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<br />TOPPING ET AL: COLORADO RIVER SEDIMENT TRANSPORT, 2
<br />
<br />flux of sediment. In this case, as the upstream supply of sedi-
<br />ment is decreased, the sediment-transport rate at a cross sec-
<br />tion will be higher than the sediment-transport rate at the cross
<br />section upstream, causmg slreamwise divergence in the sedi-
<br />ment flux. Thus, in both situations, the bed at a cross section
<br />will stop degrading only when the sediment transport rates equil-
<br />ibrate between that cross section and the cross section upstream.
<br />Cross sections surveyed in the Grand Canyon gage reach the
<br />day before the 1996 flood experiment provide a good example
<br />of the control of reach geometry on bed aggradation and scour
<br />(Figure 16). In the Grand Canyon gage reach, as the water-
<br />surlace stage increases, the cross-sectional area of downstream
<br />flow at the cableway increases faster than at the upstream end
<br />of the reach. In this usage, cross-sectional area of downstream
<br />flow excludes thai portion of the channel occuppied by lateral
<br />recirculation eddies. Because the water-surface stage ranged
<br />from 2.1 m 10 3.4 m during the 2 weeks prior to the survey, the
<br />bed topography was probably in equilibrium with a water-
<br />surface stage in this range. Indeed, because the cross-sectional
<br />areas of downstream flow at the two cross sections were ap-
<br />proximately equal at a water-surface stage of about 3.4 m, the
<br />bed topography the day before the 1996 flood experiment was
<br />probably in equilibrium with a water-surface stage of about
<br />3.4 m. At higher water-surface stages the cross-sectional area
<br />of downstream flow would increase faster at the downstream
<br />(cableway) cross section than at the upstream cross section (as
<br />observed during the 1996 flood experiment). Thus, by conser-
<br />vation of mass, the mean velocity at the downstream cross
<br />section would increase more slowly than at the upstream cross
<br />section. Because suspended-sediment transport scales as
<br />roughly the second to third power of the boundary shear stress
<br />[e.g., Engehuul and Hansen, 1967] and the boundary shear
<br />stress scales approximately as the square of the mean velocity,
<br />this effect produces substantial convergence in the flux of sed-
<br />iment, driving deposition at the cableway cross section. This
<br />deposition would continue, given an adequate upstream supply
<br />of sediment and enough time, until the area of the cableway
<br />cross section decreased enough to remove the streamwise con-
<br />vergence in the boundary shear stress field, resulting in a new
<br />equilibrium bed topography. By the same process, subsequent
<br />decreases in water-surface stage would cause erosion at the
<br />cableway cross section.
<br />To extend these results to cross sections with geometries
<br />different from that at the Grand Canyon cableway and to
<br />further illustrate the influences of both local reach geometry
<br />and upstream sediment supply on bed elevation during a flood,
<br />we constructed and applied a simple one-dimensional model
<br />using (I). This model was applied to two different types of
<br />cross sections (both with sandy beds), first for the case in which
<br />the upstream sediment supply is in equilibrium with the sedi-
<br />ment-transport capacity throughout a flood and then for the
<br />case in which the upstream sediment supply decreases (result-
<br />ing in an additional streamwise increase in sediment flux
<br />through the cross section) during a flood. In the first type of
<br />cross section (with a geometry similar to that at the Grand
<br />Canyon cableway), convergence occurs in the boundary shear
<br />stress field as the water-surface stage increases during a flood,
<br />causing deposition a[ [his cross section. In [he second type of
<br />cross section, divergence occurs in the boundary shear stress
<br />field as [he water-surface stage increases during a flood, caus-
<br />ing erosion a[ this cross section.
<br />To make these calculations simple, the model was first run
<br />lIsing a critical assumption. This assumption is (hat (he time-
<br />
<br />565
<br />
<br /> 5
<br /> I 0 l /
<br /> W
<br /> '" -5
<br /> ;0
<br /> <II
<br /> -10
<br /> .15
<br />a) 0 2. 40 60 B. 100
<br /> DISTANCE FROM LEFT BOLT (m)
<br /> 5
<br /> I .
<br /> W
<br /> '" -5
<br /> ~
<br /> -10
<br />b) .15
<br /> . 20 40 6. 6. 100
<br /> DISTANCE FROM LEFT BOLT (m)
<br />"" 1200
<br />~ E
<br />0-
<br /><3: 1000 -158mlJ>S'TREAMFftClt.lCNJU'WAY
<br />Wo (CROSS-S€CTION ') ~
<br />a:~ - - - AT CABlEWAY
<br /><~ BOO
<br />W" (CROSS SECTION ~
<br />....< 600
<br />c W
<br />"'a: 400 ~
<br />x....
<br />0"' , ' SrAQE MEN TOPOGRAPHY WAS SURVEYED
<br />a:Z 200
<br />"3:
<br />"0
<br /><0 "-3
<br />c) -2 .1 0 1 2 3 4 5 6 7 B . 10
<br /> WATER-SURFACE STAGE (m)
<br />
<br />
<br />
<br />Figure 16, (a) Cross section 0 (at the Grand Canyon cable-
<br />way) surveyed on March 26, 1996. Except for in narrow wnes
<br />near the banks, downstream flow occurs over most of this cross
<br />section. (h) Cross section 4 surveyed on March 26, 1996. The
<br />cross-hatched regions indicate the approximate areas of a lat-
<br />eral recirculation eddy (see Figure 2 for the planform view of
<br />this eddy), in whicb downstream flow is balanced by upstream
<br />flow. (c) Approximate area of downstream flow as a function of
<br />water-surface stage at both cross sections based on the topog-
<br />raphy surveyed on March 26, 1996. The faster increase in the
<br />cross-sectional area of downstream flow with increasing water.
<br />surface stage drives deposition at the cableway during floods
<br />(see text for discussion).
<br />
<br />scale of bed-topographic adjustment is much sho"er than the
<br />timescale of the rising limb of a flood. Thus, in this first round
<br />of calculations, given a stable upstream supply of sediment, the
<br />bed topography is always in equilibrium with the flow. To make
<br />the physical linkage between changes in the upstream sediment
<br />supply and the topographic response of the bed clear, the
<br />results of this simple model (Figure 17) are discussed first.
<br />Then, to make these calculations general, the results of the
<br />simple model are extended to situations where this assumption
<br />does not apply, that is, when the timescale of bed-topographic
<br />adjustment is substantial with respect to the timescale of the
<br />rising limb of a flood.
<br />As shown in Figure 17, given an equilibrium upstream sup-
<br />ply of sediment during a flood, the time of maximum or min-
<br />imum bed elevation (in both types of cross sections) should
<br />occur simultaneously with the peak water-surface stage. In this
<br />case the bed at the cross section may aggrade or scour, de-
<br />pending on the local reach geometry. In contrast, if the up-
<br />stream supply of sediment decreases during a flood, a lag will
<br />occur between the time of the flood peak and the time of either
<br />maximum or minimum bed elevation at a cross section. In the
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