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566 TOPPING ET AL: COLORADO RIVER SEDIMENT TRANSPORT, 2 -WATER SURFACE - BeD WITH EQUIUBAIUM UPSTREAM SAND SUPPLY - - . BED wm-t DECREASING UPSTREAM SAND SUPPlY w '" < .... <II MAX. ------ ---- --- ---- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 8) TIME NORMAUZED BY FLOOD DURATION w '" < .... <II --- ---- ----- ---- 0.0 0.1 0.2 0.3 0.4 0.6 0.6 0.7 0.8 0.9 1.0 b) TIME NORlIAUZEO BY FlOOO DURATION Figuft 17. ResulIs of calculations based on equation (I) for cases where the upstream supply of sediment does not change with time and decreases linearly with time during the cowse of a flood for cross sections where (a) geometrically driven con- vergence in the sediment flux occurs with increasing flow depth and (1)) geometrically driven divergence in the sediment flux occurs with increasing flow depth. case of a decreasing upsiream supply of sediment, (I) maxi- mum bed elevation will lead the flood peak at a cross section where the bed aggrade5 with increasing water-swface stage (Figure 17a), and (2) the time of minimum bed elevation will lag the flood peak at a cross section wbere the bed SCOUTS with increasing water-swface stage (Figure 17b). If the assumption used in Figure 17 does not apply and the timesca1e of bed-topographic adjustment is sqbstantial wiih respect to the timescaIe of the rising limb of a flood, the effect is to delay the time of either maximum or minimum bed ele- vation. Therefore, in the first type of cross section (Figure 17a), the time of maximum bed elevation does not ~ave to occur prior to, but could occur either with or after, the time of a flood peak. In the second type of cross section (Figures 17b) the time of minimum bed elevation will still occur after the lime of a flood peak, and the bed may still .ggrade during the receding limb. Thus, in this case the effect of a slower response of the bed topography to changing flow conditions produoes a result that cannot be distinguished from the effect of sand supply limitation. In summary. at a cross section that aggrades with increasing water-surface stage, the observation that the time of maximum bed elevation occurs prior to either the peak or the receding limb of a flood indicates a decrease in the upstream supply of sand during a flood (i.e., sand supply limitation). However, al a cross section that scours with increasing water-surface stage, the observation that the time of minimum bed elevation occurs after the peak of a flood does not necessarily indicate the presence of sand supply limitation. Therefore, because of the strong control of reach geometry on bed topography and the possibly delayed response of bed topography to changing flow conditions, the best type of cross section to use in deducing the presence of sand supply limitation is one tbat aggrades with increasing water-surface stage (e.g., the Grand Canyon cable- way cross section). Observations made during the 1996 flood experiment sug- gest that changes in bed elevation during floods in the Colo- rado River in Grand Canyon are primarily driven by reach geometry and only secondarily by depletion of the upstream sand supply. Depletion of the upstream supply of sand does not necessarily prevent bed aggradation when a strong streamwise convergence exists in the boundary shear stress field. Measure- menIs of suspended-sand concentration and bed grain size at the Grand Canyon cableway during the 1996 flood indicate that the upstream supply of sand was being depleted as early as day 1 of the flood, 4 days before the bed stopped aggrading. Depletion of the upstream sand supply therefore only bas the effect of creating or modifying a lag between the time of a flood peak and the lime of either maximum or minimum bed elevation. At a cross section where convergence occurs in the boundary shear stress. field with increasing flow, the time of maximum bed elevation in a supply-limited case will occur prior to that in a nOlHUpply-limited case; and, at a cross section where divergence occurs in the boundary shear stress field with increasing flow, the lime of minimum bed elevation in a supply- limited case will occur after that in a non-supply-limited case. 6.2, Sediment Grain-Size Evolutioa The concentration and grain size of sediment in suspension is tightly coupled 10 the grain-size distribution of the bed. For steady, uniform flow, and an upstream supply of sediment that is in equilibrium with the flow conditions, the concentration of each size class of sediment in suspension depends mainly on (I) the proportion of the fine sediment on the bed composed of that size class, (2) the settling velocity of that size class, and (3) the median size of the fine sediment on the bed. In this framework the grain-size distnbution of the bed is treated as an independent variable. However, in riveTS in which the ilp- stream supply of sediment is not in equilibrium with the flow conditions, the grain-size distribution of the bed is a dependent variable (as in either sediment-feed flumes or the Colorado River) and evolves over time as a function of changes in the sediment supply. Before adding thi6 degree of complexity to the problem, it is useful to review the coupling between the suspended and bed sediment in tbe Colorado River tbrough solution of the foUowing equations for suspended-sediment concentration derived for multiple size classes in steady, uni- form flow and an upstream supply of sediment that is in equi- librium with the flow conditions. In steady, uniform flow, when the effects of bedfonns and density stratification are excluded, (C) (C ) [(a)(h-Z)]' ~ = ~ - - z<02h 1 - cJ J 1 - cJ" Z h - a - . C ~mJ, ~ C ~mJJ (O~h)(:'~:) r (2a) 'exp [ -p ~ (z - O.2hl] These equations were derived using the two-part eddy viscosity of Rnltray and MitswUl [1974]; see McLean [1992] for the basis of the derivation. In 2, em is the volumetric concentration of sediment in size class m I C s is the total concentration of sed- iment in all size classes, z is the vertical dimension, h is the flow Z > O.2h. (2b)