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356 GRANT: CRITICAL FLOW CONSTRAINS FLOW HYDRAULICS The relation between Froude number and flow resistance is more complex but may result in maximization of friction factor at critical flow. In general, the Froude number is inversely related to the Darcy- Weisbach friction factor,f, by v' 8 Fr' = - = - S gd f where G),,=:. Flume studies with flow over obstructions on fixed beds have shown that at low relative submergences (ylD5o < 0,5), the drag coefficient tends to be maximized at critical flow [Flammer el ai" 1970J, At higher relative submergences the peak drag occurs at lower Froude numbers. This effect appears to be due to the increased free surface drag resulting from hydraulic jumps and free surface distortions over submerged roughness elements at critical flow [Balhursl el aI., 1979]. Step-pool chan- nels retain this highly dissipative morphology at less-than- formative flowN becaulle of the immobility of clasts forming steps. Recent work demonstrates that step spacing obsel'Yod In natural channels may correspond with maximum flow resis- tance at formative [Ashida et a/., 1986] and less-than-formative [Abrahams el ai" 1995] discharges, By the model proposed here, maximization of friction factor must be viewed as a cor- ollary to channel and flow adjustments rather than as a cause of them, as some extremal value theories have argued [Davies and Slither/and, 1980, 1983; Abrahams el a/., 1995J, Flow may still be critical at less than effective discharges; tumbling flow over static large roughness elements tends to be at or near critical [Morris. 1968]. Under these conditions, however, flow and channel geometry are not necessarily in equilibrium. so channel geometry is a relict of some more formative discharge. In contrast to step-pool channels, which preserve bed mor- phologies developed at critical flow, antidune structures gen- erated near critical flow in sand,bed systems are typically de- stroyed as flows wane, although an internal antidune stratification may be preserved [Middlelon, 1965; Hand el ai" 1969]. This erasure of bed forms occurs because sand-bed streams tend to be competent over a much wider range of flows; hence bed morphology continues to adjust. In some cases the standing wave/antidune flow structure is maintained 'as flows recede, although the amplitude and wavelength dimin- ish mMkedly. I have observed tiny sequences of standing waves in very shallow flows on the Oregon beach. Coarser gravel channels are in between sand-bed and boulder-bed streams, and a distinct relict antidune structure may be preserved as gravel lenses [Foley, 1977] or transverse ribs [Boothroyd and Ashley, 1975; Church and Gilbert, 1975; Rusland G051in, 1981J, Jia [1990J argued that the equilibrium bed morphology for sand channels tends to minimize Froude number. This conclu- s;on was derived from computer simulations of channels in the subcritical flow regime, mostly below the 0.01 slope threshold, In these simulations, where the slope equaled or exceeded 0.01. the minimum Froude number for the equilibrium channel approached 1.0 (cf, Figures 3 and 4 and Table 1 offia [1990]), The clustering of empirical points close to the threshold, as opposed to the active-bed lines, in Figure 4 similarly suggests that the Froude number wiJI not exceed one for a given set of flow conditions. In sand.bed streams, oscillation around critical flow occurs (15) in both time (i,e" Figure 2b) and space (i,e" longitudinal se- quences of standing waves), but in boulder,bed and bedrock channels the oscillation occurs primarily in space, as the result of longitudinal accelerations and decelerations through the critical threshold. This is well illustrated in the paleohydraulie reconstructions of the Bonneville Floods [O'Connor, 1993J, Kieffer's [1987] detailed study of the hydraulics of rapids in the Grand Canyon also demonstrates this spatial variation, al- though she points out that flow may, vary temporally because of surges and pulses similar to those observed in sand-bed streams. Release of water from breaking standing waves may produce unsteady flow, even if bed forms are not destroyed. Further validation of this hypothesis poses logistical but not technical problems. Needed are instantaneous measurements of velocities and depths for cross sections representing a range of stream gradients and types, under high flow conditions when bed load transport is active. Instantaneous measurements are required because of the strongly unsteady nature of high flows, The primary logistical problems are access to high flows in steep channels and accurate measurements of velocity and depth in channels undergoing rapid bed deformation and sed- iment tninsport. Examination of lItream JjA\lge flilQOn-'lt U\ I~"n. t1ty sitos with high Proude number flows provides a first cut at caodidate sites [Wahl, 1993J, From a practical standpoint an assumption of critical flow would dramatically simplify the many problems associated with predicting how floods behave in steep channels. Assumption of critical flow would essentially replace the standard resistance equations, such as Manning's or Chezy's, with the equation v = (gd)O,'- Establishing the discharge of paleofloods in steep channels for hazard assessments, climate change analy- ses, or land-use planning could then proceed by calculating the velocity directly from the depth of flow, as determined from flood deposit stratigraphy or shear stress considerations. In fact, this assumption has already been proposed on empirical grounds alone, with good results [Ai/en, 1982, p, 411J, (16) Conclusions Interactions between the water surface an-d bed forms are proposed to maintain competent flows in mobile channels to Fr :5 1, Competent, high-gradient streams with beds ranging from sand to boulders typically achieve an equilibrium adjust- ment between the flow, sediment transport, and channel mor- phology at or near critical flow, Under this unique hydraulic condition the maximum amount of water can be transmitted downstream for the available energy, and the tendency for flow acceleration is balanced by development of bed forms and Row structures that offset that tendency by dissipating flow energy, The adjustment to a critical flow tbresbold is a dynamic one, involving feedbacks between tbe free surface,' bed configura- tion, flow resistance, and the sediment transport rate, In ,and- bed channels this dynamic equilibrium generates pulsating flow with a concomitant nonuniform and unsteady flow field. In coarser~grained systems this oscillatory flow pattern is -sup- pressed somewhat by the lower bed load transport rates, more cbaotic structure of both the bed and the flow, and the impor- tance of particle interactions; that is, collisions. imbrication. and lodging of very large particles influence the hcd forms, The hypothesis that high-gradient streams adjust their hydraulics according to a common principle deserves further empirical and laboratory testing. If validated, it offers a simple. useful means of predicting flow hydraulics in mountain stream"