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Detailed knowledge of the flow and boundary shear stress fields near the banks of natural channels is essential for making accurate calculations of rates of near-bank sediment transport and geomorphic adjustment. This paper presents a high-resolution laboratory data set of velocity and boundary shear stress measurements and uses it to test a relatively simple, fully predictive, numerical method for determining these distributions across the cross-section of a straight channel. The measurements are made in a flume with a fairly complex cross-section that includes a simulated, cobble-roughened floodplain. The method tested is that reported by Kean and Smith in Riparian Vegetation and Fluvial Geomorphology in 2004, which is modified here to include the effects of drag on clasts located in the channel. The calculated patterns of velocity and boundary shear stress are shown to be in reasonable agreement with the measurements. The principal differences between the measured and calculated fields are the result of secondary circulations, which are not included in the calculation. Better agreement with the structure of the measured streamwise velocity field is obtained by distorting the calculated flow field with the measured secondary flow. Calculations for a variety of narrower and wider configurations of the original flume geometry are used to show how the width-to-depth ratio affects the distribution of velocity and boundary shear stress across the channel.

In this paper we develop a simple, physically based method for predicting what fraction of sediment moving over the crests of dunes will bypass their lee faces. The bypass fraction is found by calculating a characteristic excursion length for every grain size making up a particular train of dunes. All particles with excursion lengths greater than the downstream span of the lee face of the average dune are assumed not to contribute to dune propagation. Bypass fractions based on distributions of excursion lengths account for the discrepancies between total sediment discharge and dune migration rate measured by Stein [1965] and Guy et al. [1966] in laboratory flumes, as well as by us in the North Loup River of Nebraska. Calculations and these data agree over sediment-transporting conditions associated with the entire stability field for dunes composed of medium sand. The two laboratory studies show that commonly 30–60% of all sand moving over fully developed dunes is not deposited on slip faces. Measurements from the North Loup River reveal that of all sediment moving over the dune crests there, roughly 45% is not being captured on lee faces even though 99% of sediment is transported within 2 cm of the bed. The method developed herein successfully estimates measured values for the bypass fraction from 0 to 80%. Our analysis indicates that the division between grains that are deposited versus those that are bypassing falls within the range of sand sizes making up the suspended load at dune crests.

[1] A method for calculating stage-discharge relations (rating curves) in gravel bedded streams is presented and applied to five reaches at U.S. Geological Survey (USGS) gaging stations. The approach, which builds on the work of Kean and Smith (2005), uses a fluid-mechanically based model to convert measurements of stage into flow and boundary shear stress fields as appropriate for determining water discharge. The model does not use site-specific empirical roughness coefficients, such as the Manning coefficient, but rather determines channel roughness from field measurements of the channel geometry and the dominant physical and biological roughness elements in the modeled reach. The approach is fully compatible with current USGS-style river gaging procedures and can be used in conjunction with or instead of the standard empirical gaging methods. When used in parallel, the theoretical rating curves produced by the model are in good agreement with direct measurements of discharge made by the USGS. The results of our analyses indicate that the theoretical rating curve approach has the potential to substantially reduce the number of measurement visits, and therefore costs, required to develop rating curves for gaging stations. Owing to this lower cost, it is well suited for regional hydrologic studies in which rainfall distributions are measured in space and time and are compared with measured discharges on the links of the river network. In addition, our method is ideally suited for sites, such as remote locations, where it is difficult or impossible to define a complete rating curve using conventional methods alone.

This paper has two purposes. The first is to use tidal-monthly variations in the density and velocity fields and the salt and water transports as key to understanding the circulation of the Columbia River Estuary and other river estuaries. The Columbia River Estuary is a good natural laboratory in this regard, because the flushing time of the system (a few days) is short relative to the tidal month during all seasons. This allows the occurrence of distinct transitions from a strongly to a weakly stratified water column (and back) during the tidal month. Furthermore, because atmospheric processes are secondary to riverflow and tidal influence in determining the circulation, most of the energy in circulatory phenomena is confined to distinct tidal, tidal-monthly and seasonal frequency bands. Observations of salt transport and neap-spring transitions reported herein should provide important constraints on future theoretical studies of estuarine circulation. The second purpose is to describe the circulation and density field of the Columbia River Estuary as background for understanding the geologic and biological investigations discussed in other papers in this volume. Previous investigations have focused on seasonal variations in riverflow as governing the turbidity maximum and biological productivity. Studies reported in this volume show that tidal monthly variations in circulatory processes are of comparable importance. With regard to the velocity field, the influence of stratification causes the tidal flow to show the largest vertical variations in phase and amplitude in the lower estuary. The vertical distribution of the mean current is controlled by the ebb-flood asymmetry in the time-dependent flow, vertical mixing processes, the baroclinic pressure gradient, and interaction of the flow with topography. Net upstream bottom flow is weak or absent during periods of weak stratification; it is substantially only when the system is highly stratified. Net upstream bottom flow tends to occur downstream of, and net downstream bottom flow upstream of, topographic highs. This pattern of convergences in the mean flow tends to preserve these topographic highs and lows and is not consistent with the traditional view that the baroclinic mean flow and mean salinity distribution structure one another; the former results instead primarily from ebb-flood asymmetry in the time-dependent flow. Comprehensive salt transport calculations were carried out, because knowledge of salt transport variations during the tidal month is vital to understanding the causes of the observed transitions in the density and velocity fields that occur between extremes of tidal range. These calculations show that neap-spring and seasonal variability of the gross features of the salinity intrusion in the lower estuary is limited by compensating adjustments of the tidal advective and mean-flow salt transports under most riverflow conditions. In contrast, the absence of such balancing mechanisms in combination with substantial neap-spring changes in vertical mixing allow large variations in the density field and salinity intrusion length in the upper estuary. Neap-spring variability is much higher during the low-flow season than during the high-flow season. Under the lowest riverflow conditions permitted by the present dam system, large tidal monthly changes in stratification occur throughout the system. The salt transport calculations also show that salt is carried into the estuary near mid-depth by tidal mechanisms acting primarily in the North Channel. Salt is transported out of the estuary closer to the surface of the South Channel by the strong mean flow. Mean velocities near the bottom are weak, and near-bottom, upstream salt transport by the mean flow is small. It is likely that inward tidal transport of salt occurs at mid-depth in many estuaries, as this is where salinity variations during the tidal cycle are usually greatest. Finally, the dynamical observations presented here, together with the known input of sediment from the river to the South Channel suggest that the concentration of fine sediment by the tidal asymmetry in a turbidity maximum should be stronger in the South than the North Channel. Limited observations confirm this.

This chapter focuses on the role that shrub, carrs play in protecting upland stream systems from damage caused by floods, the role that beaver play in maintaining willow carrs, and the interplay of beaver ponds and beaver dams with floods. These issues, however, cannot be addressed effectively without considering their ecological settings and ramifications, and particularly, the effects that humans and overpopulations of ungulates have in disrupting these ecological settings. From a human point of view, the issue may be phrased as an inquiry into whether we are making streams more vulnerable to catastrophic flooding by continually encroaching onto the floodplains of headwater tributaries. To examine the issues quantitatively, the chapter studies mathematical models for flow through shrub, carrs. The first type of model permits the stems to be rigid and tall relative to the flow depth. The second type of model allows the shrubs to be overtopped by the flow. The general conclusions from these two separate studies are of concern in this chapter.

[1] The size, shape, and spacing of small-scale topographic features found on the boundaries of natural streams, rivers, and floodplains can be quite variable. Consequently, a procedure for determining the form drag on irregular sequences of different-sized topographic features is essential for calculating near-boundary flows and sediment transport. A method for carrying out such calculations is developed in this paper. This method builds on the work of Kean and Smith (2006), which describes the flow field for the simpler case of a regular sequence of identical topographic features. Both approaches model topographic features as two-dimensional elements with Gaussian-shaped cross sections defined in terms of three parameters. Field measurements of bank topography are used to show that (1) the magnitude of these shape parameters can vary greatly between adjacent topographic features and (2) the variability of these shape parameters follows a lognormal distribution. Simulations using an irregular set of topographic roughness elements show that the drag on an individual element is primarily controlled by the size and shape of the feature immediately upstream and that the spatial average of the boundary shear stress over a large set of randomly ordered elements is relatively insensitive to the sequence of the elements. In addition, a method to transform the topography of irregular surfaces into an equivalently rough surface of regularly spaced, identical topographic elements also is given. The methods described in this paper can be used to improve predictions of flow resistance in rivers as well as quantify bank roughness.

[1] A new method for generating stage-discharge relations (rating curves) for geomorphically stable channels is presented and applied to two streams in the Whitewater River basin, Kansas. The approach converts measurements of stage into discharge using a fluid mechanically based model. The model does not use empirical roughness coefficients, such as Manning coefficients, but rather determines channel roughness from field measurements of the (1) channel geometry, (2) the physical roughness of the bed, banks, and floodplain, and (3) the vegetation density on the banks and floodplain. These measurements are used to calculate explicitly the drag on the small-scale topographic features on the boundary, the drag on the stems and branches of woody vegetation, and the friction on the bed, banks, and floodplain. The theoretical rating curves produced by the model for two study reaches, which are near U.S. Geological Survey (USGS) streamflow-gauging stations, are in good agreement with direct measurements of discharge made by the USGS. Our method has the potential of providing accurate estimates of stream flows less expensively than conventional gauging methods. In addition, the method can be used to obtain more accurate discharge estimates than conventional indirect methods for determining discharge, which are based on estimates of Manning's coefficient.

Received 26 February 2004; revised 15 September 2004; accepted 5 November 2004; published 22 January 2005. [1] Increased erosion is a well-known response after wildfire. To predict and to model erosion on a landscape scale requires knowledge of the critical shear stress for the initiation of motion of soil particles. As this soil property is temperature-dependent, a quantitative relation between critical shear stress and the temperatures to which the soils have been subjected during a wildfire is required. In this study the critical shear stress was measured in a recirculating flume using samples of forest soil exposed to different temperatures (40� –550� C) for 1 hour. Results were obtained for four replicates of soils derived from three different types of parent material (granitic bedrock, sandstone, and volcanic tuffs). In general, the relation between critical shear stress and temperature can be separated into three different temperature ranges ( 275� C), which are similar to those for water repellency and temperature. The critical shear stress was most variable (1.0–2.0 N m � 2 ) for temperatures 2.0 N m � 2 ) between 175� and 275� C, and was essentially constant (0.5–0.8 N m � 2 )f or temperatures >275� C. The changes in critical shear stress with temperature were found to be essentially independent of soil type and suggest that erosion processes in burned watersheds can be modeled more simply than erosion processes in unburned watersheds. Wildfire reduces the spatial variability of soil erodibility associated with unburned watersheds by eliminating the complex effects of vegetation in protecting soils and by reducing the range of cohesion associated with different types of unburned soils. Our results indicate that modeling the erosional response after a wildfire depends primarily on determining the spatial distribution of the maximum soil temperatures that were reached during the wildfire.

The Reynolds stress field was measured near the bank of the Powder River in southeastern Montana. The measurements were made from the bank using an aluminum I-beam cantilevered over the water to support a carriage system for positioning an acoustic doppler velocimeter in a vertical plane perpendicular to 1) the bank and 2) the streamwise velocity field. During quasi-steady flow at the peak (71 ms) of the spring snowmelt runoff in May 1996, turbulent velocities were measured at 25 Hertz along six vertical locations spaced 0.5 m apart and within about 3.5 m of the riverbank. When the turbulent velocities are transformed to the ray-isovel coordinate system appropriate for this two-dimension problem, the turbulent characteristics near the bed are consistent with similar field measurements made by others for the one-dimensional problem of uniform flow over a horizontal bed far from lateral boundaries. The three turbulent intensities, 2 1 2 2 1 2 ) ( , ) ( v u ′ ′ and 2 1 2 ) ( w ′ , normalized by the local shear velocity, *, u were essentially constant with distance above the bed along a ray and the average values were 2.1, 1.4, and 1.2. Future turbulence measurements could be improved by measuring the streamwise flow first, then determining the approximate location of the rays and isovels so that the turbulence measurements could be made along the approximated rays rather than along verticals. In addition, to improve the possibility making turbulence measurements during steady, uniform flow, the site should be carefully selected to minimize local flow accelerations caused by spatial variability of the riverbank. Also, the measurements should be made at times when the stage is constant, no local erosion or deposition of sediment occurs, and when wind velocities are small. 1

Point measurements of flow speed and suspended-sand concentration were made from a cableway 293-km downstream from Glen Canyon Dam during the 1996 controlled flood. The data demonstrate a systematic fining of the suspended load in the Colorado River, a reduction in near-bed sand concentration with time, and a strong secondary circulation that very effectively transported suspended sand toward the channel margins. In the center of the river, the primary flow was well represented by steady, horizontally uniform flow theory, with a shear velocity of 0.081 m/s and a sand grain related roughness parameter of 4.5.10 -6 m; at the channel margins the primary flow exhibited a distinct internal boundary layer with a shear velocity of approximately 0.081 m/s and an outer boundary layer with a shear velocity of approximately twice that value. The secondary circulation was caused by long wavelength irregularities in the rockfall-produced sloping banks of the approximately trapezoidal channel. The primary flow was forced upward and toward the river center by these topographic features causing a fully 3-dimensional circulation. The upward forced vertical velocities apparently interacted with turbulence in the primary flow to produce boils. Consequently, the upwelling zone degraded to an irregular, bank-parallel boil line. Downwelling occurred over a broad region in the center of the river, but also was concentrated along well-defined convergence zones over which woody debris concentrated. This secondary circulation was very effective in transporting suspended sand toward the channel margins at the bottom, then lifting it in the boils and depositing it inshore of the boil line on the riverbanks.

The beds of fluvial channels in sandy depositional systems are mantled with numerous periodic or quasiperiodic topographic elements that can be assigned to one of two hydrodynamic groups: bed form or bar form. Bed forms (ripples and dunes) arise from convergences in the sediment transport field that are associated with boundary shear stress variations produced by vertical adjustment of the flow over periodic, small-scale morphological features; bar forms are the result of adjustments in this field caused by deflection of the flow around larger scale topographic elements. Owing to these differences, the former has heights that are only a fraction of the water depth (less than half), whereas the latter closely approximates the formative flow depth. Sandy channel fills are often characterized by stacked sets of foresetted beds. Each of these cross-stratified layers documents bar form or bed form migration and deposition. To develop a precise hydraulic reconstruction of channel flow the bed form/bar form distinction must be made for each layer and the relationships of successive layers must be carefully accounted for. Their work on the fills of modern and ancient low sinuosity channels indicates that bar form cross-stratification is readily distinguishable from dune stratification by both the frequentmore » preservation of topset-related strata and a systematic decrease in grain size along the lower portions of foreset laminae. Only when the salient features of the channel-bed morphologies are recognized and characterized in a proper hydrodynamic model can a reliable paleohydrologic reconstruction be made.« less

Long-wave theory is used herein to analyze circulation in weakly stratified and partially mixed estuaries. Unlike the highly stratified systems considered in part 1, the flows considered here have only a minimal tidal-frequency internal wave component. These estuaries may therefore be modeled as the sum of weakly interacting barotropic and baroclinic modes. The dominant factors driving the residual flow are finite amplitude barotropic effects in weakly stratified estuaries and a combination of barotropic effects and steady horizontal density gradient forcing in partially mixed estuaries. The dominant vertical exchange mechanism in the weakly stratified case is bottom boundary-induced turbulence, and that in partially mixed estuaries is believed to be random internal wave interactions. A model of the Columbia River Estuary under weakly stratified conditions accurately predicts the observed residual velocity and salinity fields, and the tidal amplitude at which a transition to a highly stratified state takes place. The partially mixed state is unstable in the Columbia River Estuary because the tides and buoyancy input are both too strong.

Long-wave theory and simple turbulence closures have been used to show that three distinct types of circulation (highly stratified, weakly stratified, and partially mixed) arise in narrow, shallow estuaries from the finite amplitude of the tide and the interaction of stratification with vertical mixing. Each type has a different dominant process causing the vertical exchange of salt and fresh water on the flood, and each gives rise to a characteristic residual circulation. The tidal circulation in highly stratified shallow estuaries, described herein, is the result of a finite amplitude internal motion driven by the barotropic tide; shear instabilities at the interface are the major vertical exchange mechanism. The residual circulation is caused primarily by ebb-flood asymmetry in interface position and thickness. A model based on our theoretical analysis shows that the interface thickness on flood is much less than the depth of flow up to a critical tidal amplitude at which the two-layer flow is destroyed, and correctly predicts the tidal amplitude of the neap-spring transition in the Columbia River Estuary. Tidal frequency internal wave motion is not found in weakly stratified and partially mixed estuaries. Because of the reduced tidal shear, shear instabilities are weak or absent, and the residual circulation assumes a very different character, as described in part 2.

When non‐cohesive sediment is set in motion by a unidirectional flow, waves of sand often result; moreover, these waves typically are asymmetrical with steep lee faces that produce flow separation. Behind each wave, a wake region forms which grows in height and decays in strength with distance downstream, producing a near‐bottom acceleration. The no‐slip condition at the bed, however, requires an internal boundary layer to form beneath the wake region which retards the flow there. The interaction of these flow processes produces a local maximum in the boundary shear stress downstream of a bump, even over an otherwise flat bed. Because the erosion rate is proportional to the stress divergence for bed load and weak suspended load transport, erosion will occur upstream of this point and desposition will occur downstream of it, thus influencing the bottom shape. Herein, the equations of motion are solved for the accelerating internal boundary layer beneath a wake. The resulting velocity and boundary shear str...

Hydrographic data from T-3 are analyzed to illustrate the behavior of the Arctic mixed layer. The mixed layer depth fluctuates 11 m annually and mixed layer salinity fluctuates 0.32‰. The fluctuations in total salt content are consistent with theoretical work by Maykut and are in phase with mixed layer depth, indicating changes in the mixed layer are controlled by salt flux. Deepening of the mixed layer brings 0.3 kcal cm−2 yr−1 of heat to the surface from the ocean below.

The boundary shear stress pattern and the superelevation of the water surface in a meander on a small stream are predicted from two simple equations representing a frictionally dominated force balance. This comes about even though inertial forces due to local bed topography delay crossing of the boundary shear stress maximum to a position farther downstream than would otherwise be expected. Measured bedform migration rates reflect the boundary shear stress field, and in this river, bedform crest orientations respond to gradients in the shear stress field by becoming oblique to the general flow direction. When so aligned the bedforms interact with the helical flow in the curved channel and induce a near bottom secondary current which, in the upstream part of the bend, is outward above the crests and inward along the troughs. At the downstream end of the bend this pattern is reversed and significant quantities of sediment are transported along troughs from one point bar to the next. This near bottom flow pa...

Measurements of the profiles of salinity and velocity at a station in the Duwamish River estuary indicate that the turbulent mixing through the density interface is highly time dependent with the most intense mixing occurring at the maximum current speed during the ebb. Additional measurements of the turbulent kinetic energy and the turbulent flux of salt by the eddy correlation technique verify that interpretation. The vertical turbulent salt flux and the turbulent kinetic energy vary by an order of magnitude over the tidal cycle and approximately half of the total vertical salt transport takes place during a short time centered about the maximum ebb. Different mechanisms of turbulence generation operate at various times during the tidal cycle and the intense mixing periods are shown to coincide with conditions favorable for the formation of an internal hydraulic jump.

A bed load transport model based on the mechanics of sediment moving by saltation yields predicted values of bed load flux as a function of boundary shear stress, grain diameter and density. The parameters required to calculate bed load transport (particle velocity, bed load sediment concentration, and the height of the bed load layer) can all be determined from our model, which computes sequences of trajectories of individual saltating grains as well as the concentration of moving grains that the flow can support. The latter is related to the momentum the accelerating grains extract from the flow. Predicted curves of bed load transport vs. boundary shear stress agree well with transport data collected by Gilbert Meyer-Peter \Iet al.,\N and the Waterways Experimental Station; measured shear stress was corrected for pressure drag when bed forms were present. Comparison of predicted bed load transport with common bed load equations reveals considerable similarity among the relationships. The best agreement with the data is produced by relationships in which the transport rate vanishes as the shear stress approaches the critical Shields’ number, as in our model and Yalin’s equation.

Shear instability is found to be the principal mechanism of vertical exchange within the pycnocline of a salt wedge estuary. A field program involving high-resolution velocity and density measurements, as well as high-frequency acoustic imagery, allowed direct comparison of instantaneous Richardson number distributions to the occurrence of shear instability. The theoretical stability threshold of 0.25 is consistent with the measurements, based on estimates of gradients that contain the mean as well as fluctuations due to internal waves. An effective stability threshold based on mean gradients is found to be approximately one-third, reflecting a significant contribution of internal wave shear. The integral effect of the mixing process is to homogenize the gradients of velocity and density, producing linear profiles of these quantities across the pycnocline. A turbulent Prandtl number of unity is suggested by the vertical distributions of velocity and density during periods of active vertical mixin...

An expression for the critical shear stress noncohesive sediment is derived from the balance of forces on individual particles at the surface of a bed. The resulting equation, for a given grain size and density, depends on the near-bed drag force, lift force to drag force ratio, and particle angle of repose. Calculated values of the critical shear stress for uniformly sized sediment correspond closely to those determined from Shields' diagram. The initial motion problem for mixed grain sizes additionally depends on the relative protrusion of the grains into the flow and the particle angle of repose. The latter decreases when the diameter of a moving grain, D, is larger than the length scale of the bed roughness, ks (D/ks > 1), and increases when D/ks < 1, producing a corresponding decrease or increase in critical shear stress. Using the Miller and Byrne experimental relationship between D/ks and particle angle of repose, which is consistent with Shields' definition of initial motion, we obtain results that are in good agreement with the available experimental critical shear stress data for heterogeneous beds.

Current observations over the continental shelf at locations off central Oregon and southern Washington had the period from July 18 to September 18, 1972, in common. Low-frequency fluctuations (less than one cycle per day) in the currents are compared by means of visual display, linear regression, and spectral analysis. The currents are found to be highly coherent over an alongshore separation of 200 km. Coherent signals occur at 0.16, 0.3, and 0.44 cpd. The signal at 0.16 cpd occurs with high mutual coherence in wind, current, and sea level and may be a forced shelf wave driven by the wind. The signal at 0.3 cpd has high coherence between current and sea level and may be a free shelf wave generated by the wind. Correlation between observations is higher for separations in the alongshore direction than in the offshore direction in spite of greater separations.

Observations of tidally driven flow of stratified water over a sill in Knight Inlet, British Columbia, have revealed a broad variety of different interactions depending upon the degree of stratification and the strength of tidal forcing. The flows are described on the basis of a Froude number dependence relating the barotropic tidal velocity to the long internal wave speed over the sill crest. For flows critical or supercritical with respect to all internal modes, an internal hydraulic jump occurs. However, typical summer conditions result in flow that is subcritical with respect to the lowest mode but supercritical with respect to higher modes; the interaction in this case results in a massive mode 2 lee wave or jump accompanied by instabilities up to 50 m high. Early in the tidal cycle the boundary layer separates from the sill and apparently suppresses lee wave formation, but this effect is inhibited at higher speeds. A further, unexpected finding is the generation of nonlinear mode 1 internal wavetrains upstream of the sill during strong tides. While these waves are similar in appearance to those predicted by Lee and Beardsley (Journal of Geophysical Research, 79, 453–462, 1974), there are important differences in the relative location and timing of their formation. The observations reveal the way in which lee waves and jumps relax as the tidal flow slackens, subsequently evolving into travelling internal bores or surges. The observations are compared with recent laboratory studies.

In a channel with bed topography that does not vary in the downstream direction, a secondary circulation composed of outward flow at the surface and inward flow near the bottom extends across the entire width. If the curvature is constant, the cross-stream velocity component near the bed and the pattern of boundary shear stress can be estimated by assuming fluid accelerations to be small. Unfortunately, this procedure cannot be used in analyzing the flow through natural river meanders, or through channels with downstream constant bottom topography but with rapidly changing curvature. In these latter cases, effects arising from bed- and bank-induced momentum changes must be accounted for. Evidence for a substantial topographically induced alteration in the cross-stream flow pattern relative to that for the analogous constant bottom topography case is provided through new analyses of several sets of laboratory and field data. Shoaling over the point bar in the upstream part of the bend is shown to force the high-velocity core of the flow toward the pool. This is accomplished by a convective acceleration-caused decrease in the cross-stream water surface slope and a resulting dominance of the vertically averaged centrifugal force. The primary effect is a velocity component toward the outside or concave bank throughout the flow depth over the upstream, shallow part of the point bar and an outward component of boundary shear stress in this region. The channel curvature-induced inward component of boundary shear stress consequently is confined to 20 or 30% of the channel width at the pool. Outward transfer of momentum over the point bar, as manifested by a rapid crossing of the high-velocity core from the inside bank to the outside one, contributes to an enhanced decrease in boundary shear stress along the convex side of the stream as the top of the bar is approached. Forces arising from topographically induced spatial accelerations are of the same order of magnitude as the downstream boundary shear stress and water surface slope force components, so they must be modeled as zero-order, not first- or second-order, effects.

An understanding of the mechanics of nonuniform flow is important in a variety of ecological and geophysical fluid mechanical problems. Moreover, the ability to predict local boundary shear stress on an uneven bed is essential in erosion and sediment transport problems. In order to elucidate the important fluid mechanical phenomena active over natural quasi-two-dimensional bed forms a series of detailed flow measurements were made above 60- to 100-m-long 1- to 3-m-high dunes in the Columbia River. In this paper, velocity profiles obtained by averaging these flow data along lines of constant distance above the riverbed are examined, and it is shown that they can be constructed from well-known uniform flow results used in conjunction with a hypothesis about the structure of internal boundary layers. This approach permits skin friction as well as total boundary shear stress, averaged over one wavelength of the bed form, to be determined from spatially averaged velocity profiles and, conversely, provides a mechanism whereby zero-order velocity profiles can be constructed for two-dimensional nonuniform channel flows. Corrections for changes in the bottom roughness parameter caused by bed load transport and for flow stratification induced by suspended load transport are derived and applied in order to make the results consistent with the measured spatially averaged shear stress field.

Erosion and deposition in streams and tidal channels depend on the divergence of the sediment-flux field, and this is governed by the distribution of boundary shear stress. As a consequence, erosion arid deposition patterns in such systems are sensitive to spatial variations in boundary shear stress, which in turn, can be induced by the complex interplay between the flow and bed and bank topography. A method whereby variations in boundary shear stress can be calculated is presented herein for flows that are broad in relation to their depth and for which the bed slopes are small. The formal scheme employs a regular perturbation expansion around a zero-order state that includes vertically integrated, topographically induced convective accelerations. Use of a zero-order velocity field that includes these accelerative effects yields a model that is applicable to streams with typical bed slopes and channel curvatures, one that can be employed in the majority of situations of interest to geomorphologists, sedimentologists, and hydraulic engineers concerned with fluvial and estuarine systems. In order to verify the model, our calculations are shown to reproduce, with reasonable accuracy, the free-surface topography and boundary shear stress distribution measured by Hooke (1975) when applied to a channel of the geometry that he investigated. The latter part of the paper describes how the interplay of the various components of flow through a curved channel with bar-pool topography is affected by the parameters of the problem.

Bed load transport in Muddy Creek, Wyoming, a sand-bedded meandering river with equilibrium bottom topography, was found to consist of a zone of maximum sediment flux that shifted across the channel from near the inside bank in the upstream part of the bend toward the pool at the minimum radius of curvature. Significant net cross-stream transport continued even through the crossings between the bends. The downstream bed load transport field for the bend which was studied in greatest detail was the same as that computed from bed form migration measurements and can be predicted from appropriate boundary shear stress data and the Yalin bed load equation. The zone of maximum bed load transport followed an outward-shifting region of maximum boundary shear stress, although in the downstream end of the bend the sediment transport maximum tended to stay closer to the centerline than the boundary shear stress maximum due to particle size influences. Net cross-stream transport varied with particle size and was effected by three major processes: topographically-induced, near-bed, cross-stream flow; trough wise flow along obliquely oriented bed forms; and rolling or avalanching of particles on bed form lee faces plus rolling or mass sliding on a steep cross-stream point bar side slope. Coarse particles were carried outward over the top of the point bar by a near-bed cross-stream flow that was induced by downstream shoaling. These particles then rolled and slid on the point bar side slope and eventually were carried toward the outer bank by troughwise transport along oblique dunes. In the upstream part of the bend, fine particles were carried inward by the channel curvature-induced, near-bed flow and by troughwise transport along lee faces of oblique bed forms. Fine particles and coarse particles crossed paths on top of the point bar, where weak troughwise flow moved just the finer particles toward the convex bank, and on the point bar face, where coarse particles rolled against the secondary circu- lation that was carrying the finer particles. Net cross-stream bed load transport was toward the pool and was on average about 10% of the downstream bed load transport. The Engelund equation predicts reasonably well the general pattern of net cross-stream transport through the meander, but it does not account for the substantial troughwise transport caused by oblique bedforms at several locations in the bend. Our observations suggest that equilibrium bed topography occurs when there is net outward transport into the outward shifting zone of maximum boundary shear stress.

The mean and turbulent flow structure under pack ice was measured during the 1972 AIDJEK pilot study with small mechanical current meter triplets at eight levels in the planetary boundary layer. CTD profiles showed a well-mixed layer of nearly neutral stability to about 35 m, bounded below by a strong pycnocline. The skin friction velocity u* was determined by measuring the Reynolds stress at 2 and 4 m below the ice (beyond the surface layer) and from consideration of other terms in the mean momentum equation. Local pressure gradients and advective acceleration due to topography could not be ignored; when an estimate of the effect was included, u* was 1.0±0.1 cm s−1 when the ice velocity relative to the ocean was 24 cm s−1. With the proper coordinate transformation, the planetary boundary layer of the ocean resembles that of the atmosphere. Composite averages of non-dimensional Reynolds stress and mean flow in the ocean, when compared with recent models of a neutrally buoyant, horizontally homoge...

Models for the combined effect of waves and currents on the continental shelf developed by Smith and by Grant and Madsen predict near-bottom velocity profiles and values of boundary shear stress that agree reasonably well with reanalyzed data collected byCacchione andDrake (Journal of Geophysical Research,87, 1952–1960, 1982) using the Geoprobe system in Norton Sound, Alaska. The data set includes velocity profiles recorded during a two-day storm, sequences of which were averaged to indicate the mean flow conditions during two periods of relatively steady flow. Estimates of the reduction in vertical diffusion of momentum and mass as a result of near-bottom suspended sediment indicate that no suspended sediment stratification correction is required in this case. The values of the shear velocity associated with the mean current, determined from the averaged velocity profiles after making a zero-plane adjustment, are better predicted by the wave-current interaction models than by a similar model that neglects the wave contribution. The results of theSmith (The sea,6, 539–577, 1977) model, presented for two different forms of the eddy viscosity, indicate a significant enhancement of the boundary shear velocity due to the current and the waves, compared to the slope of the velocity profile above the wave boundary layer, due to the current only.

A Preston tube and an array of miniature current meters for use in rivers and estuaries are described. Procedures are presented for making a Preston tube determination of boundary shear stress on hydraulically smooth, transitional, and rough surfaces in such natural waterways. Limited field results available appear to substantiate the method.

The problem of sand bed stability is formulated and solved for a model in which the local slope of the free surface is small relative to the slope of the perturbed lower boundary. The solution shows the bed to be unstable to perturbations of wavelength greater than the wavelengths for which the inertia of the sediment grains is important. Wave speeds and growth rates from numerical calculations are presented and compared with long- and short-wave asymptotic expressions. Although the fluctuating velocity field in the nonuniform flow is approximated by a constant eddy viscosity, a general picture of the relationship between the flow over a perturbed boundary and the observed geometrical and dynamical parameters of the resulting boundary waves can be developed. It is shown that the sand bed instability is due to local accelerations of a shear flow caused by the nonuniform boundary. If either the shear or the acceleration is eliminated from the analysis, the instability cannot be found. On the other hand, no wavelength is defined by such an analysis, and it is shown that the wavelength is determined by spatial adjustments in the turbulent velocity field.

Estuaries usually are regions in which mixing of miscible fluids with differing densities is occurring; thus they are regions in which turbulent processes are of particular importance. Moreover, the nature of these turbulent processes, although highly variable, has a profound effect on the circulation. As a first approximation, theoreticians often write the diffusion of mass and momentum in terms of a tensor eddy coefficient having spatially varying components. In some cases this mathematical procedure closely approximates the physics of the problem whereas in others, it is reasonable neither from an oceanographic nor mathematical point of view. Without a physical understanding of the basic processes that ultimately result in mixing in systems of various types, it is impossible to judge the validity of such simplifications for specific embayments. Nevertheless, surprisingly little basic research has been done on the small scale processes that are responsible for turbulence production in such environments.

It is now well known that tidally induced flow of stratified water over bottom irregularities can generate trains of internal waves whose envelope is of tidal frequency but which contain significant higher frequency components (Lee, 1972, Bell, 1975). Moreover, the existence of wave trains that might have been generated in this way has been demonstrated both by satellite imagery and direct observation in several places; recent examples include Massachusetts Bay (Haury, Briscoe and Orr, 1979) and the Andaman Sea (Osborne, Burch and Gordon, 1978). Nevertheless detailed measurements in the area of generation have been largely lacking. We describe here some observations of tidally induced flow over a sill in Knight Inlet, B.C. and identify certain similarities between these observations and recent laboratory experiments. The study of such flows is of value not only from the point of view of providing a better understanding of fjord circulation, but also because it provides a geophysical example of phenomena that are of basic fluid mechanical interest.

An accurate knowledge of flow in the immediate vicinity of the sea bed is important in marine geological, benthic ecological, geochemical, and sediment transport studies. However in many cases, the velocity field is complicated by the presence of ripples and dunes on the sea bed and suspended sediment-induced stratification in the flow. Recently techniques for handling these factors were developed by the authors, but they were applied only to a situation where the sea bed was comprised of a single size and specific gravity class. In this paper these techniques are extended to the case where the suspended material is characterized by an ensemble of settling velocities and critical shear velocities. The results are applied first to a flat sea bed and then to spatially averaged flow over a wavy boundary such as might be produced by natural bed forms. These calculations indicate that the sediment transport process can have a significant effect on flow near the sea bed and that proper account must be taken of the settling velocity distribution comprising the suspended sediment concentration field.

Experiments carried out by Partch and Smith (1977) at anchor stations in the Duwamish River in Seattle, Washington showed a period of intense vertical salt flux during the ebb tide. In order to examine the longitudinal structure of these mixing events, as well as to procure data relevant to the overall dynamics of the estuary, a system was developed to permit measurement of the velocity and density fields from a moving vessel. The new system was used in an experiment in March 1977. Data obtained during this experiment indicate that the intense mixing events may be triggered by a hydraulic jump that occurs at a sharp change in river depth. The relatively high salinity water formed at the jump advects downstream, but vertical mixing continues with a salt flux of at least 0.5 gm/m 2 .sec. It is postulated that this continued mixing is related to an internal hydraulic instability. As it occurs during periods of minimum shear, shear instability is ruled out as a source of energy. In addition to the intense mixing events, an internal hydraulic jump that forms at the 16th Avenue bridge around the time of maximum ebb is described. While this apparently is not related to the intense mixing events, it is of interest in its own right, and indicates the importance of internal hydraulics to the dynamics of salt wedge estuaries.

Many fjords along the British Columbia and Alaskan coasts are characterized by both a high freshwater discharge and a large tidal range. They also commonly have sills that penetrate well into the stratified near-surface layers, constricting the tidal flow and causing it to accelerate in the region of strong density gradient. An investigation of the nature of mixing in one such fjord, Knight Inlet, began in 1977. The observations were made with at least two vessels, using density and velocity profiling instruments together with high frequency echo sounders. A general description of the processes involved is given by Farmer & Smith (in press).

A pilot study in Knight Inlet, British Columbia has yielded measurements of large amplitude, nonlinear internal waves in the strongly stratified surface layer. A train of several waves with amplitudes exceeding 10 meters and periods of a few minutes, passed the vessel on each flood tide. Observations with profiling and fixed depth instruments showed that the waves travelled in a strongly sheared flow. The results are interpreted using Benjamin's (1967) approximate model for nonlinear waves in deep fluids, suitably modified to allow for shear. This model then provides the basis for an examination of conditions leading to shear flow instability. A solution to the eigenvalue problem for exponential stratification and shear is used to examine the interaction of the waves with the background flow; critical amplitudes corresponding to conditions of marginal stability are found to be comparable to those observed in Knight Inlet. The waves are discussed in terms of their likely contribution to vertical exchange processes in fjords.

The equations of motion for a sediment grain near a noncohesive bed and those for the local fluid flow are combined to produce a set of differential equations that can be solved numerically to describe the trajectory of a saltating grain as a function of time. The lift coefficient is set based on a reanalysis of data produced by Chepil, and the other parameters of the problem are set using standard fluid mechanical relationships; the initial velocity and position are specified with a separate model. The heights of the resulting trajectories are found to be significantly lower than available measurements would indicate; therefore two extensions of the model are examined: the effect that spin lift and form lift have on a series of trajectories, and the effect of partially elastic collisions between a moving grain and the bed over which it is traveling. A model that includes the second process is less complicated than one that includes the first, and it yields trajectories that agree very well with available experimental measurements when a rebound coefficient that varies around 0.5 and depends on the impact momentum is used. We conclude that the model provides a reasonable representation of saltation in water and that both fluid-solid and solid-solid interactions are required to reproduce the parameters of measured trajectories.

To assist in elucidating the turbulent structure of flow over regular sets of bed forms, data procured with small mechanical current meters above 2 m high 96 m long sand waves in the Columbia River are analyzed. The high-frequency resolution of the flow sensors and the half hour records that were collected with them are shown to be sufficient to yield reliable Reynolds stresses. In addition, an expression from which the sampling error in Reynolds shear stresses can be estimated from the sampling frequency, the local flow speed, and the distance from the boundary is presented. Although the κ−5/3 region of the energy spectrum was found, the conditions for isotropy were not satisfied anywhere within the measured band. Spatial structure of the Reynolds stress components was found to be complicated in character but explicable in basic fluid mechanical terms. Owing to the near boundary effects of form drag induced by the sand waves, the spatially averaged shear stress increased from 20 dyn/cm2 at 35 cm to 40 dyn/cm2 at 100 cm. Furthermore, the Reynolds shear stress was found to vary with the structure of the variance of the vertical velocity component rather than with that of turbulent kinetic energy as assumed in the simple closure models that have been used so far in theories for flow over such bed forms. The data also were examined for flow intermittency, and the results of Gordon and Witting [1977] were not confirmed.

A simple predictive model for velocity and boundary shear stress fields over two-dimensional bedforms of finite amplitude is presented herein, and the results of that model are compared with flume data. The production of a wake due to separation of the flow near the bedform crest and the subsequent interaction of this wake region with a developing nonuniform boundary layer downstream of the flow reattachment point are treated using the method of McLean and Smith (1986). Their model is modified in order to take account of the flow response to short-wavelength obstructions more accurately and to include the effects of wakes from upstream bedforms as these wakes coalesce in the interior. The theory is also extended to include the region between the bedform crest and the reattachment point, thereby providing a spatially complete field of velocity and shear stress. Results of the calculations are found to be in good agreement with laboratory measurements taken over immobile bedforms using a laser-Doppler velocimeter. Our finite amplitude flow model is shown to yield accurate predictions of the fluid dynamical effects ultimately responsible for the morphology of well-developed natural bedforms.