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350 GRANT: CRITICAL FLOW CONSTRAINS FLOW HYDRAULICS I' Froude number equals I at critical flow; when Fr > 1, the flow is termed supercritical. and when Fr < 1 the flow is subcritical. The hypothesis therefore states that supercritical flow should be rather uncommon in mobile-bed channels, except over short distances (i.e., tens of meters) and timescales (Le., seconds to minutes), For the purposes of this paper, mohile,hed channels are defined as those competent to transport their bed material, that is, where total boundary shear stress, Tn' is greater than or equal to Tcr' the critical shear stress for entraining grains on the 'treom hed, While this hypothe,is opplies to all mohile-hed chnnncls, it is most relevant to hydraulically steep streams (Le., those with gradients in excess of 0.01), since most low-gradient alluvial streams, such as the Mississippi River, arc quite com. petent but have insufficient velocities for their depths to achieve near.critical flow. The proposition that supercritical flow in channels is rare is not in itself new, and in recent years there has been consider- able debate over whether the Froude number criterion can be used to constrain indirect flood and paleoflood measurements. Much of this dehote hos focused on finding empirical examples of subcritical or supercritical flood flows, with mixed results. For example. Trieste (1992. I994J argues that many paleodis- charge 'estimates that report supercritical flow are due to un- derestimates of roughness in mountain streams; similar con- c1osions are reached hY Jarrell [1984 j, On the other hand, Wahl [1993J, Simon and Hardison [1994J, and HOllse and Peartl"ee [1995J, among others, document examples of near,critical and supercritical flow in steep channels. Little attention has been paid, however, to either the underlying physical mechanisms that determine the flow regime or to the fact that critical flow itself may represent both a threshold condition and a common flow state. As will be shown, the tendency for Fr to increase with increasing slope under conditions of- active sediment transport results in many steep. mobile-bed channels having a Froudc numher very close to unity. Furthermore, while it is quite common for flows to exceed criticality for short distances and periods of time, average Froude numbers are typically less than 1. Hence both sides in this debate may be correct. Observations in Sand-Bed Streams To test this hypothesis, I measured two small, sand-bed channels flowing over the seaward dipping, planar backshore of a heach on the Oregon coast (Figure Ia), Both channels : were 6-7 m wide and had gradients of 0,012-0,018; the bed material had a median grain size (Dsn) of 0.18 mm and a sorting coefficient (rr.l of 0.2 [Broome and Komar, 1979J, The longitudinal profiles of the two channels are independently imposed hy wave swash during storms: tidal fluctuations ch:lnge the hase level to which these streams are adjusted, however, so slope can vary slightly over the course of several hours. The comhination of constant discharge. high gradients <1t which the homogeneous. noncohesivc fine sand can he readily transported, and absence of external controls (e.g.. bedrock) mean that thcse channels arc neither supply.limited nor energy-limited and hence have unlimited freedom to adjust hed forms and cross-sectional dimensions, within the con- straint of the imposed gradient. Both of the measured channels and all otllc! streams ob- served on the heach displayed "pulsating Aow" resulting from <lltcrnate formation and destruction of standing waves and antiduncs. Cross-sectional and temporal variations in the Froudc numhcr wcre measured through a tidal cycle. Froude numbers were calculated by (1) using instantaneous measurc- ments of depth and velocity and assuming ct = 1. Depth was measured with a stadia rod calibrated and read to the nearest 0.5 cm; velocity was measured at 0.6 times depth with a re- cently ealihrated Montedoro Whitney model PVM-2A elec- tronic current meter with digital readout to the nearest 0.01 m/s and accuracy ::t 1 %. Depth of the velocity probe was con- stantly adjusted to maintain 0.6d. To get instantaneolls mea- sured pairs of depth and velocity, 1 lashed the digital readout of the current meter to the stadia rod, videotaped both. and then sampled the videotape at 5-s intervals. All measured cross sections had the same general pattern: <lverage Froutle numbcrs rangetl from subcritical (Fr < I) in the slower flow near the channel margins to near 1.0 at the center of the channel, with the average Froude number for the entire cross section slightly less than 1 (Figure 2a). The range of Froude numbers measured at a vertical section through time typically ranged from SUhcritical to supercritical (Fr > I), At the channel center and thalweg the Froude number ranged between 0.7 and 1.3 around a mean of 1.0, over cycles that ranged from 20-35 s (Figure 2h), These oscillations COffe, sponded to changes in bed and surface wave configurntionK as surface waves built, broke, and washed out the bed forms, returning the channel to a plane-bed condition with Fr < 1.0. These flow patterns persisted through channel incision, bank erosion, and planform changes accompanying the tidal cycle. These channels illustrated the physical mechanism, also noted hY others [Kennedy, 1963; Foley and Vanoni, 1977; Bean, 1977; Schllmm et ai" 1982], underlying the propo,ed hypothe, sis (Figure 3). Accelerating, near-critical flow deformed an initially plane bed (Figure 3a) into a series of antidunes and in-phase surface waves (Figure 3b). Increasing velocity and decreasing depths and corresponding scour in the wave troughs caused the surface waves and associated antidunes to steepen, become unstable as the flow became supercritical in the troughs, and break upstream as hydraulic jumps (Figure 3c). This transition to supercritical flow in the troughs was accentuated by the tendency of the antidunes to migrate up- stream, thereby becoming out of phase with the surface waves [Kennedy, 1963J, The downward flux of momentum from the breaking hydraulic jumps caused intense, localized bed scour, eroded the antidunes. restored the plane bed, and abruptly reduced the velocity (Figure 3d). Lowered velocities coupled with increased depths, as water previously stored in the sta- tionary waves was released, caused the flow to become suh. ...critical again and the cycle to repeat itself (Figure 3e). The key point is that the high-amplitude hed contigmation caused by increasing flow velocity induced flow instability at now slightly above critical. leading to very rapid energy dissipation ami erosion of bed forms. This feedback resulted.in unsteady, non- uniform flow around Fr = I (Figure 2h) ond a cyclical cre- ation-destruction sequence of bed forms [Alle11, 1976]. Release of wave-stored water as the standing waves break typically results in the formalinn of a downstream migrating hore [Foley and Vanani, 1977; Bean, 1977; Schumm el ai" 1982J. I observed bores due to wave collapse in the coastal channels with periodicities of 30 to 60 s, bore front heights of 2-4 em, and speeds of 1-2 mis, The hores contrihuted to the unsteady, nonuniform flow dynamics. The arrival of a l10re from upstream into a train of standing waves caused disruption of the flow pot/ern and either accelerated the regular cycle hY initiating breaking waves or else w<lshcd out the bed forms. reinitiating the wave-building sequence (Figure J).