Application of the meta-channel concept - Wiley Online Library

HYDROLOGICAL PROCESSES, VOL. 9, 485-505 (1995)

APPLICATION OF THE META-CHANNEL CONCEPT: CONSTRUCTION OF THE META-CHANNEL HYDRAULIC GEOMETRY FOR A NATURAL CATCHMENT JOHN SNELL AND MURUGESU SIVAPALAN Centre for Water Research, Department of Environmental Engineering, University of Western Australia, NedIand.9, 6009, Australia

ABSTRACT This is the second in a series of three papers about the meta-channel concept which illustrates the derivation of the principles behind the concept, the construction of the hydraulic geometry and the application of the concept to flood routing, respectively. It was shown in the first of these that a channel network in a catchment can be conceptualized into a single ‘effective channel’ representation: a meta-channel. This study uses this conceptualization to show how such a meta-channel can be constructed. The techniques derived are applied to one catchment in New Zealand. We derive hydraulic geometries expressed as functions of flow distance throughout this catchment based on the Leopold and Maddock power laws. This derivation uses classical published values for hydraulic geometry coefficients and exponents, regional parameterization of the index flood relationship for New Zealand as a whole, together with local knowledge regarding the order of magnitude of the channel roughness. Conservation principles derived from the continuity and mechanical energy balance equations are used to construct the hydraulic geometry of the meta-channel of this catchment. A meta-channel long profile is established and compared against the mainstream long profile. The effectiveness of the Leopold and Maddock power law assumptions is tested by comparing the derived hydraulic geometry against available field cross-sectional data for the gauging site at the outlet of the catchment.

KEY WORDS

Meta-channel,

Hydraulic geometry, Networks, Geomorphology

INTRODUCTION Since the original work of Sherman (1932), the concept of a unit hydrogaph has underlain much of flood routing work in surface hydrology. Implicit within this usage is an assumption that the instantaneous unit hydrograph (IUH) provides sufficient approximation to the instantaneous response function for a catchment. Instantaneous response functions (Wang et a/., 198l), derived for hillslopes and small catchments, show considerable non-linearity and are strongly dependent on antecedent conditions and rainfall intensity (Robinson and Sivapalan, submitted). In contrast, channel response functions tend to be weakly non-linear (Beven and Wood, 1993); consequently, routing of floods is often performed by solving the convective-diffusion equation, Equation (l), in which a diffusion coefficient, D , and celerity, c, are considered independent of discharge, Q

One solution to this equation is the inverse Gaussian function (Troutman and Karlinger, 1985). It is this function which is often taken to be the IUH of individual channels, (Mesa and Mifflin, 1986). Suitably weighted by a geomorphological function such as the width function, it produces a geomorphological IUH (GIUH), or channel network response function (Troutman and Karlinger, 1985; 1986; Mesa and Mifflin, 1986; Naden, 1992). It should be stressed that this approach to the catchment response problem is CCC 0885-6087/95/040485-21 0 1995 by John Wiley & Sons, Ltd.

Received 4 May 1994 Accepted 30 November 1994

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scale dependent in that the linear or weakly non-linear behaviour of the channels only dominates the nonlinearity of the hillslope response functions for sufficiently large catchments. The St Venant equations for single channel flow, which can be derived from the Navier-Stokes equations for incompressible fluid flow (Strelkoff, 1969), are non-linear and scale independent. Manipulation of these equations into more generalized forms (Sivapalan and Larsen, submitted) still maintains the non-linearities and scaling properties inherent in the original equations. Although it is often convenient to make assumptions about the extent of linearity within specific physical processes such as the routing of floods through networks, non-linear functions can be integrated using numerical techniques. The main practical reasons for using linearization is in the convenience of the mathematical techniques that are allowed and the ease of the resulting computations. To date, there are few tools which provide insight into either the issues of network response as against single channel response, or the effects of increasing catchment size on the non-linearities within the flood routing process. The advent of modern computing techniques and digitized elevation models (DEMs) has made it possible to develop tools for the extraction and manipulation of components of network systems (O’Callaghan and Mark, 1984; Band, 1986; Jenson and Domingue, 1988). Snell and Sivapalan (1994a) have extended the work on the extraction of the catchment width function, the number of channels at a given distance from the outlet, to an automated extraction of the catchment area-distance function, the catchment area convergent to a specific flow distance from the outlet. Differences between the width and areadistance functions for a catchment can be thought of as providing a measure of the underlying variability in the constant of channel maintenance throughout a catchment, which in turn is expressing an underlying spatial variability of the catchment’s geology and pedology. However, a catchment functions as an integrator in which the underlying spatial variability is effectively overshadowed or dispersed by variability in flow paths to the catchment outlet. Current models tend to be either conceptually lumped or physically distributed. We feel that, intrinsically, there is a place for physically based lumped models in which the certainty of distributed knowledge is replaced by the probability of lumped statistical distributions of physical parameters without loss of confidence in the end-product - a hydrograph. Motivated primarily towards the development of techniques of representing channel networks which preclude assumptions regarding linearities and independence of coefficients, we have defined a meta-channel, representing a collapsing of the channel network structure onto the single dimension of flow distance. This collapsing or lumping process is based on principles designed to conserve both mass and essential elements of the mechanical energy balance equation as originally derived by Strelkoff (1 969). The ‘effective’ channel or meta-channel concept is justified on the basis of an inability of a catchment to maintain a history of the origins of variations induced in travel times to its outlet. Our long-term goal is to provide a mechanism of routing flow which is independent of linearity assumptions and which addresses a fundamental scaling issue in catchment response in that it provides more accurate estimations of discharge over a larger range of catchment scales than is currently possible with an IUH approach. Before being able to route flow through this meta-channel we have to be able to derive the ‘effective’ physical characteristics of this channel. What does this channel look like? This paper briefly examines the principles behind the collapsing process [see Snell and Sivapalan (submitted) for details] and shows an application of the principles in the construction of a meta-channel for one catchment in New Zealand. We demonstrate in detail how the single channel representation of the catchment is achieved, paying particular attention to the derivation of geometrical form of this meta-channel. We show the effect of this geometry on an eventual single ‘effective’ channel representation of a network. It should be noted that, in principle, the meta-channel we derive is purely concerned with the routing of flow which reaches the network. We take the pragmatic approach that networks only extend out to some support area threshold. We are cognizant that this support threshold area is highly dependent on geology, pedology and antecedent conditions, implying in turn that this support is both spatially and temporally variable. Important questions can therefore be asked regarding the determination of this support area. It is not our intention to answer this question, nor is it relevant to the meta-channel what the short-term spatial variations are - these factors being absorbed by some form of loss modelling. Routing by the meta-

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I

I

Figure 1. (a) Single channel control volume; (b) meta-channel control volume

channel model is potentially adaptable to handle long-term variability in rain fields and known spatial variability of pedology through modulations of the area-distance function. PRINCIPLES GOVERNING THE COLLAPSE OF THE NETWORK Control volume Figure l a depicts a control volume associated with one channel. Such a control volume can be extended across a set of n(s) such channels (Figures l b and 2). The extended version of the single channel control volume delimits the control volume of a meta-channel. Each channel segment building the meta-channel control volume extends between s and s 6s, where s is the flow distance between the outflow of the catchment and the outflow of each channel segment. To specifically remove all junctions and junction effects from subsequent analysis, we assume that the number of channels within this distance is invariant. Fluid travelling through a channel has local velocity v'(2).The control volume is deformable such that its volume, v ( t ) ,is a function of time, t , only, and each point of its surface has a local velocity of G,a different

+

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catchment boundary

/

c .channel sections of equal flow I 6s

:from the outflow

Figure 2. Collapsing of the network: a plan view

physical parameter to the fluid velocity, 5. There is a volume flux, Q, into the channel entrance and a lateral influx of q1 per unit length of channel. The control volume is contained by four surfaces: Channel entrance - normal to the predominant flow direction at that point. Fixed in space with respect to the flow distance, s, with no surface velocity, G = 0, and subject to pressure, Pi,. Channel exit - with similar properties to the entrance, but subject to pressure Po,,. Wetted surface - primary source of drag in the system. It is fixed in position with respect to the flow distance. Both flow and surface velocities are zero (.' = I;i; = 0). In practice, this surface acts as a source and/or sink of fluid mass by interaction with the groundwater system. In this analysis, all such inputs are part of the lateral flows and this boundary is considered impervious. Free surface - fixed in space relative to the flow distance, but variable with respect to the bed normal. This surface expresses the deformable nature of the control volume. It provides a mass source by means of rainfall. This source is also lumped with the lateral flows. The free surface is also a kinetic energy source either from (i) rainfall - in which case it is lumped with the lateral influxes, or (ii) wind effects working on large free surface areas of wide channels. Fluid and surface velocities are generally different (17 # G) but components of these velocities normal to the surface are equal, (5.Z = w' Z). The free surface is subject to atmospheric pressure, Pa. Channel entrances and exits have maximum fluid depth, y ( s ) ,top width, B(s),and bed height above datum of c0(s). Any point C(q,() on this surface is spatially characterized by its q and ( coordinates as shown in Figure 1. Equation of continuity For continuity through a meta-channel control volume as described above (Snell and Sivapalan, submit ted)

where A ; is the flow cross-sectional area of an individual channel, also known as the channel capacity of that channel, Qiis the discharge for that channel and qi is the lateral flow of that channel at that point; n(s) is the width function at that point. The first term on the left is the rate of change of total channel capacity with time at flow distance, s,

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where the total channel capacity is the aggregate of the individual channel capacities. The second term represents the net change of total mass flux through an infinitesimally thin section stretching across all the channels constructing the meta-channel at flow distance, s. The right-hand side represents aggregation of all lateral inflows into all channels. In applying the continuity principle to the meta-channel, the individual fluxes, channel capacities and lateral inflows for each channel have to be conserved across the set of channels making up the meta-channel at flow distance, s, from the catchment outflow. Mechanical energy balance Subject to a set of assumptions, the mechanical energy balance equations for the control volume consisting of a set of channel segments, is (Snell and Sivapalan submitted)

where iii is the cross-sectionally averaged flow velocity for channel i, Hi is a total energy head for that channel consisting of a velocity head, a potential head due to the position of the channel above datum, a potential head due to the mass of water above the bed of the channel and a pressure head due to the imbalance of pressure force at the entrance and exit of the control volume. T ~is, the average stress induced at the wall of the channel, Pi is the wetted perimeter of the channel and HI! is a total head term contributed by the lateral inflows. The four terms contributing to this equation are, respectively, from left to right: (1) Kinetic energy storage. (2) Kinetic energy flux and work done by pressure/gravity. (3) Energy losses. Currently, these energy losses are considered to be a function only of the frictional effects of the wetted surface. Energy losses due to transverse flows as caused by meandering, or three-dimensional turbulence effects of mixing at junctions or of hydraulic jumps are explicitly ignored. (4) Energy head due to lateral inflows. Contains similar terms to the total energy head of the main flow. For conservation of the mechanical energy balance for a meta-channel, we preserve the same energy balance for the individual channels across the set of channels composing the meta-channel. Specifically we preserve the terms, a l ,Qiiii,QiHi,H,,qi, (Qi,=fo,P,)/(-yAi) in Equation (3). APPLICATION OF THE PRINCIPLES IN CONSTRUCTING A META-CHANNEL The meta-channel concept is applicable to any network - either natural, as extracted from DEMs, or artificially derived, as for example, Peano basins (Rinaldo et al., 1991), and optimum channel networks (Rigon et al., 1993). In this work, we limit ourselves to one naturally occurring network in New Zealand. Assumptions The generalized conservation equations become more amenable for deriving the geometry of a catchment meta-channel on making the following assumptions: a

Steady state. We assume a steady state system when determining hydraulic geometry of the metachannel from hydraulic geometries of constituent channels. Consequently, discharge is related linearly to the area contributing to that discharge through a recharge parameter. This recharge parameter, R, contains the non-linearity and spatial variability inherent in both the rainfall and the soil characteristics within the catchment. The relationship is expressed as

where M ( s ) is the area of the catchment convergent at flow distance s. It forms a cumulative distribution function. Initially, we consider the recharge to be uniform throughout the catchment - in effect, we ignore the spatial variability inherent within it. We reiterate that steady state is only assumed in forming the hydraulic

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Figure 3. Effect of relative width and depth exponents on the shape of channels

geometry of the meta-channel; however, the meta-channel can still be used to model unsteady flow situations as in runoff-routing. Gradually variedflow. The gradually varied flow assumption neglects the effects of sharp changes in the channel geometry which bring about sharp changes in the flow height, e.g. hydraulic jumps. No sediment transport. Negligible variation in velocity over the cross-section. This sets all shape factors to 1. This is in reasonable agreement with Van Driest (1946), but Henderson (1966) warns that these parameters can reach values of 2 in meandering rivers. Incompressibility One-dimensional flow. Resistance coejicients. We assume resistance coefficients determined for steady, uniform and turbulent flow, hold for unsteady, non-uniform flow situations as in the case of flood routing.

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0

-

Hydraulic geometry In routing flow through the meta-channel, we need to derive the hydraulic geometry of the single channel representation. These hydraulic geometry relationships determine both celerity, c( Q), and diffusion coeffcient, D ( Q ) (Henderson, 1966)

We usually refer to two distinct types of hydraulic geometry - at a station - identified with flow geometry and downstream - identified with channel geometry. This hydraulic geometry expressed in the form of power laws (Leopold and Maddock, 1953) is examined in depth in Appendix A. It should be noted that at-a-station hydraulic geometry characteristically shows a depth exponent, fi , slightly greater than width exponent, bl (Leopold and Maddock, 1953), hence providing concave cross-sections with increasing discharge. In contrast, downstream hydraulic geometry characteristically shows the reverse effect (Leopold and Maddock, 1953), producing convex cross-sections and implying a flattening of downstream crosssection relative to upstream. Figure 3 illustrates these effects.

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Hydraulic geometries as defined by Leopold and Maddock (1953) are smoothly varying power functions of discharge. For at a station hydraulic geometry this produces difficulties as the cross-section at any site normally shows one or more discontinuities as channel capacity is exceeded and discharge starts to enter the floodplain. Downstream geometries are affected by constrictions and expansions in the stream produced by the underlying spatial heteogeneity in both geological and pedological structures of the catchment. No attempt is made to handle these latter problems - they are ignored under the assumption of gradually varied flow and by treating the hydraulic geometry as stochastic rather than deterministic functions. For the meta-channel, ideally, we would consider the downstream hydraulic geometry of the system. However, these are typically established at mean annual flood or bankfull discharge. Such relatively high discharge display a non-linear relationship with contributing area, principally through heterogeneity in rainfall across the catchment, leading to power law relationship of the form Q = +M*

where M is the catchment area contributing discharge to the network. In such situations, bankfull discharge across the network becomes disconnected in terms of the recurrence period required to produce the bankfull discharge. This leads to the paradox that conservation of mass no longer appears to hold. As continuity is one of the fundamental principles of the meta-channel, we make the steady-state assumption mentioned previously. We establish continuity by maintaining a linear relationship between contributing area and discharge, the spatial heterogeneity mentioned earlier being absorbed by the recharge term. For the initial investigation, we assume a uniform recharge rate throughout the catchment. Future investigations will relax this constraint. We consider the hydraulic geometry of individual channels as being of at a station type, with sites of equal flow distance from the outlet being aggregated such that the total channel capacity and total discharge weighted flow velocities and depths are conserved. Although at a station scaling exponents can remain effectively constant as we travel downstream, there can be large variations in the scaling coefficients. We show in Appendix A that at a station hydraulic geometry scaling coefficients are determined from knowing the hydraulic geometry exponents typical of at a station geometry in a region together, with hydraulic geometry exponents for downstream hydraulic geometry; these latter exponents are again taken to be characteristic of a region in which a catchment occurs. Thus, for each position on the network, we determine the width, mean and maximum depths, and velocity. Hydraulic geometry can only provide the mean flow depth. In determining the hydraulic geometry parameters for a meta-channel, we need also to know the maximum flow depth in a channel and the wetted perimeter of that channel. We show (Appendix A) that the maximum fluid depth in a stream for any recharge R and at any flow distance s, y = y ( R ,s),for each position is related to mean flow depth as follows y = y(1 + + - I )

+

= -(1 CI at

+ +-l)B$

(7)

where y is mean depth, = f i / b l is the ratio of depth to width exponent, al is width scaling constant, cI is depth scaling constant and B is top width. It is trivial to derive (Appendix A) the wetted perimeter by line integration across the top width of the channel as follows

where J and q are dummy variables expressing the width and depth of a channel, respectively. Aggregation across the width function By using the conservation equations described earlier, we collapse the geometries based on the conservation of discharges, cross-sectional areas, flow depths, heights above datum and energy losses due to stresses induced in the fluid by the solid boundary. We now require one further assumption in preserving the wetted perimeter across a set of channels.

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The mechanical energy balance equation provides both a kinetic energy storage, Qii, and a kinetic energy flux term, Qii2,by which we can preserve discharge velocity - see Appendix B. The flux conserved velocity, uf, is used in all subsequent calculations in this application as, in the light of our steady-state assumption, the storage term is neglected. However, in routing problems we will be dealing with non-steady situations. To meet this problem, we need to establish a storage conserved velocity, us, from which we derive, A, a parameter used for converting between the two velocities. This parameter becomes in effect a shape factor and can be considered to maintain all the variability usually conveyed by individual channel shape factors but neglected in our aggregation process A="

U

(9)

u1

Assuming all shape factors

KZ

1 , we formulate the aggregation of hydraulic parameters as

and P

=

c

c

i= 1

i= 1

P; ii; fP =

Ei; f P i

Determination of the friction factor of each stream in the network The shear stress, ro,at the bed of the channel is related to the Darcy-Weisbach friction factor, f,through 70

2

= pu, = 0.125pfii2

where u, is shear velocity. For fully turbulent flow this friction factor is related to roughness height, (1938) and extended by Bray (1979)

f -4 w 0.248 + 2.28 log

(12) E,

as established by Keulegan

c1 -

The variable, d , is some depth scale. Keulegan (1938) used hydraulic radius, Bray (1979), mean depth; we use maximum depth, y . We consider E to be an effective roughness height of the size of the 50th percentile in the sediment size distribution. We further assume roughness height to be uniform throughout the catchment. Determination of slopes It is feasible to establish local slopes from pixel heights in a DEM. However, we found this produced errors due to random fluctuations in the digitizing process. The diffusion coefficient is particularly sensitive to the slope parameter - see Equation (5). In this work we apply a polynomial fitting to the aggregated elevation data. The degree of the polynomial is estimated as the minimum value providing a reasonable fit by eye. We consider this point again in the discussion. Determination of meta-channel parameters Appendix B outlines the algorithm used to determine each of the meta-channel parameters from the hydraulic geometries of the individual streams. As shown there, discharge, cross-sectional area and wetted perimeters are conserved by direct aggregation across the width function. Discharge velocity is established from aggregating a discharge weighted kinetic energy flux term. Maximum flow depth and bed height are established in analogous procedures. The friction factor is established from the aggregated energy loss term, together with the meta-channel parameters of discharge velocity and wetted perimeter. The top width is determined as the derivative of the power law function between cross-sectional area and manimum

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493

Figure 4. Network extracted from the Hutt catchment DEM at 46 875 m2 resolution

depth. Cross-sectional area and top width then provide mean depth. Celerity, c(Q), is determined by differentiating the power law function determined from the regression between discharge and cross-sectional area. D ( Q ) is determined from the slope curve obtained by polynomial fitting the meta-channel long profile and discharge and top width according to Equation (5). Finally, hydraulic radius, obtained from crosssectional area and wetted perimeter, is used with the friction factor to obtain Manning's n. RESULTS In this section, we provide an example of how the concept was used for a naturally occurring catchment in the North Island of New Zealand. The Hutt catchment occurs on the southern edge of the island and has a gauging station at the Kaitoke site. This site drains an area of approximately 90 km2 and shows considerable relief, with a height drop of over 1 km in a maximum flow distance of 17.5km. The water course is boulder-strewn in the upper reaches and gravel-bedded throughout the channels above the gauging station. According to Hicks and Mason (1991), the sediment size profile shows a 50 centile value of 90mm. At the Taita Gorge site on the same river, but draining an area of 555 km2, the 50th percentile value is 86 mm, illustrating uniformity in the ability of the system to maintain its sediment load throughout a major part of its length. Figure 4 shows the channel network extracted using a steepest gradient approach of determining flow paths (Band, 1993) from the DEM of the catchment. A threshold area of 0.31 km2 was chosen for the initiation of a channel. This network was used as a basis for constructing the width function (Figure 5a) and the area-distance function (Figure 5b) for the catchment and accumulated area functions for the total network and for each individual link in that network. From this network of area functions, meta-channel parameters were derived according to the algorithm outlined in Appendix B. Eight recharge values were used ranging uniformly between 1.8 and 7.2mm h-'. At a station and downstream hydraulic geometry exponents were taken from classical published work on hydraulic geometry. Regional downstream hydraulic geometry scaling coefficients and the coefficient and exponent of the discharge-catchment area power law, all corresponding to mean annual flood, were obtained from Mosley (1992). Roughness height was obtained from Hicks and Mason (1991). Parameters are shown in Table I.

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Figure 5 . (a) Hutt catchment width function; (b) area function

In Figures 6-9, we present each parameter determined for the meta-channel as a function of flow distance, but defer discussion of these results to the next section. Polynomial fitting of the meta-channel bed elevations was performed to the sixth order with a coefficient of determination of 0.995. The fit obtained is shown in Figure 10a. For interest, we compare meta-channel bed height with main stream bed height in Figure lob. Table I1 shows meta-channel parameters derived from the model compared with published values as provided by Hicks and Mason (1991) for Kaitoke gauging station.

DISCUSSION Figures 6-9 imply that although hydraulic geometries of individual channels making up a meta-channel follow the Leopold-Maddock power law formulation, the hydraulic geometry behaviour of the metachannel itself is not necessarily that described by those same power laws. Instead of smoothly increasing functions expressed by those laws, we find that all the derived hydraulic geometry relationships display a different form, which can be traced back to a distinct dependency on the network width function. This

Table I. Hydraulic geometry parameters for the Hutt catchment Parameter At a site velocity exponent At a site width exponent At a site depth exponent Downstream velocity exponent Downstream width exponent Downstream depth exponent Downstream velocity scaling coefficient Downstream width scaling coefficient Downstream depth scaling coefficient Regional discharge-area exponent Regional discharge-area scaling coefficienI t Roughness height

Value 0.34 0.33 0.33 0.1

0.5 0.4 0.61 7.09 0.23 0.8 2 90 mm


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Figure 6. (a) Discharges against flow distance for a set of recharges. (b) contributing area against Row distance. (c) cross-sectional area against flow distance for a set of recharges. (d) discharge velocity against flow distance for a set of recharges

dependency itself can be attributed to the non-linearity inherent in these laws being expressed through discontinuities in the relevant meta-channel parameters. The diffusion coefficient particularly shows a marked impact from discontinuities in the top width. Both the top width and wetted perimeter are particularly affected to the point where they exhibit an underlying envelope characteristic of the width function, n(s). To some extent this is to be anticipated in that, for the chosen downstream hydraulic geometry exponents, the top width exponent, bZ, was slightly greater than the depth exponent, fi. Both the celerity and the diffusion coefficients for this catchment show marked non-linearity throughout the flow distance and discharge range. This introduces reasonable doubt as to whether flood routing models involving constant parameters would perform adequately in such a catchment. The lack of requirement of the meta-channel concept to be tied down to basic assumptions of linearity allows this inherent nonlinearity to express itself freely through the c ( Q ) and D ( Q ) relationships. Friction factors and Manning's n show considerable instability at low discharge, but become smoother (a) (b)

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Figure 7. (a) Mean flow depth against flow distance for a set of recharges. (b) maximum flow depth against flow distance for a set of recharges. (c) top width against flow distance for a set of recharges. (d) wetted perimeter against flow distance for a set of recharges

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(a) Ep 600 8 m

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Figure 8. (a) Meta-channel bed elevation against flow distance for a set of recharges. (b) meta-channel bed slope against flow distance for a set of recharges. (c) diffusion coefficient against flow distance for a set of recharges. (d) celerity against flow distance for a set of recharges

(apart from junction effects) at higher discharges. This is a result of the relationship between roughness height and depth of flow and the relative values of these two parameters, especially for links draining small areas. The assumption of uniform roughness height is open to question, although as indicated from the 50th percentiles of the two different sites widely separated on the same river, there is some observational basis for this assumption. However, roughness height will obviously vary and Hack (1957) provides an empirical relationship of the bed slope with contributing area and roughness height from which this value may be estimated. Similarly, the resistance law used in the determination of friction factors as based on the work of Keulegan (1938) and Bray (1979) has underlying problems. The law used, Equation (13), explains only about 39% of the variation is the friction factor. In fact, better correlations were obtained by Bray (1979) between the friction factor and bed slope ( r 2 of 0.53). We prefer the relationship with the roughness height as being more consistent with the physical bases underlying the conservation equations.

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Figure 9. (a) Friction factor against flow distance for a set of recharges. (b) Hydraulic radius against flow distance for a set of recharges. (c) Manning's n against flow distance for a set of recharges. (d) velocity correction parameter against flow distance


497

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(a)

Hutt River Catchment Polynomial fitting of slope

-meta-channel bed elevetlon poiymnial fined bed elevation

E

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Hutt River Catchment main stream elevatia versus meta channel elevation

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Figure 10. (a) Polynomial fit of meta-channel long profile; (b) Meta-channel long profile compared with mainstream long profile

Stress at the solid boundary T,,,

= yASo

implies from Equation (12) that the friction factor is related to bed slope

f =-

U2

This is a strong indication, as mentioned previously, that the estimation of friction effects will be facilitated with more satisfactory techniques of determining local bed slopes. We have found the use of local slopes as calculated from DEMs not to be advisable due to the extreme variability from inaccuracies inherent in the digitization process. Rather, a local slope determined from a power law relationship with catchment area as derived by Tarboton et al. (1989) or a multiscaling slope exponent approach (Gupta et al. submitted) would be preferable. This is a natural extension to work performed in this study. The approximation of bed slope by a polynomial function shows a good fit in the lower reaches, but a poor fit in the highest reaches. This could possibly be improved by, for instance, piecewise exponential fitting, as indicated by Richards (1982). This would also provide a more realistic basis to this process. The polynomial fit produces a false value of zero slope at the outflow, which in turn produces a spuriously high value of the diffusion coefficient. For the Hutt catchment, the upper reaches are extremely Table 11. Cross-sectional characteristics at the Kaitoke gauging site Characteristic

Units

Discharge Discharge velocity Cross-sectional area Top width Mean depth Hydraulic radius Friction factor Manning's n

m3s-' m s-' m2

m m m

Calculated value

108 2.05 52.9

434 1.22 1.21 0.076 0.032

* Estimated from diagram and supplied cross-sectional area

Published value 104 2.08 50.5

37* 1.36* 1.45 N/A 0,047

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steep, falling well outside basic assumptions underlying the analysis leading to the generalized conservation equations as applied to the meta-channel. Results show very small diffusion coefficients (- 1m2 s-I) in the upper part of the meta-channel, indicating that kinematic wave solutions of the routing problem would be adequate within these reaches. The lower reaches show appreciable increases both in diffusion coefficients and celerities, which indicates that solutions of the convective-diffusion equation with constant coefficients are potentially open to doubt. The conversion parameter between flux-derived and storage-derived velocity, A, shows maximum deviation from a value of 1.0 of 1.5% and can safely be ignored in further work on this catchment. Its magnitude is dependent on the width function. The mainstream long profile and meta-channel long profile are very similar. Once again, the largest deviations appear at larger values of the width function. Apparent differences in meta-channel long profile between Figures 10a and 10b result from the different sampling lengths used in these two plots - every 1 m in Figure 10a and every lOOm in Figure lob. We concede that the geometries selected to illustrate this construction of a catchment meta-channel are simplistic in nature based, as they are, purely on Leopold and Maddock type power laws. No attempt has been made to include floodplains within these geometries. It would be expected that at very high discharges, discrepancies would occur by flow maintained in the channel that would naturally have entered the floodplain. Our assumption of Leopold and Maddock power laws is to a certain extent vindicated by the good agreement between the predicted and actual hydraulic geometries for the Kaitoke site (see Table 11). This is especially so, considering that the only point knowledge of the site required was the roughness height and the area of catchment draining to that point - all other parameters used are either classical published generalizations for hydraulic geometry exponents (Leopold and Maddock, 1953), or regional parameters for the whole of New Zealand. Even for roughness height, the estimation only needs to be order of magnitude rather than exact as the relationship with the friction factor is logarithmic in nature. Generalization of this geometry to a series of power laws using different scaling coefficients and exponents, switchable on recharge value, or indeed to situations where series of cross-sections have been established through field study are trivial extensions to techniques already described, being easily incorporated within the model. In this work several assumptions have been made. We examine the effects of the more important of these as follows:

-

Steady-state assumption - made to obtain meta-channel downstream hydraulic geometry. The assumption being made here is that a linear relationship exists between the discharge at a point and the catchment area converging to that point. There is empirical evidence that the main variability expressed in the discharge-catchment power law relationship is governed by rainfall processes (Gupta et a/., submitted). Ibbitt (1994) has found a linear relationship between discharge and contributing area under normal flow, steady-state conditions for the Ashley catchment in New Zealand. Our assumption implies that, ignoring rainfall variability, discharges of channel-forming significance are linearly related to contributing area. Backwater effects - neglect of these effects becomes apparent in discontinuities evident in metachannel hydraulic geometry parameters. Dynamic storage implicit in these back water controls is expected to induce smoothing in transitions across junctions. No sediment transport - this impacts to some extent on the hydraulic geometry that forms. Certainly, Richards (1982) indicates that both at a station and downstream geometries are affected by the sedimentology of stream banks. This effect can be seen in power law regression plots whose slopes are not constant with discharge. Our assumption of a linear relationship for the power law regression leads to inaccuracy in hydraulic geometries, especially at higher flows. Uniform roughness height - as roughness height is used only within a logarithmic relationship, it would not be expected to markedly affect the hydraulic geometry demonstrated by a metachannel.


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CONCLUSIONS Starting with generalized equations of continuity and mechanical energy balance as established for a meta-channel control volume (Snell and Sivapalan, submitted), a technique for the construction of such a meta-channel involving DEMs, regional parameterizations of hydraulic geometry parameters and a working knowledge of local roughness heights has been developed. The results of the prediction of the hydraulic geometry at a single channel outflow from a catchment channel network system are encouraging. This single channel representation of a catchment network provides a powerful tool for the subsequent routing of flows through that network. It allows any single channel routing method to be applied to the channel network as a whole. The meta-channel concept addresses scaling in two regards. Firstly, it demonstrates the technique of upward scaling in which spatially distributed information is lumped into a statistically distributed system. Secondly, it allows the routing of flows to be investigated through a range of catchment sizes. The advantages of representing the network in this way lies in the absence of linearity assumptions implicit in unit hydrograph approaches. As non-linearity is fundamentally handled by this technique, we have a more powerful tool available for examining the scaling behaviour of catchments. Furthermore, results to date, based on the Leopold and Maddock power law relationships for hydraulic geometries of individual channels, indicate that some of the work performed with an assumed linearity of response for larger catchments may be open to reasonable doubt. The technique has the potential for modelling at large catchment scales in that it provides a natural linkage between hydrological and general circulation models. A set of metachannels representing primary drainage basins can be used to route water between general circulation model grid cells and hence improve water balance ‘accounting’. Many questions still remain unresolved concerning this meta-channel concept - not the least of which are how well it deals with field-measurement hydraulic geometries compared with the power law distributions currently modelled, and how well does the routing of flow through a meta-channel compare against routing through distributed network systems. Both junction effects and meandering, important agents of energy dissipation in river systems, are neglected. These need to be addressed in future. We are currently assessing the meta-channel construct for routing of flow through a catchment using variable parameter Muskingum-Cunge techniques. ACKNOWLEDGEMENTS

Thanks are expressed to the officers of National Institute of Water and Atmospheric Research Ltd of New Zealand for providing a suitable analysis of DEM data for the Hutt River catchment. This research was supported in part by a special Environmental Fluid Dynamics grant awarded by the University of Western Australia to the second author. J. D. Snell was supported by a University of Western Australia Research Scholarship, a Centre for Water Research Scholarship and an Australian Research Council Grant (Small Grants Scheme, Grant Ref. 04/15/031/254). Centre for Water Research Reference No. ED 886 JS.

REFERENCES Band, L. E. 1986. ‘Topographic partition of watersheds with digital elevation models’, War. Resour. Res., 22, 15-24. Band, L. E. 1993. ‘Extraction of channel networks and topographic parameters from digital elevation data’ in Beven, K. and Kirkby, M. J. (Eds), Channel Network Hydrology, Wiley, Chichester. pp. 13-42. Beven, K., and Wood, E. F. 1993. ‘Flow routing and the hydrological response of channel networks’ in Beven, K. and Kirkby, M. J. (Eds), Channel Network Hydrology, Wiley, Chichester. pp. 99- 128. Bray, D. 1. 1979. ‘Estimating average velocity in gravel-bed rivers’, J. Hydr. Dh. ASCE, 105(HY9) 1103-1 122. Gupta, V. J., Mesa, 0.J., and Dawdy, D. R. ‘Multiscaling theory of flood peaks: regional quantile analysis’, War. Resour. Res., submitted. Hack, J. T. 1957. ‘Studies of longitudinal stream profiles in Virgina and Maryland, Prof. US Geol. Surv. Pap. 294B. Henderson, F. M. 1966. Open Channel, Flow. MacMillan, New York. Hicks, D. M., and Mason, P. D. 1991. Roughness Characteristics of New Zealand Rivers. Water Resources Survey. DSIR Marine and Freshwater, Wellington. Ibbitt, R. P. 1994. ‘Optimal channel networks: validation in a small catchment’, NIWA Misc. Rep. No. 197, 32 pp.

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Jenson, S. K., and Dominigue, J. 0. 1988. ‘Extracting topographic structure from digital elevation data for geographic information system analysis’, Photogr. Engin. Remore. Sensing, 54, 1593-1600. Keulegan, G. H. 1938. ‘Laws of turbulent flow in open channels’, J . Res. Nut. Bur. Standards, Res. Pap. RPf151,21,707-741. Leopold, L. B., and Maddock, T. 1953. ‘The hydraulic geometry of stream channels and some physiographic implications’, U.S. Geol. Surv. Prof. Pap. 252, 9-16. Mesa, 0 .J . , and Muffins, E. R. 1986. ‘On the relative role of hillslope and network geometry in hydrologic response’ in Gupta, V. K., Rodriguez-Iturbe, I., and Wood, E. F. (Eds), Scale Problems in Hydrology, D. Reidel, Dordrecht. pp. 1-17. Mosley, M. P. 1992. ‘River morphology’ in Waters of New Zealand. New Zealand Hydrological Society. pp. 285-304. Naden, P. S. 1992. ‘Spatial variability in flood estimation for large catchments: the exploitation of channel network structure’, Hydrol. S C ~37, . 53-71. O’Callaghan, J. F., and Mark, D. M. 1984. ‘The extraction of drainage networks from digital elevation data’, Comp. Vision Graphics Image Process. 28, 323-344. Richards, K. 1982. Rivers, Form and Process in Alluvial Channels. Methuen, London. Rigon, R., Rinaldo, A,, Rodriguez-Iturbe, I., Bras, R. L., and Ijjasz-Vasquex, E. 1993. ‘Optimal channel networks: a framework for the study of river basin morphology’, War. Resour. Res., 29, 1635-1646. Rinaldo, A. Marani, A,, and Rigon, R. 1991. ‘Geomorphological dispersion’, Wut. Resour. Res., 27, 513-525. Robinson, J. S., and Sivapalan, M. ‘Instantaneous response functions of overland flow and subsurface storm flow for catchment models’, Hydrol. Process., submitted. Sherman, L. K. 1932. ‘Streamflow from rainfall by the unit-graph method’, Eng. News Rec. 108, 501-505. Sivapalan, M., and Larsen, J. E. ‘A generalized non-linear diffusive wave equation’, Waf. Resour. Res., submitted. Snell, J . D., and Sivapalan, M. 1994. ‘On geomorphological dispersion in natural catchments and the geomorphological unit hydrograph’, W a f .Resour. Res., 30, 2311-2323. Snell, J. D., and Sivapalan, M. ‘The meta-channel - a basis for the modelling of rainfall-runoff at the drainage basin scale’, War. Resour. Res., submitted. Strelkoff, T. 1969. ‘One-dimensional equations of open-channel flow’, J . Hydr. Div. ASCE, 95(HY3), 861 -876. Tarboton, D. G., Bass, R. L., and Rodriguez-Iturbe, L. 1989. ‘Scaling and elevation in river networks’, War. Resour. Res., 25, 20372051. Troutman, B. M., and Karlinger, M. R. 1985. ‘Unit hydrograph approximations assuming linear flow through topologically random networks’, War. Resour. Res., 21, 743-754. Troutman, B. M., and Karlinger, M. R. 1986. ‘Averaging properties of channel networks using methods in stochastic branching theory’ in Gupta, V. K., Rodriguez-Iturbe. I., and Wood, E. F. (Eds), Scale Problems in Hydrology. D. Reidel, Dordrecht. pp. 185-216. Van Driest, E. R. 1946. ‘Steady turbulent-flow equations of continuity, momentum, and energy for finite systems’, J . App. Mech. ASME. 3. - ,A-231-A-238. Wang, C. T., Gupta, V. K., and Waymire, E. 1981. ‘A geomorphologic synthesis of non-linearity in surface runoff, Waf.Resour. Rex, 17, 545-554. 1

APPENDIX A: HYDRAULIC GEOMETRY

Reconciliation of at a station and downstream hydraulic geometries At a station hydraulic geometry expresses the temporal variation in geometrical parameters at a fixed point in space. For any site at a given flow distance from the outlet, this hydraulic geometry can be represented by Top width, B( t ) = a lQf;!( t )

Mean flow depth, j ( t ) = c1 Q,fi ( t )

(16)

Mean flow velocity, ii( t ) = k l (2,”’ ( t ) From Leopold and Maddock (1953) it can be seen that the scaling coefficients, a l lc1 k l , by implication, are functions of space only and not of time, whereas the scaling exponents, bl ,f1, m l , are independent o f both time and space. Similarly, downstream hydraulic geometry expresses the spatial variation for the same parameters at a fixed return period B(s) = a2Qh2(s)


50 1

In this instance, the scaling coefficients, a2,c 2 ,k2, by implication, are functions of time expressed through the return period only and not of space. Once again, the scaling exponents, b2,f2, m2, are independent of both time and space. For a fixed return period at a fixed site, then the two hydraulic geometries can be equated. Considering the top width as being typical of the hydraulic geometry parameters, then a, = a2Qb2-bI

(18)

where Q = Q A = QD Furthermore, the regional variation of discharge with contributing area for a constant return period, either mean annual flood or bankfull discharge, is Q ( s ) = +M(s)'

(19)

This equation is only meaningful in the sense of downstream hydraulic geometry. The coefficient $ is a function of return period, but not space. From Equations (18) and (19) = a2~(bz-bl)M(s)B(bz-bl)

(20)

Analogous equations are derived for both mean flow depth and mean flow velocity. As ul and M are both independent of the return period, then, by implication, so is the product term u2$b2-bl,even though individually a2 and +b2-b1 are functions of the return period. By estimating this product term for a given return period, such as the mean annual flow, this term is established for all sites and return periods. As typically b2 > b l , the implication is that the downstream scaling coefficient for top width, u2, should decrease monotonically with return period. For the mean flow velocity, then typically m2 < m l , implying that as $ increases, +"?-'"' decreases montonically with the return period. This in turn implies that the downstream velocity scaling coefficient, k2, will increase montonically with the return period. These conclusions are demonstrated by Leopold and Maddock (1953). We are interested in deriving the set of at a station hydraulic geometries throughout the flowpath domain, s. For reasons previously stated, we have made a steady-state approximation, such that Q A ( ~ )= M R ( t )

(21)

Consequently, by suitably rearranging Equations (16), (17), (19) and (21), we obtain hydraulic geometry relationships reconciled in time and space

where s refers to any spatial point on a network and t refers to any generalized return period. Note constants K B = a2+(t)(b2-bl)1 KY -- c2 +(IZ-h)l and K, = k2$(mz-ml). Determination of maximum depth offlow from mean depth offlow Hydraulic geometry considerations only supply mean flow depths. Stage-discharge relationships require maximum flow depths. For hydraulic geometries expressed in terms of Leopold and Maddock power laws, we determine maximum depth of flow, y , from mean flow depth, y, as follows

Cross-sectional area, A = By

(23)

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Eliminating, discharge, Q A , between the top width and mean flow depth expressions from Equation (16), we obtain

Letting (b =fi /bl, we obtain for cross-sectional area

Differentiating with respect to top width, this relationship now becomes

We can relate the derivative dA/dB to J , mean flow depth, through dB B = - dA = - . - dA dy dB dy dB dY

= J ( 4 + 1)Substituting and rearranging

Integrating flow depth over q, a dummy variable, between 0 and the maximum flow depth, y , and width over (, a dummy variable, between 0 and top width, B, we obtain (30)

Thus we form a simple relationship between mean and maximum flow depths based only on width and depth hydraulic geometry exponents. Determination of wetted perimeter The wetted perimeter is required for the determination of energy losses associated with stress induced by the bed of each stream. It is derived from a line integration of the top width across the depth domain. Let the domain of possible depths be represented by q and the domain possible top widths associated with those depths by €., An infinitesimal elemental increase in wetted perimeter, dP, is engendered by an incremental elemental increase in flow depth, dq. For a symmetrical cross-section

[

+ { d (:)}*]'

d P = (dq)2

(5)

We use d because of symmetry in the cross-section. Integrating this expression across total flow depth in the channel and multiplying by two, because of symmetry, we obtain P=2

[[I + - 1 (di)?]i 4 drl

d7


Using Equations (29) and (22) and recalling that

503

4 =fi / b l , this integration becomes (33)

APPENDIX B: ALGORITHM FOR THE DETERMINATION OF META-CHANNEL HYDRAULIC GEOMETRY, CELERITY AND DIFFUSION COEFFICIENT

L-I

A A(Y) Q Determination of hydraulic geometry 'dRi +

+

Q(A)

= 0.248

(see note 1)

+ 2.3610glO(4)

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J . SNELL A N D M. SIVAPALAN

y c -A

B

c+-

dQ dA

Dt-

Q 2BSO

1 Note: 1. Functional form is obtained by a power law fitting of the values obtained over the set of recharge values. LIST OF SYMBOLS Cross-sectional area Width scaling constant in hydraulic geometry power law functions Top width of the channel Width scaling exponent in hydraulic geometry power law functions Celerity in routing equations: depth scaling constant in hydraulic geometry Diffusion coefficient A representative depth scale Darcy-Weisbach friction factor; depth scaling exponent in hydraulic geometry power law functions Acceleration due to gravity Head Head introduced by lateral inflow The ith component in a set of channels Regional coefficient scaling top width to contributing area and recharge Regional coefficient scaling mean flow depth to contributing area and recharge Regional coefficient scaling mean flow velocity to contributing area and recharge Discharge velocity scaling constant in hydraulic geometry power law functions Contributing area Discharge velocity scaling exponent in hydraulic geometry power law functions A vector component normal to a surface The number of channels at flow distance s from the outflow of the network Manning's n Wetted perimeter Atmospheric pressure Volume flux = SA udA or discharge Lateral inflow Recharge coefficient Hydraulic radius = A / P Bed slope Flow direction Dimension of time Shear velocity ith component of the velocity vector ii Discharge velocity Discharge velocity from conservation of kinetic energy flux Discharge velocity from conservation of kinetic energy storage Velocity vector Local velocity at any point on the surface of the control volume Position vector Depth of water in the channel Mean depth of water in the channel Control volume Energy storage shape factor


Regional flood index scaling coefficient Length of control volume Effective roughness height of channel Height of an arbitrary point in the flow, normal to the bed Specific weight Conversion parameter between storage and flux determined velocities Density of the flow Shear stress at the solid boundary Regional flood index scaling exponent Lateral position of a point P across the width of a channel Height of the bed of the channel above a datum point Ratio of depth exponent to width exponent

505