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Adjustment of stream-channel shape to hydrologic regime
Journal of Hydrology, 30 (1976) 365--373
365
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
ADJUSTMENT OF STREAM-CHANNEL SHAPE TO HYDROLOGIC REGIME
G. PICKUP
Department of Geography, University of Papua New Guinea, Port Moresby (Papua New Guinea) (Received December 11, 1975; accepted for publication December 29, 1975)
ABSTRACT
Pickup, G., 1976. Adjustment of stream-channel shape to hydrologic regime. J. Hydrol., 30: 365--373. Bed-load channels tend to adjust their cross-sections so that given slope, roughness and sediment load, channel shape approaches the optimum for bed-load transport. The extent to which any one shape is the optimum varies with discharge, so four Cumberland Basin stream channels have been investigated to determine the discharges at which their present cross-sections represent the optimum for bed-load transport. These discharges have return periods ranging from 1.1 to 1.5 years on the annual series. The return periods closely correspond with return periods for the discharge at which, over a period of time, the most bed-load is transported. These return periods vary from 1.15 to 1.45 years when the same bed-load equation is used. The close correspondence between sets of return periods suggests that bed-load channels tend to adjust their cross-sections to become the optimum shape for bed-load transport at or close to the discharge at which the most bed-load transport is accomplished. NOTATION
List o f symbols A
Ai B D
Di G Q Qbf Qe
channel area c o n s t a n t in t h e M e y e r - P e t e r a n d Mtiller bed-load equation c o n s t a n t in t h e M e y e r - P e t e r a n d Mtiller bed-load equation mean depth of flow stage a b o v e t h e b e d bed-load discharge discharge bankfull discharge the discharge at which, over a period of t i m e , t h e m o s t b e d - l o a d is t r a n s p o r t e d
ms
m m tonnes/day ma/sec ma/sec ma/sec
366
NOTATION (continued)
Qopt
Ql.ss Qs/Q
R S Wb Wi
c dm g g~ kb
km kr x ~w 7s
the discharge at which a particular channel shape is the o p t i m u m for bed-load transport given slope, roughness and the characteristics of the sediment load the 1.58-year flood on the annual series bank friction correction factor in the Meyer-Peter and Mi]ller equation hydraulic radius slope of the energy grade line bed-width channel width at stage D i a coefficient mean size of the bed-material the gravity constant bed-load discharge measured under water the Strickler bed-roughness coefficient the Strickler mean roughness coefficient the Strickler particle-roughness coefficient the width--depth exponent specific weight of water specific weight of sediment shear stress
m3/sec m3/sec
m m m
m m/sec 2 tonnes/sec per m of width
INTRODUCTION
Within the constraints imposed by perimeter sediment, the size and shape of a river channel is determined b y the whole sequence of sediment-bearing flows which it experiences. Each flow produces its own particular channel response which depends on the magnitude of the event and the initial state of the channel. The condition of the channel at any time is therefore the product of all previously occurring events. Over a period of time, channel response, like streamflow, corresponds with a particular magnitude--frequency distribution. The mean of this distribution represents the average condition and the flow "associated" with it is the socalled dominant discharge. The linkage b e t w e e n the channel condition and the dominant discharge has usually been assumed rather than demonstrated. Furthermore, it does not appear to have been explained in the light of classical principles of river behaviour. This paper is a preliminary a t t e m p t to remedy these deficiencies b y presenting a case study of adjustment to hydrologic regime in four bedload channels of the Cumberland Basin, New South Wales.
367
T H E N A T U R E OF C H A N N E L A D J U S T M E N T T h e channels described in this paper have highly cohesive bank material and bed-material consisting mainly of fine to m e d i u m gravel. Bed forms such as dunes were not observed and values of the roughness coefficient did not vary greatly with discharge. The following statements on the nature of channel adjustment apply specificallyto these conditions and not necessarily to others. Subject to certain constraints, rivers have a tendency to adjust their channel cross-sections towards equilibrium in at least two ways. Firstly,as R u b e y (1952) states: "The stream makes for itselfthe channel of maximum hydraulicradiusor maximum efficiencyof flow that itisable to maintain under itsconditionsof characterof bed, amount of load and other controllingvariables." Secondly, the cross-section becomes adapted to the sediment load carried by the stream or as Griffith (1927) puts it: "A riverfullycharged with siltmust.., tend to adopt that form of sectionwhich will give ita maximum silt-carryingcapacity." Whether this state actually develops depends on the characteristicsof the bank sediment for as Mackin (1948) states: "For any one set of slope--debris--chargefactorstherewillbe one criticaldegree of erodibilityof the bank-forming materialssuch that an originallytoo narrow channel will quickly develop a cross-sectionwith depth--widthrelationsthat provide "the maximum silt-carryingcapacity".If the bank-forming materialsare lesserodiblethan thiscritical degree the stream may tend to develop the maximum efficiencysectionover a period of time; that is,the finalstablechannel section,however long delayed may approximate the idealform for the givenslope--debris--chargefactors.But ifthe bank-forming materials are more erodiblethan the criticaldegree the stream willadopt and maintain a sectionthat iswider and shallowerthan the idealtransportationsection." In a bed-load channel, the conditions of " m a x i m u m efficiency of flow" and " m a x i m u m sflt~carryingcapacity" m a y be mutually exclusive. T h e channel with the m a x i m u m hydraulic radius has a semi-circular cross-section, but as Lane (1937), Sundborg (1956) and S c h u m m (1963) have noted, bed-load channels tend to have cross-sections which are wide and shallow. T h e two optima therefore do not seem to coincide at the same point. Where a substantial part of the sediment load is carried as bed-load, the rate of sediment transport mainly depends on the flow conditions and the width of the channel bed. These factors operate in the following manner. T h e rate of bed-load transport per unit-width is functionally related to the flow velocity or to the tractive force, r, where: r =Tw'D'S
368 The width of bed determines the number of unit-widths over which that rate of bed-load transport occurs. All other things being equal, the expression for total bed-load transport in a rectangular channel in which bed-width, Wb, equals top-width may therefore be written:
V = f(Wb,~,w,D,S)
(2)
If discharge, slope and resistance to flow remain constant, channel width and flow depth are inversely related. This may be illustrated using the Strickler equation:
Q = AkmR2/3Sln
(3)
In a wide rectangular channel, depth of flow and hydraulic radius are approximately the same and A = Wb" D. Rearranging eq. 3 therefore gives the following expression for flow depth:
-WbkmSll2
(4)
Since the channel cross-section variables which affect bed-load transport are inversely related, it is likely that increasing bed-width will result in an increase in bed-load transport until a m a x i m u m is reached. Any subsequent increase in bed-width will result in a reduction in bed-load transport. This t e n d e n c y has been described and experimentally verified by Gilbert (1914) who defines the d e p t h / w i d t h ratio at the point where bed-load transport is maximised as the o p t i m u m form ratio. Field observations indicate that the t e n d e n c y for a channel to develop a cross-section corresponding with the optimum-form ratio overrides the tendency for it to develop a cross-section with the m a x i m u m hydraulic radius. This is apparent from the fact t h a t as the bed-material becomes coarser, and the proportion of total load carried as bed-load increases, channel cross-sections become wider and flatter (Sundborg, 1956; Schumm, 1963). It is worth noting, however, t h a t the o p t i m u m form ratio corresponds with the maxim u m hydraulic radius the channel is able to attain under the circumstances. Any increase in bed-width b e y o n d the o p t i m u m form ratio involves a decrease in hydraulic radius. RELATIONSHIP OF THE OPTIMUM BED-LOAD TRANSPORTING FORM AND HYDROLOGIC REGIME Given slope, roughness and the size of load, the shape of the o p t i m u m bedload transporting channel cross-section depends on discharge. The o p t i m u m shape at one discharge is therefore not the same as the o p t i m u m at another. In some channels, the shape of the cross-section changes with discharge, presumably as the o p t i m u m shape associated with the new discharge is sought This does not seem to be the case in the Cumberland Basin, where the author has observed only small changes in channel shape over a large range of dis-
369
charges. Because changes in shape are small and bank material is highly cohesive, it is possible that the cross-section of many channels in the Cumberland Basin corresponds with the o p t i m u m bed-load transporting form for a narrow range of discharges. This range of flows can be considered to represent the averaged effect of all the other flows experienced by the channel and it may be classed as the " d o m i n a n t discharge". Estimates of the magnitude and frequency of this flow have been obtained for four sites previously investigated by Pickup and Warner (1976): Spring Creek, Oakey Creek, Thompsons Creek and Kemps Creek. The following m e t h o d was used. Each reach was assumed to have a fixed slope and fixed bed- and bankroughness coefficients but its cross-section was assumed to be systematically varied b y increasing bed-width. Estimates of bed-load transport rates in the stream were then obtained using the Meyer-Peter and Miiller (1948) equation: 7w
\kr /
A i ( T s - - T w ) dm
/~/s
The characteristics of the channel cross-section associated with a particular bed-width were determined as follows. The profile of a cross-section may be described by the equation:
Wi = c" Di x
(6)
where Wi is the channel width at stage, i, and Di is the maximum depth at stage, i. Given the exponent, x, values of bed-width, Wb, and the coefficient, c, may be calculated. In Cumberland Basin streams, the width of the bed corresponds fairly closely with channel width at a stage 0.3 m above the lowest point of the bed so Wb and W0.3 have been taken to be the same. Wb and x are related b y the empirical equation: log Wb = log W0.3 = 0.953 -- 0.775 x
(7)
while c may be calculated as:
c = Wb/0.3 x
(8)
The bed-width may therefore be varied by changing the value of x and using eqs. 6--8 to determine the characteristics of the resultant channel. The results of bed-load discharge calculations b y eq. 5 for Spring Creek, Oakey Creek, Thompsons Creek and Kemps Creek are presented in Fig. 1. Each of the unbroken lines on the graphs indicates the variation of bed-load discharge with bed-width at a particular discharge. The broken line on each graph passes through the maximum of each of the bed-load discharge--bedwidth curves. The relationships shown in Fig. 1 confirm that, given slope, resistance to flow, and the size of the load, there is an o p t i m u m bed-load transporting form for each channel. They also confirm the statement made at the beginning of this section, that the o p t i m u m form varies with discharge. Thus, as
370
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~ } ' 4 )
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Fig. 1. The relationship between bed-width and the rate of bed-load transport at various discharges in Spring Creek, Oakey Creek, Thompsons Creek and Kemps Creek. Figures in parentheses refer to discharge in m3/sec. bed-width increases, it represents the o p t i m u m bed-load transporting form at progressively larger discharges. On each graph of Fig. 1 there is a vertical arrow, marked (A), which indicates the present bed-width of t he reach. T he discharge at which the present channel f o rm represents th e o p t i m u m for bed-load transport, (Qopt), occurs where the arrow intersects t he b r o k e n line drawn t hr ough the m a x i m u m of each of the bed-load discharge--bed-width curves. These discharges are listed in Table I,
371 TABLE I Values of Qopt compared with estimates of dominant discharge and their annual series return periods presented by Pickup and Warner (1976) Site
Qopt Q (m3/sec)
2.20 0.85 Thompsons Creek 0.75 Kemps Creek 1.20 Spring Creek Oakey Creek
Most effective discharge (Qe)
1.58-year
Natural bankfull
flood
discharge
return period (years)
Q (m3/sec)
return
Q
period (years)
(m 3/sec)
Q (m 3/sec
return period (years)
1.50 1.20 1.10 1.10
1.30 1.10 1.30 2.40
1.45 1.35 1.37 1.20
3.11 3.89 3.89 28.32
198.3 25.2 12.1 69.3
>50 5.6 2.5 1.9
together with their return periods on the annual series. The discharges vary between 0.75 and 2.20 m3/sec, and the return periods range from 1.10 to 1.50 years. It is interesting to compare these data with estimates for alternative definitions of d o m i n a n t discharge for the same four streams. Pickup and Warner (1976) have presented values for the 1.58-year flood on the annual series, the bankfull discharge and the most effective discharge, Qe. The most effective discharge is defined as the midpoint of that range of flows, which, over a period of time transports a greater proportion of the bed-material load than any other flow range, and it was calculated using the Meyer-Peter and Miiller equation and a regional flow duration curve. These d o m i n a n t discharge estimates are also presented in Table I. Bankfull discharges are very much larger than values of Qopt even in the three channels which are not incised. This suggests that although channel size may be related to large flows, channel shape is not. Values of the 1.58-year flood are also greater than Qopt, especially in the case of Kemps Creek. The closest correspondence is between Qopt and the most effective discharge. The return period of this flow varies between 1.20 and 1.45 years on the annual series compared with Qopt return periods from 1.1 to 1.5 years. An important inference may be drawn from the similarity of Qopt and Qe return periods. It was suggested by Pickup and Warner (1976) that those characteristics of the channel which are shaped by bed-material transport are likely to be adjusted more closely to the discharge at which most bed-load is transported than to any other flow. It has also been suggested in an earlier section of this paper, that bed-load channels tend to have a cross-section which, under prevailing conditions of load, slope and roughness, is the most efficient one for bed-load transport. By linking these two ideas, it is possible to derive a third one which may be stated as follows. A bed-load channel tends to adjust its shape so it tends towards the o p t i m u m or bed-load transport maximising form at the discharge at which, over time, the most bed-load is transported. This statement appears to be verified by the similarity between
372
Qopt and Qe return periods, providing a demonstrable link between one definition of d o m i n a n t discharge and the form of channel. The relationship between Q o p t a n d Qe may be close, but exact correspondence is unlikely for a number of reasons. Firstly, some error may be introduced by the use of the Meyer-Peter and Mtiller equation which, although it is one of the more reliable bed-load equations, is still far from perfect. The precise nature of this error is difficult to assess. Pickup and Warner (1976) have discussed possible errors in the estimates of Qe and have concluded that the estimates are of the correct order of magnitude because two different bedload equations produced similar Qe values. If the values of Q o p t a r e in error, then it is likely that they are too large, for the Meyer-Peter and Mtiller equation tends to produce conservative results. Even if the estimates of Q o p t a r e too large, the true values would still be closer to the Qe values than to the other estimates of d o m i n a n t discharge. Another, more important reason for the lack of exact correspondence between Q o p t a n d Qe is the fact that channel cross-sections may not quite be the optimum for bed-load transport because other energy expenditure conditions have to be satisfied at the same time. These conditions may include equable distribution of energy expenditure per unit width (Leopold and Langbein, 1962, Langbein and Leopold, 1964) and the least time rate of energy expenditure (Yang, 1971a, b}. With these complexities, deterministic explanations of river behaviour may not be adequate. Future, more sophisticated models of river response to variable discharge are therefore likely to contain a stochastic element in which the most probable condition of the channel occupies an intermediate position between opposing constraints. ACKNOWLEDGEMENTS
The author would like to thank Dr. R.F. Warner and Prof. M.G. Wolman for their advice and assistance. The work was carried out while the author was a research student at the University of Sydney. Financial assistance was supplied by the University of Sydney and the Department of Education and Science. Secretarial assistance and drafting services were supplied by the Department of Geography, University of Papua New Guinea.
REFERENCES Gilbert, G.K., 1914. The transport of debris by running water. U.S. Geol. Survey, Prof. Pap., 86. Griffith, W.M., 1927. A theory of silt and scour. Proc. Inst. Cir. Eng., 223: 243--263. Lane, E.W., 1937. Stable channels in erodible material. Trans. Am. Soc. Civ. Eng., 102: 123--142. Langbein, W.B., 1964. Geometry of river channels. J. Hydraul. Div., Am. Soc. Civ. Eng., 90: 301--312. Langbein, W.B. and Leopold, L.B., 1964. Quasi-equilibrium states in channel morphology. Am. J. Sci., 262: 782--794.
373
Leopold, L.B. and Langbein, W.B., 1962. The concept of entropy in landscape evolution. U.S. Geol. Surv. Prof. Pap., 500-A. Mackin, J.H., 1948. Concept of the Graded River. Bull. Am. Geol. Soc., 59: 463--512. Meyer-Peter, E. and Miiller, R., 1948. Formulas for bed-load transport. Int. Assoc. Hydraul. Struct. Res., 2nd Meet., Stockholm, pp. 39--64. Pickup, G. and Warner, R.F., 1976. Effects of hydrologic regime on magnitude and frequency of dominant discharge. J. Hydrol., 29: 51--75. Rubey, W.W., 1952. Geology and mineral resources of the Hardin and Brussels Quadrangles (in Illinois). U.S. Geol. Surv., Prof. Pap., 218. Schumm, S.A., 1963. A tentative classification of alluvial river channels. U.S. Geol. Surv., Circ., 477. Sundborg, A., 1956. The River Klaralven, a study of fluvial processes. Geogr. Ann., 38: 127--316. Yang, C.T., 1971a. Potential energy and stream morphology. Water Resour. Res., 7: 311--322. Yang, C.T., 1971b. Formation of riffles and pools. Water Resour. Res., 7: 1567--1574.
365
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
ADJUSTMENT OF STREAM-CHANNEL SHAPE TO HYDROLOGIC REGIME
G. PICKUP
Department of Geography, University of Papua New Guinea, Port Moresby (Papua New Guinea) (Received December 11, 1975; accepted for publication December 29, 1975)
ABSTRACT
Pickup, G., 1976. Adjustment of stream-channel shape to hydrologic regime. J. Hydrol., 30: 365--373. Bed-load channels tend to adjust their cross-sections so that given slope, roughness and sediment load, channel shape approaches the optimum for bed-load transport. The extent to which any one shape is the optimum varies with discharge, so four Cumberland Basin stream channels have been investigated to determine the discharges at which their present cross-sections represent the optimum for bed-load transport. These discharges have return periods ranging from 1.1 to 1.5 years on the annual series. The return periods closely correspond with return periods for the discharge at which, over a period of time, the most bed-load is transported. These return periods vary from 1.15 to 1.45 years when the same bed-load equation is used. The close correspondence between sets of return periods suggests that bed-load channels tend to adjust their cross-sections to become the optimum shape for bed-load transport at or close to the discharge at which the most bed-load transport is accomplished. NOTATION
List o f symbols A
Ai B D
Di G Q Qbf Qe
channel area c o n s t a n t in t h e M e y e r - P e t e r a n d Mtiller bed-load equation c o n s t a n t in t h e M e y e r - P e t e r a n d Mtiller bed-load equation mean depth of flow stage a b o v e t h e b e d bed-load discharge discharge bankfull discharge the discharge at which, over a period of t i m e , t h e m o s t b e d - l o a d is t r a n s p o r t e d
ms
m m tonnes/day ma/sec ma/sec ma/sec
366
NOTATION (continued)
Qopt
Ql.ss Qs/Q
R S Wb Wi
c dm g g~ kb
km kr x ~w 7s
the discharge at which a particular channel shape is the o p t i m u m for bed-load transport given slope, roughness and the characteristics of the sediment load the 1.58-year flood on the annual series bank friction correction factor in the Meyer-Peter and Mi]ller equation hydraulic radius slope of the energy grade line bed-width channel width at stage D i a coefficient mean size of the bed-material the gravity constant bed-load discharge measured under water the Strickler bed-roughness coefficient the Strickler mean roughness coefficient the Strickler particle-roughness coefficient the width--depth exponent specific weight of water specific weight of sediment shear stress
m3/sec m3/sec
m m m
m m/sec 2 tonnes/sec per m of width
INTRODUCTION
Within the constraints imposed by perimeter sediment, the size and shape of a river channel is determined b y the whole sequence of sediment-bearing flows which it experiences. Each flow produces its own particular channel response which depends on the magnitude of the event and the initial state of the channel. The condition of the channel at any time is therefore the product of all previously occurring events. Over a period of time, channel response, like streamflow, corresponds with a particular magnitude--frequency distribution. The mean of this distribution represents the average condition and the flow "associated" with it is the socalled dominant discharge. The linkage b e t w e e n the channel condition and the dominant discharge has usually been assumed rather than demonstrated. Furthermore, it does not appear to have been explained in the light of classical principles of river behaviour. This paper is a preliminary a t t e m p t to remedy these deficiencies b y presenting a case study of adjustment to hydrologic regime in four bedload channels of the Cumberland Basin, New South Wales.
367
T H E N A T U R E OF C H A N N E L A D J U S T M E N T T h e channels described in this paper have highly cohesive bank material and bed-material consisting mainly of fine to m e d i u m gravel. Bed forms such as dunes were not observed and values of the roughness coefficient did not vary greatly with discharge. The following statements on the nature of channel adjustment apply specificallyto these conditions and not necessarily to others. Subject to certain constraints, rivers have a tendency to adjust their channel cross-sections towards equilibrium in at least two ways. Firstly,as R u b e y (1952) states: "The stream makes for itselfthe channel of maximum hydraulicradiusor maximum efficiencyof flow that itisable to maintain under itsconditionsof characterof bed, amount of load and other controllingvariables." Secondly, the cross-section becomes adapted to the sediment load carried by the stream or as Griffith (1927) puts it: "A riverfullycharged with siltmust.., tend to adopt that form of sectionwhich will give ita maximum silt-carryingcapacity." Whether this state actually develops depends on the characteristicsof the bank sediment for as Mackin (1948) states: "For any one set of slope--debris--chargefactorstherewillbe one criticaldegree of erodibilityof the bank-forming materialssuch that an originallytoo narrow channel will quickly develop a cross-sectionwith depth--widthrelationsthat provide "the maximum silt-carryingcapacity".If the bank-forming materialsare lesserodiblethan thiscritical degree the stream may tend to develop the maximum efficiencysectionover a period of time; that is,the finalstablechannel section,however long delayed may approximate the idealform for the givenslope--debris--chargefactors.But ifthe bank-forming materials are more erodiblethan the criticaldegree the stream willadopt and maintain a sectionthat iswider and shallowerthan the idealtransportationsection." In a bed-load channel, the conditions of " m a x i m u m efficiency of flow" and " m a x i m u m sflt~carryingcapacity" m a y be mutually exclusive. T h e channel with the m a x i m u m hydraulic radius has a semi-circular cross-section, but as Lane (1937), Sundborg (1956) and S c h u m m (1963) have noted, bed-load channels tend to have cross-sections which are wide and shallow. T h e two optima therefore do not seem to coincide at the same point. Where a substantial part of the sediment load is carried as bed-load, the rate of sediment transport mainly depends on the flow conditions and the width of the channel bed. These factors operate in the following manner. T h e rate of bed-load transport per unit-width is functionally related to the flow velocity or to the tractive force, r, where: r =Tw'D'S
368 The width of bed determines the number of unit-widths over which that rate of bed-load transport occurs. All other things being equal, the expression for total bed-load transport in a rectangular channel in which bed-width, Wb, equals top-width may therefore be written:
V = f(Wb,~,w,D,S)
(2)
If discharge, slope and resistance to flow remain constant, channel width and flow depth are inversely related. This may be illustrated using the Strickler equation:
Q = AkmR2/3Sln
(3)
In a wide rectangular channel, depth of flow and hydraulic radius are approximately the same and A = Wb" D. Rearranging eq. 3 therefore gives the following expression for flow depth:
-WbkmSll2
(4)
Since the channel cross-section variables which affect bed-load transport are inversely related, it is likely that increasing bed-width will result in an increase in bed-load transport until a m a x i m u m is reached. Any subsequent increase in bed-width will result in a reduction in bed-load transport. This t e n d e n c y has been described and experimentally verified by Gilbert (1914) who defines the d e p t h / w i d t h ratio at the point where bed-load transport is maximised as the o p t i m u m form ratio. Field observations indicate that the t e n d e n c y for a channel to develop a cross-section corresponding with the optimum-form ratio overrides the tendency for it to develop a cross-section with the m a x i m u m hydraulic radius. This is apparent from the fact t h a t as the bed-material becomes coarser, and the proportion of total load carried as bed-load increases, channel cross-sections become wider and flatter (Sundborg, 1956; Schumm, 1963). It is worth noting, however, t h a t the o p t i m u m form ratio corresponds with the maxim u m hydraulic radius the channel is able to attain under the circumstances. Any increase in bed-width b e y o n d the o p t i m u m form ratio involves a decrease in hydraulic radius. RELATIONSHIP OF THE OPTIMUM BED-LOAD TRANSPORTING FORM AND HYDROLOGIC REGIME Given slope, roughness and the size of load, the shape of the o p t i m u m bedload transporting channel cross-section depends on discharge. The o p t i m u m shape at one discharge is therefore not the same as the o p t i m u m at another. In some channels, the shape of the cross-section changes with discharge, presumably as the o p t i m u m shape associated with the new discharge is sought This does not seem to be the case in the Cumberland Basin, where the author has observed only small changes in channel shape over a large range of dis-
369
charges. Because changes in shape are small and bank material is highly cohesive, it is possible that the cross-section of many channels in the Cumberland Basin corresponds with the o p t i m u m bed-load transporting form for a narrow range of discharges. This range of flows can be considered to represent the averaged effect of all the other flows experienced by the channel and it may be classed as the " d o m i n a n t discharge". Estimates of the magnitude and frequency of this flow have been obtained for four sites previously investigated by Pickup and Warner (1976): Spring Creek, Oakey Creek, Thompsons Creek and Kemps Creek. The following m e t h o d was used. Each reach was assumed to have a fixed slope and fixed bed- and bankroughness coefficients but its cross-section was assumed to be systematically varied b y increasing bed-width. Estimates of bed-load transport rates in the stream were then obtained using the Meyer-Peter and Miiller (1948) equation: 7w
\kr /
A i ( T s - - T w ) dm
/~/s
The characteristics of the channel cross-section associated with a particular bed-width were determined as follows. The profile of a cross-section may be described by the equation:
Wi = c" Di x
(6)
where Wi is the channel width at stage, i, and Di is the maximum depth at stage, i. Given the exponent, x, values of bed-width, Wb, and the coefficient, c, may be calculated. In Cumberland Basin streams, the width of the bed corresponds fairly closely with channel width at a stage 0.3 m above the lowest point of the bed so Wb and W0.3 have been taken to be the same. Wb and x are related b y the empirical equation: log Wb = log W0.3 = 0.953 -- 0.775 x
(7)
while c may be calculated as:
c = Wb/0.3 x
(8)
The bed-width may therefore be varied by changing the value of x and using eqs. 6--8 to determine the characteristics of the resultant channel. The results of bed-load discharge calculations b y eq. 5 for Spring Creek, Oakey Creek, Thompsons Creek and Kemps Creek are presented in Fig. 1. Each of the unbroken lines on the graphs indicates the variation of bed-load discharge with bed-width at a particular discharge. The broken line on each graph passes through the maximum of each of the bed-load discharge--bedwidth curves. The relationships shown in Fig. 1 confirm that, given slope, resistance to flow, and the size of the load, there is an o p t i m u m bed-load transporting form for each channel. They also confirm the statement made at the beginning of this section, that the o p t i m u m form varies with discharge. Thus, as
370
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(t2)
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90 ~
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~.
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Bed width, m.
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Bed width, m.
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(08)
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I0- ~ ~ ( 1 " 0 ) ~t ~ ,
~
3 4 5 Bed width, m.
6
(1"2)
~ } ' 4 )
~ 2o-
-~
2
3
4
~06)''(0'81, 5
~
4'
8
Bed width, m.
Fig. 1. The relationship between bed-width and the rate of bed-load transport at various discharges in Spring Creek, Oakey Creek, Thompsons Creek and Kemps Creek. Figures in parentheses refer to discharge in m3/sec. bed-width increases, it represents the o p t i m u m bed-load transporting form at progressively larger discharges. On each graph of Fig. 1 there is a vertical arrow, marked (A), which indicates the present bed-width of t he reach. T he discharge at which the present channel f o rm represents th e o p t i m u m for bed-load transport, (Qopt), occurs where the arrow intersects t he b r o k e n line drawn t hr ough the m a x i m u m of each of the bed-load discharge--bed-width curves. These discharges are listed in Table I,
371 TABLE I Values of Qopt compared with estimates of dominant discharge and their annual series return periods presented by Pickup and Warner (1976) Site
Qopt Q (m3/sec)
2.20 0.85 Thompsons Creek 0.75 Kemps Creek 1.20 Spring Creek Oakey Creek
Most effective discharge (Qe)
1.58-year
Natural bankfull
flood
discharge
return period (years)
Q (m3/sec)
return
Q
period (years)
(m 3/sec)
Q (m 3/sec
return period (years)
1.50 1.20 1.10 1.10
1.30 1.10 1.30 2.40
1.45 1.35 1.37 1.20
3.11 3.89 3.89 28.32
198.3 25.2 12.1 69.3
>50 5.6 2.5 1.9
together with their return periods on the annual series. The discharges vary between 0.75 and 2.20 m3/sec, and the return periods range from 1.10 to 1.50 years. It is interesting to compare these data with estimates for alternative definitions of d o m i n a n t discharge for the same four streams. Pickup and Warner (1976) have presented values for the 1.58-year flood on the annual series, the bankfull discharge and the most effective discharge, Qe. The most effective discharge is defined as the midpoint of that range of flows, which, over a period of time transports a greater proportion of the bed-material load than any other flow range, and it was calculated using the Meyer-Peter and Miiller equation and a regional flow duration curve. These d o m i n a n t discharge estimates are also presented in Table I. Bankfull discharges are very much larger than values of Qopt even in the three channels which are not incised. This suggests that although channel size may be related to large flows, channel shape is not. Values of the 1.58-year flood are also greater than Qopt, especially in the case of Kemps Creek. The closest correspondence is between Qopt and the most effective discharge. The return period of this flow varies between 1.20 and 1.45 years on the annual series compared with Qopt return periods from 1.1 to 1.5 years. An important inference may be drawn from the similarity of Qopt and Qe return periods. It was suggested by Pickup and Warner (1976) that those characteristics of the channel which are shaped by bed-material transport are likely to be adjusted more closely to the discharge at which most bed-load is transported than to any other flow. It has also been suggested in an earlier section of this paper, that bed-load channels tend to have a cross-section which, under prevailing conditions of load, slope and roughness, is the most efficient one for bed-load transport. By linking these two ideas, it is possible to derive a third one which may be stated as follows. A bed-load channel tends to adjust its shape so it tends towards the o p t i m u m or bed-load transport maximising form at the discharge at which, over time, the most bed-load is transported. This statement appears to be verified by the similarity between
372
Qopt and Qe return periods, providing a demonstrable link between one definition of d o m i n a n t discharge and the form of channel. The relationship between Q o p t a n d Qe may be close, but exact correspondence is unlikely for a number of reasons. Firstly, some error may be introduced by the use of the Meyer-Peter and Mtiller equation which, although it is one of the more reliable bed-load equations, is still far from perfect. The precise nature of this error is difficult to assess. Pickup and Warner (1976) have discussed possible errors in the estimates of Qe and have concluded that the estimates are of the correct order of magnitude because two different bedload equations produced similar Qe values. If the values of Q o p t a r e in error, then it is likely that they are too large, for the Meyer-Peter and Mtiller equation tends to produce conservative results. Even if the estimates of Q o p t a r e too large, the true values would still be closer to the Qe values than to the other estimates of d o m i n a n t discharge. Another, more important reason for the lack of exact correspondence between Q o p t a n d Qe is the fact that channel cross-sections may not quite be the optimum for bed-load transport because other energy expenditure conditions have to be satisfied at the same time. These conditions may include equable distribution of energy expenditure per unit width (Leopold and Langbein, 1962, Langbein and Leopold, 1964) and the least time rate of energy expenditure (Yang, 1971a, b}. With these complexities, deterministic explanations of river behaviour may not be adequate. Future, more sophisticated models of river response to variable discharge are therefore likely to contain a stochastic element in which the most probable condition of the channel occupies an intermediate position between opposing constraints. ACKNOWLEDGEMENTS
The author would like to thank Dr. R.F. Warner and Prof. M.G. Wolman for their advice and assistance. The work was carried out while the author was a research student at the University of Sydney. Financial assistance was supplied by the University of Sydney and the Department of Education and Science. Secretarial assistance and drafting services were supplied by the Department of Geography, University of Papua New Guinea.
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Leopold, L.B. and Langbein, W.B., 1962. The concept of entropy in landscape evolution. U.S. Geol. Surv. Prof. Pap., 500-A. Mackin, J.H., 1948. Concept of the Graded River. Bull. Am. Geol. Soc., 59: 463--512. Meyer-Peter, E. and Miiller, R., 1948. Formulas for bed-load transport. Int. Assoc. Hydraul. Struct. Res., 2nd Meet., Stockholm, pp. 39--64. Pickup, G. and Warner, R.F., 1976. Effects of hydrologic regime on magnitude and frequency of dominant discharge. J. Hydrol., 29: 51--75. Rubey, W.W., 1952. Geology and mineral resources of the Hardin and Brussels Quadrangles (in Illinois). U.S. Geol. Surv., Prof. Pap., 218. Schumm, S.A., 1963. A tentative classification of alluvial river channels. U.S. Geol. Surv., Circ., 477. Sundborg, A., 1956. The River Klaralven, a study of fluvial processes. Geogr. Ann., 38: 127--316. Yang, C.T., 1971a. Potential energy and stream morphology. Water Resour. Res., 7: 311--322. Yang, C.T., 1971b. Formation of riffles and pools. Water Resour. Res., 7: 1567--1574.