Kinematic dispersion in stream networks

2. Scale issues and self-similar network organization

Patricia M. Saco and Praveen Kumar

Environmental Hydrology and Hydraulic Engineering, Department of Civil and Environmental Engineering, University ofIllinois, Urbana, Illinois, USA

Received 30 May 2001; revised 27 March 2002; accepted 16 April 2002; published 21 November 2002.

[1] When the flow parameters, such as celerity and hydrodynamic dispersion coefficient,are allowed to vary spatially within a basin, three mechanisms, namely, geomorphologic,kinematic, and hydrodynamic dispersion, contribute to the variance of the instantaneousresponse function. The relative contributions of the three dispersion mechanisms as afunction of scale, or Strahler order of the basin, are studied. This analysis is performed fortwo study basins, the Vermilion and theMackinaw river basins, in central Illinois. Log lineartrends for all the dispersion coefficients as a function of scale are observed. These trendscan be cast in the form of Horton law type of relations. The asymptotic behavior of thedispersion coefficients of basins with self-similar network structure is consistent with theobservations. INDEX TERMS: 1860 Hydrology: Runoff and streamflow; 1824 Hydrology:

Geomorphology (1625); 1848 Hydrology: Networks; KEYWORDS: hydrograph, geomorphologic instantaneous

unit hydrograph, instantaneous response function, hydraulic geometry, self-similarity, scaling

Citation: Saco, P. M., and P. Kumar, Kinematic dispersion in stream networks, 2, Scale issues and self-similar network organization,

Water Resour. Res., 38(11), 1245, doi:10.1029/2001WR000694, 2002.

1. Introduction

[2] During the last two decades, several contributionslinking the network structure and the flow dynamics haveappeared in the literature [Rodriguez-Iturbe and Valdes,1979; Gupta et al., 1980; Wang et al., 1981; Mesa andMifflin, 1986; van der Tak and Bras, 1990; Rinaldo et al.,1991; Jin, 1992; Naden, 1992; Snell and Sivapalan, 1994;Robinson et al., 1995; Yen and Lee, 1997]. An importantcontribution in this direction corresponds to the analysisperformed by Rinaldo et al. [1991], who derived an ana-lytical expression, in the form of a dispersion coefficient,which quantifies the portion of the basin’s hydrographvariance that is due to the influence of the river networkorganization. They called this influence as geomorphologicdispersion. They found that under the assumption of spatiallyinvariant celerity and hydrodynamic dispersion coefficientthroughout the basin, hydrodynamic and geomorphologicdispersion are the two mechanisms contributing to thevariance of travel times.[3] However, there is empirical evidence that shows that

flow velocities vary nonlinearly with flow discharge both intime and space along the river network [Minshall, 1960;Pilgrim, 1976]. Saco and Kumar [2002] relaxed theassumption of spatially invariant hydrodynamic parametersand considered spatially varying, as a function of Strahlerorder, celerity and hydrodynamic dispersion coefficients bycharacterizing the flow using hydraulic geometry relations.It was found that the presence of spatially varying celeritiesinduces a third contribution to the variance of travel times,referred to as kinematic dispersion.

[4] Flow through the river system exhibits different char-acteristics at various scales, from small headwater streams tomajor rivers. Low-order streams near the headwater aretypically characterized by moderate to steep slopes andconstitute the primary conduits of water and sediment tothe higher-order streams. In higher-order streams, gradientdecreases, channels widen, transport of large sedimentparticles decreases, while total volume of sediment in-creases. The focus of this research is to analyze how thethree dispersion mechanisms change as a function of scale,that is Strahler order of the basin, as a reflection of theprocesses that prevail at each scale. The scale analysis of thedispersion contributions is performed for two study basins,the Vermilion and the Mackinaw river basins, in centralIllinois. This analysis shows the presence of log linear trendsfor all the dispersion coefficients as a function of scale thatcan be cast in the form of Horton’s law type of relations.[5] The behavior of the dispersion coefficients of basins

with self-similar network structure is also analyzed. Thecharacterization of a stream network in a self-similar frame-work allows us to specify the state-to-state transitionsparsimoniously. It is found that as scale increases, thedispersion coefficients asymptotically show trends akin toHorton’s laws. The theoretical Horton’s ratios predicted forthe study basins using the equations for topological self-similar networks are close to the ratios estimated from thelog linear trends.[6] The rest of the paper is organized as follows. Section 2

provides a brief review on the concept of kinematic disper-sion and presents the relevant equations used in this research.A brief description of the study basins is given in section 3.Section 4 describes the underlying hypothesis and method-ology for the computation of the spatially variant celeritiesand hydrodynamic dispersion coefficients. The scale analysisof the relative contributions of the different dispersion

Copyright 2002 by the American Geophysical Union.0043-1397/02/2001WR000694

27 - 1

WATER RESOURCES RESEARCH, VOL. 38, NO. 11, 1245, doi:10.1029/2001WR000694, 2002

mechanisms is presented in section 5. In section 6 we deriveexpressions for the dispersion coefficients of basins withtopological self-similar networks and we study their asymp-totic behavior as the scale becomes large. Summary andconclusions are presented in section 8. The Appendicesdescribe some technical derivations pertinent to this research.

2. Review of Kinematic Dispersion

[7] The theory of the geomorphologic instantaneous unithydrograph (GIUH) [Rodriguez-Iturbe and Valdes, 1979;Gupta et al., 1980] postulates that the distribution of arrivaltimes of water drops at the outlet of a basin depends on thetopological structure of the river network. A basin of order� is then considered as a collection of paths, where a path g

is defined as consisting of the sequence of states, or ordersw, that a water drop follows from its injection into the basinto the outlet. For each path in the network one can obtain atravel time distribution, and the travel time distribution atthe basin’s outlet, that is, the unit hydrograph, is obtained byrandomizing over all possible paths. By relaxing theassumption of spatial invariance of the coefficient of hydro-dynamic dispersion and the kinematic wave celerity, andassuming that they are a function of the Strahler order w ofthe stream, an equivalent celerity ug and hydrodynamicdispersion coefficient DLg, which preserve the mean andvariance of the travel times for each path g, can becomputed as [Saco and Kumar, 2002]:

ug ¼ Lg

EðTgÞ¼ LgP

w2gLwuw

ð1Þ

where Lw is the mean length of a stream of order w, and Lg =�w2gLw is the mean path length; and

DLg ¼u3g

Lg

VarðTgÞ2

¼u3g

Lg

Xw2g

LwDLw

u3w

!: ð2Þ

The parameters uw and DLw are the celerity and thehydrodynamic dispersion coefficient for the stream of orderw given as:

uw ¼ 3

2vw* ð3Þ

and

DLw ¼ uwhw*

3Swð4Þ

where vw* and hw* are the flow velocity and depth in the statew, respectively. The asterisk signifies that they correspond toreference steady state uniform flow conditions which shouldbe meaningful to the problem analyzed [Rinaldo et al.,1991]. Sw is the average bed slope for the state w.[8] An equivalent network celerity, that preserves the

mean travel time over the network, can be defined in termsof the mean path length of the network (L(�)) and thenetwork’s mean travel time (E(Tn) = Eg(E(Tg))) as follows:

un ¼Lð�ÞE Tnð Þ ¼

Pg2� p gð ÞLgPg2� p gð Þ Lg

ug

: ð5Þ

where p(g) is the probability of a water drop following aparticular path g [Rodriguez-Iturbe and Valdes, 1979; Sacoand Kumar, 2002] (see also Appendix 6). Using thisequivalent network celerity, the geomorphologic dispersioncoefficient (DG) is obtained as [Rinaldo et al., 1991]:

DG ¼ un

2L �ð ÞÞXg2�

p gð Þ Lg� �2 � X

g2�p gð ÞLg

!28<:

9=;: ð6Þ

The kinematic-geomorphologic dispersion coefficient,which captures the combined effects of heterogeneity offlow velocities and paths, is obtained as:

DKG ¼ un3

2Lð�ÞXg2�

pðgÞ Lg

ug

�2

�Xg2�

pðgÞLgug

!28<:

9=;: ð7Þ

The contribution due to the existence of different celeritiesalong different paths which will be addressed as ‘‘kine-matic’’ dispersion can be therefore isolated as:

DK ¼ DKG � DG: ð8Þ

The contribution due to hydrodynamic dispersion will bedesignated as DD and can be obtained as:

DD ¼ un3

Lð�ÞXg2�

pðgÞLgDLg

u3g: ð9Þ

The variance of the travel time distribution f (t), that is, of theinstantaneous response function, is therefore induced by theeffects of the three dispersion mechanisms defined aboveand can be obtained as:

Var Tnð Þ ¼ 2L �ð Þu3n

DD þ DG þ DKð Þ: ð10Þ

[9] A closed form solution for the travel time distributionunder the assumption of spatially variant, as a function ofStrahler order, celerity and hydrodynamic coefficients is stillnot available. A second-order approximation to the networkIRF is obtained by the use of equivalent network parametersestimated to preserve the first two moments of the networktravel time distribution [Saco and Kumar, 2002]. For thiscase, the equivalent network hydrodynamic dispersion DLn,computed to preserve the variance of the network IRF givenby equation (10), is obtained as:

DLn ¼ DD þ DK ; ð11Þ

and the equivalent network celerity un (computed topreserve the mean) is given by equation (5). These networkequivalent parameters un and DLn are then used as spatiallyinvariant values in the expression for the network traveltime distribution obtained by Rinaldo et al. [1991]:

f tð Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pDLn t

3p X

g2�p gð ÞLg exp �

Lg � unt� �2

4DLn t

( ): ð12Þ

It is important to note that although DLn is used as a spatiallyinvariant parameter, it accounts for the contribution of both

27 - 2 SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2

hydrodynamic and kinematic dispersions, the latter appear-ing because of the existence of spatially variant celerities.

3. Study Basins

[10] The relative contribution of the different varianceproducing mechanisms has been analyzed for the Vermilionand the Mackinaw river basins, both tributaries of theIllinois river system (Figure 1). The Vermilion river basinhas an area of 3449 km2 and a total relief of 362 meters. TheMackinaw river basin has an area of 2871 km2 and a totalrelief of 426 meters. These basins are among the 18 majorriver basins in Illinois for which a convenient form ofhydraulic geometry relations exists that allows for theestimation of celerities and hydrodynamic coefficients forstreams of varying Strahler order [Stall and Fok, 1968].[11] The river networks for the two basins were derived

from 7.5-minute digital digital elevation models (DEMs)that have horizontal resolution of 30 m. Flow directionswere identified using the imposed gradients method [Gar-brecht and Martz, 1997]. This method produces realisticand topographically consistent drainage patterns. It signifi-cantly improves the identification of flow directions in flatareas as compared to other existing methods [Saco, 2002]. Itwas found for both basins, that the channel networksderived using this method match very well with the digitalstream data contained in the National Hydrography Data set(NHD) [Saco, 2002]. A pruning criteria was used to definethe beginning of the channel network [Peckham, 1995b,1998]. This method consists of removing all of the exteriorlinks (or leaves) in the space-filling network obtained fromthe flow grid which contains the flow directions for everypixel in the network. The pruning criteria is based on theidea that the branching structure of a given river networkreflects the flow convergence in the underlying topography.Therefore, in terrains with higher relief, the length of theexterior links is shorter than those of flatter areas, becausethe topography in higher relief areas is more intricated. Thevalues of the number of streams (Nw), mean drainage area(Aw), mean along-channel length (Lw), and mean bed slope(Sw) were obtained using the networks extracted from theDEM. Horton plots of the stream numbers, areas, lengthsand slopes show that the Horton laws for all those variableshold very well in both basins (Figure 2 and Table 1).

4. Spatially Variant Hydrodynamic Coefficients

[12] The celerities uw (equation (3)) and the hydrody-namic dispersion coefficients DLw

(equation (4)) areobtained for reference steady uniform flow conditions withflow rate Q*. The value of the reference flow rate Q* in thisstudy is allowed to vary in space and time. Given a spatiallyuniform rate of rainfall excess I, steady state flow conditionsare obtained when all the basin saturated area is contributingrunoff to the control section. Therefore the reference flowQ* can be related to the rate of rainfall excess I through thefollowing relation:

Qw* Ið Þ ¼ I k Aw ð13Þ

where k is the percentage of saturated area and Aw is themean drainage area for the basins of order w. Invoking

Horton’s law of drainage area (Aw = A1Raw�1), equation (13)

can be rewritten as:

Qw* Ið Þ ¼ Q1* Ið ÞRw�1

a ð14Þ

where Q1*(I ) = I k A1.

[13] To obtain the mean flow velocities v* and flowdepths h* that correspond to the reference flow Q*, weused the hydraulic geometry relations obtained by Stall andFok [1968]. They empirically verified that mean velocitiesand depth at a channel cross-section are a function of thefrequency (F ) of flow discharge and Strahler order (w) as:

ln v ¼ av � bvF þ lvw ð15Þ

ln h ¼ ah � bhF þ lhw ð16Þ

where av, bv, lv, ah, bh, and lh are empirical regressioncoefficients. The frequency of flow discharge (F ) is relatedto the flow discharge Q through [Stall and Fok, 1968]:

lnQ ¼ aQ � bQF þ lQw ð17Þ

where aQ, bQ and lQ are also empirical regressioncoefficients. The regression coefficient values, hereafterreferred to as hydraulic geometry coefficients, for the studybasins are summarized in Table 2.[14] For a given value of the rainfall excess rate I, which

corresponds to a reference flow condition Qw* (equation(14)) we compute F using equation (17), and then we obtain

Figure 1. Location of the Vermilion and the Mackinawriver basins in the state of Illinois (IL). The neighboringstates shown are Wisconsin (WI), Indiana (IN), andMichigan (MI).

SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2 27 - 3

the reference velocities (vw*) and depths (hw*) from equations(15) and (16). Finally vw* and hw* are replaced into equations(3) and (4) to get uw and DLw. For the steady state referenceflow conditions used in this study, and under the validity ofequations (15), (16) and (17), it can be shown that uw andDLw follow a Horton’s law type of relation (see Appendix Afor details). That is, the celerity and the hydrodynamicdispersion coefficient can be obtained as:

uw Ið Þ ¼ u1 Ið ÞRw�1u ð18Þ

and

DLw Ið Þ ¼ DL1 Ið ÞRw�1DL

ð19Þ

respectively. In equation (18)Ru is the celerity ratio andu1(I) is the celerity for the stream of order 1. In equation

Figure 2. Horton plots and corresponding Horton ratios (from top to bottom) for stream numbers (Nw),drainage area (Aw), along-channel length (Lw), and bed slope (Sw).

Table 1. Stream Ratio (Rb), Area Ratio (Ra), Length Ratio (Rl),

Slope Ratio (Rs), Celerity Ratio (Ru), and Hydrodynamic Disper-

sion Ratio (RDL) for the Vermilion and the Mackinaw River Basins

Vermilion Mackinaw

Rb 3.55 4.55Ra 3.64 4.87Rl 2.00 2.54Rs 1.53 1.60Ru 1.30 1.21RDL

3.19 3.54

Table 2. Empirical Hydraulic Geometry Coefficients From Stall

and Fok [1968] Used to Compute the Celerity and the

Hydrodynamic Dispersion Coefficienta

Vermilion Mackinaw

aQ 1.606 0bQ 6.28 7.52lQ 1.04 1.34av �0.093 0.255bv 2.19 2.26lv 0.175 0.121ah �0.60 �1.006bh 2.71 3.13lh 0.361 0.469

aThe units for aQ, bQ, and lQ are ln(cfs); for av, bv, and lv, they areln(ft/s); and for ah, bh, and lh, they are ln(ft).

27 - 4 SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2

(19)RDLis the hydrodynamic dispersion ratio, and DL1

(I) isthe hydrodynamic dispersion coefficient for the stream oforder 1.[15] The celerity and hydrodynamic dispersion coeffi-

cient for each Strahler order w in the Vermilion and theMackinaw river basins were computed for two differentrates of rainfall excess (1 and 5 mm/hr), and for an assumedpercentage of saturated area k = 1%. The values of thereference flow rate Q*, exceedance probability F, celerity uand hydrodynamic dispersion DL for each Strahler order ware presented in Table 3 for the Vermilion and the Mack-inaw river basins respectively. The estimated values of Ru

and RDLfor the Vermilion and the Mackinaw river basins

are presented in Table 1.[16] The range of values of the rainfall excess rate I which

are meaningful for the analysis of these basins is restricted.This is because the hydraulic geometry relations developedby Stall and Fok [1968] (equations (15), (16) and (17)) havebeen obtained from linear regressions with observed data

with exceedance probability F in the range [0.1, 0.9].Consequently, the regression equations are strictly validwithin this range. When Q becomes large (Table 3) thevalues of F become very small with increasing basin orderand may fall outside the range of validity.

5. Scale Analysis

[17] The relative contribution of the different dispersioncoefficients (DG, DK, and DD) is computed for the completeVermilion and Mackinaw river networks, which are of order6, and for their mean nested subbasins of order 5, 4 and 3. Itis important to mention here that the scale analysis of thedispersion coefficients, as measured by the basin order, canbe performed in two different ways. First we can analyze therelative contribution of the different dispersion coefficientsobtained for a fixed rainfall excess rate (I ). Alternatively wemay study the dispersion coefficients across nested subba-sins of different Strahler orders for a fixed frequency of flowdischarge (F ). The results presented in this section corre-spond to a fixed rainfall excess rate. The analysis isextended for the fixed frequency case in section 7.

5.1. Equivalent Network and Dispersion Coefficients

[18] To compute the dispersion coefficients (DG, DK andDD) as well as the equivalent network hydrodynamicparameters (un and DLn, it is necessary to estimate theprobabilities ( pg), lengths (Lg), and equivalent hydrody-namic parameters (ug and DLg) for all the possible paths inthe network. These values are computed for the complete(� = 6) Vermilion and Mackinaw river basins, and for theirsubbasins of order � = 5, 4 and 3.[19] Table 4 shows the equivalent network celerity (un,

equation (5)), and the geomorphologic (DG, equation (6)),kinematic (DK, equation (8)), hydrodynamic (DD,equation(9)), and equivalent network hydrodynamic (DLn, equation(11)) dispersion coefficients obtained for a rainfall excessrate of 1 and 5 mm/hr in the Vermilion and Mackinaw riverbasins. All coefficients increase with both basin order (�)and rainfall excess rate (I ). Figures 3 and 4 show that thelogarithms of all the coefficients follow a remarkable lineartrend with basin order. This provides evidence for theexistence of power law behavior as a function of the areaof the Strahler order subbasins. The slopes corresponding to

Table 3. Values of Reference Flow Conditions (Q*) for Each

Strahler Order and for Rainfall Excess Rates of I = 1 mm/hr and I =

5 mm/hr and Values of Frequency (F ), Celerity (u), and

Hydrodynamic Dispersion (DL) for the Reference Flow conditions

in the Vermilion and Mackinaw River Basins

Order

I = 1 mm/hr I = 5 mm/hr

Q*,m3/s F

u,m/s

DL,m2/s

Q*,m3/s F

u,m/s

DL,m2/s

Vermilion1 0.01 0.55 0.15 0.3 0.06 0.29 0.26 1.22 0.05 0.51 0.19 1.1 0.23 0.25 0.34 3.73 0.17 0.47 0.25 3.4 0.85 0.21 0.44 11.84 0.62 0.43 0.33 10.7 3.1 0.17 0.58 37.65 2.26 0.39 0.43 34.2 11.3 0.13 0.75 1206 8.24 0.35 0.56 109 41.2 0.09 0.98 382.5

Mackinaw1 0.002 0.54 0.2 0.1 0.01 0.32 0.32 0.32 0.009 0.51 0.24 0.4 0.05 0.29 0.39 1.13 0.05 0.47 0.29 1.3 0.23 0.26 0.47 44 0.22 0.44 0.35 4.5 1.1 0.23 0.57 14.25 1.08 0.41 0.43 15.9 5.39 0.19 0.7 50.36 5.27 0.37 0.52 56.2 26.35 0.16 0.85 178.1

Table 4. Equivalent Network Celerity and Hydrodynamic Dispersion Coefficients for a Rainfall

Excess Rate of 1 and 5 mm/hr in the Vermilion and the Mackinaw River Basins

�

un, m/s DG, m2/s DD, m

2/s DK, m2/s DLn, m

2/s

I = 1 I = 5 I = 1 I = 5 I = 1 I = 5 I = 1 I = 5 I = 1 I = 5

Vermilion3 0.22 0.39 38.8 68 1.9 6.8 20.6 36.1 22.5 42.94 0.27 0.48 87.4 153.3 5.1 17.8 43.1 75.6 48.2 93.45 0.34 0.6 312.5 547.7 14.3 50.3 126.7 222 141 272.46 0.45 0.78 1370.2 2401.8 47.7 167.3 783 1372.5 830.7 1539.8

Mackinaw3 0.27 0.44 21 34.1 0.9 2.8 8.2 13.3 9.1 164 0.32 0.53 62.5 101.4 2.9 9 25.5 41.3 28.3 50.35 0.39 0.63 137.2 222.6 9.3 29.6 50.3 81.6 59.7 111.26 0.47 0.77 785.4 1273.9 34.6 109.8 321.5 521.5 356.1 631.3

SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2 27 - 5

un, DG, DK and DD are shown in Table 5 (columns 2 and 4).It is evident from Figures 3 and 4 that changes in the rainfallexcess rate affect the intercept (by introducing a shift inordinate) but not the slopes of the linear trends. This isfurther explained in section 5.2. A physical explanation forthe existence of these linear trends is given in section 6.2and is related to the inherent self-similar character of thegeometric (as captured by Horton’s laws of stream lengths,areas and slopes) and topologic self-similar characteristicsof the network.[20] Figure 5 shows the relative contributions of the

different dispersion coefficients as percentages of the totaldispersion for different basin orders. The total dispersionwas computed as the sum of the three different contributionsDG, DD, and DK. The relative contributions do not changesignificantly with basin order. The relative contribution ofthe hydrodynamic dispersion is very small (less than 10%for both basins).[21] In particular, it is of interest to analyze the relative

contribution of the kinematic dispersion coefficient DK

which arises due to the spatially varying celerities. Therelative contribution of DK with respect to DD depends onthe rainfall excess rate I (see also section 5.2). The relativecontribution of DK is less than that of DG, but its contribu-tion to the total dispersion is significant for both basins. Inthe Vermilion river basin, the contribution of DK corre-sponds to about 35% of the total dispersion for both rainfallrates analyzed. In the Mackinaw river basin, DK contributesabout 30% of the variance.[22] The combined contribution of DK and DD, which

constitutes the equivalent network hydrodynamic dispersionDLn, is very important for both basins and is significantly

larger than that of DD alone at all scales. The equivalentnetwork parameters (un and DLn) are used to evaluate thenetwork response function using the network approximationmethod (equation (12)). The network IRFs correspond tothe derivative of the step response function (S-hydrograph)obtained for a step input, that is, a continuous rainfall excessrate of I units. Figures 6 and 7show the nonlinear character-istics of the network IRF which strongly depend on both therainfall excess rate and basin order. For a basin of givenorder �, as I increases, the hydrograph’s peak valueincreases, whereas the time to peak and the spread decreasebecause the celerity increases. For a basin rainfall excessrate I, as the order of the basin increases, the hydrograph’speak value decreases, whereas the time to peak and thespread increase.

5.2. Dependence on Rainfall Excess Rate

[23] The celerity varies with the rainfall excess rate as apower law with exponent

bvbQ

(Table 2), that is, we canrewrite equation (18) as (see also Appendix A):

uw Ið Þ ¼ IbvbQuw 1ð Þ: ð20Þ

The hydrodynamic dispersion exhibits the same type ofpower law relation, but with exponent

bvþbhbQ

(Table 2), thatis, equation (19) can be rewritten as:

DLw Ið Þ ¼ IbvþbhbQ DLw 1ð Þ ð21Þ

where uw(1) and DLw(1) are the celerity and hydrodynamic

dispersion coefficient for a channel of order w and for a unit

Figure 3. (left) Equivalent network celerity (un) and (right) hydrodynamic dispersion coefficient (DLn)obtained for a rainfall excess rate of 1 and 5 mm/hr in the Vermilion and Mackinaw river basins. Thepoints in the plots correspond to the values estimated from equations (5) and (11). The lines represent theregression lines for orders 3 to 6.

27 - 6 SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2

rate of rainfall excess. As mentioned in section 4, uw(equation (18)) and DLw (equation (19)) follow a Horton’slaw type of relation. This means that when plotting thelogarithms of uw and DLw

against order w, the points follow alinear trend in which the slopes define Horton type of ratiosRu and RDL

. Equations (20) and (21) show that as I changes,the slope of these lines remains unchanged. Changes in Iresult in a shift along the ordinate that is proportional to

IbvbQ ðor I

bvþbhbQ Þ.

[24] The equivalent celerity un varies with rainfall excessrate following a power law relation similar to that of uw. Thederivation of this relation is straightforward using equations(1), (5) and (20), that is:

un Ið Þ ¼ IbvbQun 1ð Þ ð22Þ

where un(1) is the network equivalent celerity for thenetwork obtained for a unit rate of rainfall excess.[25] Similarly, using equations (6), (8) and (22), it is

straightforward to show that DG and DK vary with the rateof rainfall excess following a power law similar to that of unand uw. That is,

DG Ið Þ ¼ IbvbQDG 1ð Þ ð23Þ

and

DK Ið Þ ¼ IbvbQDK 1ð Þ ð24Þ

where DG(1) and DK(1) are the dispersion coefficients for aunit rainfall excess rate. A similar power law relation isobtained for the hydrodynamic dispersionDD from equations(2), (9) and (21):

DD Ið Þ ¼ IbvþbhbQ DD 1ð Þ ð25Þ

whereDD(1) is the hydrodynamic dispersion coefficient for aunit rainfall excess rate.

Figure 4. Geomorphologic (DG), kinematic (DK), and hydrodynamic (DD) dispersion coefficients for arainfall excess rate of 1 and 5 mm/hr in the Vermilion and Mackinaw river basins. The points in the plotscorrespond to the values estimated from equations (6), (8), and (9). The lines represent the regressionlines for orders 3 to 6.

Table 5. Empirical and Theoretical Values for the Horton’s Ratios

Associated With the Equivalent Network Celerity (un) and the

Dispersion Coefficients (DG, DK, and DD)a

Vermilion Mackinaw

Empirical Theoretical Empirical Theoretical

un 1.27 1.30 1.20 1.21DG 3.31 2.60 3.21 3.07DK 3.31 2.60 3.21 3.07DD 2.91 3.19 3.36 3.54

aThe empirical values were obtained from the slopes of the lines inFigures 3 and 4. The theoretical values were predicted using equations (39),(43), (45), and (46).

SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2 27 - 7

[26] Finally, using equations (11), (24) and (25), DLn isfound to vary with the rainfall excess rate as:

DLn Ið Þ ¼ IbvbQDK 1ð Þ þ I

bvþbhbQ DD 1ð Þ

¼ IbvbQ DLn 1ð Þ þ DD 1ð Þ I

bhbQ � 1

�� �ð26Þ

where DK(1), DD(1) and DLn 1ð Þ are the kinematic,hydrodynamic and equivalent network dispersion coeffi-cients for a unit rate of rainfall excess.[27] As mentioned in section 5.1, Figures 3 and 4 show

that the logarithms of un, DG, DK, DD and DLn follow alinear trend with basin order. Equations (22), (23), (24), (25)and (26), explain why the slopes of these lines are notaffected by changes in the rainfall excess rate. Changes in Iresult in a shift, along the ordinate, of the lines obtained forany of these coefficients. As seen in section 5.1, the relativecontribution of DK with respect to DG does not vary withrainfall excess rate because the exponents of the power lawrelations given by equations (23) and (24) are equal. Then,the ratio of their respective contributions remainsunchanged and do not affect the trends observed acrossdifferent scales. On the other hand, the relative contributionof DK with respect to DD does depend on the rainfall excess

rate I. Given that bh can only take positive values, DD

increases with increasing I at a faster rate than DG and DK.Then for larger values of I than those used in this analysis,the contribution of DD could become larger than that of DG

and DK (i.e., DK/DD decreases with increasing I). Never-theless, this conclusion is only valid when the value of I iswithin the range of physically meaningful values discussedat the end of section 4.

6. Topological Self-Similar Networks

6.1. Dispersion Coefficients for TopologicalSelf-Similar Networks

[28] The estimation of the equivalent network celerity(equation (5)) and dispersion coefficients (equations (6),(8), (9), and (11)), involves the computation of differentpath characteristics, such as probabilities, lengths, equiv-alent celerities and equivalent hydrodynamic dispersioncoefficients for each path in the network, where thenumber of paths increases with basin of order � as 2(��1).In this section, first we show that the equivalent networkcelerity (un) and the hydrodynamic dispersion coefficient(DD) are independent of the characteristics of the individ-ual paths in the network and can be obtained directly as afunction of the number of streams (Nw), mean area (Aw),

Figure 5. Contributions from geomorphologic dispersion (DG), hydrodynamic dispersion (DD), andkinematic dispersion (DK), given as percentages of the total dispersion for the (top) Vermilion and(bottom) Mackinaw river basins and corresponding to a rainfall excess rate I of (left) 1 mm/hr and (right)5 mm/hr.

27 - 8 SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2

mean length (Lw), mean celerity (uw) and mean hydro-dynamic dispersion coefficient (DLw

) of the subbasins ofdifferent Strahler orders (w). This implies that it is possibleto estimate un and DD without computing the probabilities,lengths, celerities and hydrodynamic dispersion coeffi-cients for each path. Second, we show that for basinswith topologically self-similar networks, the geomorpho-logic (DG) and kinematic (DK) dispersion coefficients arealso independent of the characteristics of the individualpaths in the network. Third, we study the asymptotic

behavior of the dispersion coefficients, as the order ofthe basin becomes large.[29] For any basin of order �, it can be shown that the

mean path length of the network can be written as (seeAppendix B):

Lð�Þ ¼ 1

A�

X�w¼1

AwNw;�Lw ð27Þ

where Nw,� is used to represent the number of streams oforder w in a basin of order �.

Figure 6. Network instantaneous response function for the Vermilion river basin for the two rainfallexcess rates (I = 1and I = 5 mm/hr) and for different basin orders (� = 3 and 6).

Figure 7. Same as 6 but for the Mackinaw river basin.

SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2 27 - 9

[30] The equivalent network celerity (un) of a basin oforder � is given by equation (5). The numerator can bereplaced by equation (27) and the denominator, Eg(E(Tg)),can be replaced by an expression that looks similar to (27)but with Lw substituted by Lw

uw(its derivation is analogous to

that of (27)). Then, an alternative form for the equivalentnetwork celerity of a basin of order �, which is independentof the path characteristics, is obtained as:

un½ ��¼P�

w¼1 AwNw;�LwP�w¼1 AwNw;�

Lwuw

: ð28Þ

[31] An alternate equation for the hydrodynamic disper-sion coefficient DD can also be derived. First, an expressionthat looks similar to (27) can be derived for Eg(Var(Tg)) bysubstituting Lw with LwDLw

u3w. Then, this expression and equa-

tions (27) and (28) can replaced into equation (9) to get:

DD½ ��¼

P�w¼1 AwNw;�Lw

� �2P�

w¼1 AwNw;�Lwuw

� �3 X�

w¼1

AwNw;�LwDLw

u3w: ð29Þ

Expressions that do not involve path’s characteristics, thatis, similar to equations (27), (28) and (29), can be obtainedfor the geomorphologic (DG) and kinematic-geomorpholo-gic dispersion (DKG) coefficients for basins with topologi-cally self-similar networks.[32] Peckham [1995a, 1995b] defines a deterministic

self-similar tree of order � as one which can be madeidentical to any of its complete Strahler subtrees of order kby removing all of its Strahler streams that have ordersless than � � k. The self-similar property for this treeconstruction is therefore obtained as an asymptotic result,that is, in the limit as � tends to infinity. Peckham [1995a,1995b] explores some of the properties of deterministicself-similar trees in terms of tributary structure of thenetworks. The side tributary structure of a basin of order� can be described by a generator matrix (T ), a lowertriangular matrix containing the side tributaries numbersTw, k which denote the average number of side tributariesof order w entering a stream of order k,

T ¼ Tw;k� �

w¼1;...;��1;k¼2;...;�ð Þ: ð30Þ

For topologically self-similar trees the constraint Tw, w + k =Tk should be satisfied for all values of w. This implies thatthe matrix should have constant values along the diagonals.[33] In this case the number of tributaries (both upper and

side tributaries) of order w entering a stream of order k canbe obtained as:

nw;k ¼ ðTw;k þ 2lÞ ð31Þ

where l = 1 for k = w � 1, and l = 0 otherwise. Then the self-similarity constraint implies that nw,w+k = nk should besatisfied for all values of w.[34] For basins with topological self-similar networks it is

possible to derive an expression for Varg(Lg) which does not

involve the computation of path probabilities and lengths as(see appendix (B)):

½Varg Lg

� ��� ¼ 1

A�

X�w¼1

2LwNw;�1

2AwLw þ

Xw�1

j¼1

N��wþj;�AjLj

!( )

� 1

A�

X�w¼1

AwNw;�Lw

!2

: ð32Þ

The geomorphologic dispersion coefficient DG for a basinof order � can be obtained by replacing equations (27), (28)and (32) into equation (6):

DG½ ��¼1

2P�

w¼1 AwNw;�Lwuw

X�

w¼12LwNw;�

1

2AwLw

�

þXw�1

j¼1N��wþj;�AjLj

�� 1

A�

X�

w¼1AwNw;�Lw

� �2�:

ð33Þ

An equation similar to (32) is obtained for [Varg(E(Tg))]�by substituting Lw with Lw

uw. This expression, and equations

(27) and (28) are replaced in equation (7) to get thekinematic-geomorphologic dispersion coefficient DKG for abasin with a topologically self-similar network:

DKG½ ��¼

P�w¼1 AwNw;�Lw

� �22P�

w¼1 AwNw;�Lwuw

� �3 (X�

w¼12Lw

uwNw;�

1

2Aw

Lw

uw

þXw�1

j¼1N��wþj;�Aj

Lj

uj

�� 1

A�

X�

w¼1AwNw;�

Lw

uw

�2):

ð34Þ

[35] Equations (29), (33) and (34) can be used to computethe dispersion coefficients corresponding to the mean sidetributary structure of basins with stochastic self-similarnetworks. It should be noticed that no particular form ofthe generators Tk is needed to derive the results presented inthis section. Therefore these results are valid for basins withany type of self-similar network structure, such as, Toku-naga trees [Tokunaga, 1978], recursive replacement trees[Peckham, 1995a, 1995b], trees with linear generators[Peckham, 1995a, 1995b], and structurally Hortonian trees[Scheiddeger, 1970]. Cui et al. [1999] tested the validity ofa stochastic self-similar network model for several largebasins and concluded that the model is not inconsistent withthe tributary data of the basins tested. Some preliminarytesting of the stochastic self-similar property using thestochastic Tokunaga model [Cui et al., 1999] show thatthe study basins are also not inconsistent with the model.This study is completed and the results will be reported inthe future. Equations (29), (33) and (34) have the advantageof being computationally much simpler than the equationsfor the dispersion coefficients given in section 2. Also, theseequations allow for the analysis of scaling behavior of thedispersion coefficients as will be shown next.

6.2. Asymptotic Behavior of the Dispersion Coefficients

[36] In this section we analyze the asymptotic behavior ofthe dispersion coefficients (DG, DK, and DD) for basins withself-similar network topologies. For a deterministic self-

27 - 10 SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2

similar tree, the stream numbers of the basin of order� (Nw, �with w = 1. . .�) are related to the stream numbers of any ofthe subbasins of order � through the following relation:

Nw;� ¼ Nw�ð���Þ;� ð35Þ

where w < � < �. Equation (35) is derived by inductionfrom equation (B11) given in appendix B, and noticing thatN�,� = N�,� = 1 (this derivation is straightforward). Usingequation (35), and assuming that the Horton’s laws ofstream lengths and stream areas hold exactly, (that is, Lw =L1Rl

w� 1 and Aw = A1Raw� 1, where Rl and Ra are the length

and area ratios respectively) equation (27) can be rewrittenas:

L �ð Þ ¼ 1

A�

A1N1;�L1 þ RaRl

X��1

w¼1AwNw;��1Lw

h i

¼ A1

A�

N1;�L1 þ RlL �� 1ð Þ: ð36Þ

Then, the asymptotic behavior of L(�) as � ! 1 is ob-tained as:

lim�!1

L �ð ÞL �� 1ð Þ

¼ lim�!1

A1

A�N1;�L1

Ra

Rb

A1

A�N1;�L1 þ Rb

RaRlLð�� 2Þ

h iþ Rl

8<:

9=;

¼ Rl ð37Þ

where the Horton’s law of stream numbers is used as anasymptotic result for topologically self-similar networks (thatis Nw,�� cRb

��w, where ‘‘�’’ denotes asymptotic equality as(�� w) grows large) and therefore we have that N1,�/N2,� =Rb for large � [Peckham, 1995a, 1995b].[37] By invoking equation (18), a derivation similar to the

previous can be followed to obtain the asymptotic behaviorof [Eg(E(Tg))]� (that is the denominator of equation (28)) as� ! 1:

lim�!1

½Eg E Tg� �� �

��½Eg E Tg

� �� ����1

¼ Rl

Ru

: ð38Þ

The asymptotic behavior of the equivalent celerities is thenobtained by combining equations (37) and (38):

lim�!1

un½ ��½un���1

¼ Ru: ð39Þ

Following a derivation that is similar to that of equation (36)(and using the same assumptions), we can write:

½EgðL2

gÞ�� ¼ 1

A�

2A1L1X�w¼2

LwNw;�Nw�iþ1;� þ A1L2

1N1;�

!

þ R2l EgðL

2

gÞh i

��1: ð40Þ

To derive its asymptotic behavior we use again the Horton’slaw of stream numbers as an asymptotic result fortopologically self-similar networks. We assume that forlow values of (� � w), where the asymptotic limit is still not

valid, ln Nw,�} versus w is concave upward (Shreve [1966]).Under this assumption Nw,�Rb Nw+ 1,�, and therefore:

lim�!1

Eg L2

g

� �h i�

Eg L2

g

� �h i��1

¼ R2l : ð41Þ

The asymptotic limit for the variance of path lengths,obtained using (37) and (41), is:

lim�!1

Varg Lg� �� �

�

Varg Lg� �� �

��1

¼ R2l : ð42Þ

Using equations (37), (39) and (42) we can derive anasymptotic expression for the geomorphologic dispersioncoefficient as:

lim�!1

DG½ ��DG½ ���1

¼ RlRu: ð43Þ

An equation that is similar to (41) can be derived for[Eg(Tg

2)]� and used to get lim�!1([Varg(Tg)]�/[Varg(Tg)]

��1) = Rl2/Ru

2. This limit, and those defined by equations (37)and (39) are used to get:

lim�!1

DGK½ ��DGK½ ���1

¼ RlRu: ð44Þ

The asymptotic behavior of kinematic dispersion coefficientis obtained from equations (43) and (44) as:

lim�!1

DK½ ��DK½ ���1

¼ RlRu: ð45Þ

By invoking equations (18) and (19) and following aderivation similar to that for the asymptotic behavior ofL(�) (equations (36) and (37)), we get the asymptotic

behavior of EgLgDLg

u3g

� �h ithat is used in combination with

equations (39) and (37) to get the asymptotic behavior ofhydrodynamic dispersion coefficient:

lim�!1

DD½ ��DD½ ���1

¼ RDL: ð46Þ

[38] That is, for basins with self-similar networks, thedispersion coefficients (for equilibrium flow conditionsestimated using a uniformly distributed rainfall input) obeyHorton’s law type of relations as [DG]� � cGRG

��1, [DK]� �cKRK

��1, and [DD]� � cDRD��1, where ‘‘�’’ denotes asymp-

totic equality as � grows large, and RG = RlRu, RK = RlRu

and RD ¼ RDLare the ratios given by equations (43), (45)

and (46) respectively. Using equations (A3), (A5) and (A8)(see appendix A) and recognizing that Ru = Rv, the aboveratios may be further simplified as functions of Horton’sratios and hydraulic geometry coefficients. They can there-fore be written as:

RG ¼ RK ¼ xRuaRl ð47Þ

and

RD ¼ xj mþuð ÞRa

Rs ð48Þ

SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2 27 - 11

where x ¼ e �lQbv=bQþlvð Þ; u ¼ bv=bQ; j ¼ e �lQbh=bQþlhð Þand m ¼ bh=bQ.[39] These results show that the ratio of equivalent

celerities and dispersion coefficients for subbasins of con-secutive orders nested within topological self-similar net-works, will asymptotically approach a constant as the scalebecomes large. Consequently the logarithms of the equiv-alent celerity and dispersion coefficients for basins of largeorders will show a linear trend whose slope can be predictedusing these equations. The values of the ratios predicted forthe study basins are shown in columns 3 and 5 of Table 5.The values for the Mackinaw river basin are closer to theratios obtained from the slopes of the regression lines inFigure 4 (columns 2 and 4 in Table 5) than those obtainedfor the Vermilion river basin, but we should expect to getlarge uncertainty due to the limited number of points for theregression. We can therefore infer that the results plotted inFigure 4 are a reflection of topological self-similarity in thenetwork structure. That is, the linear trend present in theplots of the logarithms of DG, DK and DD versus basin orderis very well explained if the mean side tributary structure isself-similar, with the small deviations from the trend linepossibly caused by the deviations present in the ‘‘real’’ sidetributary matrix as compared to the mean side tributarystructure.

7. Scale Analysis for Fixed Flow Frequencies

[40] As mentioned at the beginning of section 5, the scaleanalysis across subbasins of increasing Strahler order can beperformed by comparing the dispersion coefficientsobtained using either the same rainfall excess rate (I ) orthe same frequency of flow discharge (F ) for all thesubbasins. The asymptotic analysis described in section6.2 corresponds to the fixed rainfall excess rate (I ) scenario.In this section we analyze the asymptotic behavior of thedispersion coefficients for basins with topologically self-similar networks by considering a fixed frequency orexceedance probability of flow discharge (F ).[41] As seen in section (5.2), the dispersion coefficients

for a basin of given order depend on the rainfall excess ratethrough equations (23), (24) and (25). An expression thatrelates the frequency of flow discharge (F ) and the rainfallexcess rate (I ) is obtained by combining equations (14) and(17) as:

IðFÞ½ �w¼ I Fð Þ½ ��R��wI ð49Þ

where

ln I Fð Þ½ ��¼ aQ � bQF þ lQ�� ln kA�ð Þ ð50Þ

and RI is a rainfall rate ratio obtained as:

RI ¼ e�lQRa: ð51Þ

[I(F )]w and [I(F )]� give the uniform rainfall excess ratesnecessary for producing equilibrium flow discharges withthe same exceedance probability F at the outlet of basins oforder w and � respectively. Equation (49) can then be usedto obtain the rates of rainfall excess ([I(F )]w) in each of thenested subbasins (w = 1. . .�), needed to produce an

equilibrium flow discharge with the same exceedanceprobability F. These rainfall excess rates, uniformlydistributed over each subbasin, can be used to get thespatially varying hydrodynamic coefficients from equations(18) and (19). Finally, the dispersion coefficients for eachsubbasin are obtained from equations (6), (8) and (9).Alternatively, if the dispersion coefficients for a unit rainfallexcess rate are known, it is possible to use equationsequations (23), (24) and (25), to get the values for therainfall excess rates [I(F )]w in each subbasin. That is, thegeomorphologic coefficient for each subbasin w, is obtainedas:

DG I Fð Þð Þ½ �w¼ I Fð Þ½ �bvbQw DG 1ð Þ½ �w; ð52Þ

the kinematic dispersion coefficient as:

DK I Fð Þð Þ½ �w¼ I Fð Þ½ �bvbQw DK 1ð Þ½ �w ð53Þ

and the hydrodynamic dispersion coefficient as:

DD I Fð Þð Þ½ �w¼ I Fð Þ½ �bvþbhbQ

w DD 1ð Þ½ �w: ð54Þ

[42] When considering a fixed exceedance probability offlow discharge (F ), the asymptotic behavior of the disper-sion coefficients for equilibrium flow conditions in basinswith topologically self-similar networks, can be derived bycombining equations (52), (53), and (54) and equations(43), (45) and (46). That is, the asymptotic behavior of thegeomorphologic coefficient is obtained as:

lim�!1

DG I Fð Þð Þ½ ��DG I Fð Þð Þ½ ���1

¼ lim�!1

I Fð Þ½ ��I Fð Þ½ ���1

�bvbQ DG 1ð Þ½ ��

DG 1ð Þ½ ���1

¼ R�bv

bQI RlRu: ð55Þ

The kinematic dispersion coefficient has the same asymp-totic behavior as the geomorphologic coefficient. Thehydrodynamic dispersion coefficient behaves asymptoti-cally as:

lim�!1

DD I Fð Þð Þ½ ��DD I Fð Þð Þ½ ���1

¼ R�bvþbh

bQI RDL

: ð56Þ

As RI is a positive value larger than 1, the Horton ratios RG,RK and RDL

are smaller than those obtained for a fixedrainfall excess rate (I ). This is because the rainfall excessrate needed to produce an equilibrium flow discharge withgiven frequency F at the outlet of a subbasin of order w,decreases as w increases (see equation (49)).

8. Summary and Conclusions

[43] The relative contributions of the geomorphologic,kinematic and hydrodynamic dispersion coefficients acrossdifferent scales are studied under the assumption of hydro-dynamic coefficients varying spatially as a function of thebasin’s Strahler order. Two basins in the Illinois Riversystem are used as case studies. Hydraulic geometry rela-

27 - 12 SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2

tions developed specifically for these basins [Stall and Fok,1968] are used to obtained the hydrodynamic coefficients.The network properties are obtain using DEMs. It is foundthat the logarithms of all three coefficients increase withbasin order following a remarkable linear trend suggestingthe existence of power law behavior as a function of thebasin’s area.[44] Changes in the rainfall excess rate affect the inter-

cept, that is, they introduce a shift in the ordinate, but notthe slopes of the linear trends. The shift in ordinate is apower law function of the rainfall excess rate with exponentthat is the same for DG and DK, and slightly larger for DD.Consequently, the relative contribution of DK with respect toDG does not vary with rainfall excess rate, and the relativecontribution of DK (or DG) with respect to DD decreasesslightly with increasing I. It is also found that for a basin ofgiven order �, as I increases, the hydrograph’s peak valueincreases, whereas the time to peak and the spread decreasebecause the celerity increases. For a uniform rainfall excessrate I, as the order of the basin increases, the peak flowdecreases, whereas the time to peak and the spread increase.[45] The behavior of the dispersion coefficients of basins

with self-similar network structure is also analyzed. It isfound that in basins with self-similar network structure inwhich Horton’s laws of stream lengths and stream areashold exactly and for equilibrium flow conditions the dis-persion coefficients asymptotically obey Horton’s law typeof relations. The Horton type ratios for DG, DK and DD

predicted for the study basins using the asymptotic analysisfor self-similar networks are close to those obtained withoutinvoking the self-similarity hypothesis. This suggests theexistence of topological self-similarity in the study basins ina mean sense. That is, the linear trend present in the plots ofthe logarithms of DG, DK and DD versus basin order is verywell explained if the mean side tributary structure is self-similar, with the small deviations from a straight line beingcaused by the deviations from the mean side tributarystructure present in the ‘‘real’’ side tributary matrix. Asmentioned previously, Horton’s law type of relations forthe equivalent celerity and the dispersion coefficients as afunction of the basin’s order indicate the existence ofpower law behavior as a function of the area of the Strahlerorder subbasins. That is, the equivalent celerity and thedispersion coefficients, and consequently, the mean andvariance of the IRF are scale invariant or self-similar. Itwas shown that the self-similarity in the first two momentsof the IRF is inherited from the self-similar characteristicsof the network tributary structure. The results presented forthe asymptotic analysis of the dispersion coefficients ofbasins with self-similar networks are valid for basins withany type of self-similar network structure, such as, Toku-naga trees [Tokunaga, 1978], recursive replacement trees[Peckham, 1995a, 1995b], trees with linear generators[Peckham, 1995a, 1995b], and structurally Hortonian trees[Scheiddeger, 1970].[46] The asymptotic results for the scaling properties of

the dispersion coefficients in self-similar networks indicatethat DD increases at a faster rate than DG with increasingbasin order. This result is contrary to the postulate ofRinaldo et al. [1991] who argued that the geomorphologicdispersion may increase at a faster rate than the hydro-dynamic dispersion with increasing basin order. We should

note though, that DD is significantly smaller than DG at allscales of interest.

Appendix A: Horton’s Law Type of Relations forCelerities and Hydrodynamic DispersionCoefficients

[47] This appendix presents the details of the derivationof the Horton’s law type of relations for uw and DLw whicharise when using the hydraulic geometry relations derivedby Stall and Fok [1968] (equations (15), (16) and (17)) forsteady state reference flow conditions.[48] The mean flow velocities v* and flow depths h* are

obtained using the hydraulic geometry equations (15) and(16). The exceedance probability (F ) that we need to specifyin equations (15) and (16) corresponds to the reference flowQ* which is obtained from equations (17) and (14) as

Fw Ið Þ ¼ c1 þ c2 ln I þ c3w; ðA1Þ

where c1 = (ln(Ra/A1k) + aQ)/bQ, c2 =�1/bQ, and c3 = (lQ �lnRa)/bQ. Equation (A1) can be replaced in equations (15)and (16) to obtain Horton’s law type of relations for both thereference flow velocity (v*) and depth (h*). The resultingexpression for v* is:

v*w Ið Þ ¼ v*1 Ið ÞRw�1v ¼ v*1 1ð ÞI

bvbQRw�1

v ðA2Þ

where v1*(1) is the reference velocity for a unit rainfallexcess rate in a stream of order 1, computed as ln v1*(1) =av + lv + (bv/bQ) (ln(kA1b)� aQ � lQ). The velocity ratio isobtained as:

Rv ¼ xRua: ðA3Þ

where x ¼ e �lQbv=bQþlvð Þ and u = bv/bQ. Similarly h*becomes:

hw* Ið Þ ¼ h1* Ið ÞRw�1h ¼ h1* 1ð ÞI

bhbQRw�1

h ðA4Þ

where h1*(1) is the reference depth for a unit rainfall excessrate in a stream of order 1, given as ln h1*(1) =ah+lh+ (bh/bQ)(ln(kA1b) � aQ � lQ). The depth ratio is obtained as:

Rh ¼ jRma� ðA5Þ

where j ¼ e �lQbh=bQþlhð Þ and m = bh/bQ.[49] Equation (A2) is then replaced into equation (3) to

obtain a Horton’s law type of relation for the celerity as afunction of the stream order, that is:

uw Ið Þ ¼ u1 Ið ÞRw�1u ¼ u1 1ð ÞI

bvbQRw�1

u ðA6Þ

where u1(1) = 3/2v1*(1) and Ru = Rv = xRau.

[50] The expression for the hydrodynamic dispersioncoefficient results from replacing equations (A4), (A6),and the Horton’s law of stream slopes, Sw = S�Rs

��w

[Strahler, 1964], in equation (4):

DLw Ið Þ ¼ DL1 Ið ÞRw�1DL

¼ DL1 1ð ÞIbvþbhbQ Rw�1

DLðA7Þ

SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2 27 - 13

where DL1(1) = u1(1)h1*(1)/(3S1) and

RDL¼ RuRhRs ¼ xjR mþuð Þ

a Rs: ðA8Þ

Appendix B: Mean and Variance of Path Lengthsfor Self-Similar Networks

[51] This appendix presents the derivation of the equa-tions for the mean and variance of path lengths for a basinwith self-similar network topology. The mean path length ofthe network in a basin of order � is given by L(�) =�g2�p(g)Lg. The probability p(g) of a water drop followingany particular path to the outlet, in a basin of order �, is justthe probability, px1

, of starting out in the appropriate state xi,times the probabilities of making each transition to streamsof higher order along the path pi, j [Rodriguez-Iturbe andValdes, 1979]. The probabilities pi, j of a transition fromstate i to state the j of higher order can be computed as:

pi;j ¼ ni; j* Nj;�=Ni;� ðB1Þ

where ni, j is the average number of streams of order idraining into each stream of order j, and Ni,� and Nj,� arethe total number of streams of order i and j in a basin oforder �. The probability px1

of starting out in theappropriate state of order x1 = w, can be computed as[Gupta et al., 1980]:

p1 ¼N1A1

A�

ðB2Þ

pw ¼ Nw;�

A�

Aw �Xw�1

j¼1

Aj

Nj;�pj;w

Nw;�

!ðB3Þ

for 2 � w � �, where � is the order of the complete basin,A� is the area of the total basin and Aw is the mean area ofsubbasins of order w.[52] We will first show the derivation of the mean path

length for a basin of order 3. The collection of paths in athird-order basin is � = {g1, g2, g3, g4}. The networkportion (that is, the overland state is not considered) of eachof these paths is given by:

g1 : w1 ! w2 ! w3 ! outlet

g2 : w1 ! w3 ! outlet

g3 : w2 ! w3 ! outlet

g4 : w3 ! outlet

Then the mean path length is:

Lð� ¼ 3Þ ¼Xg2�

pðgÞXw2g

Lw

¼ pg1ðL1 þ L2 þ L3Þ þ pg2ðL1 þ L3Þ þ pg3ðL2 þ L3Þþ pg4ðL3Þ ðB4Þ

where the path probabilities are:

pg1 ¼A1

A3

n1;2n2;3

pg2 ¼A1

A3

n1;3

pg3 ¼A2

A3

� A1

A3

n1;2

� �n2;3

pg4 ¼A3

A3

� A1

A3

n1;3 � A2

A3

n2;3

ðB5Þ

If we replace equation (B5) into equation (B4) and realizingthat:

Nw;� ¼P��w

k¼1 nw;wþkNwþk;� if w < �

1 w ¼ �:

8<: ðB6Þ

we obtain:

Lð� ¼ 3Þ ¼ 1

A3

X3w¼1

AwNw;�Lw: ðB7Þ

The general equation for the expected value of the pathlength for a basin of order � (L(�)) is derived using aninductive procedure to get:

Lð�Þ ¼ 1

A�

X�w¼1

AwNw;�Lw: ðB8Þ

This relationship has been numerically verified for basins ofarbitrary order �.[53] The variance of the distribution of path lengths for a

basin of order 3 is:

Varg Lg

� �� ��¼3

¼X

g2� p gð Þ Lg� �2� X

g2� p gð ÞLg� �2� �

�¼3

¼ pg1 L1 þ L2 þ L3� �2þpg2 L1 þ L3

� �2n

þ pg3 L2 þ L3� �2þpg4 L3

� �2o�npg1 L1 þ L2 þ L3� �

þ pg2 L1 þ L3� �

þ pg3 L2 þ L3� �

þ pg4 L3� �o2

:

For basins with topological self-similar networks (that is,networks whose tributary structure satisfy the constraintnw,w+ k = nk for all w) we can write equation (B5) as:

pg1 ¼A1

A3

n21

pg2 ¼A1

A3

n2

pg3 ¼A2

A3

� A1

A3

n1

�n1

pg4 ¼A3

A3

� A1

A3

n2 �A2

A3

n1:

The stream numbers for topological self-similar networkssatisfy [Peckham, 1995a, 1995b]:

Nw;� ¼P��w

k¼1 nkNwþk;� if w < �

1 w ¼ �:

8<: ðB11Þ

(B10)

(B9)

27 - 14 SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2

After replacing equation (B10) into equation (B9), rearran-ging terms and using the result given by equation (B11) weobtain:

Varg Lg

� �� ��¼3

¼ 1

A3

(X3

w¼12LwNw

1

2AwLw

þXw�1

j¼1N3�wþjAjLj

!)

� 1

A3

X3

w¼1AwNwLw

�2

: ðB12Þ

The expression for a basin of order � is obtained using aninductive procedure and numerical verification as:

Varg Lg

� �� ��¼ 1

A�

(X�w¼1

2LwNw;�

1

2AwLwþ

Xw�1

j¼1

N��wþj;�AjLj

!)

� 1

A�

X�w¼1

AwNw;�Lw

!2

: ðB13Þ

[54] Acknowledgments. This research was supported by NSF grantEAR 97-66121. We thank Efi Foufoula-Georgiou and Andrea Rinaldo fortheir review comments.

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�������������������������P. Kumar and P. M. Saco, Environmental Hydrology and Hydraulic

Engineering, Department of Civil and Environmental Engineering,University of Illinois, Urbana, IL 61801, USA. (kumar1@uiuc.edu)

SACO AND KUMAR: KINEMATIC DISPERSION IN STREAM NETWORKS, PART 2 27 - 15