d~Q 0.4 , y~Q 0.1 .The first of these is an observation of the slope-area law. A theoretical framework was put forward by RodriguezIturbe et al. [3] that was able to account for some of these observations. Using two unproven principles an expression was derived for the total rate of expenditure of the mechanical potential energy of all the water in the network (E):The unsubstantiated suggestion was then made that natural river networks evolve in such a way as to make E a global minimum. Computed networks that obey this rule are called optimal channel networks, and these networks have been found to obey Hack's law, Horton's laws, Moon's law, Melton's law, Strahler ordering, and the drainage-basin area law to the same extent as natural networks [4][5][6][7][8]. Different global minimization principles were also suggested by Yang, Song, and Davies and Sutherland [9-11], but in each case a full justification for the method proved difficult to obtain. Rinaldo et al. [12,13], and latterly Sun, Meakin, and Jossang [14], performed a direct minimization of E by computer simulation, using a simulated annealing approach. In this model, a channel network was placed on a square lattice such that the water present at each site in the lattice could drain through the network to the perimeter of the lattice. This network was then rearranged, until a drainage network with a minimum value of E was found. This value was remarkably constant, using a wide variety of initial conditions. The identification s~Q 20.5 was then made [15], enabling a value of height to be allotted to each site in the lattice, and it was discovered that the scalar field of heights was continuous over basin boundaries.In this Letter basic hydrodynamics is used to establish an erosion equation, which is numerically shown to erode the landscape to give steady drainage networks. We then show analytically that this implies a particular generalization of the slope-area law. This law is then revealed to be equivalent to the minimization of a global quantity L, which explains how local erosion can act to minimize a global qu...

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“…1. The straight line on this graph indicates that the drainage area has a power-law distribution, and the gradient agrees well with empirical data[18].…”

supporting

confidence: 67%

“…A theoretical framework was put forward by RodriguezIturbe et al [3] that was able to account for some of these observations. Using two unproven principles an expression was derived for the total rate of expenditure of the mechanical potential energy of all the water in the network (E):…”

mentioning

confidence: 97%

Mechanism for Global Optimization of River Networks from Local Erosion Rules

Sinclair

Ball

1996

Phys. Rev. Lett.

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“…There is accumulating evidence that many hydrologic and geomorphic systems may indeed behave chaotically in the temporal domain (Afouda 1989;Rodriguez-Iturbe et al 1989, 1991Willgoose et al 1991;Malanson, Butler, and Georgakakos 1992;Renwick 1992;Phillips 1987Phillips , 1990Phillips , 1992a. Thus deterministic chaos must be considered a likely candidate to explain, at least in part, spatially complex, irregular landscapes.…”

Section: Discussionmentioning

confidence: 99%

“…Both the amount of data and sampling interval needed to detect chaos in correlation dimension analyses and the reliability of such analyses are a subject of ongoing research and debate in the geosciences (cf. Rodriguez-Iturbe et al 1989; Ghilardi and Rosso 1990;Chouet and Shaw 1991;Elsner and Tsonis 1992).…”

Section: Jonathan D Phillips / 105mentioning

confidence: 99%

Spatial‐Domain Chaos in Landscapes

Phillips¹

1993

Geographical Analysis

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Landscapes are often extremely complex. Earth surface features and processes are typically characterized by a number of components related to each other in a dense web of interactions, and may exhibit irregular, even apparently random, spatial patterns. The behavior of these biophysical landscape systems is typically nonlinear, in that a change in inputs (such as precipitation, insolation, or tectonic uplift) does not produce the same proportional response across the range of the inputs. For these reasons a systems-oriented approach to physical geography, which emphasizes or at least recognizes the complex web of interrelationships in the landscape, has become prominent in recent decades in physical geography and related environmental sciences (Chorley and Kennedy 1971;Huggett 1985;Gregory 1985). Likewise, conceptual models of landscape systems have emphasized the inherent nonlinearities via frameworks such as the threshold concept (Schumm 1979;Bull 1979;Chappell 1983;Muhs 1984). This paper seeks to expand the concept of deterministic complexity (complexity arising from internal dynamics of deterministic systems) of nonlinear dynamical systems-which has been applied almost entirely in the temporal domain-to the spatial domain. Specifically, this paper attempts to show that deterministic complexity in the spatial domain is, in at least some cases, an inevitable outcome of deterministic complexity in temporal Jonathan D. Phillips is associate professor of geography at East Carolina University.

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“…We assume a separation of mechanical and electrical timescales and update ij according to the Laplacian relaxation algorithm between each mechanical timestep in Equation (3). The relaxation iteration obtained from (1) and (2), is given by…”

mentioning

confidence: 99%