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Journal of Statistical Physics. Vol 91. Nos. 1/2. 19'JH s

Models of Fractal River Basins

Marek Cicplak,1 Achille Giacometti,2 Amos Maritan,3 Andrea Rinaldo,'1Ignacio Rodriguez-Iturbe,3 and Jayanth R. Banavar6

Received July 24, I'997; final January 16. 1998

Two distinct models for sell-similar and self-affine river basins are numericallyinvestigated. They yield fractal aggregation patterns following nontrivial powerlaws in experimentally relevant distributions. Previous numerical estimates onthe erilieal exponents, when existing, are confirmed and superseded. A physicalmotivation for both models in the present framework is also discussed.

KEY WORDS: Dynamical erilieal phenomena; growth process; rivers; runofand stream How; erosion and sedimentation; aggregation patterns.

I. INTRODUCTION

Experimental analyses of river networks0* have shown clear examples ofbehavior analogous to erilieal phenomena characterized by the absenceof a single well-defined length seale reflected in a power-law behavior ofvarious quantities. A fundamental question thai arises from these observations is whether, in analogy with conventional erilieal phenomena, onemay fruitfully classify this behavior into universality classes that are characterized by different sels of exponents. A related point is whether there existscaling relationships between the various exponents of a given universalityclass. Another vital issue is the elucidation of simple models amenable to

1 Polish Academy of Science, 02-668 Warsaw, Poland.- INI M Ullila di Vene/.ia. Diparlimenlo di Scienze Ambientali, 1-30123 Venice, llaly.1IN1-M Trieste and International School for Advanced Studies (SISSA), 1-34014 Orignano diTrieste, Italy.

4 Islitulo di Idraulica "O. Poleni," Unvcrsita di Padova, 1-35131 Padua, Italy.5 Department of Civil Engineering, Texas A&M University, College Station, Texas 77843.6 Department of Physics and Center for Materials Physics, Pennsylvania State University,

104 Davcy Laboratory, University Park, Pennsylvania 16802.1

0O22-47l5/98/O4O0-O00l$15.0O/0 O IMS Plenum Publishing Corporation

2 C i e p l a k e t a l .i

analysis that nevertheless capture some of the features of fractal fluvialpatterns.

Within this framework a theoretical description of the system needsto address two basic issues.'2 9) First, a careful characterization of thetopological properties of the networks is essential for understanding thebasic transport mechanism in the basin. References 2 and 3 are recentattempts in this direction. Optimization principles have been exploited bothnumerically'4-7' and analytically'5' to explain Ihe tendency of naturaldrainage networks to evolve toward an optimal stable topology. Generalscaling arguments can be found in ref. 8.

Second, a study of the dynamical evolution of ihe landscape (ongeological time scales) as a result of interaction with external agents (rain,wind etc.) would be desirable.'6,91

The present work will address only the fust point. We will argue that,despite much progress in the past few years, the problem is not yet fullyunderstood and deserves further analysis. To this aim we will discuss, basedon physical arguments, two toy models of river networks. The fust modelleads to a self-similar river basin, and is relevant when the erosionalproperties of the surface soil are slrongly heterogeneous. The second modelconsiders the homogeneous basin case and results in a self-affine rivernetwork.

Although the overwhelming majority ol" the observational data areconsistent with a self-affine description (i.e., networks display a privilegeddirection), the marked self-similarity of the basins with their own subbasins suggests a crossover from a self-similar character above some lengthscale. This is one of the reasons for considering models with both characters.It should also be emphasized that although some of the features of the modelspresented here were previously discussed in the literature (see below), webelieve that both the physical motivations and the analysis carried out hereare essentially new.

The plan of the paper is as follows. In the next section few definitionsand scaling relations will be recalled. In Section III results for self-similarare presented. Section IV is dedicated to the self-affine counterpart. InSection V few relevant experimental results will be briefly reminded for thesake of completeness. Section VI will summarize our findings along withsome future perspectives.

II. DEFINITIONS AND SCALING LAWS

We define a river network as a spanning (loopless) tree on a lattice oflinear size L."0) Each site has exactly one output bond to one of ilsneighbors and no restriction on Ihe number of input bonds (three at most

Models of Fractal River Basins

on a square lattice). In a river basin, the area at any site is defined as Ihenumber of sites upstream of the site connected by the network. From thecomputational point of view, it can also be regarded as a measure of theflow rate if a unit weight is assigned to each source thus simulating a unitconstant precipitation.

The equation for s,, the area at a given site /, is

•v.- = I »Vv,+ ljenn( i )

(1)

where uy is I if/ collects water from ils nearest-neighbor (im)silcj and 0o t h e r w i s e . i

It is experimentally observed'" and theoretical explained'7-^1 that inriver basins the probability density p(s, /_) of a site hit-vnigp-afea s in asystem of size L, has the settling form

p(s, L) -TF{1? (2)

where F(x) is a scaling function which lakes into account finite size effectsand (/> is the finite size exponent.

Similarly the distribution of tipstream lengths has been also predicted'8'and confirmed by field observation"" to display the universal form:

n(l,L) = l-vf _l_L'1' (3)

where /(.v) is the analogue of F(x) and d, coincides with the stream fractaldimension. The upstream length is defined as follows. At a given site theareas (see Eq. (I)) of the nearest-neighbours are checked. The site with thelargest value leads to ihe outlet. The site with the next-largest value isdefined lo be an upstream site—it indicates the longest path towards thesource. If two (or more) equal areas are encountered, one is randomlyselected. Alternatively, a burning algorithm'3' could be also employed.

In natural basins, the drainage area s and the stream length / arerelated by Hack's law"2'

/ i / * (4)

The sub-basin from any site defined as all the upstream sites connectedlo it is characterized by typical longitudinal and transverse lengths £,,and fx. For self-affine river networks one defines the Hurst (or wandering)exponent as ^x-Ci' with H^ 1. Note that for self-similar river networks

. C i e p l a k e t a l .

(in which each rivulet originating from any site and proceeding lo theglobal outlet is a fractal characterized by the same fractal dimension d,),the Hurst exponent //= I.

As one might expect the exponents are not independent. For self-similar networks (./,> 1, /■/=!) d, determines all the other exponents'78'

0 = 2 , hm*h& r = 2-2 / . / „ y = 2 /d l (5)

For self-affine networks (.// = I, //< I ) it is // which defines all ihe othere x p o n e n t s ' 5 ' V _ _ _ _ ^ C

< / > = ] + H , / / = 1 / ( H - / / ) , r =I +2//1 + // ' )• = I -l- //

Two features of the above relations are worth mentioning, First thereis experimental evidence in the observed data that //,< 1 and d,> 1. Thisapparent contradiction might be explained with the crossover between thetwo regimes occurring at some length scale, as mentioned in the introduction. Secondly it turns out from (5) and (6) that identical values of theexponents are obtained from both cases if <//=2/( I + //). This means thaiknowledge of the exponents other than d, and // cannot discriminate theself-similar or self-affine character of the basin. In this respect a directmeasure of.// and // appears lo be crucial for ils characterization.

III. SELF-SIMILAR RIVER NETWORK MODEL

We first discuss the model of self-similar river networks. Consider anetwork which is a square lattice of size Lx L where the links of the riversare identified with the bonds of ihe lattice. Periodic boundary conditionsare assumed in the left-right direction. The bottom side of the square isdefined to be the (fixed) outlet which collects the water that is flowing out.Independent random numbers in the range (0, I) are assigned lo the differentbonds representing the erodability P'ts, of ihe surface soil of ihe bond /.

The physical situation we have in mind leads lo river network formation based on an invasion percolation like mechanism."1' The weakesterodable link is selected and assumed lo be a part of the network. Thesecond-ranking weakest link is then selected and so on. The process isiterated in the ensemble of the remaining links until till sites are connected,i.e., they all have a route to the outlet. Loops are excluded since once apreferred route is selected, alternative routes formed due lo Ihe presence ofa loop would be energetically unfavourable. Operationally, one thusobtains the network by incorporating the regions in order of increasing

M o d e l s o f F r a c t a l R i v e r B a s i n s 5

strength so that no loops arc formed, yet all sites on the lattice are connectedto the outlet sites. A variant of the above procedure leading to the samestructure, consists of starting from the links connected to the outiels, selectingthe weakest one and proceeding invasively (i.e., always choosing the newweakest link) in the new ensemble of the interfacial links. This model,which was originally introduced by Stark'21 was subsequently rediscoveredby Manna and Subramanian.'" This is a model of headward growth ofstreams away from a rift, the weakest bond corresponding to Ihe pointmost susceptible lo bank failure. The motivations which led the aboveauthors to the model were however completely different from ours. Onemight suspect that a variant of the above model having (statistically)spherical geometry would lead to a different different universality class. Wecheeked that this is not Ihe case by starling with a central outlet andproceeding as above until ihe whole domain is drained.

A typical river obtained by our procedure is shown in Fig. 1. We havecarried out detailed studies of the scaling properties of the networks. Ournumerical simulations involved sizes up to L= 192 with a typical number

HIff.

Fig. I. Typical self-similar river network on a 128 x 128 lattice obtained by our optimizationprocedure. Only the largest river is shown. Periodic boundary condition are used only in thedirection transverse lo ihe dominant How. The size of the circles is a measure of the valueof s.

5 . C i e p l a k e t a l .

of different configurations of the order of 500. A summary of our results ispresented in Table I along with the results of observational data."1 Inorder lo get a precise estimate of the r exponent we used two differentmethods. The first consists in plotting the local slope (which is triviallyrelated to r) of the cumulate area distribution (see Fig. 2)

P(s, L)=\ ds' p(s', 1. 7!

as a function of s. This is reported in Fig. 3. An average over all thesevalues yields r= 1.406±0.021. On the other hand we performed a finitesize analysis of the exponent extracted at the various sizes (see Fig. 4). Anextrapolation then yields r= 1.404±0.001. The value reported in Table Iis then the arithmetic average of these two values. The exponent </' canbe determined by plotting the universal function as defined in (2). This isalso shown in Fig. 5. In a similar way we computed ihe exponent )• itsdefined in (3) obtaining y= 1.612 + 0.049. In Fig. 6 the universal functionf(x) is computed yielding a good collapse for d,= 1.22. The value for d, canbe confirmed by an independent compulation of the typical length

< / " )< / . - « >

= w> (8

Table I. The Exponents Predicted by the Scaling Arguments, Measured in OurSimulations and for River Basins"

Self-similar Scll'-alllne

Scaling predictions Measured(with „,= 1.21 ±0.02)

Scaling predictions Measured Uivcr basins(with //= \)

U - 1 . 2 1 + 0 . 0 2 1.22 ±0.04

r 1.395 ±0.01 1.38 ±0.03h I 0.605 ±0.01 0.62 ±0.02y 1 .65 ±0.03 1.60 ±0.05dt- 1.21 ±0.02

-1.21 ±0.02

I .-III I 0.02

II ±0.211.75 O.St)1.43 ±0.020.57 0.60

IS |.o

" dy is the fractal dimension of the river basin boundary. Nole ihe inconsistency in ihe observational data—d, is greater than I suggesting a self-similar network, whereas // < I iiuli-

____-|p_.3 self-affine structure.

Models of Fractal River Basins

SELF-SIMILAR

c

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L0G10stig. 2. A log-log plot of Ihe cumulate area distribution /'(.v. /.) vs. s, for lengths ranging from/.= |(, io /. = %. The value of the exponent r= 1.395±0.01 can be compared with r= 1.33corresponding to the Scheidegger and t- 1.5 corresponding lo the Mean Field model."""

1.5

T 1.4

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14 Cieplak et al.

Two recent investigations arc pertinent to our work. In ref. 2-1 theinterplay between ciuenehecl disorder and non-linearity in the landscapeevolution was shown lo be relevant for the interpretation of the real riversHeld data. On the other hand, a even more recent renormalizalion groupanalysis of a continuum equation'251 suggests thai the Iwo disorder-dominated networks studied here, the unweighted spanning-lrees studied inref. 3 and the aforementioned landscape model of ref. 24, all belong lodifferent universality classes albeil will) very close exponents. We believethat this scenario is intriguing and deserves further attention.

ACKNOWLEDGMENTS

We are indebted to Deepak Dhar, S. S. Manna and Joachim King foruseful discussions. This work was supported by grants from KBN (grantnumber 2P302-127), NASA, NATO and the Center for Academic Computingat Penii State.

REFERENCES

1. D. O.Tarboton, R. L. Bras, and I. Rodriguez-Iturbe, Water Resour. Res. 24:1317 (I98HJ;I). Lavallee, S. Lovejoy and I). Schertzer, Fractals in Geography, N. S. Lam and I DcCola, eds. (Prentice Hall, Englewood ClifTs, 1993), 159; 1 Rodriguez-Iturbe, M. Marani.R. Rigon, A. Rinaldo, Water Resour. Res. 30:3531 H994): I). R. Montgomery and W. L.Dietrich, Nature 336:232 (1988); Science 255:826 (1992); S. I\ breycr and R. S. Snow.Geomorphology 5:143 (1992).

2. C. I\ Stark, Nature 352:423 (1991).3. S. S. Manna and B. Subramanian, Phys. Rev. Lett. 76:3460 (1996).4. I. Rodriguez-Iturbe, A. Rinaldo, R. Rigon, R. L. bras, E. Ijjasz-Vasquez, A. Marani,

Water Resour. Res: 28:1095 (1992); R. Rigon, A. Rinaldo, I. Rodriguez-Iturbe, E. Ijjasz-Vasquez, R. L. bras, Water Resour. Res. 29:1980 (1993); A. Rinaldo, I. RodrigllC7.-lltl.be,R. Rigon, E. Ijjasz-Vasquez and R. L. Bias. Phys. Rev. Lett. 70:822 (1993); lor earlierstudies linking optimization principles lo drainage networks, see A. I). Howard, WaterRes. Res. 7:863 (1971); 26:2107 (1990) and references (herein.

5. A. Maritan, F. Colaiori, A. flammini, M. Cieplak and J. R. banavar, Science 272:984(1996); F. Colaiori, A. Flammini, A. Maritan and J. R. banavar. Phys. Rev. E 55:1298(1997).

6. Lattice models of river basin evolution are discussed, eg., by S. Kramer and M. Mauler,Phys. Rev. Lett. 68:205 (1992); R. L. Lcheny and S. R. Nagcl, Phys. Rev. Lett. 71:1470( 1 9 9 3 ) . /

7. T. Sun, P. Meakin and T. Jussang, Phys. Rev. E 4M&65 119941: 51:5353 (1995); Hater/Res. Res. 3(^2599 (1994); l>. Meakin, J. Feder and T. Jossangr/'/n'.v/c</ A 176:409 ( 1991 )y/

8. A. MarilanTA~~Rin;ildo, R. Rigon, A. Giacometti and I. Rodriguez-Iturbe. Phys. Rev. E\/ 53:1510(1996).

9. J. R. banavar, F. Colaiori, A. Flammini, A. Giacometti, A. Maritan and A. Rinaldo, Phys,Rev. Lett. 78:4522 (1997).

Models of Fractal River Basins 1b

10. S. S. Manna. I). Dhar and S. N. Majumdar. Phys. Rev. li 46:4471 (1992).11. R. Rigon, I. Rodriguez-Iturbe, A. Mantan. A. Giacometti, I). Ci. Tarboton and A. Rinaldo,

Water Resour. Res. 32:3367 (1996).12. I. I. Hack. U.S. Ceo! Surv. Prof. Paper 294:1 (1957).13. R. Chandler. .1. Koplik. I.erman and J. Willcmsen. J. Fluid Meeli. 119:249 (1982);

R. I.enormand. C. A'. Seances. Acad. Sci. Sci: li 291:279 (1980).14. See e.g., II. Takayasu. M. Takayasu, A. Provala and G. Huber, ./. Stat. Phys. 65:725

(1991).15. A.-L. Uarabasi. Phys. Ret: Lett. 76:3750 (1996).16. An optimal channel network is the spanning lice ihal minimizes £,/V. where the sum

over i runs over all the bonds of ihe tree. /', is ihe erodabilily of the ;-lh bond and .v, isdefined in liq. (I ). Our model corresponds to a heterogeneous basin with non uniform P,and ;• = <>.

17. C. M. Newman and 1). L. Stein. Phys. Rev. Lett. 72:2286 (1994).18. M. Cieplak. A. Marilan and J. R. banavar, Phys. Rev. Lett. 72:2320 (1994).19. .1. Feder, Fractals (Plenum, New York, 1988).20. We arc grateful lo Deepak Dhar for correspondence on this point.21. I'. Meakin. Phys. Sir. 45:69 (1992); P. Meakin, J. Phys. A 20:1.1113 (1987).22. .1. King and P. Meakin. Phys. Rev. A 40:2064 (1989).23. M. Cieplak. A. Marilan, and .1. R. banavar, Phys. Rev. Leu. 76:3754 (1996).24. G. Caldarclli. A. Giacometti. A. Marilan, I. Rodrigues-lturbe and A. Rinaldo, Phys. Rev. E

55:R4865 (1997); A. Rinaldo. I. Rodrigues-lturbe, R. Rigon, F. Ijjazs-Vasques and R. L.bras. Phys. Rev. Lett. 70:822 (1993).

25. b. Tadic, Phys. Rev. Lett, (in press) (1997).