Transcript CHAPTER 19: EFFECT OF THE HYDROGRAPH ON MORPHOLOGY OF GRAVEL-BED STREAMS
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004
CHAPTER 19:
EFFECT OF THE HYDROGRAPH ON MORPHOLOGY OF GRAVEL-BED STREAMS
Up to this point the effect of a flood hydrograph has been folded into a characteristic flood flow and an intermittency. This, of course, is only intended to be shorthand for a full flood hydrograph. The effect of a full flood hydrograph on the morphology of a gravel-bed river is explored in this chapter.
Elbow River, Alberta, Canada at low flow Elbow River, Alberta, Canada during 100-year flood.
Images courtesy Alberta Research Council, Canada 1
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 NONLINEARITY IN THE SEDIMENT TRANSPORT RELATION
The sediment transport rate is in general a nonlinear function of e.g. the Shields number. This means that doubling the Shields number more than doubles the sediment transport rate. To see this, consider a simplified sediment transport relation q* = q*( *) of the form
q
(
)
t
(
)
n t where q* denotes the Einstein number, * denotes the Shields number, t constant and n t is a is equal to an exponent (1.5 for Meyer-Peter and M üller, 2.5 for Engelund-Hansen). It follows that where is any multiplicative factor,
q
(
)
t
(
)
n t n t
[
t
(
)
n t
]
n t
q
(
)
In the above simple example, a value of n t of 1.5 means that doubling the Shields number trebles the Einstein sediment transport rate.
Because of this nonlinearity, the sediment transport rate
averaged over a hydrograph
is biased toward the higher flows in the hydrograph, and is thus
greater
than the sediment transport rate at the
average flow
of the hydrograph.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004
Q
SPECIFICATION OF THE HYDROGRAPH
discharge below which sediment transport is negligible t In order to perform a numerical calculation with a flow hydrograph, the actual flood hydrograph (black line) must be discretized into a series of constant discharges Q w , each extending for time duration t w , w = 1..W. The river is assumed to be morphologically inactive when the river is not in flood.
Now let t f denote the (short) flood time step used to to calculate morphodynamic evolution (via the Exner equation of sediment continuity). It is convenient to choose each time duration t w to be an integral number n step,w of flood time steps t f , so that t w n step , w t f 3
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 SPECIFICATION OF THE HYDROGRAPH contd.
Once t f is specified along with Q w and n step,w for w = 1..W, it is possible to determine a discharge Q p for every time step p = 1..P in the hydrograph, where Q discharge below which sediment transport is negligible P = P w W 1 n step , w An example is given below.
Q w , m 3 /s 3 30 20 7 3 n step,w 2 1 2 3 5 13 Q p , m 3 /s 3 3 30 20 20 7 7 7 5 5 5 5 5 p 5 6 7 8 9 10 1 2 3 4 11 12 13 t 4
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 NORMAL FLOW APPROXIMATION
In principle it is necessary to model the evolution of a hydrograph in the downstream direction by means of flood routing. In a relatively short reach of a gravel-bed stream without tributaries, however, the hydrograph can be approximated as the same in time at every point in space. Again in a steep mountain stream, the flow can be approximated as satisfying the normal flow approximation.
Now let q w,p (= Q p /B p , where B p denotes the channel width at the pth flow) denote the flow discharge per unit width for the pth flow in the hydrograph. Generalizing from Chapter 17, the boundary shear stress of the pth flow in the hydrograph at the kth spatial node b,p,k is given as b , p , k
u
2 , p , k
k
1 / 3 s , p , k 2 r
q
2 w , p 3 / 10
g
7 / 10
S
p 7 / , k 10 where u *,p,k , k s,p,k and S p,k denote the current shear velocity, roughness height and bed slope at the kth node and the pth step in the hydrograph. This boundary shear stress can then be used in a sediment transport formulation of the form of Parker (1991), Powell-Reid-Laronne (2001) or Wilcock-Crowe (2003) to compute the gravel transport rate at any node and at any point in the hydrograph. An implementation of Exner for 5 each time step then allows modeling of the evolution of the bed.
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 INTRODUCTION TO
RTe-bookAgDegNormGravMixHyd.xls
The code in the workbook RTe-bookAgDegNormGravMixHyd.xls computes aggradation and degradation in a gravel-bed river subject to a repeating hydrograph. The bedload transport formulation is that of Parker (1990). The input is fairly extensive, and is thus explained in some detail in the next few slides.
The user first specifies a flood time step:
t
f
Input time step
0.25 time step in flood days
The user then inputs the number of steps in the flood hydrograph and clicks a button to set up a table for input: ndisc
Input number of discharges in hydrograph
10 ndisc must be between 2 and 16 Click to set up table for hydrograph input 6
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 INPUT FOR
RTe-bookAgDegNormGravMixHyd.xls
Water discharge per unit width q w , sediment feed rate per unit width q bTf of each flow in terms of number of time steps n step and duration are then input into the table: 7
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 INPUT FOR
RTe-bookAgDegNormGravMixHyd.xls
contd.
The spreadsheet computes the duration of the flood hydrograph in days. The user must specify the number of times per year this hydrograph is to be cycled. The spreadsheet then computes the flood intermittency I f .
I f
The above hydrograph has a duration of the following number of days:
days 30
Input the number of times this hydrograph is to be cycled per year
ncyc 1
(E.g. a 12-day hydrograph cycled 3 time per year implies that the river is morphologically active for 12 x 3 = 36 days per year) The flow intermittency
Inter
(no. of days per year the river is morphologically active/365.25) is:
Inter 0.0821355
The above number must be less than unity; if not adjust
dt
,
nstep
and
ncyc
to achieve this
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1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 INPUT FOR
RTe-bookAgDegNormGravMixHyd.xls
contd.
Grain size distributions are input next. The user specifies the number of grain sizes defining the input distributions, clicks a button to set up a table for input, and fills in grain sizes and percents finer for the gravel feed, initial surface size distribution and substrate. npp
Input number of grain sizes defining distributions, i.e.
number of grain sizes for which a percent finer is specified.
8 npp must be between 2 and 9 Click to set up input for grain size distributions
Input grain size distributions in terms of grain size and percents finer.
Grain Percents finer Initial size, mm Feed 256 128 64 32 16 8 4 2 100 95 80 50 25 10 5 0 Surface 100 95 80 50 25 10 5 0 Scroll down for more input Substrate 100 95 Put in grain size distributions for sediment feed, initial 80 surface and substrate, starting from coarsest size.
50 25 In using the Parker (1990) formulation for gravel transport, 10 the grain size distributions must be renormalized so that there 5 is no content below 2 mm.
0 9
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 INPUT FOR
RTe-bookAgDegNormGravMixHyd.xls
contd.
The user then inputs the dowstream bed elevation d , reach length L, initial bed slope S fbI , number of spatial intervals M, number of hydrograph cycles until a printout Mcycprint and number of printouts after the initial one Mprint. The user clicks the indicated button to perform a calculation.
Note: the calculation may be rather slow!
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1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 OUTPUT OF
RTe-bookAgDegNormGravMixHyd.xls
The output is quite extensive, and includes the following plots.
1. A plot including a) the flow hydrograph, b) time variation of sediment feed (summed over all grain sizes) within the hydrograph and c) the time variation of total bedload transport rate (summed over all grain sizes) over the final hydrograph cycle of the run, at the farthest downstream node.
2. Long profiles of bed elevation. 3. Long profiles of bed slope. 4. Long profiles of surface geometric mean size.
5. Long profiles of bedload geometric mean size.
6. Long profiles of the ratio of the bedload transport rate to the feed rate (equal to unity everywhere at equilibrium).
7. Long profiles of a) the flow depth H at the end of the maximum flow of the final hydrograph of the run; b) the flow depth H at the end of the final flow of the final hydrograph of the run; c) Shields number sg * based on surface geometric mean size at the end of the maximum flow of the final hydrograph of the run: d) Shields number sg * based on surface geometric mean size at the end of the final flow of the final hydrograph of the run. 11
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 MORE OUTPUT OF
RTe-bookAgDegNormGravMixHyd.xls
Output plots continue as follows.
8. Long profiles of a) surface geometric mean size D sg at the maximum flow of the final hydrograph of the run, b) surface geometric mean size D sg at the final flow of the final hydrograph of the run, c) bedload geometric mean size D lg at the maximum flow of the final hydrograph of the run; d) bedload geometric mean size D lg at the final flow of the final hydrograph of the run. 9. Grain size distributions of a) the sediment feed, as well as the following distributions at the farthest downstream node; b) bed surface at the end of the maximum flow of the final hydrograph, c) bed surface at the end of the final flow of the final hydrograph, d) bedload at the end of the maximum flow of the final hydrograph, e) bedload at the end of the final flow of the final hydrograph. 12
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 APPLICATION OF THE PROGRAM: EFFECT OF THE HYDROGRAPH ON EQUILIBRIUM STREAM MORPHOLOGY
No matter what the input parameters are (as long as there is some upstream feed of gravel), if the program is run for a long enough time the river reach will approach a mobile-bed equilibrium (averaged over the hydrograph). It is interesting to compare the
equilibrium reached with a full hydrograph
with the
equilibrium reached by a constant flow equal to the average flow of the hydrograph, continued for the same duration as the hydrograph.
To this end we consider a 30-day hydrograph with a minimum value of q w of 3 m 2 /s, a maximum value of 16 m 2 /s and an average value of 6 m 2 /s, as compared with a flow run constant at 6 m 2 /s for the duration of each hydrograph.
The hydrograph and its average value are given in the next slide.
13
18 16 14 12 10 8 6 4 2 0
0
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 THE HYDROGRAPH
average flow of hydrograph = 6 m
2
/s
5 10 15
days
20 25 14 30
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 SEDIMENT FEED AND GRAIN SIZE DISTRIBUTIONS
The sediment infeed rate q bTf is kept constant at 5x10 -5 m 2 /s at each step of the hydrograph. In the case of the full hydrograph, this leads to a final equilibrium at the upstream node with strong cyclic fluctuations; the bed will degrade significantly at the high flows and aggrade significantly at the low flows. These fluctuations will be damped out farther downstream as the river becomes free to increase its bedload transport rate in response to increasing discharge. Thus at the final equilibrium the sediment transport rate increases or decreases in accordance with the flow discharge, and bed elevation shows only minor cyclic fluctuations, at all nodes except those near the upstream end. The grain size distributions of the feed, initial surface and substrate are taken to be identical, as shown in the plot to the right.
100 90 80 70 60 50 40 30 20 10 0
Grain Size Distributions
Feed Initial Surface Substrate 1 10 100 1000 15
Size mm
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 OTHER INPUT PARAMETERS
Time step: 1 day Number of flows in the hydrograph: 10 Number of times per year the hydrograph is cycled: 1 Downstream bed elevation: 0 m Reach length: 10,000 m Initial bed slope: 0.00262
Number of spatial intervals: 10 Mcycprint: 200 Mprint: 6 The duration of the calculation is 1200 years, enough to insure that a mobile-bed equilibrium is obtained.
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30 25 20 15 10 5 0 0
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH THE FULL HYDROGRAPH Downstream Variation in Bed Elevation
The long profile has reached equilibrium.
2000 4000 6000
Distance m
8000 10000 0 yr 200 yr 400 yr 600 yr 800 yr 1000 yr 1200 yr final w.s.
17
0.003
0.0025
0.002
0.0015
0.001
0.0005
0 0
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH THE FULL HYDROGRAPH Downstream variation in bed slope
2000 The bed slope has equilibrated to a value S = 0.00151.
4000
x
6000 8000 10000 0 yr 200 yr 400 yr 600 yr 800 yr 1000 yr 1200 yr 18
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH THE FULL HYDROGRAPH Downstream Variation in Surface Geometric Mean Size
100 10 1 0 2000 The surface geometric mean size has coarsened from an initial (unarmored) value of 28.8 mm to an equilibrium mobile armor of 49.2 mm 4000 6000
Distance m
8000 10000 0 yr 200 yr 400 yr 600 yr 800 yr 1000 yr 1200 yr 19
100
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH THE FULL HYDROGRAPH
Downstream variation in bedload geometric mean size
10 1 0 2000 By the end of the run, the bedload geometric mean size is everywhere almost identical to that of the feed, i.e. 28.8 mm.
4000 6000
Distance m
8000 10000 0 yr 200 yr 400 yr 600 yr 800 yr 1000 yr 1200 yr 20
100 90 80 70 60 50 40 30 20 10 0 1
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH THE FULL HYDROGRAPH Grain size distributions: sediment feed; downstream (ds) surface at end of maximum flow of final hydrograph (hg); ds surface at end of final flow of final hg; ds bedload at end of maximum flow of final hg; ds bedload at end of final flow of final hg
10 By the end of the run, the surface size distribution at the node farthest downstream hardly varies with flow in the hydrograph.
feed ds final surface max flow ds final surface end flow ds final load max flow ds final load end flow The distribution of the load at the node farthest downstream fluctuates strongly with flow in the hydrograph, averaging over the hydrograph to the feed value.
100 1000 21
Grain size mm
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH THE FULL HYDROGRAPH
18 By the end of the run, the sediment transport rate at the downstream node (red line) is tracking the hydrograph, varying strongly with discharge.
0.0005
16 14 q w m 2 /s q bTf m 2 /s q bT m 2 /s 0.00045
0.0004
0.00035
12 0.0003
10 0.00025
8 0.0002
6 0.00015
4 0.0001
2 0.00005
0 0 5 10 15
days
20 25 30 0 22
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH CONSTANT FLOW FOR DURATION OF HYDROGRAPH: q W = AVERAGE VALUE OF ORIGINAL HYDROGRAPH Downstream Variation in Bed Elevation
30 25 Again a mobile-bed equilibrium is attained after 1200 years, but the upstream elevation of 26.2 m is much higher than the value of 15.1 m attained with the full hydrograph.
20 15 10 0 yr 200 yr 400 yr 600 yr 800 yr 1000 yr 1200 yr final w.s.
5 0 0 2000 4000 6000
Distance m
8000 10000 23
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH CONSTANT FLOW FOR DURATION OF HYDROGRAPH: q W = AVERAGE VALUE OF ORIGINAL HYDROGRAPH Downstream variation in bed slope
0.003
0.0025
0.002
0.0015
0.001
0.0005
0 0 The slope equilibrates to a value of 0.00262, i.e. much higher than the value of 0.00151 obtained with the full hydrograph. Since a steeper slope is required to carry the same load, it can be concluded that the
constant average flow of the hydrograph is less efficient at moving sediment than the full hydrograph.
2000 4000 6000 8000 10000 0 yr 200 yr 400 yr 600 yr 800 yr 1000 yr 1200 yr 24
x
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 RESULTS WITH CONSTANT FLOW FOR DURATION OF HYDROGRAPH: q W = AVERAGE VALUE OF ORIGINAL HYDROGRAPH Downstream Variation in Surface Geometric Mean Size
100 10 1 0 The surface geometric mean size equilibrates to a value of 66.0 mm, i.e.
much coarser than the value of 49.2 mm attained with the full hydrograph.
2000 4000 6000
Distance m
8000 10000 0 yr 200 yr 400 yr 600 yr 800 yr 1000 yr 1200 yr 25
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 FINAL SURFACE GRAIN SIZE DISTRIBUTIONS
The final surface size distribution with the full hydrograph is markedly coarser than the feed sediment (mobile-bed armor), but markedly finer than the final surface size 100 90 80 70 60 50 40 30 20 10 0 1 feed final surface with full hydrograph 10
D mm
100 final surface with constant-flow hydrograph 1000 26
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 A QUESTION
In the case of the constant hydrograph of q w sediment feed rate q bTf of 5x10 -5 m 2 = 6 m 2 /s for the entire 30 days, and a /s, the bed equilibrated at a slope S = 0.00262.
What would the feed rate have to be increased to in order for the bed to equilibrate at the same slope, when subjected to the full hydrograph?
The answer can be found by trial and error. In performing the calculations, it proved necessary to change the time step from 1 day to 0.25 days in order to prevent numerical instability.
The required sediment feed rate q bTf is 5.6x10
-4 m 2 /s, or 11.2 times higher than if the constant flow is used. This calculation illustrates how much more efficient hydrographs are at transporting sediment than equivalent constant flows.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 SLOPE PROFILE AT FINAL EQUILIBRIUM WITH A HYDROGRAPH
Downstream variation in bed slope
0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0 0 The final slope profile with the full hydrograph and a feed rate q bTf = 5.6x10
-4 m 2 /s shows variation at the upstream end. This variation represents the consequence of a sediment feed rate that does not vary with flow. In fact, the profile is a snapshot of the bed at the end of the last flow of the last hydrograph; at the end of the highest flow of the last hydrograph, the upstream slope would be below 0.00262. This cyclic time variation vanishes downstream of the third node.
2000 4000 6000 8000
x
10000 0 yr 400 yr 800 yr 1200 yr 1600 yr 2000 yr 2400 yr 28
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 A REMARKABLE TRADEOFF
The figures of Slides 21, 22 and 28 reveal an interesting feature concerning the case of a cycled hydrograph (Parker, 2004). In the simulation with a hydrograph, the flow discharge varies according to the cycled hydrograph of Slide 14, but the sediment feed rate at the upstream is held constant over the hydrograph.
When mobile-bed equilibrium (averaged over the hydrograph) is reached, at and near the upstream node (M = 1) the bed degrades and coarsens at the high flows of the hydrograph (when the sediment supply is insufficient) and aggrades and becomes finer at the low flows of the hydrograph (when the sediment supply is excessive). This feature is evident in Slide 28, where the bed slope at the upstream end of the reach is high at the end (low) flow of the hydrograph.
Not far downstream of the feed point, however, a remarkable tradeoff is evident at mobile-bed equilibrium: the bed slope, bed elevation and surface grain size distribution become invariant with point on the hydrograph, but the bedload transport rate increases in magnitude and becomes coarser at the high flows, and decreases in magnitude and becomes finer at the lows flows. These features are evident in Slides 21, 22 and 28, and are illustrated in more detail in the next two slides.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 A REMARKABLE TRADEOFF contd.
18 Once mobile-bed equilibrium is 16 reached (averaged over the hydrograph), bed slope S and 14 12 surface grain size distribution (including geometric mean size D sg ) vary cyclically near the upstream end, but obtain constant values over the hydrograph everywhere else.
10 8 6 4 2 0 0 5 10 15
days
high flow D sg constant sediment feed 20 low flow S x x = 0 x = L 25 30 30
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 A REMARKABLE TRADEOFF contd.
Once mobile-bed equilibrium is 18 16 reached (averaged over the hydrograph), total bedload 14 12 transport rate q bT and bedload geometric mean size D lg are constant over the hydrograph only 10 8 6 in a short region near the feed point; farther downstream both 4 2 increase at high flow and 0 decrease at low flow.
high flow 0 5 10 15
days
constant sediment feed D lg low flow q bT 20 25 30 31 x x = 0 x = L
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 CAVEAT!
The calculations presented in this chapter are subject to some error. This is best explained in terms of the cyclic aggradation and degradation that occurs near the upstream end of the reach when mobile-bed equilibrium is reached.
Formally, the program should store the grain size distribution of the bed material as it aggrades, and then access this same grain size distribution as it degrades. An execution of this step requires storage of the deposited substrate in memory. This has not been implemented in RTe-bookAgDegNormGravMixHyd.xls
, largely because this storage would greatly slow the execution of a program that already takes a substantial amount of time to run from Excel. Instead, when the bed degrades the substrate is assumed to have the same size distribution as specified initially as input to the program (e.g. as shown Slide 9).
While the above simplification introduces some modest error into the results, the qualitative results presented here are verified by a full model that does not employ the simplification.
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 REFERENCES FOR CHAPTER 19
Parker, G., 1990, Surface-based bedload transport relation for gravel rivers.
Journal of Hydraulic Research
, 28(4): 417-436.
Parker, G., 2004, Response of the gravel bed of a mountain river to a hydrograph.
Proceedings
, 2004 International Conference on Slopeland Disaster Mitigation, Taipei, Taiwan, October 5-6, 11 p; also downloadable at http://cee.uiuc.edu/people/parkerg/conference_reprints.htm
.
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