Modelling bedload yield in braided gravel bed rivers

Geomorphology 36 Ž2000. 89–106 www.elsevier.nlrlocatergeomorph

Modelling bedload yield in braided gravel bed rivers A.P. Nicholas Department of Geography, Exeter UniÕersity, Rennes DriÕe, Exeter, EX4 4RJ, UK Received 19 February 2000; received in revised form 6 July 2000; accepted 9 July 2000

Abstract This paper outlines an approach for estimating the annual bedload yield of a braided channel. This procedure is based on the extension of theory of flow and sediment transport in braided rivers recently presented by Paola Ž1996.. The revised approach accounts explicitly for the relationship between increasing discharge and varying channel hydraulics, and is suitable for use in obtaining bedload transport rate estimates over a range of discharges. Integration of such estimates using flow-duration data allows annual bedload yield to be determined. Model parameterisation is achieved using topographic survey data for the Waimakariri River, New Zealand. Comparison of modelled bedload yield with values estimated from repeated topographic surveys indicates that the model is able to accurately predict both the medium-term Žc. 30 years. mean annual bedload yield of the Waimakariri at Crossbank Žthe section 17.8 km upstream of the river mouth., and also short-term fluctuations in bedload yield associated with varying annual flow statistics. Streamwise patterns of volumetric erosion and deposition determined for a 45-km length of the Waimakariri using the model are also in broad agreement with trends identified in topographic survey data for the period 1961–1997. However, significant deviations between modelled and surveyed volumes of cut and fill are evident at some locations. Comparison of model performance with conventional applications of bedload transport equations, which tend to underestimate transport rates for braided channels, suggests that the approach presented here may represent a significant improvement. This is the case because the model quantifies the relationship between braid intensity and spatial variability in flow hydraulics at a cross-section. Output from the model suggests that braided rivers may transport a significant proportion of their annual bedload at lower discharges than those indicated by earlier theoretical approaches. Results also provide quantitative support for the argument that increased intensity of braiding may promote higher rates of bedload transport in gravel bed rivers. q 2000 Elsevier Science B.V. All rights reserved. Keywords: braided river; bedload transport; numerical model

1. Introduction A sound understanding of the relationship between bedload transport capacity, channel pattern and river width is fundamental to the development of improved management strategies in braided rivers. In particular, engineers have sought to understand

whether conversion of a wide braided river to a single-thread channel will result in a change in bedload yield ŽCarson and Griffiths, 1987; Davies and Lee, 1988; Griffiths, 1989.. A review of previous work on this topic illustrates that much uncertainty remains. Conventional channel pattern continuum studies associate braided rivers with relatively steep

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A.P. Nicholasr Geomorphology 36 (2000) 89–106

slopes, abundant sediment supply, non-cohesive bank materials and higher rates of bedload transport than straight or meandering channels ŽSchumm and Khan, 1972; Schumm, 1985; Ferguson, 1987.. Furthermore, studies of dynamic river behaviour suggest that cycles of increasing and then decreasing sediment supply, linked to sediment wave propagation, are often accompanied by a transition from a single-thread river to a braided network during stream bed aggradation, followed by reversion to a single channel as supply declines and degradation ensues ŽChurch, 1983; Knighton, 1989.. However, whether the process of stream braiding results in an increase in bedload transport, thus enabling the increased load to be conveyed, or whether it is merely a consequence of excess sediment supply is an issue that has yet to be resolved. Recent laboratory flume experiments have shed much light on the dynamic behaviour of braided rivers, but they have not answered this fundamental question. For example, studies by Ashmore Ž1991., Hoey and Sutherland Ž1991., and Warburton and Davies Ž1994. have each identified the fluctuating nature of bedload transport by braided streams and demonstrated its relationship to morphological change at the scale of the individual bar-pool unit. Furthermore, they have shown that although these phenomena are associated with river disequilibrium over short spatial and temporal scales, when averaged over the duration of flume experiments, relationships between bedload yield and hydraulic variables may be described by conventional equilibrium sediment transport equations. Despite this progress the role played by channel pattern indices Že.g. channel width to depth ratio and braid intensity. in such relationships remains unclear. For example, Ashmore Ž1988. and Davies and Lee Ž1988. conclude that a reduction in braid intensity andror channel width is associated with increased bedload transport for equilibrium braided streams. However, Warburton and Davies Ž1994. identify the opposite trend in their data, with bedload transport rates being directly proportional to braid intensity. The difficulty of identifying a consistent relationship between transport rate and channel pattern in the results of these experiments may reflect the role of other variables, such as stream gradient and the degree of bed armouring, or the problem of quantifying channel morphology

using simple, descriptive parameters ŽWarburton, 1996.. Carson and Griffiths Ž1987. provide a detailed review of theoretical approaches to this problem and also identify conflicting results between different studies. They examine regime approaches based on channels with simple rectangular or parabolic section shapes, which suggest that transport rates either decrease ŽHenderson, 1966. or increase ŽBagnold, 1977; Parker, 1979. with channel width to depth ratio. Instead, they conclude that an optimum channel width exists at which the bedload transport rate is maximised, thus providing some support for attempts to maximise transport rates by restricting river width, at least in ‘over wide’ channels. However, they also recognise the limitations of representing braided channels using simple cross-section shapes, and conclude that braided streams may be more efficient transporters of bedload than single-thread channels, and that this possibility should be further investigated. Theoretical approaches based upon rectangular channel sections are of course, of limited utility in braided rivers where hydraulic and sedimentological parameters show marked spatial and temporal variations. In these environments bedload transport may be confined to narrow active bed zones ŽDavoren and Mosley, 1986. so that the use of section-averaged variables results in significant underestimation of actual bedload transport rates ŽCarson and Griffiths, 1987.. Furthermore, regions of maximum bedload transport may switch location as discharge rises, both within individual channel cross-sections ŽAshworth et al., 1992; Laronne and Duncan, 1992. and between cross-sections of differing hydraulic geometry ŽNordin and Beverage, 1965.. This phenomenon has been proposed as a possible mechanism driving the propagation of gravel waves in braided streams ŽGriffiths, 1993. and introduces additional complexity to the question of whether transport rates are positively or negatively correlated with stream braiding, since any such relationship may be stage-dependent. Distributed computer simulation models of flow and sediment transport in braided streams provide an improved understanding of the spatial complexity of braided river mechanics Že.g. Murray and Paola, 1994.. Furthermore, recent advances in two-dimen-

A.P. Nicholasr Geomorphology 36 (2000) 89–106

sional ŽLane and Richards, 1998. and three-dimensional ŽNicholas and Sambrook Smith, 1999. modelling of flow hydraulics in multi-channel systems may in future allow a physically based treatment of these processes. However, while such schemes may be appropriate for use in modelling process–form interactions at the scale of the bar-pool unit, they are both computationally expensive and demanding in terms of their data requirements. Consequently, onedimensional models may remain more effective tools for use in the derivation of bedload flux estimates at the reach-scale, provided that such models are modified to incorporate the effects of the spatial variability in flow and sediment transport processes that characterise braided streams Žcf. Paola, 1996.. The potential for using such an approach to quantify the relationship between bedload yield and stream morphology is investigated herein with reference to the Waimakariri River, New Zealand.

2. Theory Paola Ž1996. outlined a model of flow and bedload transport by braided channels that forms a starting point for the analysis presented below. In Paola’s model, water and bedload discharge are represented respectively using the fluid continuity equaŽ1948. sedition and the Meyer-Peter and Muller ¨ ment transport relation: Q s WH0 V Ž 1 q g . qs s

Qs W

s

Ž 1.

K Ž lt y tc .

3r2

'

g Ž rs y r . r

tc s u Ž rs y r . gD50 ,

Ž 2.

ment particle size and l is the ratio of grain shear stress to total shear stress. In Eq. Ž1., the term Ž1 q g . accounts for the fact that in braided rivers depth and depth-averaged velocity are spatially variable and are, typically, positively correlated ŽPaola assigns a value of 0.3 to the correlation coefficient g on the basis of the field data of Mosley Ž1982... Local boundary shear stress is related to flow velocity using a drag law of the form, Vs

t

(

,

Cf

Ž 4.

where C f is an empirical drag coefficient. Eqs. Ž1. – Ž4. represent standard relationships that may be used to estimate bedload transport rates for channels of simple geometry Žwith g s 0 in Eq. Ž1... Paola extends this approach to incorporate the effects of spatial variability in braided river hydraulics by modelling the resulting boundary shear stress distribution using a gamma probability density function Ž fg ., fg Ž t . s

a at)ay1 eyat) t0 G Ž a .

,

Ž 5.

where t 0 is the cross-section mean boundary shear stress,

t 0 s r gH0 S

Ž 6.

t) s trt 0 ,

Ž 7.

S is the longitudinal stream gradient, t) is a dimensionless boundary shear stress, a is a non-dimensional measure of the width of the stress distribution and G Ž a . is the standard gamma function. Eqs. Ž2. and Ž4. can now be written as:

Ž 3.

where, Q is the fluid discharge, W is the water surface width, H0 is the mean flow depth, V is the mean velocity, Qs and qs are respectively the section integrated and unit bedload transport rate, u is the dimensionless critical shear stress, K is an empirical Ž1948. assign coefficient ŽMeyer-Peter and Muller ¨ these two parameters values of 0.047 and 8, respectively., g is the acceleration due to gravity, rs and r are the sediment and fluid density, t and tc are the local boundary shear stress and critical shear stress for sediment entrainment, D50 is the median sedi-

91

Qs s

WK s

`

Vs

`

H g Ž r y r . 'r t

H0

(

Ž lt y tc .

3r2

fg Ž t . dt

Ž 8.

c

t Cf

fg Ž t . dt .

Ž 9.

These equations can be used to predict the total bedload transport rate and water surface width for a braided channel carrying a given discharge. However, to solve these equations the boundary shear stress distribution at a cross-section must be defined by specifying values for a and t 0 . Paola determines

A.P. Nicholasr Geomorphology 36 (2000) 89–106

92

a by fitting the data obtained by Mosley Ž1982. for the braided Ohau River to Eq. Ž5.. He then determines the mean boundary shear stress using the theory of Parker Ž1978. which describes the equilibrium width of straight gravel channels and can be used to relate the mean boundary shear stress to the critical boundary shear stress for entrainment Žand, hence, the bed sediment size. by: t 0 s Ž 1 q ´ . tc ,

Ž 10 .

where ´ takes a value of 0.2 according to arguments based upon lateral transport of longitudinal momentum. A number of modifications are made here to the model outlined above. First, the drag coefficient Ž C f . in Eq. Ž9., which Paola treated as a constant, is replaced by the following roughness relationship,

r

1

( ( s

f

8C f

s 2.21

V

ž( / gD50

0.34

.

Ž 11 .

Griffiths Ž1981. found that this expression accounted for 38% of variance in friction factor for a large data set of mobile gravel bed rivers in New Zealand. Second, the ratio of grain shear stress to total boundary shear stress Ž l., which Paola assumed to be unity, is determined as a function of section-average hydraulic variables. This can be achieved by using a Strickler type relation to estimate the grain roughness component, 1r6 n s CD50 ,

Ž 12 .

where n is a manning grain roughness coefficient and C is an empirical constant. Converting this to a Darcy–Weisbach type grain friction factor yields, f g s 8 gC

2

D50

ž / H0

1r3

,

Ž 13 .

which can be combined with the total roughness relation ŽEq. Ž11.. to give the following simple expression for l written in terms of the Froude number ŽFr.:

l s b Fr 2r3 ,

Ž 14 .

where b is an empirical constant. Carson and Griffiths Ž1987. estimate grain roughness using C s

0.048, which implies b s 0.88. Alternatively, b can be calculated where l and Fr are known. Prestegaard Ž1983. presents these data for six divided reaches of gravel bed rivers in the USA, that suggest b values in the range of 0.5–1, with a mean value of 0.65. Finally, an alternative approach must be used to define the boundary shear stress distribution at a cross-section. This is necessary because although the method employed by Paola yields promising results when compared with both field and flume data for laterally unconstrained channels at a single discharge ŽPaola, 1996., where discharge varies or where channel width is imposed the method has a number of limitations. Eq. Ž10. implies that mean boundary shear stress is controlled by bed sediment size and is independent of discharge. Furthermore, changes in discharge will be accommodated largely by changes in width, since mean depth will remain constant ŽEq. Ž6.. while increasing velocity occurs only due to the declining width of the stress distribution Ž a increases with discharge.. Eq. Ž10., therefore, seems best suited to modelling the mean boundary shear stress for a particular dominant discharge, which is in fact, the condition that it is intended to represent ŽParker, 1978.. However, determination of mean annual bedload yield involves integration of sediment transport rates over a range of discharges. Arguments against the use of Eq. Ž10. might also be raised on the basis that it applies to straight gravel channels rather than braided streams, and represents a stability condition under which lateral bank erosion will not occur. For boundary shear stresses in excess of this threshold value, lateral sediment transport leads to river widening and hence, a reduction in flow depth and shear stress. Carson and Griffiths Ž1989. have shown that bank erosion within individual anabranches and downstream transfer of material to sites of bar deposition represents a fundamental mechanism of bedload transport in braided rivers. Furthermore, Murray and Paola Ž1997. state that processes observed in a two-dimensional, distributed model of stream braiding indicate that flow convergence and divergence in braided channels may invalidate such threshold functions as a means of predicting anabranch width. The boundary shear stress distribution is defined here using an alternative approach that involves specifying the relationship between flow discharge,

A.P. Nicholasr Geomorphology 36 (2000) 89–106

channel geometry and the degree of variability exhibited by flow depths at a channel cross-section. This is accomplished using survey data quantifying channel topography at 36 cross-sections along the Waimakariri River, New Zealand. These data were collected by the Canterbury Regional Council over the period 1995–1997. The boundary shear stress distribution at a cross-section is assumed here to be a product of spatial variations in flow depth. The latter are quantified by fitting normalized flow depths Ž H ) s H r H0 . for a given water level and cross-section to a gamma distribution of the form: fg Ž H . s

a a H)ay1 eya H) H0 G Ž a .

.

Ž 15 .

This procedure yields values of the parameter a as a function of water level for each cross-section. Flow depth distributions are converted to boundary shear stress distributions using,

t s r gHS,

Ž 16 .

where t is the local boundary shear stress, H is the local flow depth and S is the section–mean longitudinal bed slope. This approach provides a means of incorporating the effects of spatial variability in boundary shear stress. However, it must be recognised that the latter will also be a function of variability in the local energy slope and roughness Žand hence, bed sediment size.. Although no data exist with which to quantify these variables in the current application, an assessment of the importance of local variability in energy slope can be made from the results of numerical simulations carried out by the author using the hydraulic model Žhydro2de. in a separate application. This is a finite volume code which solves the two-dimensional depth-averaged shallow-water equations Žcf. Connell et al., 1998. and is, therefore, capable of providing estimates of the local energy slope and shear stress. Preliminary results from simulations conducted for a short reach Ž100 = 50 m. of the braided Harper River, New Zealand, suggest that spatial variability in local energy slope is substantial Ženergy slope varies across three orders of magnitude.. Consequently, Eq. Ž15. does not provide reliable point estimates of local boundary shear stress where S is defined as the section-averaged longitudinal bed slope. Despite this,

93

Eqs. Ž15. and Ž16. do provide a reliable means of defining the boundary shear stress distribution at a cross-section. Shear stress distributions determined for the Harper River using local depths and the mean channel slope yielded a mean within 5% of that determined for the same data using local predictions of both depth and energy slope. In addition, the variance of the modelled shear stress distribution was only 10% lower when determined using local energy slopes compared to that calculated for the mean channel slope. This is the case because areas of high flow depth Že.g. scour pools. tend to be associated with low energy slopes, while areas of shallow flow Že.g. bar margins. tend to be associated with high energy slopes. Consequently, extremes of flow depth are compensated by local energy slope leading to reduced spatial variability in boundary shear stress. Ignoring the effects of spatial variability in bed roughness, application of the approach outlined above might be expected to lead to slight underestimation of a . Eqs. Ž1., Ž9., Ž11., Ž15. and Ž16. can be used to calculate river discharge for any given water level. The section-integrated bedload transport rate can then be estimated from Eqs. Ž3., Ž8. and Ž14.. In the following section this approach is applied using available data for the Waimakariri River. The aims of this are threefold. First, to assess the ability of the model to derive estimates of bedload yield for the Waimakariri; second, to compare the performance of the model to conventional sediment transport relations that ignore spatial variability in flow and sediment transport rates in braided rivers; and third, to investigate the relationship between bedload yield and channel morphology.

3. Bedload yield estimates for the Waimakariri river at crossbank The braided rivers of the Canterbury Plains, New Zealand Žsee Fig. 1Ža.. have been modified by extensive engineering works over the past 50 years. On the Waimakariri and Ashburton Rivers this has involved the construction of stopbanks and groynes designed to reduce flood risk associated with channel bed aggradation ŽReid and Dick, 1960; Griffiths, 1979; Reid and Poynter, 1982.. These structures

94

A.P. Nicholasr Geomorphology 36 (2000) 89–106

Fig. 1. Ža. The main braided rivers of the Canterbury Plains, South Island, New Zealand. Žb. The Waimakariri at Crossbank Ž17.8 km upstream of the river mouth.. Flow is from left to right and the braidplain width is approximately 1 km.

A.P. Nicholasr Geomorphology 36 (2000) 89–106

serve to contain large floods and seek to prevent aggradation by restricting channel width and maximising sediment transport rates ŽNevins, 1969.. However, this approach has been only moderately successful and in-channel aggradation remains a problem ŽDavies and Lee, 1988. that must be managed by continuous gravel extraction from affected reaches ŽGriffiths, 1991.. The approach outlined above was applied initially to calculate the mean annual bedload yield of the Waimakariri at Crossbank Žthis cross-section is lo-

95

cated 17.8 km upstream of the mouth of the Waimakariri, Fig. 1Žb... This location was selected because bedload yield estimates based upon repeated cross-section surveys are available at this site. Furthermore, a number of previous attempts have been made to apply conventional transport relationships at this location Žcf. Carson and Griffiths, 1987, 1989.. Fig. 2Ža. shows the modelled relationships between total water surface width Ži.e. excluding exposed bars., mean flow depth and discharge at Crossbank. Fig. 2Žb. shows the cross-section topog-

Fig. 2. Ža. Modelled relationships between discharge, total water surface width and mean flow depth at Crossbank. Žb. Modelled relationship between discharge and the gamma distribution shape parameter Ž a . at Crossbank. Inset shows cross-section topography at this location.

96

A.P. Nicholasr Geomorphology 36 (2000) 89–106

raphy at this location and the relationship between discharge and the depth distribution shape parameter Ž a .. Depth data fitted to gamma distributions for four discharges ranging from the mean annual flow Ž120 m3 sy1 . up to 3000 m3 sy1 are shown in Fig. 3. These data indicate that spatial variations in flow depth are reduced as discharge rises and bed topography is drowned out Ži.e. a increases with discharge.. Although Paola employed a constant value of a , he identified a similar trend with increasing discharge in the data of Mosley Ž1982.. Comparison of a values determined here Žfor all 37 sections. with those obtained by Paola for the Ohau River indicates that the former are somewhat higher. For example, Paola found a s 1.14 for a discharge close to the mean annual flood whereas, values determined for the Waimakariri at an equivalent discharge typically lie within the range 1 - a 4. This contrast may be attributed to two factors.

First, differences in the approaches used to calculate values of a ŽPaola employed the hydraulic data of Mosley Ž1982. which incorporate the effects of spatial variability in bed roughness.; second, differences in channel morphology Že.g. intensity of braiding. and grain size characteristics between the two rivers. Seal and Paola Ž1995. argue that a should scale with the variability in bed sediment grain size. The ratio D 85 rD50 is approximately 3 for much of the Waimakariri ŽGriffiths, 1979; Carson and Griffiths, 1987. while the equivalent parameter for the Ohau River exceeds 4 ŽMosley, 1982.. This may also account for higher values of a in the current study. Annual bedload yield estimates were derived for the river at Crossbank by applying the procedure outlined above using flow duration data available for the Waimakariri over the period 1967–1997. These data were obtained downstream of Crossbank. This is acceptable as no significant tributaries enter the

Fig. 3. Gamma probability density functions fitted to dimensionless depth distributions at four discharges for the Waimakariri at Crossbank Ž H is the local flow depth and H0 is the section-averaged flow depth..

A.P. Nicholasr Geomorphology 36 (2000) 89–106

Waimakariri over the course of its lower 50 km and loss of discharge into river bed gravels is minimal ŽLockington, personal communication.. Flow and transport rate calculations were carried out at 5 m3 sy1 intervals up to the maximum annual discharge. Fig. 4Ža. shows temporal variations in the mean annual flow Ž Q0 . and peak annual flow Ž Q1 . over the period of record. Fig. 4Žb. shows estimated annual bedload yield derived with b s 0.65 " 0.05. Transport rate estimates indicate a weak tendency towards an increase over time, with substantial fluctuations in bedload yield corresponding to variations in flow Žmean annual flow in particular.. Volumetric

97

changes in sediment storage based on repeated cross-section surveys suggest a mean annual bedload yield at Crossbank of 260,000 " 10,000 m3 over the past few decades ŽGriffiths, 1991.. This compares favourably with mean estimated transport rates for the period 1967–1997 between 223,000 Ž b s 0.6. and 435,000 Ž b s 0.7. m3 yeary1. Data quantifying temporal fluctuations in transport rates at Crossbank are sparse. However, Carson and Griffiths Ž1989. present bedload yield estimates based on survey data of 187,000 m3 yeary1 for the period 1967–1973, and 277,000 m3 yeary1 for the period 1973–1983. This represents a more substantial increase in bed-

Fig. 4. Ža. Temporal variations in mean and peak annual flow monitored at the downstream end of the Waimakariri for the period 1967–1997. Žb. Modelled mean annual bedload transport at Crossbank for the period 1967–1997. The solid line shows estimates for b s 0.65. Dotted lines show higher and lower estimated transport rates derived using values of b s 0.7 and b s 0.6, respectively.

98

A.P. Nicholasr Geomorphology 36 (2000) 89–106

load yield than that estimated using the model Že.g. for b s 0.6, the average annual yield is 195,000 m3 for 1967–1973 compared to 225,000 m3 for 1973– 1983.. Carson and Griffiths Ž1989. also present an estimate of transport rates for the period 1986–1987 based on morphological changes over a short Ž- 1 km. reach at Crossbank. This estimate of 154,000 m3 yeary1 also compares favourably with the flux derived using the procedure outlined above Že.g. for b s 0.6, the estimated transport rate for this period is 162,000 m3 yeary1 .. These data indicate that the sediment transport model generates reasonable estimates of bedload yield, both in terms of the average annual yield over the last three decades and shortterm Žannual. temporal changes in transport rates that appear to be associated with fluctuations in annual flow conditions. Comparison of the results of this model with those of previous approaches to estimating bedload yield on the Waimakariri highlights the advantages of the procedure outlined here over conventional methods. Carson and Griffiths Ž1987. reported predictions of mean annual bedload yield on the Waimakariri at Crossbank of only 18,000 m3 derived using the approach of Bagnold Ž1980.. Specification of bed sediment size Žand hence entrainment thresholds. is one source of error in these calculations. Additionally, failure to account for the role of channel morphology in controlling spatial variability in hydraulic variables will lead to substantial errors in bedload yield estimates. Carson and Griffiths Ž1987. noted the difficulty of applying bedload transport laws where the relationship between discharge and river width is unknown and where flow and sediment parameters are spatially variable. Transport rate estimates based on rectangular channels of fixed width involve over-estimation of the critical discharge for entrainment and hence, considerable underestimation of bedload transport rates. Carson and Griffiths Ž1989. addressed this problem when estimating gravel transport at Crossbank by assuming that sediment transport is confined to the main anabranch channels, which were estimated to convey 75% of the flow and were assigned a fixed width of 100 m. This approach generates reasonable estimates of bedload yield when compared with values derived from topographic surveys and aerial photographs. However, as they observe, specification of width and discharge values for

the main anabranches is somewhat arbitrary and the method is therefore difficult to apply elsewhere in a systematic and consistent manner. Fig. 5Ža. shows the relationship between estimated transport rates and discharge calculated using three approaches; first, using the model outlined above Žreferred to hereafter as the non-uniform flow model.; second, using this model, but for the case where a s ` Ži.e. depth and boundary shear stress across a section are considered uniform, but channel width is a function of discharge., referred to as the uniform flow model; and third, using the fixed width method proposed by Carson and Griffiths Ž1989.. Fig. 5Žb. shows the proportion of the total bedload yield transported by flows below a given discharge, again estimated using these three approaches. Estimates derived using the uniform flow model indicate no bedload transport below a discharge of 1600 m3 sy1 , and substantially lower transport rates than those determined using the other methods for the majority of flows. Consequently, total bedload yield predicted using this method is two orders of magnitude lower than actual transport rates estimated from survey data by Griffiths Ž1991.. This occurs despite the use of realistic channel width values and results from the under-prediction of peak boundary shear stress due to the assumption of uniform flow depth. Estimated annual bedload yields derived using the other two methods are similar Že.g. 318,000 and 286,000 m3 yeary1 , respectively, for the non-uniform flow model and the fixed width method.. The most significant difference between these two methods is the relative contribution of low and high flows to the total bedload yield. Results derived using the fixed width method indicate zero sediment transport below a threshold discharge of 200 m3 sy1 and substantially higher rates above 300 m3 sy1 than those determined using the non-uniform flow model. Indeed, 50% of the total bedload yield estimated using the non-uniform flow model occurs for discharges below 200 m3 sy1 . The equivalent discharge for the fixed width method is approximately 600 m3 sy1 . The smaller relative contribution to the total annual yield of low flows reflects two features of the fixed width method. First, since width remains constant mean depth Žand boundary shear stress. will be overestimated for high discharges and underestimated for low discharges. Second, this approach ignores the

A.P. Nicholasr Geomorphology 36 (2000) 89–106

99

Fig. 5. Ža. Relationship between estimated bedload transport rate and discharge determined using three different approaches. Žb. Relationship between discharge and cumulative proportion of the total bedload transported at or below that discharge, determined using three different approaches.

importance of spatial variations in flow depth. Inspection of frequency distributions shown in Fig. 3 and mean depth data in Fig. 2Ža. indicates that mean depth increases more rapidly with discharge than maximum depth. Consequently, even at low discharges, a small proportion of the total channel width is characterised by deep water and relatively high boundary shear stresses Žwhere a simple relationship between depth and shear stress is assumed.. The non-uniform flow model provides a more realistic treatment of the boundary shear stress distribution and suggests that substantial bedload transport may occur at lower discharges than predicted previously. For example, Carson and Griffiths Ž1987. report critical discharges for entrainment on the Waimakariri in the range of 600–1100 m3 sy1 . These were determined using a range of bedload transport laws where the river was treated as a simple, rectangular channel. Although the results of

the non-uniform flow model indicate that substantial bedload transport may occur at lower discharges, this approach neglects a number of other controls on sediment transport that introduce uncertainty into these results. In particular, a single value of the critical shear stress is employed here. In reality, particle size and bed structure Žand hence, entrainment thresholds. will vary across the braidplain. Furthermore, although Eqs. Ž15. and Ž16. provide a reliable means of approximating the boundary shear stress distribution at a cross-section, Eq. Ž15. will not provide accurate predictions of local boundary shear stress unless data exist to quantify the local energy slope. Systematic errors in bedload transport estimates will consequently result where entrainment thresholds and local shear stress are correlated spatially. Lack of data means that these factors cannot be evaluated here. Despite this limitation, the nonuniform flow model employed here appears to offer

100

A.P. Nicholasr Geomorphology 36 (2000) 89–106

a number of advantages over the fixed width method presented by Carson and Griffiths Ž1989.. It provides a more realistic treatment of spatial variability in boundary shear stress and changes in channel width with increasing discharge. In addition, it is relatively simple to constrain using topographic cross-section data, avoids the use of arbitrarily determined parameters and is, consequently, simple to apply where such data are available.

4. Downstream variations in bedload transport Griffiths Ž1979. and Carson and Griffiths Ž1989. presented data quantifying volumetric rates of cut and fill along the Waimakariri indicating considerable non-uniformity of bedload transport rates. However, Carson and Griffiths Ž1989. noted that streamwise variations in channel characteristics Že.g. slope, braid intensity, width and particle size., which are believed to control sediment transport, appear to be relatively insignificant. They also stated that the data required to estimate bedload yield for the Waimakariri upstream of Crossbank, using their approach based upon an arbitrary fixed channel width, were not available. In contrast, the model presented here can be implemented using topographic survey data obtained between 1995 and 1997 for 36 channel cross-sections located between 10 and 56 km upstream of the river mouth. Grain size statistics for this length of the Waimakariri were estimated using data presented by Griffiths Ž1979. and Carson and Griffiths Ž1987., by assuming that D50 s 3D 84 . Fig. 6 shows channel topography at six representative cross-sections. Fig. 7Ža. – Žc. illustrate systematic changes in channel characteristics along the Waimakariri. Data in Fig. 7Ža. – Žc. are presented for nine reaches between 10.1 and 55.2 km upstream of the river mouth Žaverage reach length is 5 km.. These data represent average values of morphological variables determined for the four surveyed cross-sections within each of the nine reaches. Averaging was carried out in this way because individual sections are unlikely to be representative of channel hydraulic geometry for the reach as a whole. A number of clear trends in channel morphology are evident in these figures. Median particle size and

reach averaged bed slopes generally decline in the downstream direction. In contrast, other morphological variables exhibit more complex patterns. Maximum values of braidplain width and high braid intensity ŽFigs. 2Žb. and 7Žb.. occur around Crossbank Ž17.8 km.. Braidplain width declines downstream of this point, where the channel is constrained by stopbanks ŽFig. 6Ža. and Žb... Further upstream Ž32.2 km. incision has occurred, also promoting a reduction in channel width ŽFig. 6Žd... Upstream of this braidplain width and braid intensity increase rapidly and remain high until the point where the Waimakariri leaves its lower gorge Ž58 km from the river mouth.. Longitudinal variations in the gamma function shape parameter Ž a . are shown in Fig. 7Žc. for three discharges. Generally, a decreases in an upstream direction indicating greater spatial variability in bed topography Žassociated with more intense braiding in upstream reaches., although for reaches above 30 km spatial variations in a are small. It is noticeable that longitudinal trends in both a and flow width ŽFig. 7Žb.. are more marked at peak discharges than for the mean annual flow of 120 m3 sy1 . This suggests that the degree and pattern of non-uniformity exhibited by bedload transport is strongly stage-dependent, as hypothesized by Griffiths Ž1993.. Fig. 8Ža. shows the measured volumetric change in sediment storage along the Waimakariri between 1961 and 1997, based on repeated cross-section surveys by the Canterbury Regional Council Žand previously the North Canterbury Catchment Board.. These data could be used to estimate the minimum rate of bedload transport within each reach Že.g. Martin and Church, 1995.. However, transport rates estimated in this way are prone to errors from a number of sources Žcf. Lane, 1997.. Instead, the surveyed patterns of erosion and deposition were compared with estimated volumes of cut and fill derived by calculating the sediment transport rate within each of the nine channel reaches using the non-uniform flow model. The balance between the volume of sediment entering each reach and the calculated volumetric bedload flux within that reach indicates the volume of erosion or deposition within the reach. Transport rates were calculated using the morphological information presented in Fig. 7Ža. – Žc., a value of b s 0.65, and flow-duration data for this time period

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Fig. 6. Surveyed cross-sectional geometry at six representative locations along the Waimakariri. Distances are upstream of the river mouth.

made available by the Canterbury Regional Council. Estimates of volumetric changes in sediment storage determined using survey data may include some error, however, consistent patterns of change, both spatially across several individual cross-sections and temporally throughout a number of survey periods,

suggest that such errors do not affect the gross longitudinal patterns of erosion and deposition. Fig. 8Žb. shows the mean annual bedload yield for each reach estimated using the model. A clear maximum occurs in these estimates between 25.8 and 30.2 km upstream of the river mouth. The maximum

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Fig. 7. Longitudinal variations in channel morphology along the Waimakariri; Ža. channel slope and median bed sediment size; Žb. water surface width at three discharges; and Žc. the gamma distribution shape parameter Ž a . determined from surveyed cross-section topography for three discharges.

estimated transport rate of approximately 500,000 m3 yeary1 represents about 1.6 times the value calculated for Crossbank. Fig. 8Žc. shows a comparison of the annual volumetric rates of erosion and deposition derived from the survey data and those determined using the estimated bedload transport rates. A degree

of agreement is evident in these results, in that both modelled and survey-based estimates of channel change indicate aggradation in the reach downstream of 25.8 km and degradation upstream of this point. The transition from erosion to deposition marks the location of maximum bedload yield and highlights

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Fig. 8. Ža. Patterns of total volumetric erosion and deposition along the Waimakariri for the period 1961–1997, determined from repeat cross-section surveys. Žb. Modelled mean annual bedload yields for the period 1961–1997. Žc. Modelled and surveyed annual volumetric erosion and deposition for the period 1961–1997.

the need for engineering work and gravel extraction in the lower reaches of the river to reduce the risk of flooding. Despite broad agreement between the patterns of modelled and surveyed cut and fill, substantial discrepancies exist between the two within most reaches. Indeed, in five out of eight reaches the

difference between the two values is at least 50% of the higher figure. In spite of these deviations between modelled and surveyed volumetric channel changes, the results shown here are a considerable advance on those presented in previous studies. They suggest that the

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procedure employed to determine sediment fluxes is capable of predicting the overall streamwise trends in bedload transport resulting from complex, and visually subtle Žwhen viewed from aerial photographs. variations in river morphology. Identification of simple relationships between calculated bedload yields Žin Fig. 8Žb.. and morphological variables Žin Fig. 7Ža. – Žc.. is not possible. Longitudinal variations in bedload yield must be controlled by three main factors: the ratio of channel slope to bed sediment size, braidplain width, and the degree of variability exhibited by bed topography across a section Ža measure of the intensity of braiding.. A reduction in river width will, other things being equal, lead to an increase in bedload transport within a reach. However, increased braid intensity will also achieve this by enhancing topographic variability across the braidplain Žreducing a ., leading to an increased frequency of locations characterised by deep water Žand high boundary shear stresses., even at relatively low discharges. Paola Ž1996. associated these regions of high shear stress with scour pools at channel confluences. They might also be linked to zones of high shear on the outer banks of individual meandering anabranch channels. It has been suggest that bank erosion on the outside of bends is an important control on bedload transport in wide gravel bed rivers ŽCarson and Griffiths, 1987; Laronne and Duncan, 1992., and that increased braiding leads to a greater number of such sites and, hence, enhanced sediment transport. The results presented here are consistent with this view and provide a quantitative basis for suggesting that conversion of braided rivers to single-thread channels of lower width may not necessarily lead to an increase in bedload transport capacity if the width of the flow depth Žor shear stress. frequency distribution is also reduced substantially.

5. Summary and conclusions A procedure for estimating mean annual bedload yield in braided rivers has been shown to provide reasonable estimates of annual bedload yield for the Waimakariri River, New Zealand. The approach is based on an extension of the theory of Paola Ž1996., which can be used to predict channel width and

bedload transport for a braided river characterised by a particular dominant discharge. The revised model outlined here includes an explicit treatment of the relationships between discharge, total water surface width, and the degree of spatial variability exhibited by flow depth at a cross-section. Model parameterisation is carried out using topographic survey data available for 36 cross-sections along the Waimakariri. This revised model generates improved estimates of bedload yield when compared with conventional approaches, which are known to overestimate the critical discharge for gravel entrainment and underestimate bedload transport rates ŽCarson and Griffiths, 1987.. In addition, the approach developed here provides a more realistic representation of the complexity exhibited by flow depth and boundary shear stress within braided streams. In doing so, it avoids the need to make arbitrary judgements about channel dimensions and conveyance capacities when attempting to overcome the limitations of conventional sediment transport approaches ŽCarson and Griffiths, 1989.. Spatial variability in flow and bed topography within braided rivers exerts a significant influence on bedload discharge. This variability is generally highest at low flows and is reduced as discharge rises and bed topography is drowned. As a consequence, braided rivers may be capable of transporting considerable quantities of bedload even at relatively low discharges. This is the case because such flows occur frequently and because, although mean flow depths are shallow at low discharges, narrow zones of deep water still occur. The results presented here also illustrate that the relationship between bedload transport and channel pattern is influenced both by overall braidplain width and braid intensity. Conversion of wide, braided rivers to narrow single-thread channels is, therefore, not guaranteed to result in an increase in bedload transport capacity. Furthermore, in the case of the Waimakariri, it is clear that the relationship between braidplain width and the parameter a is not a simple one Žsee Fig. 7Žb. and Žc... Consequently, it is difficult to predict the effect of imposed changes in width on either a or bedload yield. Ultimately, aggradation in downstream reaches of the Waimakariri is a product of the declining river gradient and is a problem that is unlikely to be solved by a simple change in channel width.

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Acknowledgements This work was funded by the Nuffield Foundation. Jonathan Laronne and Colin Thorne are thanked for their reviews which led to some valuable modifications to the manuscript. Cross-section survey data and flow duration information for the Waimakariri River were supplied by the Canterbury Regional Council. The assistance of Trish Lockington in providing these data is gratefully acknowledged. Fig. 1Žb. was provided by Air Logistics.

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