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470 Topographic Response of a Bar The assumption of quasi-steadiness in the flow model does not preclude treatment of cases where discharge and bed topography vary in time. Rather, this assumption limits the range of applicability of the algorithm to cases in which the time scales associated with discharge variations and bed elevation changes are long compared to the time scales of the local flow (e.g., the time required for a fluid parcel to traverse the reach of interest). For the case of discharge variations, this condition requires that the unsteadiness in the flow field makes a negligible contribution to the local momentum balance or, equivalently, that the contribution of the flood wave slope to the local surface slope is small. For temporal changes in bed elevation, model validity is maintained as long as the rate of change in bed elevation is much less than the flow velocity, a condition that is almost always satisfied. Thus, although the discharge and bed elevation may vary slowly in time, each may be treated as constant in the flow computation for a majority of natural flows. These approximations allow simplification of the equations expressing momentum balance for the flow, and also lead to a straightforward iterative technique for predicting bed evolution and the response of the bed to discharge variation, as discussed below in greater detail. As mentioned above, the flow model requires that the spatial distribution of roughness length, Zo, be specified as an input. The roughness lengths input to the model must be representative of all sources of momentum extraction from the flow, including grain saltation and form drag on bars and bedforms. When the sediment size and bedform and bar geometries are known, the value of the overall roughness p8!ameter may be calculated using the .method described by Nels.on and Smith. (1988a]. Where some areas of the bed are much different than others (e.g., duneS versus upper plane bed), this detailed. approach should be usetl. When bedform geometries are not well-known, but bed material and sediment transport rates are somewhat similar over much of the bed, a simple alternative is to use the value of (0 ( = zo/h) that yields the observed elevation drop over the reach of interest. This approach distributes the roughness length in proportion to depth which, at least for channels with duned beds, is a reasonable approximation. Choosing the value of (0 such that the observed elevation drop through the reach of interest matches that predicted by the model ensures that the total channel drag is accounted for in an int~al sense. The potential error incurred using this technique is due to the spatial distribution of roughness, rather than the magnitude. The simpler technique was employed in all calculations presented herein. Sediment Transport Model The goal for the sediment transport component of a bed evolution model is an equation or set of equations that enable one to calculate the sediment flux over each point on the bed. In general, this requires the treatment of both suspended load transport and bedload transport. In the case of bedload transport, any of several different bedload equations may be employed. These equations predict the sediment flux in the bedload layer given particle size, skin friction boundary shear stress, and the critical shear stress necessary to initiate motion. These equations typically contain empirical coefficients which are set using flume or field measurements~ As a . result, each of the various equations typically works best Qver the range in which it was calibrated, and the best bedload equation to employ in a given situation depends upon the flow conditions and sediment size. Various bedload ~uations and their ranges of applicability are discussed in detail by Wiberg and Smith (1989). Where sediment is transported as both bedload and suspended load or pnmarily suspended load, the fluxes of sediment over the bed depend upon the boundary shear stress and sediment size, as well as the local flow velocities and turbulent diffusivities. In general, the suspended sediment concentration in the flow depends upon a balance between downward particle settling, upward turbulent diffusion, and "