Kinematic dispersion in stream networks 1. Coupling hydraulic and network geometry

Authors

  • Patricia M. Saco,

    1. Environmental Hydrology and Hydraulic Engineering, Department of Civil and Environmental Engineering, University of Illinois, Urbana, Illinois, USA
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  • Praveen Kumar

    1. Environmental Hydrology and Hydraulic Engineering, Department of Civil and Environmental Engineering, University of Illinois, Urbana, Illinois, USA
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Abstract

[1] We investigate the coupling of river network structure and hydraulic geometry thereby relaxing the assumption of spatially invariant celerities and hydrodynamic dispersion coefficients. The presence of spatially varying celerities induces a dispersion effect, referred to as kinematic dispersion, on the network travel time distribution. Its contribution to the total dispersion is comparable to that due to the heterogeneity of path lengths, that is, geomorphologic dispersion, and significantly larger than the hydrodynamic dispersion. If this contribution is ignored, the hydrograph shows a higher peak flow, a shorter time to peak, and shorter duration.

1. Introduction

[2] Significant insight gained toward the understanding of the impact of the network geomorphology on the streamflow response has been achieved using the theory of geomorphologic instantaneous unit hydrograph (GIUH) [Rodriguez-Iturbe and Valdes, 1979; Gupta et al., 1980] in which the basin's hydrologic response was for the first time coupled to the geomorphologic structure of the river network. During the last two decades, several contributions linking the network structure and the flow dynamics have appeared in the literature [Wang et al., 1981; Mesa and Mifflin, 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]. Rinaldo et al. [1991] derived an analytical expression, in the form of a dispersion coefficient, which quantifies the portion of the basin's hydrograph variance that is due to the influence of the river network organization. They called this influence as geomorphologic dispersion.

[3] These developments provide a significant leap forward in our understanding of the physical basis of hydrologic response. However, the characteristics of the network geometry have not been effectively coupled to hydraulic geometry relations which implicitly characterize the nonlinearity of flow dynamics. As a result simplified assumptions of spatially invariant hydrodynamic characteristics over the entire network are invoked. For example, in the work of Rinaldo et al. [1991] an analytical expression for the travel time distribution for individual reaches is obtained by assuming that the mean travel time within each reach can be computed using a velocity that is invariant within the basin boundaries. However, flow through the channel network is inherently nonlinear and shows a strong dependence on scale, i.e., the size of the basin. Pilgrim [1976], using tracer studies, showed that the average flow velocities are a nonlinear function of the discharge, but reach an asymptotic value at high flows. Carlston [1969] empirically investigated Leopold and Maddock's [1953] hydraulic geometry relationships, for streams with natural channels, and concluded that width (w), depth (d) and velocity (v) are proportional to powers of the discharge Q, i.e., wQζ, ϑhQϑ, and vQη. He found that the mean values of ζ, ϑ and η were 0.46, 0.38 and 0.16 indicating a slowly increasing velocity. This was an indication that the increase in discharge in the downstream direction is accommodated through either an increase in flow depth and/or an increase in width. However, most studies linking flow and network geometry do not account for this dependence on hydraulic geometry. In addition, although these studies support the contention that flow through a stream network is generally nonlinear, the argument of slowly increasing velocity has been used to support the development of linear or quasi-linear models (i.e., models applicable to a certain range of flow conditions) by using a spatially uniform celerity and hydrodynamic dispersion over the entire basin. Consequently these models are valid for small basins thereby limiting their applicability.

[4] Nonlinearity has been investigated by relaxing the hypothesis of constant velocity through the channel network in the works of Valdes et al. [1979], Wang et al. [1981], Robinson et al. [1995], and Yen and Lee [1997]. However, none of these studies deals specifically with the coupling of the hydraulic geometry with the stream network organization which is the subject of this paper.

[5] In this research we show that the presence of spatially varying celerities induces a dispersion effect, referred to as kinematic dispersion, on the network travel time distribution. Its contribution to the total dispersion is comparable to that due to the heterogeneity of path lengths, that is geomorphologic dispersion, and significantly larger than the hydrodynamic dispersion. If this contribution is ignored the hydrograph shows a higher peak flow, a shorter time to peak and shorter duration.

[6] The rest of the paper is organized as follows. Section 2 provides a brief review of the current state of the art on which this research is based. Analytical equations for the various dispersion mechanisms that arise in this study are derived in section 3. In section 4 we present two different approximations that can be used to estimate the network instantaneous response function. Section 5 summarizes the hydraulic geometry relations used in this research. A case study is presented in section 6, in which the relative contributions of the various dispersion mechanisms is analyzed. Summary and conclusions are given in section 7. Appendix A describes some technical results pertinent to this research.

2. Review of Geomorphologic Dispersion

2.1. State-Space Model

[7] The GIUH theory postulates that the distribution of arrival times of water drops at the outlet of a basin depends on the topological structure of the river network. Using the Strahler ordering scheme [Strahler, 1957] a pathway can be defined as a set of transitions between the initial order of a droplet (the order of the stream into which the droplet is initially injected), and the higher order streams until the outlet is eventually reached. For example, in a third order basin the collection of all paths Γ = {γ1, γ2, γ3, γ4} is given as

equation image

where oω denotes the overland state which contributes directly to stream of order ω. Note that this proposition can be easily extended to a basin of arbitrary order Ω.

[8] Let the path γ be defined by the collection of states γ = {x1, x2, … xk} where x1 = oω, x2 = ω with ω one of {1, … Ω}, xj with j = 3, …, k − 1 is one of {ω + 1, … Ω − 1} and xk = Ω. The probability p(γ) of following any path to the outlet, in a basin of order Ω, is just the probability, πx1, of starting out in the appropriate state x1, times the probabilities of making each transition to streams of higher order along the path:

equation image

where pxi, xj is the transition probability from the state xi to xj. The travel time through a path is the sum of the travel times through each of its individual states: Tγ = Tx1 + … + Txk. The travel time distribution fb(t) at the basin's outlet when the rainfall is instantaneously and uniformly distributed over the entire basin is obtained by randomizing over all possible paths:

equation image

where * denotes convolution, fxi(t) is the travel time distribution through each of the individual states of network path γ.

[9] Alternatively, the above formulation can be separated into hillslope and network responses as

equation image

where fh = fx1 is the hillslope (or overland) response and the network response is given as

equation image

Note that fγ(t) = fx2 * fx3 * … * fxk(t) is the travel time distribution through the network portion of each individual path γ. There are several expressions in the literature to obtain the hillslope instantaneous unit hydrograph [Henderson and Wooding, 1964; Kirkby, 1976; Mesa and Mifflin, 1986; van der Tak and Bras, 1990; Naden, 1992; Robinson et al., 1995; Lee and Yen, 1997]. The focus in what follows will be on the estimation of the the river network response function f(t).

[10] From the above formulation it is clear that in order to completely characterize the GIUH through the travel time distribution fb(t) we need to determine the following: (i) The initial probabilities πx1 of starting out in a particular state; (ii) The state-to-state transition probabilities pxi, xj; and (iii) The residence time distribution in each state fxi(t).

[11] The initial probability of starting in a particular state xi = oω is simply the fraction of the basin area that is in that overland state oω. The expressions for state-to-state transition probabilities are more involved and can be derived using the Horton ratios or directly from the network tributary structure [see Rodriguez-Iturbe and Valdes, 1979].

2.2. Residence Time Distribution

[12] Different methodologies can be used to derive the residence time distributions. In the original development by Rodriguez-Iturbe and Valdes [1979] the residence time distributions in each state were assumed to have an exponential distribution with the parameter λω = V/equation imageω where V is a constant velocity characteristic of the basin and equation imageω is the mean length of the state ω. Gupta et al. [1980] used a uniform distribution and van der Tak and Bras [1990] proposed a gamma distribution. Rinaldo et al. [1991] developed a physically based model for the travel time distribution. They used an advection-dispersion equation to describe the flow through individual streams given as [Henderson, 1966]:

equation image

where hω, DLω and uω are the flow depth, the coefficient of hydrodynamic dispersion (m2/s) and the kinematic wave celerity (m/s) for the state ω, respectively. The latter two can be computed as:

equation image

and

equation image

where v*ω and h*ω are the “reference” flow velocity and depth in the state ω, respectively. They correspond to reference steady state uniform flow conditions which should be meaningful to the problem analyzed [Rinaldo et al., 1991]. equation imageω is the mean bed slope for the state ω.

[13] The solution for the travel time distribution for state ω is obtained using Laplace transforms [Rinaldo et al., 1991]. Let equation imageω(s) be the Laplace transform of fω(t), then [Rodriguez-Iturbe and Rinaldo, 1997]:

equation image

where

equation image

The inverse Laplace transform of equation (8) gives the travel time distribution for state ω:

equation image

This expression for the travel time distribution of the individual channels can be used to obtain the network response (f(t)). That is, the Laplace transform of the travel time distribution for each path (equation imageγ(s)) can be computed using equation (8) as:

equation image

Equation (11) can be used to get the Laplace transform of the travel time distribution for the network (equation (4)):

equation image

However, in general this expression has no analytical inverse.

2.3. Geomorphologic Dispersion

[14] For the special case in which the kinematic celerity uω and the hydrodynamic dispersion DLω coefficient can be considered as spatially invariant (u and DL respectively) for all the states ω, the inverse Laplace transform of equation (12) has an analytical expression [Rinaldo et al., 1991]:

equation image

where the mean length of path γ is

equation image

Equation (13) represents the network's travel time distribution.

[15] The above formulation was used by Rinaldo et al. [1991] to obtain the total variance of the travel time distribution (Var(T)):

equation image

where

equation image

is the mean path length of the network. To quantify the roles of the different variance-producing processes they defined a geomorphologic dispersion coefficient (DG) as follows:

equation image

The first term in the right hand side of equation (15) represents the contribution induced by the presence of hydrodynamic dispersion along a path of length equal to equation image(Ω). The second term, which corresponds to the geomorphologic dispersion, appears because of the presence of the river network and accounts for the variance of travel times induced by the existence of paths of different lengths equation imageγ.

[16] Equation (17) can be expressed in terms of the first two moments of the distribution of path lengths [Snell and Sivapalan, 1994]:

equation image

so that equation (15) can be rewritten as:

equation image

where the subscript γ will be used to denote a moment which is computed over all possible paths γ.

[17] Using the above, Rinaldo et al. [1991] concluded that there are two mechanisms contributing to the variance of travel times: (1) Part of the variance of the water drops' travel times is due to the dispersion along the individual paths which is induced by the hydrodynamic effects. If all paths had the same length this would constitute the only mechanism contributing to the variance since the rate of arrivals through different paths would coincide. (2) The remaining variance is due to the heterogeneity of path lengths in the stream network, which produces the spread in the arrival rates referred to as geomorphologic dispersion.

3. Kinematic Dispersion

3.1. Background

[18] The analysis performed by Rinaldo et al. [1991] and Snell and Sivapalan [1994] on the different mechanisms which contribute to the variance of the streamflow response considers only the particular case in which the hydrodynamic parameters (u and DL) can be considered as spatially invariant along the river network. It is noted, however, that due to the nonlinear nature of the momentum equations, the hydrodynamic parameters depend on the local flow conditions. This dependence can be quantified using equations (6) and (7) which are stated in terms of local reference flow conditions. The linearity condition invoked in the earlier analysis [Rinaldo et al., 1991; Snell and Sivapalan, 1994] assumes that the reference flow conditions do not change along the river network. However, there is empirical evidence which specifically refutes that assumption. Pilgrim [1976] found important evidence of nonlinearity in the rainfall-runoff process. His study was based on tracer measurements of travel times and average velocities on different reaches of a basin. From the analysis of these measurements he concluded that the response of a basin is not only nonlinear in the lumped sense (i.e., considering variations of runoff for varying rainfall at a given control section) but also it is highly nonlinear in the space domain, with mean velocities increasing in the downstream direction. He, however, found that for the higher flow conditions, especially those corresponding to over-bank-full conditions, the linearity assumption provided a good approximation.

[19] The work by Stall and Fok [1968] constitutes another empirical study that demonstrates the existence of nonlinearity in the flow dynamics. In their analysis, they developed separate sets of equations for the hydraulic geometry of 18 rivers in Illinois using data from a total of 166 stream gaging stations. They found that the logarithm of flow discharge, velocity and depth is a linear function of the frequency of discharge and Strahler order. This leads to the conclusion that reference flow conditions and, therefore the hydrodynamic parameters do change along the river network.

[20] The nonlinearity of the hydrologic response in the time domain can be accounted for by the use of instantaneous response functions (IRF) [Valdes et al., 1979; Wang et al., 1981; Robinson et al., 1995; Yen and Lee, 1997] which describe a time variant basin response. The IRF formulation is similar to that of the IUH in that the system response is still linear and given by the convolution of the effective rainfall input and the instantaneous response function (IRF); but the IRF is allowed to vary with time as a function of the rainfall history. The IRF may or may not account for the nonlinearity that takes place in the space domain. For example, in the work by Valdes et al. [1979], the computation of residence time distributions is performed by considering a time varying velocity which is spatially invariant over the complete basin (that is, the velocity varies with the effective rainfall but is spatially invariant for streams of different Strahler orders). Based on Pilgrim's [1976] observations regarding the spatial variability of velocities across the basin for over bank-full flow conditions, many authors have used the convenient assumption of considering spatially invariant velocities (or alternatively spatially invariant hydrodynamic parameters). However, for lower flows, the velocity cannot be considered spatially invariant and it is necessary to study the effect that the spatially varying dynamic parameters (velocities, celerities and/or hydrodynamic dispersion coefficients) have over the basin response.

[21] Temporal and spatial variation of the basin's response function have been considered in the works by Wang et al. [1981] and Lee and Yen [1997]. Wang et al. [1981] derived expressions for the mean travel time over both hillslopes and channels of varying Strahler order which implicitly account for changes in velocities. The velocity changes are based on numerical experiments performed with the linear geomorphic model proposed by Gupta et al. [1980]. Lee and Yen [1997] applied kinematic-wave theory and derived explicit equations to compute the mean travel time of flows in the overland and channels of different Strahler order for equilibrium flow conditions. However, none of these studies deal specifically with the identification of the different dispersion mechanisms which contribute to the variance of streamflow response, which we address here.

[22] We will show here that when considering spatially varying celerity and hydrodynamic dispersion coefficient, there are three mechanisms that contribute to the variance of the network response function. Two of these mechanisms correspond to the ones identified when considering spatially invariant hydrodynamic coefficients, that is geomorphologic and hydrodynamic dispersion, and the third mechanism arises due to the spatially varying celerity. Note that the geomorphologic dispersion arises due to the difference in arrival times because of the difference in path lengths to the outlet. In addition, if we further impose that the celerities in each state and, consequently, for each path are different, it will further alter the arrival time distribution. We refer to the mechanism that introduces spread in the travel time distribution due the spatially varying celerity as kinematic dispersion. Figure 1 illustrates the concepts of geomorphologic and kinematic dispersion. Figure 1a shows a system of channels with varying lengths connected to a single outlet, in which hydrodynamic effects are considered negligible and the celerities are the same for all channels. The spread of travel times for water instantaneously and uniformly poured in this system will be completely induced by geomorphologic dispersion effects. On the other hand, in the system shown in Figure 1b, the channels have all the same length but different celerities, and again hydrodynamic effects are assumed negligible. The spread of travel times in Figure 1b will be completely induced by differences in celerities termed as kinematic dispersion. In a stream network (such as the one represented in Figure 1c) the geomorphologic and kinematic dispersions act together and give rise to what we refer as kinematic-geomorphologic dispersion. We also show here that the spatially varying celerities alter the expression for the hydrodynamic dispersion as well.

Figure 1.

Schematic to illustrate the concepts of geomorphologic and kinematic dispersion. (a) System of channels with varying lengths connected to a single outlet in which hydrodynamic effects are considered negligible and the celerities are the same for all channels. The spread of travel times is completely induced by geomorphologic dispersion. (b) System of channels with identical lengths but different celerities in which hydrodynamic effects are considered negligible. The spread of travel times is completely induced by kinematic dispersion. (c) In a river network the effects of geomorphologic and kinematic dispersion act together and give rise to the kinematic-geomorphologic dispersion. (d) Partition of the total dispersion when the geomorphologic dispersion (DG), the kinematic dispersion (DK), and the hydrodynamic dispersion (DD) act together.

3.2. Dispersion Coefficients

[23] We start with the same assumptions as Rinaldo et al. [1991], i.e., GIUH framework of the state-space approach and validity of the advection-diffusion equation for the flow through a channel in state ω, but we relax the assumption of spatially invariant celerity and hydrodynamic dispersion coefficient throughout the basin's network. Instead, we assume that they are functions of the Strahler order. The theoretical derivations given below are not dependent on any particular form of this dependence. In general they can be derived from the hydraulic geometry variation in a basin. The first two moments of the path's travel time distribution can be derived from the moment generating function given in equation (11). For a generic path γ the mean and the variance of the travel time distribution are:

equation image

and

equation image

[24] The first two moments of the network travel time distribution can be derived from its moment generating function obtained using equation (12). The expected value of the network's travel time distribution is:

equation image

and can be generalized for an infinite number of paths γ as:

equation image

An equivalent celerity uγ, that preserves the mean travel time for each path, can be defined for each path as:

equation image

Then, equation (22) can be rewritten as:

equation image

The variance of the network's travel time distribution is:

equation image

which, in the limit for an infinite number of paths, can be generalized as:

equation image

It should be noticed that equation (26) was first derived by Rinaldo et al. [1991], however, they only analyzed its physical significance for the case of spatially invariant hydrodynamic parameters.

[25] Defining an equivalent hydrodynamic dispersion coefficient, that preserves the variance of travel times over the path, as:

equation image

we can rewrite equation (26) as:

equation image

Although this last expression is equivalent to equation (26), it has the advantage of being written in terms of the equivalent path celerity and hydrodynamic dispersion coefficient. For the analysis that follows, it should become clear that for the case of hydrodynamic parameters varying with Strahler order, different paths γ in the network have (1) different length (equation imageγ) (equation (14), this is also true for spatially invariant parameters); (2) different equivalent celerities uγ (equation (24)); and (3) different equivalent hydrodynamic dispersion coefficients DLγ (equation (28)).

[26] In the special case where the celerity and hydrodynamic dispersion parameters are considered spatially invariant, equation (29) simplifies to the form given in equation (15) (with uγ = u and DLγ = DL). In this case there are only two variance-contributing dispersion mechanisms, the hydrodynamic and the geomorphologic dispersion (equation (19)).

[27] When considering spatially varying hydrodynamic parameters, we find that three mechanisms contribute to the variance. The first term on the right hand side of equation (29) represents, as in the previous case, the contribution due to hydrodynamic dispersion. In this case, each path has a different equivalent hydrodynamic dispersion which induces different values for the variance of the IRF for each path and the resulting contribution is given by a mean value of the variances over each path. The second and third terms in (29), represent the contribution due to differences in mean travel times induced by differences in both celerity (uγ) and length (equation imageγ) over different paths γ.

[28] The contribution due to the network organization can be quantified using the geomorphologic dispersion coefficient (equation (17)) with the prior choice of a suitable equivalent network celerity un. This equivalent network celerity can be defined in terms of the mean path length (Eγ(equation imageγ) = equation image(Ω)) and the network's mean travel time (E(Tn)) as follows:

equation image

which for an infinite number of paths can be written as:

equation image

The physical interpretation of this equivalent celerity is that if all the water drops traveling to the outlet through the different paths had the same celerity un, the mean travel time would be the same as the one computed by considering spatially varying parameters, i.e., with equation (22) or (25).

[29] Then, the geomorphologic dispersion coefficient (DG) is obtained by replacing this equivalent network celerity un, instead of u, in equation (17) and corresponds to the dispersion induced by considering water drops traveling through the network at a spatially invariant velocity un, that is:

equation image

However, it is evident from the second term on the RHS of equation (29) that for spatially varying celerities the dispersion mechanism is more complex than that captured by the geomorphologic dispersion coefficient. This can be captured by defining a kinematic-geomorphologic dispersion coefficient

equation image

which when generalized for an infinite number of paths becomes

equation image

Equation (33) can be alternatively written as

equation image
equation image

where

equation image

is a “dynamical length” for path γ. It represents a stretching of the mean path length equation imageγ by a factor equation image such that the arrival time with a spatially invariant celerity un is the same as that with variable celerity uω through each state. Alternatively, we may consider this as an equivalent condition under which geomorphologic dispersion with spatially invariant celerity un will completely compensate for the dispersion due to celerity variation except for a scale factor equation image (see equation (38)). With this notation we can also write (see also equations (18) and (19)):

equation image

and

equation image

The above equations using dynamically equivalent path lengths provide important insight into the dispersion mechanism due to the interplay between the distribution of path length and the spatial variation of the celerities (Rinaldo et al. [1995], who analyzed the impact of hillslope velocities in the basins IRF, used an expression akin to (37) to characterize hillslope travel time distributions using a width function approach).

[30] It is possible to isolate the contribution due to the existence of different celerities along different paths which will be addressed as “kinematic” effect. The resulting kinematic dispersion (DK) is defined as follows:

equation image

That is, DK results from the dispersion induced by the existence of different travel times over different paths, once the contribution induced by the variance of paths lengths (i.e., DG) has been subtracted. In closed form equation (40) can be written as

equation image

[31] The contribution to the variance of the basin arrival time due to hydrodynamic dispersion will be designated as DD and is obtained as:

equation image

or, for an infinite number of paths:

equation image

When the hydrodynamic parameters are assumed to be spatially invariant (DLγ = DLω = DL), then DD becomes equal to DL, which is in agreement with the linear theory.

[32] The expressions for the three dispersion coefficients can be replaced in equation (29) (see also equation (39)) to obtain:

equation image

This is analogous to the expression found by Rinaldo et al. [1991] (see equation (19)) but accounts for the three different dispersion mechanisms (Figure 1d) which contribute to the variance of the network's response function when considering spatially variable hydrodynamic parameters.

[33] It should be noticed here that the kinematic dispersion, DK, can take positive or negative values. A negative contribution implies that the velocities over different path are such that longer paths have higher equivalent paths celerities, which tends to reduce the geomorphologic dispersion that would otherwise be caused by the longer path lengths. To illustrate this, lets consider an extreme example in which the hydrodynamic dispersion for all paths is zero, and the celerities are such that all the raindrops traveling over different paths arrive simultaneously to the control section. In that case the ratio of the path length to the equivalent path celerity (equation imageγ/uγ) is the same for all paths, and therefore the variance of the expected value of the travel time over the different paths (Varγ(E(Tγ))) is zero and DK = −DG. On the contrary, the contribution of DK becomes positive and large when the equivalent path celerities over longer paths are lower than those of the shorter paths. Finally, DK becomes zero for the case studied by Rinaldo et al. [1991] of spatially invariant celerities (in which un = uγ = uω = u), that is, DKG = DG. It is noted here that for real basins longer paths have lower equivalent celerities and consequently DK is positive (see section 6).

[34] Note that equations (25) and (44) provide the first two moments of the arrival time distribution under the assumption of spatially variant, as a function of Strahler order, celerity and hydrodynamic coefficient. However, a closed form solution for the travel time distribution f(t) is still not available. In section 4 we derive two different second order approximations of f(t) using the moments derived earlier.

4. Approximations to the Network Response Function

4.1. Path Approximation Method

[35] An analytical expression to approximate the IRF for each path, which accounts for the effect of spatially varying parameters, is obtained as described below. The instantaneous response function for each path is approximated by an inverse Gaussian distribution (i.e., same distribution as the one describing each individual stream) with spatially invariant parameters uγ and DLγ given by equations (24) and (28) respectively. The Laplace transform of this travel time distribution is:

equation image

for any particular path γ, where θγ(s) is computed using uγ and DLγ in equation (9). It follows immediately from the definition of the equivalent path parameters, that the travel time distribution for each path given by equation (45) has the same first two moments as the travel time distribution given by equation (11).

[36] This approximation for the path travel time distribution can be used to obtain the network travel time distribution. That is, the Laplace transform of travel time distribution for the network is obtained by replacing (45) into (12):

equation image

whose inverse Laplace transform leads to an analytical expression for the network IRF:

equation image

Equation (47) provides an analytical expression for the instantaneous network response function derived using an approximation for the travel time distribution for each path. It is shown in appendix A, that for channel Peclet numbers (equation image) larger than a threshold Z, where Z is of the order of 10, all the moments of the network travel distribution characterized by equation (4) are very well approximated by the moments of the network travel distribution given in equation (47) (see also section 6).

4.2. Network Approximation Method

[37] In this section, an analytical approximation of the network IRF, is obtained by the use of equivalent network hydrodynamic parameters. The equivalent network celerity un is given by equation (30). The equivalent network hydrodynamic dispersion DLn is similar to the equivalent path hydrodynamic dispersion DLγ but it preserves the variance of the network IRF given by equation (29) as shown below. The network equivalent parameters un and DLn are spatially invariant parameters for the complete network that can be replaced in equation (13) to obtain:

equation image

The expressions for the mean and the variance of (48) are:

equation image

and

equation image

It is evident that by replacing the equivalent network celerity un given by equation (30) into (49), we set [E(Tn)]eq = E(Tn). DLn is estimated so that the equivalent variance [Var(Tn)]eq (equation 50) equals the one obtained by considering spatially varying hydrodynamic parameters Var(Tn) (equation (27)). The resulting expression for DLn is:

equation image

[38] Note that although un and DLn are used as spatially invariant parameters, they are estimated to account for the nonlinear effects that arise when considering spatially varying hydrodynamic parameters. As seen from equation (51) the equivalent hydrodynamic dispersion parameter DLn accounts for the contribution of both hydrodynamic and kinematic dispersions; the latter appearing because of the existence of spatially variant celerities.

5. Hydraulic Geometry

[39] Stall and Fok [1968] assembled and presented data from 166 USGS stream gaging stations in Illinois which were used to define downstream hydraulic geometry relations for these streams. They found that the logarithms of flow discharge, velocity and depth are a linear function of the frequency of discharge and Strahler order. The discharge Q was related to the frequency of occurrence of a particular flow frequency F and to the order of the stream ω using a linear multiple regression model:

equation image

where αQ, βQ, and λQ are empirical regression coefficients. This equation is only valid for 0.1 ≤ F ≤ 0.9 due to the increased uncertainty outside this range.

[40] Utilizing the dependence of flow velocity v and flow depth h on the discharge Q, Stall and Fok [1968] developed the following relationships:

equation image
equation image

where αv, βv, λv, αh, βh, and λh are also empirical regression coefficients.

[41] These equations are not dimensionless, hence in the original work, Q has dimensions of cfs, u is in feet/second, and h has units of feet. Note that the above formulations are consistent with Leopold and Maddock's [1953] relationships hQϑ, and uQη where the flow frequency F determines the proportionality constants. These relationships are used in the case study given next.

6. Case Study

[42] The relative contribution of the kinematic dispersion as a variance producing mechanism is analyzed for the Vermilion river basin. This basin is among the 18 river basins in the Illinois River System (see Figure 2) for which a convenient form of hydraulic geometry relations exists that allows for the estimation of celerity and hydrodynamic coefficient as a function of Strahler order [Stall and Fok, 1968].

Figure 2.

Location of the Vermilion river basin in the state of Illinois (IL). The neighboring states shown are Wisconsin (WI), Indiana (IN), and Michigan (MI).

[43] The river network for the basin was derived from 7.5-minute digital elevation models (DEMs) developed by the USGS (United States Geological Survey) that have horizontal resolution of 30 m. The basin has an area of 3449 km2 and a total relief of 362 m that results in very mild slopes. Flow directions were identified using the imposed gradients method [Garbrecht and Martz, 1997]. This method was developed to produce realistic and topographically consistent drainage patterns. It significantly improves the identification of flow directions in flat areas as compared to other existing methods. In particular it was found that the channel network for the Vermilion river basin derived using this method is closer to the river network obtained from the National Hydrography Data set than the one derived using alternate methods [Saco, 2002]. The resulting network has Strahler order Ω = 6. The values for the number of streams (Nω), mean drainage area (equation imageω), mean along-channel length (equation imageω), and mean bed slope (equation imageω) as a function of the Strahler order were obtained using the extracted network.

[44] The spatially varying celerities uω and hydrodynamic dispersion coefficients DLω were obtained using the hydraulic geometry relations derived by Stall and Fok [1968] (see section 5). The reference flow discharge Q* used to compute uω (equation (6)) and DLω (equation (7)) corresponds to steady state conditions. Given a spatially uniform rate of rainfall excess I the steady state, or equilibrium, flow discharge is obtained when all the basin saturated area is contributing runoff to the control section. For a rainfall excess rate I, which corresponds to a flow discharge Q* at the outlet of the stream of order ω, we computed F from equation (52), and then the reference velocities (v*ω) and depths (h*ω) from equations (53) and (54). Finally v*ω and h*ω were replaced into equations (6) and (7) to get uω and DLω. The celerity and hydrodynamic dispersion coefficient for each Strahler order ω in the Vermilion river basin were computed for two different rates of rainfall excess (I) (1 and 5 mm/hr), and for an assumed percentage of saturated area basin k = 1%.

[45] To compute the dispersion coefficients (DG, DK and DD), as well as the network hydrodynamic parameters (un and DLn), it is necessary to estimate probabilities, lengths and equivalent hydrodynamic parameters for all the possible paths in the network. These values are shown in Table 1. The path probabilities, p(γ), are computed using equation (1). The probabilities pxi, xj of a transition from state xi, corresponding to a channel of order ω to state the xj corresponding to a channel of higher order k, are given by:

equation image

where nω, k is the number of streams of order ω draining into streams of order k and Nω is the total number of streams of order ω. Both nω, k and Nω were directly obtained from the tributary structure of the river network extracted using the DEMs. The mean tributary structure for the sub-basins of orders 5, 4 and 3, was obtained from that of the total basin. The probability πx1 of starting out in the appropriate state of order x1 = ω, was computed as the fraction of the basin area that drains directly into a stream of order ω [Gupta et al., 1980, equation (25)]. The length of each path corresponds to the sum of the lengths of streams of all orders in that path. The values of the equivalent celerity (uγ) and the equivalent hydrodynamic dispersion (DLγ) for each path γ were computed using equations (24) and (28), respectively.

Table 1. Equivalent Path Hydrodynamic Coefficientsa
Path DefinitionpγLγ, kmI = 1 mm/hrI = 5 mm/hr
uγ, m/sDLγ, m2/suγ, m/sDLγ, m2/s
  • a

    The first three columns correspond to the series of states (or channel orders ω), probabilities (pγ), and lengths (Lγ) of the different paths in the Vermilion river network (order Ω = 6). The last four columns display the values of equivalent path celerity (uγ) and equivalent path hydrodynamic dispersion (DLγ) for a rainfall excess rate of I = 1 mm/hr and I = 5 mm/hr.

 234560.125139.360.4234.80.73122
1234 60.075103.970.4134.90.72122.5
123 560.01121.660.4341.20.76144.6
123  60.09486.260.4344.20.76155
12 4560.018130.50.4341.30.76144.8
12 4 60.01195.110.44440.76154.5
12  560.033112.80.4651.20.8179.6
12   60.05977.40.47610.83214
1 34560.029134.930.4339.90.76139.9
1 34 60.01799.540.4341.90.76147.2
1 3 560.002117.220.4548.80.79171.2
1 3  60.02281.830.4656.40.81198
1  4560.019126.070.4548.30.8169.6
1  4 60.01290.680.46550.81193.1
1   560.014108.360.4862.50.85219.3
1    60.03372.970.5283.60.9293.4
 234560.068137.150.4338.40.75134.9
 234 60.041101.750.4339.90.75140
 23 560.006119.440.4546.50.78163.4
 23  60.05184.040.4652.60.8184.7
 2 4560.01128.290.4546.30.79162.4
 2 4 60.00692.890.4651.60.8181.3
 2  560.018110.580.48590.83207.1
 2   60.03275.180.5760.88266.8
  34560.024132.710.4544.50.78156.1
  34 60.01497.320.4548.70.79171
  3 560.002115.010.4755.80.82195.9
  3  60.01879.610.4969.10.86242.7
   4560.005123.860.4754.80.83192.3
   4 60.00388.460.4966.10.86231.9
    560.005106.150.5173.30.89257.4
     60.12370.750.561090.98382.5

[46] Table 2 shows the values of the equivalent network celerity (equation (30)), and the geomorphologic (equation (32)), hydrodynamic (equation (42)), kinematic (equation (40)), and equivalent network hydrodynamic (equation (51)) dispersion coefficients for the Vermilion river basin for two rainfall excess rates. As seen in Table 2, the kinematic dispersion contributes about 35% of the total dispersion of the network response function. Its contribution is significantly larger than that of the hydrodynamic dispersion. All the dispersion mechanisms (DG, DK, and DD) increase with rainfall excess rate I (see the companion paper by Saco and Kumar [2002] for a detailed scale analysis).

Table 2. Equivalent Basin Hydrodynamic Coefficientsa
 Vermilion
I = 1 (mm/hr)I = 5 (mm/hr)
  • a

    The first five rows display the values of the equivalent network celerity (un) the geomorphologic dispersion (DG), hydrodynamic dispersion (DD), kinematic dispersion (DK), and equivalent network hydrodynamic dispersion coefficients (DLn) for the Vermilion river basin. The last two rows display the values of the celerity (u) and the hydrodynamic dispersion coefficient (DL) for the stream with the highest order (ω = 6).

un (m/s)0.450.78
DG (m2/s)1370.22401.8
DD (m2/s)47.7167.3
DK (m2/s)7831372.5
DLn (m2/s)830.71539.8
u(ω = 6) (m/s)0.560.98
DL(ω = 6) (m2/s)109382.5

[47] Figures 3 and 4are used for the comparison of the exact solution of the network IRF obtained using numerical convolution over each path (equation (4)) with the network IRFs obtained using the path and the network approximation methods (equations (47) and (48) respectively). The network IRFs correspond to the derivative of the step response function (S hydrograph) obtained for a step input, that is, a continuous rainfall excess rate of I units. It is evident that the IRF obtained from the path approximation method matches that obtained using numerical convolution very closely. The reason for this is that the range of channel Peclet numbers (Peω) obtained for the Vermilion river varies between 362 (for ω = 6) and 1000 (for ω = 1) for a rainfall excess rate of 1 mm/hr, and between 180 (for ω = 6) and 489 (for ω = 1) for a rainfall excess rate of 5 mm/hr. Within this range of Peω, all the moments of equation (4) are closely approximated by those obtained from equation (47) (see Appendix A) and therefore the IRFs obtained from both approaches can be expected to match. Figures 3 and 4 also show that the flow peak, time to peak, and duration of the IRF obtained from the network approximation method provides a good approximation to those obtained from the convolution.

Figure 3.

Comparison of the network IRFs obtained using the convolution integral (equation (4)), the path approximation method (equation (47)), the network approximation method (equation (48)) and using spatially invariant hydrodynamic parameters (equation (13)). The IRFs shown in Figure 3 correspond to a rainfall excess rate of 1 mm/hr.

Figure 4.

Same as Figure 3 but for a rainfall excess rate of 5 mm/hr.

[48] The network response obtained using equation (13) with spatially invariant hydrodynamic parameters is also shown in Figures 3 and 4. There is no general guideline to select the spatially invariant celerity and hydrodynamic parameters for this case. For the comparison performed here, they correspond to that for the channel with the largest order, that is ω = 6 (see Table 2). The IRF obtained using the network approximation method, which incorporates kinematic dispersion effects, lags the IRF obtained using spatially invariant hydrodynamic parameters because the equivalent network celerity is smaller than the celerity of the channel of highest order. It also has a larger variance because the equivalent hydrodynamic dispersion is larger than the hydrodynamic dispersion of the channel of highest order. The use of spatially invariant hydrodynamic parameters tends to predict a network's travel time distribution with smaller mean travel time and variance. The corresponding IRF will therefore tend to overestimate peak flows, and underestimate time to peak and duration of the hydrograph unless the parameters are specifically chosen to avoid this problem, as is the case in the network approximation method.

[49] As explained in section 3.2 the kinematic dispersion, DK, can potentially take positive or negative values. A negative contribution implies that the velocities over different path are such that longer paths have higher equivalent path celerities, which tends to reduce the geomorphologic dispersion that would otherwise be caused by the longer path lengths. We found that for the Vermilion river basin there is general tendency for the path equivalent celerity uγ and equivalent hydrodynamic dispersion DLγ to decrease with path length equation imageγ (Table 1). Figure 5 illustrates the plot of uγ versus equation imageγ for a rainfall excess rate of 5 mm/hr. Although paths of longer length do not always have smaller velocities than shorter ones, the two variables are indeed negatively correlated. Hence, the contribution of the dispersion due to differences in celerities along different paths (DK) takes a positive value.

Figure 5.

Plot of equivalent path celerity (uγ) versus path length (equation imageγ) for all paths in Vermilion river basin (for a rainfall excess rate of 5 mm/hr).

7. Summary and Conclusions

[50] Previous research on the different mechanisms that contribute to the variance of runoff response have used spatially constant hydrodynamic parameters. However, empirical studies suggest that it is more realistic to work with spatially varying hydrodynamic parameters which depend on flow discharge, reflecting the underlying nonlinear character of the governing momentum equations. In this study, Stall and Fok's [1968] hydraulic geometry relations are used to estimate the celerities and hydrodynamic dispersion coefficients for streams of different Strahler orders as a function of the rainfall excess rate. The impact that these spatially varying hydrodynamic parameters bear on the basin's hydrologic response is analyzed.

[51] Using spatially invariant hydrodynamic parameters the total variance of the network response function can be characterized as the sum of two contributions: geomorphologic (DG) and hydrodynamic (DL) dispersions [Rinaldo et al., 1991]. It is found in this study, that spatially varying celerities over the stream network introduce a dispersion, called kinematic dispersion (DK), which is in addition to the geomorphologic and hydrodynamic dispersion. In the Vermilion river basin the kinematic dispersion contributes about 35% of the total dispersion of the network response function. Its contribution is significantly larger than that of the hydrodynamic dispersion but smaller (about half) than that of the geomorphologic dispersion. It is also found that all the three contributions increase with the rainfall excess rate.

[52] Two different approximations to the network IRF, called the path approximation method and network approximation method, have been derived. For the range of channel Peclet numbers used in this study, the travel time distribution obtained using the path approximation method gives an excellent approximation to the exact solution obtained by convolution. That is, all the moments are well approximated with estimation errors that can be considered negligible. The travel time distribution estimated using the network approximation method only preserves the first two moments of the exact solution. However, it provides a very good estimate of the peak, time to peak, and duration of the IRF. This approximation uses network equivalent hydrodynamic parameters (un and DLn) which are used as spatially invariant. The equivalent network hydrodynamic dispersion (DLn) includes the effect of both hydrodynamic and kinematic dispersion and therefore it accounts for the spatial variation of celerities. The IRF obtained using spatially invariant hydrodynamic parameters tends to overestimate peak flows, and underestimate time to peak and duration of the hydrograph unless the parameters are specifically chosen to avoid this problem.

[53] We have used the Horton-Strahler ordering scheme in the analysis as it provides a convenient way to incorporate the topologic and geometric symmetries observed in river networks. This framework has been used extensively in previous studies [Rodriguez-Iturbe and Valdes, 1979; Gupta et al., 1980; Wang et al., 1981; Rinaldo et al., 1991; Yen and Lee, 1997]. In particular, the recent developments in the study of topological self similarity in river networks which are based on the use of the Horton-Strahler ordering scheme [Peckham, 1995; Tarboton, 1996; Veitzer and Gupta, 2000] allowed us to explore the scaling and asymptotic properties of the dispersion coefficients [see the companion paper by Saco and Kumar, 2002]. This framework also allowed us to incorporate the observed scale dependence of velocity by taking advantage of the scale dependent hydraulic geometry relations derived from the field study performed by Stall and Fok [1968]. However, this scheme is not without limitations. In particular the averaging involved in the use of a Horton-Strahler ordering scheme wipes out some of the natural variability of the topologic and geometric organization present in river networks. Consequently, alternate approaches have been used in the literature such as the width function or link based approach [Gupta and Waymire, 1983; Troutman and Karlinger, 1985; Snell and Sivapalan, 1994; Rinaldo et al., 1995]. Though hillslope/channel velocity variations using the width function approach have been analyzed by Rinaldo et al. [1995], their study considers spatially invariant transport parameters (that is, celerities and hydrodynamic dispersion) throughout the network. Incorporating spatially varying transport parameters into the link based approach is a research challenge to be addressed in the future. Further extensions to this work certainly need to incorporate hillslope velocities and assess their relative contribution to the variance of the basin response.

Appendix A:: Statistical Moments of the Path Approximation Method

[54] As shown below, for channel Peclet numbers (Peω) above a certain threshold, the difference between the moments of the exact solution of the network travel time distribution (equation (4)) and those of the path approximation method (equation (47)) is negligible.

[55] The Laplace transform for the travel time distribution for each channel (equation (4)) can be written as:

equation image

where Yω = 4DLω/uω2. Equation (A1) constitutes the moment generating function of the channel's travel time distribution. The moments can be obtained as:

equation image

where 〈tfωn〉 denotes the moment of order n. Therefore the goodness of a given approximation to the travel time distribution can be analyzed by evaluating the moments generated by equation (A2). For s small enough such that, Yωs lies in the interval −1 < Yωs < 1, we can use a Taylor series expansion to get:

equation image

The channel's travel time distribution can be approximated by truncating this series expansion at an appropriate level. In particular, the approximation to the travel time distribution that results from a second order approximation for the square root in equation (A1), can be written as:

equation image

Where the prime is used to refer to the approximation. The first two moments of equation (A4) are exact (that is, 〈tfω〉 = 〈tfω〉 and 〈tfω2〉 = 〈tfω2〉). For Peclet numbers above a threshold Z, the error in the approximated moments 〈tfωn〉, with n ≥ 3, induced because of the truncation is at least one order of magnitude smaller than the value of the moment, that is (〈tfωn〉 − 〈tfωn〉) < (0.1 · 〈tfωn〉). It can be shown that for Z of the order of 10, the error in all the approximated moments can be considered negligible, that is:

equation image

for all n.

[56] Equation (A4) can be rewritten as:

equation image

Using this approximation for the Laplace transform of the channel travel time distribution in equation (11), we obtain the following approximation for the the Laplace transform of the path time distribution (equation imageγ(s)):

equation image

This approximation holds for s such that −1 < Yωs < 1 for all ω ∈ γ. Then, we can write:

equation image

Recalling that equation imageγ = ∑ω∈γequation imageω and using the definition of the equivalent path parameters uγ (equation (24)) and DLγ (equation (28)) we can rewrite equation (A7) as:

equation image

We should note here that if Peω ≫ 10 for all ω, then Peγ ≫ 10. For Peω ≫ 10, the difference between moments obtained using equation (A9) and equation (45) as moment generating functions can be considered negligible, that is:

equation image

For Peω ≫ 10 we can therefore expect the analytical expression for the inverse Laplace transform obtained from equation (45) to match closely the convolution integral defined by equation (1). Consequently, the IRF obtained using the approximation given by equation (47) should closely match that given by equation (4).

Acknowledgments

[57] This research was supported by NSF grant EAR 97-66121. We thank Efi Foufoula-Georgiou and Andrea Rinaldo for their review comments.

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