WATER RESOURCES RESEARCH, VOL. 30, NO. 12, PAGES 3541-3543, DECEMBER 1994

Power law distribution of discharge in ideal networksH. de Vries, T. Becker, and B. EckhardtFachbereich Physik und Institut fur Chemie und Biologie des Meeres der Carl von Ossietzky UniversitatOldenburg, Germany

Abstract. Data for several river networks show algebraically decaying distributionfunctions for the mean annual discharge and the dissipation rate (Rodriguez-Iturbe etal., 1992). We derive a relation between the exponent a in the integrated mean annualdischarge distribution P(Q > q) ~ q~a and the topological dimension Dt of thenetwork, a = 1 - \IDt. Using the experimentally determined value D, « 1.8(Tarboton et al., 1988) we find a = 0.45, in good agreement with the data of(Rodriguez-Iturbe et al., 1992). For the random model (Shreve, 1967, 1969; Smart andWerner, 1974) we find an exponent of 1/2.

D

1. IntroductionRiver networks satisfy a number of scaling relations, e.g.,

for the distribution of length of streams, the catchment area,the discharge etc. [Mandelbrot, 1983; Abrahams, 1984]. Inrecent years, relations between different observables werederived for ideal networks. One of the first results of thiskind is due to La Barbera and Rosso [1989, 1990]. Theyfound a relation between the topological dimension D, of thenetwork and Horton's bifurcation ratio RB and stream lengthratio RL [Horton, 1945],

, In Rt%>=^R-. (1)J

Similar results were obtained for the fractal dimension of thenetwork and the fractal dimension of individual streams[Tarboton et al., 1990; Liu, 1992]. The distribution of discharge for several river networks in North America wasanalyzed by Rodriguez-Iturbe et al. [1992] using digitalelevation maps (DEM). For all river basins the distributionfunction for the discharge q showed an algebraic decay

P ( Q > q ) ~ q (2)

with an exponent approximately equal to 0.45. Our intentionis to find a relation between the exponent a and the fractaldimensions and/or Hortonian constants by considering idealnetworks. For the sake of convenience, Strahler s [1952]ordering scheme is used exclusively in this paper.

2. Distribution of Discharge in Ideal NetworksA river network consists of several springs, streams

emanating from them, and vertices where streams merge. Alink in this network is any stream between two neighboringvertices or between a spring and the next vertex. With eachlink is associated an area which directly drains into the link.We assume an ideal network which is characterized by thefollowing properties: (1) the length of the links of the

-network and their associated areas are described by randomvariables with distributions that are independent of locationCopyright 1994 by the American Geophysical Union.Paper number 94WR02178.0043-1397/94/94 WR-02178S05.00

within the network; and (2) the networks follow the Hortonlaws with RB> Rl- Real networks show these properties,too \Horton. 1945; Shreve, 1967; Gardiner,. 1973]. Thesecond property leads to equation (1) [La Barbera andRosso, 1989, 1990]. Furthermore, we assume that the networks have many levels in Strahler's ordering scheme,ideally infinitely many. Nevertheless, the scaling laws obtained are also found in many finite networks with maximalorder less than 10.

Following Rodriguez-Iturbe et al. [1992] we assume thatthe mean annual flow is proportional to the drainage area.Using property 1 it follows for our ideal networks that theexpectation value q^ for the mean (annual) discharge in alink of order to, is proportional to the number Zu of linksdraining through this link since each link drains on averagethe same area. We assume that the mean Zu of Z^ isproportional to the mean number of links in a subnetwork oforder to. We can compute this number from the Horton lawstogether with the Strahler ordering scheme: Measuring thelength of the streams in units of the meanjink lengtlTthe~~^mean number of links in a stream of order / isT.^- . In asubnetwork of order to there are, on average, RB~' suchstreams. The total mean number of links in a subnetwork oforder to is the sum over all terms RB~'RL~l for i < to, i.e.,

Z v - ^ R S - ' r ' l ^ - ^ t - ( 3 )w R b - R lWe therefore have for the mean qu of qw the relation

q , - R b - R l - WWe now consider the entire network up to the order il.

The mean number N(to) of links of order to in the network is

N(to) = /?£"' n i l - f )RB . (5)

The mean total number Ntot(il) of links in a network withmaximal order il is thus

a rZ - r?A', »-2«W-g=_g (6)

So the probability f(qw) of drawing a link of order to withthe mean flow qw becomes

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3542 DE VRIES ET AL.: DISTRIBUTION OF DISCHARGE IN IDEAL NETWORKS

Table 1. Exponents a for Several River BasinsCalculated With the Help of (13b)

River Rb R l < * = 1 - (In RL/\n RB)

Daddys Creek, 4.1 2.2 0.44Tennessee

Allegheny River, 4.5 2.4 0.42Pennsylvania

Youghiougheny River, 4.57 2.24 0.47Maryland

Hubbard support A50 4.7 2.3 0.46

Morphometric data from Morisawa [1962] and Tarboton et al.[1988].

JXqJ =N(to) R

Ntot(n) \rbFor the distribution function P(Q > qw) we then derive

(7)

^(G^-J-Skr) =R(RJRB)W -(RJRb)

1 - (RJRb)

n+i(8)

As noted before, we assume networks with rather largeorders il. It is thus possible to find to such that both fi:» toand to :» 1 are satisfied. Since RB > RL we can thereforeapproximate

P i Q ^ q J - i R L ' R B r -Under the same conditions, (4) gives

q . - R s -

By (1), RL and RB are related by

«_ = «_"■.Substituting (10) and (11) into (9), we obtain

from which one can read off

a = 1 ~(\ID,)

or

a = 1 - (In RL/\n RB).

(9)

(10)

(11)

(12)

(13a)

(13b)

Substituting the measured value of 1.8 for D, [Tarboton etal., 1988] in (13a) one finds or ~ 0.45, in good agreement withthe measured value [Rodriguez-Iturbe et al., 1992].

In Table 1 we have listed the values of RB and RL togetherwith the calculated values for a from (13b) for several largedrainage basins. All exponents are close to the value of 0.45.The apparent universality of the exponent is therefore equivalent to the fact that river networks tend to have a topological dimension near 1.8.

3. Distribution of Discharge in the RandomM o d e l

In this section we want to compute the distribution function for the discharge for the infinite topologically random

channel networks in the random model [Shreve, 1967, 1969;Smart and Werner, 1974].

Shreve showed that these networks follow the Hortonlaws with RB = 4 and RL = 2. Using (13b) we get for theexponent a

a = 1/2. (14)This result could also be obtained in another way:

Shreve labeled the links by the number of sources drainingthrough them. This number is called the magnitude of thelink. He concluded that the probability/(n) of drawing a linkof magnitude n out of an infinite topologically randomchannel network is [Shreve, 1967]

f(n) =2-(2/1-1) (2n- iIn- 1 (15)

Again we assume the mean discharge qn in a link ofmagnitude n to be proportional to the number of linksdraining through this link. This number equals 2n - 1 sincetwo links always join together. Thus for n » 1,

qn~2n . (16)From (15) we find for the distribution function [cf. Eldon,1975]

P(Q ̂ <7 J = 2k=n

-(2A-I)

2k- 1

2k- 1

k_ j —In (2n)l

(n\)2

Using the Stirling formula

nl**enlnn-n(2irn)mand (16) we obtain

P(Q S qn) (nn) 1/2

n ^ > \

<z;"2! !

(17)

(18)

(19)

in agreement with the result of (14).The difference between the calculated exponent in

Shreve's model and the observed exponent for real rivernetworks provides further evidence that river networks arenot purely topologically random and is in agreement withother tests of the random topology model (see, for example,Werrity [1972]).

4 . F ina l Remarks

The exponent a characterizing the algebraic decay in thedistribution function for the discharge has been relateddirectly to the topological dimension D, of the network andHorton's constants RB and RL. Numerical work in progressshows this to be valid in finite networks of order less than 11so that a does not seem to be an independent characteristicquantity of networks.

The random model yields a value which is larger than theobserved values. This may be due to finite size effects suchas fluctuating Horton constants or slow rate of convergencewith size. Or it may be due to a growth mechanism whichdoes not yield a topologically random network. Numericalwork in progress shows a minor influence of the first twoeffects and seems to favor a deviation from topologically

DE VRIES ET AL.: DISTRIBUTION OF DISCHARGE IN IDEAL NETWORKS 3543

random networks. This question clearly deserves furtherattention.

Acknowledgment.of this work.

References

We thank H. J. Schellnhuber for his support

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T. Becker, H. de Vries, and B. Eckhardt, Fachbereich Physik undInstitut fur Chemie und Biologie des Meeres der C. v. OssietzkyUnive'rsitat, Postfach 25-03, D-26111 Oldenburg, Germany.

(Received January 27, 1994; revised July 29, 1994;accepted August 18, 1994.)

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