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[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.

[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.

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.

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.

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...

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.

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.

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.

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.

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.

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.