Italy

RAPID COMMUNICATIONS

PHYSICAL REVIEW E MAY 1997VOLUME 55, NUMBER 5

Randomly pinned landscape evolution

Guido Caldarelli, Achille Giacometti, and Amos MaritanIstituto Nazionale di Fisica della Materia, International School for Advanced Studies and INFN, via Beirut 2-4, Trieste I-34013,

Ignacio Rodriguez-IturbeDepartment of Civil Engineering, Texas A&M University, College Station, Texas 77843

Andrea RinaldoIstituto di Idraulica ‘‘G. Poleni,’’ Universitadi Padova, I-35131 Padova, Italy

~Received 4 March 1997!

A simple scheme for the evolution of a fluvial landscape in heterogeneous environments is critically exam-ined to capture the essential mechanism responsible for the recurrent scale-free landforms in the river basin. Itis shown that, regardless of boundary and initial conditions, geomorphological constraints in the form ofquenched randomly pinned regions play a key role in the robust emergence of aggregation patterns with ascaling behavior in agreement with that of real river basins.@S1063-651X~97!51805-8#

PACS number~s!: 05.40.1j, 68.70.1w

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Considerable efforts have been recently devoted to mels of landscape evolution@1–3# in view of newly acquiredexperimental evidence showing recurrent scale-free sttures in the geomorphology of the fluvial basin often coving over four orders of magnitude of linear size@4#. As instandard critical phenomena, the mechanism producing sstructures is expected to depend only on a few basic featcommon to all the networks, rather than on the details ofparticular system under consideration. This can be conniently shown by classifying the results into a limited number of universality classes determined by a set of critiexponents.

A crucially simple lattice model, inspired by selorganized critical@5# models, aimed at the basic mechanisfor fluvial landform evolution was recently proposed@6#. Themodel addresses only fluvial dynamics~whose imprinting isargued to dominate the morphology of river basins@1#! andis based on a dynamic threshold for activity defined byexceeding of well-defined shear stresses.

Although the dynamics of evolution is purposely etremely simple, it has been shown@6# that this choice is bothphysically meaningful and suitable as an algorithmic searing mechanism.

Here we show that the results of the original simulatio@6# were affected by finite size effects and by a limitchoice of boundary and initial conditions which all combinto yield results very close to the experimental ones. Hereuse larger scale simulations and more general boundaryinitial conditions. In this way we identify two distinct universality classes. The first one, corresponding to the origformulation @6#, fails to reproduce consistently the scalinbehavior of field observations. The second one agreesobservational data.

Since in fluvial processes inhomogeneous precipitatiand geomorphological constraints are expected to pladefinite role in the development of the fluvial basin@7#, weshow that an essential ingredient of the model should eftively take heterogeneity of the terrain into account, a featwhich was missing in the original model and which provida new universality class.

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Specifically we considered two physically based kindsrandomness. We have explicitly verified that the first onamely heterogeneous rainfall, is not effective in changthe universality class of the original model. The second owas devised to model the geomorphological constraintsthe form of random pinning of the landscape. This paphinges on this feature which proves effective and of muphysical significance and indeed defines the second cwhich consistently agrees with the one of real river basin

Our lattice model is defined as follows. Lethx be theheight of the landscape associated with every sitex of asquare lattice of sizeL3L. We assume that on the loweside~kept at heighth50) all sites are possible outlets~mul-tiple outlets! of an ensemble of rivers which are competinto drain the wholeL3L basin. A random noise is added tthe initial substrate in order to have unbiased initial contions. Each site~pixel! collects water from a distributed injection in addition to the flow which drains to the site frothe upstream connected sites. The steepest descent contions allow the assignment of drainage directions to evsite of an arbitrary landscape. The drained areaax ~which isalso a measure of the total flow rates collected at that site! isassociated with each sitex according to the equation

ax5(y~x!

ay1r x ~1!

where the sum runs over the subset of neighbor sitesy(x)whose area is actually drained byx. If no contribution toxexists, thenax5r x and it represents a source. The secoterm in Eq.~1! represents rainfall at positionx. For uniforminjection r x51. Random injection will be briefly discusseat the end. Another significant morphological indicator is t~up!stream lengthl x from any sitex to its source, which iscomputed according to straightforward procedures@8#.

The time evolution of the model follows the followinsteps.

R4865 © 1997 The American Physical Society

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R4866 55GUIDO CALDARELLI et al.

~1! The shear stresstx acting on every site is computeaccording to@6# tx5Dhxax

1/2 whereDhx is the local gradientalong the drainage direction.

~2! If the shear stress on a site exceeds a threshold vtc , then the corresponding heighthx is reduced—mimickingerosive mechanisms—in order to decrease the local gradand the shear stress and set it just at the critical threshThis produces a rearrangement of the network accordinEq. ~1! followed by a new update of the whole pattern asstep~1!.

~3! When all sites have shear stress below~or at! thresh-old, the system is in a dynamically steady state. Sincesituation is not necessarily the most stable, a perturbatioapplied to the network to increase the stability of a nsteady state. A site is picked up at random and its heighincreased in such a way that no lakes, i.e., sites whose heis lower than the heights of all neighbors, are formed. St~1! and ~2! then follow as before.

A small percentp ~typically 5%) of the heights are chose

FIG. 1. Log-Log plot of the cumulative area distributioP(a,L) versusa without ~up! and with ~down! pinning. The fullline ~slightly shifted upward for sake of clarity! has a slope corresponding tot51.38 andt51.43, respectively.

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~randomly and in uncorrelated manner! to be pinned to theirinitial value. This is equivalent to assigning a large valuetc ~infinite in our case! to a randomly chosen fractionp ofthe sites and a valuetc51 to the remaining sites. Thus thessites do not take part in the landscape evolution procNote that this is a reasonable geological feature~see Mont-gomery and Dietrich in Ref.@4#!. One expects this effect toincrease the meandering of the main river structures tproviding an additional mechanism for aggregation. The cp50 corresponds to the case studied in Ref.@6#. It is worthmentioning that for a given concentrationp of pinned sites, atypical correlation lengthl p;p21/2 ~in two dimensions! isintroduced. For system sizesL@ l p@1 ~5lattice spacing;10 m!, crossover from the pure (p50) case to the trulyasymptotic behavior (pÞ0, L→`) should be observableThis is quite typical in disordered systems and it is obserhere too.

After a suitable number of the perturbations@step~3!#, thesystem reaches a steady state which is insensitive to furperturbations and where all statistics of the networks

FIG. 2. Scaling functiona12tP(a,L) versusa/L11H in thesame order as above. The values used to obtain the collapset51.38,H50.60 andt51.43,H50.75, respectively.

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55 R4867RANDOMLY PINNED LANDSCAPE EVOLUTION

FIG. 3. ~Color!. Comparison between the evolution of two identical initial configurations of the model with sizeL5100 without~a! andwith pinning ~b!. The pinning dilution was 5%.

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stable. The resulting steady state proves scale-free, i.e.characterized by power-law distributions of the physiquantities of interest, which will be analyzed below.

Our numerical calculations were carried out on a twdimensional square lattice~where each site has eight neareneighbors! for sizes up toL5200 and a number of perturbations up to 106. This high number of perturbations involvemakes the simulations very time consuming, thus preventests on much larger sizes. For comparison the original silations @6# were carried out up toL564.

The initial height field has a nonzero average slopem,along the ‘‘main flow’’ direction, and height fluctuations noexceeding 10% ofmL. Boundary conditions are reflecting ithe direction transversal to the main flow and open~multipleoutlets are allowed! in the parallel one. Averages over fe~up to five! configurations were routinely taken@9#. Thischoice proves sufficient in view of the self-averaging natof the random perturbation.

The scaling analysis of the resulting landforms is baon P(a,L) andP( l ,L), the cumulative probability distribution of the drainage areaa and stream lengthl , respectively,in a domain of linear sizeL. The following scaling forms areexpected@8# to hold:

P~a,L !5a12tFS a

L11HD , ~2!

P~ l ,L !5 l 12gGS l

Ldl D . ~3!

HereH is the Hurst~or wandering! exponent anddl is thestream-length~or chemical distance! fractal exponent@10#.t andg are two of the various critical exponents that candefined for river networks@1#. For self-affine river basins

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(H,1, dl51) scaling relations can be derived@8# yieldingall exponents in terms ofH only @11#. One finds

t5112H

11H, g511H . ~4!

Experimental values oft and g are available from earlierexperimental analyses from basins of different size, geoloexposed lithology, climate and vegetation@1#. There it wasobserved that, while a majority of basins tend to seeminuniversal valuest51.4360.02 andg51.860.1, exceptionsare found where altered values are observed, i.e., the vachange in a concerted manner and scaling relations sucthose in Eq.~4! are still satisfied.

Scaling analysis for the cumulative area distributionsshown in Figs. 1~a! and 1~b!. Figure 2 contains the corresponding collapse plots according to Eq.~2!. Similar resultsare also obtained for the upstream lengths cumulative dibutions. Our estimates of the critical exponents are indepdently derived and they agree with the scaling laws~4!.Their values are t51.3860.02, H50.6060.05, andg51.6060.02 in the absence of heterogeneity, i.e.,p50;t51.4360.02, H50.7560.05, andg51.7060.02 for therandom pinning case~i.e., pÞ0). The errors are statisticafor t andg. ForH the error is based on a careful estimatethe collapse plot. It should be noted that the collapse in2~b! is worse for the two highest sizes considered~see Fig. 3!due to poorer statistics.

Of course the fact thatH,1 implies thatdl51 has beenused in Eq.~3! for the analysis of the upstream lengths dtribution. However, corrections to scaling appear to be strger for the lengths rather than the areas distribution. In Figthe typical network obtained using the above dynamicsreported without~a! and with ~b! random pinning start-

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R4868 55GUIDO CALDARELLI et al.

ing with the sameinitial configuration. It is worth noticingthat the randomness increases the river wandering andthe value ofH andt. The following remarks are in order.

~a! We have explicitly verified that our results do ndepend on the imposed boundary conditions, i.e., chanfrom single to multiple outlets does not affect the criticexponents. This was the basic flaw in the original testing@6#of the model.

~b! When multiple outlets are allowed the statistics bason the largest rivers give slightly low Hurst exponents bofor the homogeneous and the heterogeneous case. In allthe resulting exponents are consistent with experimentalues and with the scaling laws~4! within the error bars.

~c! The same analysis for a bigger concentration of rdom pinned regions~up to 10%) does not show significanchanges of the above values of the exponents. This indicthat very likely whenpÞ0 and less of the percolation thresold there is a single universality class different from tp50 case. We did not pursue our study up to the percolathreshold of the pinned regions since new phenomena, sas large scale lake formations, occur and hence all relescales would be altered. This was beyond the objective of

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present work. Nor did we fully analyze further values ofpsince we expect onlytwo universality classes to be prese~with or without disorder! in view of the large amount ofcomputer time involved.

~d! We explicitly checked that random precipitations boconstant in time and updated at each perturbation@step ~3!above# do not significantly change the critical exponentsboth the homogeneous and heterogeneous terrains coered here. This is consistent with rigorous results derivedRef. @12# in a related context where random rainfall has beshown to be irrelevant.

In conclusion our results indicate that real rivers arewell described by the critical exponents of the homogenemodel, i.e.,p50. However, by introducing a~small! concen-tration of quenched nonerodible regions, a new universaclass is found which agrees well with the observational daThese pinned regions surrogate the ubiquitous presencgeomorphological constraints in real landscapes andtheir important role is appealing.

Enlightening discussions with Jayanth Banavar, MaCieplak, Francesca Colaiori, and Alessandro Flamminiacknowledged.

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@1# For an overview see, e.g., I. Rodriguez-Iturbe and A. RinalFractal River Basins: Chance and Self-Organization~Cam-bridge University Press, Cambridge, England, 1997!, and ref-erences therein.

@2# R. Leheny and S.R. Nagel, Phys. Rev. Lett.71, 1470~1993!;A. Giacometti, A. Maritan, and J. Banavar,ibid. 75, 577~1995!; J.R. Banavar, F. Colaiori, A. Flammini, A. GiacometA. Maritan, and A. Rinaldo~unpublished!.

@3# T. Sun, P. Meakin, and T. Jo¨ssang, Phys. Rev. E49, 4865~1994! and references therein.

@4# I. Rodriguez-Iturbe, R.L. Bras, E. Ijjasz-Vasquez, and D.Tarboton, Water Resour. Res.28, 988 ~1992!; D.G. Tarboton,R.L. Bras, and I. Rodriguez-Iturbe,ibid. 24, 1317~1988!; 26,2243~1990!; P. La Barbera and R. Rosso,ibid. 25, 735~1989!;D.R. Montgomery and W.E. Dietrich, Nature~London! 336,232 ~1988!; Science255, 826 ~1992!; D.G. Tarboton, R.L.Bras, and I. Rodriguez-Iturbe, Water Resour. Res.25, 2037~1989!.

@5# P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett.59, 381~1987!; Phys. Rev. A38, 364 ~1988!.

@6# A. Rinaldo, I. Rodriguez-Iturbe, R. Rigon, E. Ijjasz-Vasque

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and R.L. Bras, Phys. Rev. Lett.70, 822 ~1993!.@7# M.A. Summerfield,Global Geomorphology~Longman, Sin-

gapore, 1991!; W.E. Dietrich, C. Wilson, D.R. MontgomeryS.L. Reneau, and R. Bauer, J. Geol.20, 675 ~1992!.

@8# A. Maritan, A. Rinaldo, R. Rigon, A. Giacometti, andRodriguez-Iturbe, Phys. Rev. E53, 1510~1996!.

@9# We have verified that, for samples of linear sizeL564, aver-ages up to 30 initial configurations do not change the fiestimates, within the error bars.

@10# A value ofH51/2 would correspond to a network constructaccording to a simple random-walk rule often referred to asScheidegger model in this context@see, e.g., H. TakayasuPhys. Rev. Lett.63, 2563~1989!#.

@11# Self-similar river networks (H51, dl.1), have been dis-cussed in M. Cieplak, A. Giacometti, A. Maritan, A. RinaldI. Rodriguez-Iturbe, and J.R. Banavar~unpublished!. It is note-worthy that any fractal network with stream-length exponedl yields identical exponents of a self-affine network withH ifdl52/(11H) and vice versa.

@12# A. Maritan, F. Colaiori, A. Flammini, M. Cieplak, and J.RBanavar, Science272, 984 ~1996!.