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'. and 0 is the particle size. Oc is obtained by substituting the grain shear stress (1") for l' and 0 for 0 in Equation (1), and rear- r~nging into the following form: , l' o = . c 1'*/1', - 1') (2) Reported values for l' *c for the median par- ticle size of the surface bed material range from 0.03 (Meyer-Peter and Muller 1948; Neill 1968) to 0.06 (Shields 1936). A value of 0.047 is commonly used in engineering practice based on the critical condition pre- dicted by the Meyer-Peter, Muller bedload transport equation (Meyer-Peter and Mul- ler 1948). More recent studies indicate that a value of 0.03 may be more appropriate for gravel and cobble bed streams (Parker et al. 1982; Andrews 1983). For purposes of this analysis, incipient conditions were computed using values of T*c of 0.03 and 0.047 to address the range of conditions associated with uncertainty in selection of a single appropriate value. The bed shear stress due to grain resistance (1") is a better descriptor of near-bed hydraulic energy in gravel-cobble bed streams than the more commonly used total shear stress because it eliminates the effects of flow resistance due to irregularities in the channel bound- ary, nonlinearity of the channel, variations in channel width, and other factors that contribute to the total flow depth, but not the energy available to move individual partides on the channel bed (Einstein 1950; Mussetter 1989); 1" is computed from the following relation: 1" = 'YY'S (3) where Y' is the portion of the total hy- draulic depth associated with grain resis- tance (Einstein 1950) and 5 is the energy slope at the cross section. The value of Y' is computed by iteratively solving the semilogarithmic velocity profile equation: v (Y') V*' = 5.75 + 6.25 log Ks (4) where V is the mean velocity at the cross section, Ks is the characteristic roughness height of the bed, and V.' is the shear ve- locity due to grain resistance given by: V*' = VgY'S. (5) ~ 122 61 i 5:- ., , i' 4' 8 I ~ 3 -- 1--- o l o 1_ __ 2500 ---r--- 2lXXl 500 1CXXl 1500 tiQRIZONTAL DISTANCE (ft) FIGURE 6. Computed velocity profiles for three discharges in the right branc~ chann~l, RM 16.5, Yampa River. Tertiary bar or rIffle locatIOns are also shown. The characteristic roughness height of the bed (K,) was assumed to be 3.5 084 (Hey 1979), where 084 is the particle size for which 84% of the bed material is smaller. The average 084 of the riffles or tertiary bars in the study reach is approximately 100 mm (3.9 in.) (Figure 4). Mean velocities predicted by the HEC-2 model vary from less than 1 ft/sec in the pool area upstream of the primary bar at a discharge of 50~ ds to approximately 7.5 ft/sec over the nffle at cross section 2.7 at a discharge of 10,000 ds (Figure 6). The calibrated water-sur!~ce elevations resulted in average veloCIties from the HEC-2 model that were very dose to those measured during stream gaging at 1,207 ds. Three riffles or tertiary bars are located in each of the two channel branches. They occur at cross sections 1.5, 2.8, and 5.5 on the left branch and at cross sections 0.8, 2.7, and 5.1 on the right branch. The en- ergy grade line profiles for a range of dis- charges from 500 to 10,000 ds for the right (Figure 7) and left (Figure.8) branch c~an- nels indicated that the nffles or tertiary bars are very pronounced at low discharges but are drowned out at higher discharges. The variation in incipient conditions with discharge at the riffles or tertiary bars (cross sections 0.8,2.7,5.1) (fofT*c = 0.03) is shown in Figure 9. The ordinate of the figures is the dimensionless grain shear stress (1'*') that is the ratio of the grain shear stress (1") to the critical shear stress (TJ or: 1" 'YY'S l' *' = - = (6) 1'c T*c('Y, - 1')050 When 1'*' < 1, there is insufficient shear Rivers . Volume 4, Number 2 April 1993