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<br />WATER RESOURCES RESEARCH, VOL. 33, NO.2, PAGES 349-358, FEBRUARY 1997
<br />
<br />Critical flow constrains flow hydraulics in mobile-bed streams:
<br />A new hypothesis
<br />
<br />Gordon E. Grant
<br />Pacific Northwest Research Station, USDA Forest Service, Corvallis, Oregon
<br />
<br />Abstract. A new hypothesis predicts that in mobile-bed river channels, interactions
<br />between the channel hydraulics and bed configuration prevent the Froude number (Fr)
<br />from exceeding 1 for more than short distances or periods of time. Flow conditions in
<br />many steep, competent streams appear to be close to critical. Froude numbers of steep
<br />(slope = 0,01) sand-bed streams with considerable freedom to adjust boundaries oscillate
<br />between 0,7 and 1.3 over 20- to 30-s cycles, with an average of 1.0 at the channel thalweg.
<br />Critical flow in these streams is maintained by the interaction between the mobile bed and
<br />free water surface at high Fr, which results in a cyclical pattern of creation and
<br />destruction of bed forms, Field observations support that a similar mechanism of flow-bed
<br />form interaction constrains Fr :5 1 in active-bed braided gravel rivers, step-pool streams,
<br />laboratory rills, lahar-runout channels, and even some bedrock channels, Empirical and
<br />analytical results show that as slope increases, competent flows tend to asymptotically
<br />approach critical flow. An assumption of critical flow would dramatically simplify
<br />paleohydraulic flow reconstructions and modeling of flow hydraulics in high gradient streams,
<br />
<br />Introduction
<br />
<br />The morphology of alluvial channels reflects a complex in-
<br />teraction among flow hydraulics, channel geometry, energy
<br />Ji!.sipation, and sediment transport. This relationship is at the
<br />heart of Mackin's [1948, p. 471] famous dictum that "a graded
<br />stream is one in which, over a period of years, slope is deli-
<br />cately adjusted to provide, with available discharge and pre-
<br />vailing channel characteristics, just the velocity required for the
<br />transportation of the load supplied by the drainage basin."
<br />Ever since Mackin, fluvial geomorphologists and engineers
<br />have been searching for the principles and physical mecha-
<br />nisms underlying channel adjustments. This work has taken On
<br />new urgency as direct streamflow and channel modifications,
<br />land use, and changing climate alter water and sediment trans-
<br />port regimes, sometimes changing the channel morphology
<br />and ecological functioning of streams. The many mutually de.
<br />pendent variables range over wide spatial and temporal scales
<br />in fluvial systems, however, and this variability has forestalled
<br />any comprehensive or deterministic predictions of how hydrau-
<br />lics, sediment transport, and channel morphology adjust to
<br />changes in driving variables [Schumm and Lichty, 1965; Sling,
<br />erlalld, 1981],
<br />These mutual adjustments are perhaps best understood in
<br />sand-hed streams, where hoth field and lahoratory observa-
<br />tions demonstrate the close interaction among the hydraulic
<br />free surface, channel bed forms, and sediment transport dy_
<br />namics [Kelllledy, 1963; Foley alld Vallolli, 1977; Bean, 1977;
<br />Schumm et aI., 1982], For example, as flow velocity increases,
<br />bed forms in sand channels typically undergo a transition from
<br />plane bed to dunes to antidunes, with concurrent changes in
<br />the free surface from flat to ripples to standing waves. Much
<br />less understood is the relation among flow hydraulics, hed
<br />forms, and channel morphology in steeper, coarser-grained
<br />
<br />This paper is not subject to U.S. copyright. Published in 1997 by the
<br />Aml:rican Geophysical Union.
<br />
<br />Paper number 96WR03134.
<br />
<br />channels, such as mountain streams, which commonly display a
<br />diverse array of bed forms (e.g., transverse ribs in gravel chan-
<br />nels, step-pools in houlder-bed streams) and complex hydraulic
<br />phenomena (e.g., breaking standing waves, hydraulic jumps).
<br />Differences in the abundance of these hydraulic and bed form
<br />features form the basis for many of the proposed channel
<br />classification schemes for steep channels [Le., Grant et al.,
<br />1990; Richards alld Clifford, 1991; MOlltgomery alld Buffillgtoll,
<br />1993]. But no comprehensive theory yet exists to link flow
<br />hydraulics and bed forms across a range of channel types, bed
<br />material, and slopes.
<br />As a first step toward developing such a theory, a new hy-
<br />pothesis is proposed here that argues that a similar mechanism
<br />of flow-bed form interaction as observed in sand.bed streams
<br />applies to a wide range of mobile.bed channels. Support for
<br />this hypothesis is developed in three parts: first by detailed
<br />examination of the mechanism observed in sand-bed streams,
<br />then ,through an empirical analysis of hydraulic data from a
<br />wide range of mobile~bed channels, and finally by reference to
<br />studies on diverse channels drawn from the literature. Taken
<br />together, these results suggest that the hypothesis is reasonable
<br />and consistent with the best available data; proof of the hy~
<br />pathesis awaits a more rigorous theoretical analysis of bed-flow
<br />interactions.
<br />
<br />The Hypothesis
<br />
<br />The hypothesis states that in mobile-bed river channels, in-
<br />teractions between the channel hydraulics and bed configura,
<br />tion prevent the Froude number ftom exceeding one for more
<br />than short distances or periods of time. The Froude number
<br />(Fr) is defined hy
<br />
<br />aO.sv
<br />Fr = (gd)O,'
<br />
<br />where v is flow velocity, d is flow depth, a is the kinetic energy
<br />correction factor, and g is gravitational acceleration. The
<br />
<br />(t)
<br />
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