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[1] We explore a stochastic component of topographic evolution of sandy river beds and its relationship to bed material flux. The behavior of trains of mobile bed forms can be decomposed into two independent constituents, translation and deformation. Translation is the mean downstream migration of the bed at velocity that defines the Lagrangian reference frame of the bed. Deformation is the sum of all changes to the bed's topographic profile measured from within the bed's moving reference frame. The occurrence of deformation leads to exponential decorrelation of bed topography that is in dynamic equilibrium with flow conditions. For the field and laboratory data sets used, correlation decays to 0.5 by the time the bed translates 40% and 360% of the mean bed form length, respectively. Proportions of bed material flux responsible for translation and deformation can be straightforwardly calculated. Translation flux is measured using the traditional bed form-bed load equation. Deformation flux is determined by excess topographic change scaled by the ratio of horizontal sediment velocity to fall velocity. Deformation represents the sediment exchanged between bed load and suspended load. Because deformation is a stochastic process with zero mean, the apparent rate of deformation decreases as a function of time interval between bed surveys. For the field case, deformation accounts for 40% of bed material flux while it is only 1% of the flux in the laboratory.

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[2] The transport of sand by flowing water and the emergence of bed form topography at the sediment-water interface are phenomena with a rich history of observation from Sorby [1859] and Gilbert [1914] until today. The essential problem is to understand the state and evolution of a series of sandy bed forms and its relations to both the sediment-transport and water-flow fields. In this paper we put forth a new framework motivated by the oldest type of observation, visual. Scientists who have watched bed forms migrate in a flume or in natural systems have undoubtedly noticed that each crest and trough combination is at least slightly different in size and shape from all of its neighbors. Additionally, with observation over enough time they would notice that the bed forms change their shape and arrangement as they move along, whether or not the bulk flow and sediment transport conditions are steady. These aggregate changes in sizes, shapes and arrangements of bed forms, exclusive of their mean downstream translation, define the active deformation of sandy river bottoms.

2. Context

[3] The notion of bed deformation has been discussed by many researchers in seminal papers regarding the configuration and behavior of sand beds in the context of sediment transport that include Nordin [1971], Allen [1976], and Nikora et al. [1997]. In spite of this, the movement of bed material associated with bed deformation is not presently accounted for in commonly used methods for calculating sediment flux from successive bathymetric surveys [Znamenskaya, 1963; Stein, 1965; Simons et al., 1965; Shen and Cheong, 1977; Engel and Lau, 1980; van den Berg, 1987; Nittrouer et al., 2008]. The goal of this paper is to demonstrate a framework and a method for calculating that portion of the bed material flux.

[4] Central to all theory regarding sediment transport on Earth's surface is a statement of mass conservation, nominally, the Exner equation [Paola and Voller, 2005]

This Eulerian expression for sediment continuity relates local changes in sediment surface elevation to local gradients in sediment flux and changes in total transport. The first term in (1) is the ordinary derivative of bed elevation, η, with respect to time, t, multiplied by the fraction of the bed that is sediment, 1 − p, where p is bed porosity. The second term is the local derivative of the sediment volume in the water column above the bed, for simplicity it is written here as the product of mean volume concentration, C, and flow depth, H. The third term is the local gradient in volume flux of sediment per unit width of flow, q, measured in the streamwise direction, x.

[5]Richardson et al. [1961] and Simons et al. [1965] were among the first to produce the now standard bed form-bed load equation based on the tracking of bed forms. These early workers proposed that bed forms in equilibrium with steady, uniform flows are unchanging as they migrate, i.e., of constant form in a Lagrangian reference frame

By combining (2) with the transformation between Lagrangian and Eulerian frames, generally termed the transport rule [Strang, 1986]

where V_{c} is the characteristic bed form migration rate, the following description for the ordinary, Eulerian derivative of bed elevation is produced

The right-hand side of (4) can then be substituted into (1) to produce

For the case of steady flow that we are considering here, the right-hand term in (5) must equal zero and this equation reduces to

This relation can then be integrated with respect to x to produce

which describes how sediment discharge associated with equilibrium bed forms varies as a function of bed elevation. Equation (7) is a linear relationship between local sediment flux and local bed elevation plus a constant of integration, q_{0}. For the special case of q = 0 where η = 0 in the trough of a bed form, q_{0} must also equal zero, and (7) reduces to

This equation in turn can be integrated over the mean length of an equilibrium bed form to produce

Equation (9), is an expression for the mean sediment flux, 〈q〉, averaged over a bed form length, L_{c}, as long as the bed migration rate, bulk bed porosity, and bed form height, h, are known. Equation (9) can be thought of as the product of the bed form cross-sectional area, L_{c}h/2, the bed form migration speed, V_{c}, and the proportion of the bed form cross section that is occupied by sediment, 1 − p, normalized by the mean bed form length, L_{c}. While the exact details of the derivation of (9) have been quite different in different papers [e.g., Znamenskaya, 1963; Stein, 1965; Simons et al., 1965; Engel and Lau, 1980], they all contain the essential component that the bed forms migrate without changes in size, shape or spacing. In other words, they are assumed to be constant in a Lagrangian reference frame.

[6] One other important commonality for all derivations of (9) is that q_{o}, the constant of integration from (7) equals zero. Simons et al. [1965] interpret q_{o} as that part of the bed load which does not enter into the propagation of dunes and ripples. However, they also say that q_{o} equals zero as long as the bed is covered with ripples and dunes and that the bed load is defined as the material which remains in near continuous contact with the bed. They note, however, that at high transport conditions in their experiments, when some of the sediment moves by suspension, then the load computed by dune tracking is always less than the directly measured bed material load [Simons et al., 1965]. Stein [1965] recognizes that (9) can be exactly correct when the amount of material bypassing a dune crest in suspension is equal to that which moves upstream by traction through scour within the recirculation zone between dunes. In this case there would be no net sediment movement except that caused by the advancement of the whole train of dunes. Engel and Lau [1980] develop that treatment further to say that there is an elevation somewhat above the trough depth at which there is no net flux, and it corresponds to the flow reattachment point downstream of the trough recirculation zone. In essence their treatment adds a coefficient in front of h in (9) that can correct for the residual flux between dunes.

[7] In contrast to the interpretation of Simons et al. [1965], we take the constant of integration to be the fraction of bed material load that moves intermittently in near-bed suspension. Some previous studies have documented the bed sediment that occasionally moves in suspension. It is in part the load calculated as bypass fraction by Mohrig and Smith [1996]. It is also the percent of the bed material that is deposited from suspension onto dune lee slopes found by Kostaschuk et al. [2008]. Somewhat obviously, the sediment that is exchanged between the bed load and suspended load volumes is a part of the bed material responsible for the evolution of bed topography. It is exactly the effects of this material's transport that we here label bed deformation. It is the topographic aggregate of all changes in size, shape, and spacing of traditionally defined bed forms, but its measurement is irrespective of their delineation.

[9] In addition, much work has been done to describe the nonuniform structure of trains of bed forms. Some have described deterministic aspects of spatial variability [Mohrig and Smith, 1996], but many others have sought to shed light on stochastic components of bed geometries using a variety of methods from contour crossings [Nordin, 1971] and spectral analysis [Hino, 1968; Nordin, 1971; Shen and Cheong, 1977] to structure functions [Nikora et al., 1997], roughness functions [Nikora and Hicks, 1997; Jerolmack and Mohrig, 2005], and traditional dune picking methods [van der Mark and Blom, 2007]. The growth to stable equilibrium bed forms has also been investigated in stochastic contexts with spectral methods [Jain and Kennedy, 1974] and interface width methods [Nikora and Hicks, 1997; Jerolmack and Mohrig, 2005]. A stochastic component to sandy bed evolution has been long recognized, but as of yet there exists no unifying theoretical framework with which to investigate it or its possible relation to sediment transport.

[10] We therefore propose that the bed is more appropriately described by a kinematic wave with a source term

than by the topographically constant form given by (2). In (10), Π is defined as the deformation rate and is an elevation change per unit time. Equation (10) produces a range of bed behaviors determined by the values of Π. Π = 0 is exactly the case of translationally invariant topography. When Π > 0 the bed is aggrading, and likewise when Π < 0 the bed is eroding. The case of statistical steady state or dynamic equilibrium for a bed is given by

where ΔT is an arbitrary time step. It is a condition of statistical topographic consistency but not of deterministic topographic constancy. In other words, the long-term, average local deformation rate is zero, but there are still fluctuations around that average.

[11] The dynamic bed equation of (10) in combination with the notion of stochasticity in (11) can account for variability in the topographic evolution of individual bed forms or of a whole bed profile. In (10) the source term Π can include bed form state change due to changing flow conditions as well as the changes of mean bed elevation that are both short-time variations around an equilibrium profile and that are larger excursions like climbing bed forms. From this context it can be seen that deformation as defined has great import for the creation of stratification and the preservation of bed features associated with transporting sediment. It can produce stratigraphy that encompasses the whole range of behaviors between the steady aggradation of Rubin and Hunter [1982] to the mass-conservative variability described by Paola and Borgman [1991]. By elucidating the nature of (10) for cases when (11) holds, we hope to create a path for explicitly using the statistics of Π to understand the relations between bed forms, their environments, and preserved sedimentary structures. The first step toward that goal is to develop a relation between Π in (10) and the horizontal sediment fluxes over the bed. This is empirically explored below.

4. Observational Evidence

4.1. Field Data

[12] The field data used in the analyses presented here were collected with low-altitude aerial photography over the North Loup River in Nebraska [Mohrig, 1994]. The 40 min sequence of images obtained at 1 min intervals was rectified with a grid of control points surveyed in the field. Bed elevations were calculated from the light intensity of the river bottom images by taking advantage of the attenuation of light through the water column. Light from the sun passed through the water column reflected off the bottom and was recorded by an analog camera mounted on a helium balloon 30 m above. Because the substrate was homogeneously sandy, the light intensity reaching the camera was a function of the local water depth, H, the incident light intensity, I_{0}, and the attenuation coefficient of the water, α (Figure 1). I_{0} and α values were not measured in the field. Rather, those values have been found by measuring water depth at the deepest and shallowest locations in the survey area and then solving for the parameters using the relation for depth and light intensity [Soo, 1999]

I is the gray-scale intensity for each pixel in the digitized, rectified photos. Because the field measurements were taken in the shallowest and deepest waters, the relation fit to (12) brackets the entire range of depth values and therefore the whole bed elevation field is interpolated thus excluding error growth associated with extrapolation.

[13] The resulting data represent a complete assessment of the bed elevation, η, in four dimensions: x, y, z, and t, streamwise position, cross-stream position, vertical position, and time, respectively. The length and width of the grid in x and y are 31 m and 15.5 m respectively. Grid resolution is 0.02 × 0.02 m, and the root mean square horizontal positioning error from the rectification process is 0.10 m. The vertical resolution is 0.001 m, with a nominal accuracy of ±0.002 m. For the purposes here, to compare results with the experimental data, a single profile was drawn from the middle of the bed form field. This profile has been detrended, subtracting a mean bed elevation and removing a slight downstream shallowing. The data are therefore centered around zero but still express the original topographic character of the bed forms (Figure 2a). The whole profile comprises about 10 dune scale bed forms in the streamwise dimension and many smaller bed forms.

[14] During the period that photographs were taken, measurements were made of other aspects of the relevant field conditions through the reach [Mohrig, 1994]. The mean water surface slope was 1.4 × 10^{−3}. The depth averaged flow velocity was 0.70 m/s. The median sediment size was 0.3 mm. The photos were taken over a straight reach of the river with a width of nearly 30 m and an average depth of 0.25 m. The flow was subcritical with a Froude Number of 0.45.

4.2. Flume Data

[15] Laboratory data are also used in this study and were collected by Blom et al. [2003]. During their B2 flume experiment, longitudinal bed profiles were extracted from an evolving train of sandy bed forms every 10 min. Each long profile was 28.3 m in streamwise length and had associated with it a spatial resolution of 0.01 m. It took approximately 3 min to measure topography from end to end of the profile. Because the bed forms were migrating coincidently with the surveying procedure, the bed profile is somewhat altered from that which would be obtained by an instantaneous snapshot of the bed. The bed should be scaled (either expanded or compressed depending on the direction that the survey cart travels) by the distance that the bed forms migrate in the time that it takes the cart to traverse the flume. This is equivalent to the ratio of the bed form velocity to the cart velocity, ∼0.5% in this case.

[16] The transport conditions included a mean fluid flow velocity of 0.69 m/s. The mean water surface slope was 2.2 × 10^{−3}. With a mean depth of 0.39 m the Froude Number was 0.35. The sediment used in the experiment was a trimodal mixture with well-defined modes at 0.68, 2.1, and 5.7 mm. The experiment was designed to address vertical sorting through bed forms, and by the time that the bed forms reached equilibrium, the smallest size fraction comprised 90% of bed surface and the load transmitted along the bed. The other two size groups each accounted for half of the remaining 10% of the load [Blom et al., 2003]. We therefore use 0.68 mm as the modal sediment size in subsequent calculations.

4.3. Correlation Method

[17] The effects of deformation can be measured for lab and field data in two separate ways. First, the effects of profile deformation on the statistical properties of the evolving bed can be measured and is addressed in this section. Second, and detailed in section 4.4, the flux of material associated with changes in size, shape, and spacing of bed forms can be measured directly and compared to the flux associated only with the mean translation of the bed form train.

[18] The most prominent effect of deformation on the statistical properties of a bed profile is the reduction of topographic correlation during bed evolution. This effect has been explicitly shown by both Nordin [1971] and Nikora et al. [1997]. Correlation provides a measure of similarity or dissimilarity for two data series by summing the normalized product of their values over their shared domain. Here we use correlation to quantify the similarity between members of an evolving set of bed profiles. Specifically, we use cross correlation (13) [Davis, 1986] to find the migration rate of the bed and to assess similarity of profiles in the bed's Lagrangian reference frame. Cross correlation, R, is calculated as

where the subscripts 1 and 2 denote the original and evolved profiles, respectively, d is the shared domain of the two bed profiles, l is the length by which the evolved profile is shifted relative to its original position, and σ is the standard deviation of the elevation, η. Cross correlation is performed for a range of shift lengths, l, in order to determine the maximum cross correlation for a pair of profiles. Associated with the maximum cross-correlation value, R_{max}, for each profile pair is an interval of time and the shift length. The shift length that maximizes the cross correlation is interpreted to be the mean translation distance for the bed profile as a whole (Figure 3). As long as the relation between the translation length and the duration is linear, the characteristic velocity for the bed profile is straightforward to calculate through a linear least squares regression of the covarying translation lengths and evolution times (Figure 4). The characteristic velocity can be thought of as the mean migration rate of all the largest-scale bed forms in a single profile.

[19] Inspection of Figures 2a and 2b demonstrates two salient features relevant to the evolution of sand bed topographies. First, there are subregions within each domain that can be locally characterized by greater or lesser velocities than the profile as a whole. Second, changes in bed form size, shape, and spacing occur in minor amounts everywhere and in significant amounts locally. The first type of behavior is a product of the relative translation rates for individual topographic elements. The result is a change within the overall arrangement of the bed and a decrease of the cross-correlation maximum for the bed as a whole. Added to that is the second type of deformation that encompasses relatively diminutive elevation changes, centimeter-scale bed forms, trough scours, etc., that occur along the whole profile and also contribute to decorrelation. The combined result is an evolutionary response for bed form profiles marked by a decrease and broadening of the maximal cross-correlation peak due to an increasing degree of bed deformation (Figure 3).

4.4. Correlation Decay

[20] The agglomeration of all the styles of deformation has the effect of reducing the cross-correlation maximum, and the size of that reduction can be ascribed to temporal and spatial variations in topographic change. Using the characteristic velocity to transform from evolution time to translation length, the decorrelation can be investigated as a function of distance traversed by the bed topography. An appropriate dimensionless translation length, L_{T}*, is calculated by normalizing the translation length by the mean bed form length, L_{c}

For the 39 cross-correlation pairs from the North Loup River, representing translation of just over half of a mean length, values for cross-correlation maxima, R_{max}, decrease exponentially as a function of L_{T}* (Figure 5)

Regression produces an exponential function with a decay constant, c = 1.8, that explains over 90% of the covariance between cross-correlation maxima, R_{max}, and L_{T}*. The decay constant, c, is a fundamental scale of sandy bed evolution that summarizes the deformation of the bed. We propose to call it the characteristic deformation.

[21] The magnitude of the decay constant represents the efficiency with which the bed topography becomes decoupled from previous conditions. For greater magnitudes the bed will decouple from its previous state over a shorter translation distance while lesser magnitudes of characteristic deformation will yield profiles that retain information from previous states over longer translation distances. In the spirit of radionuclide decay processes, an interfacial half-life, λ, can be defined which has no units but corresponds to the normalized translation distance, L_{T}* where the profiles have a maximum correlation of 0.5

Even though L_{T}* does not have units of time, it represents the time it takes the bed to decouple itself from an original state scaled by the bed velocity and the mean bed form length. For the North Loup case λ is 0.39. This means that by the time the bed has translated 39% of a mean bed form length, it is only 50% correlated with its original state.

[22] The B2 experimental data shown in Figure 5b have a characteristic deformation of 0.19 and the Pearson product correlation value 0.94 for the exponential relation between the cross-correlation maxima and dimensionless translation lengths. The associated interfacial half-life, λ, equals 3.6 which means that the B2 profile translated 360% of its mean length before becoming only 50% correlated to its original state. While the characteristic deformation is an order of magnitude smaller for the B2 data than the North Loup data, the B2 data have an interfacial half-life that is an order of magnitude larger. The relation of inverse proportionality can be seen in (16).

[23] Correlation decay for profiles successively drawn from a field of equilibrium bed forms should never result in correlations that are near zero. Rather, there must be a minimum effective similarity between any two profiles that have the same bulk topographic character. Given a sufficiently long observation period, the correlation should become asymptotic to a value representing that minimum similarity. However, this behavior is not evident in either data set analyzed here. For the field data, the decorrelation rate is high, but the duration of sampling is too short for the correlation to become asymptotic. In contrast the experimental data cover a period of evolution in which the bed migrates many dune lengths, but the decorrelation rate is sufficiently low that it also does not become asymptotic. Given enough time it is expected that the rate of decorrelation would become significantly reduced from its initial, exponential form, and the correlation would approach a roughly constant value. One method to calculate an expectation for that asymptotic value would be to correlate a set of randomly generated bed topographies that share a single set of characteristic height and length scales. In natural settings, we surmise that the value is generally of order 0.1.

[24] As the bed evolves from the motion of sediment, the topography demonstrates two major behaviors, translation and deformation. Each translative step is accompanied by a deformation change, and that change can be cast as an injection of new information into the bed that had not been present before. At each time step, the bed is constituted by some percent of its previous state and by some percent of a new, altered state. This alteration is partially due to the emergence and growth of grain-scale perturbations on the backs of larger bed forms. The rest is due to the interaction of larger-scale bed forms. As long as flow conditions are steady, the relative dependence upon the previous state, its memory, is constant. Memory, m, is therefore defined as the compliment of the decay after being discretized and normalized

Using small steps in L_{T}* this relation can be found from c with only small errors of discretization. For a choice of 0.1 L_{T}*, m is 83% for the North Loup data and 98% for the B2 data. The magnitude of m is clearly a function of the dimensionless translation length used in the discretization.

4.5. Calculating Deformation Flux

[25] We have built a case for a flux of bed material associated with bed deformation that can be accounted for independently of the sediment flux associated with translating bed forms. Bed deformation is specifically the difference between bed topography that is invariant in the bed's Lagrangian reference frame and the real, stochastically changing bed elevations (equation (10)). The nature of bed deformation can be elucidated by producing statistical distributions of the deformation rate, Π. These distributions summarize the local departures of the bed from a purely translational evolution by describing the relative frequency of those departures as a function of their magnitude. The laboratory data and field data used here represent very different cases in both magnitudes and styles of bed deformation as indicated by histograms of their respective deformation rates (Figure 6). We will first explore their similarities and fundamental properties in order to calculate the flux associated with deformation.

[26] The bed profiles in both the field-based and laboratory data sets display neither net aggradation nor net degradation, but rather they are in states of dynamic equilibrium. In other words the mean elevation change over the whole profile and through time was zero even though there were many local regions undergoing some amount of aggradation or degradation not caused by mean profile translation. This is the condition described by (11) which acknowledges that, while the mean must be zero for a conservative system, the local behavior contains a stochastic component that is nonzero. Furthermore it necessitates that the sum of topographic changes in the degrading regions is equal to the sum of topographic changes in the aggrading regions even though the distribution of Π itself is not required to be symmetric. The equivalence for total elevation change above and below zero (Figure 6) suggests strongly that the extra sediment placed into motion at regions undergoing degradation in excess of the change associated with pure translation is elsewhere balanced by sediment extracted from the flow in regions of similarly excess aggradation. The elevation changes tied to this deformation flux are calculated using

where D is the average of elevation changes that are independent of the translating bed and summed over the profile length. Δx is the distance between elevation measurements on the profile, Δt is the time step between the two profile surveys, and N is the total number of measurements in the profile. Equation (10) suggests a method for calculating Π. However, directly computing the sum of the first two terms in (10) will result in large errors near discontinuities in the discrete data approximation of the bed surface. A much more stable way to calculate Π is through the difference between two profiles in their Lagrangian reference frame

[27]D can be interpreted as the mean vertical flux over the profile, in both the upward and downward directions. This vertically oriented flux of bed material (an exchange between the suspended load and bed load sediment reservoirs) associated with bed deformation is related to a horizontal flux of sediment that is responsible for deformation, q_{sD}. In order to produce the deformation flux, q_{sD}, from D, the vertical flux must be scaled to a horizontal flux.

[28] The simplest way to scale the vertical to horizontal fluxes is through the ratio of the distances that a grain travels in each direction during its trajectory. Because the upward and downward vertical fluxes are balanced in magnitude, we simplify the trajectory by considering only the grain's descending path. The height from which a grain falls is the product of its transit time and fall velocity. The horizontal distance that a grain traverses during descent is the product of the transit time and its horizontal velocity. Because these two transit times are the same, the ratio of vertical to horizontal velocities is equivalent to the ratio of vertical to horizontal distances traversed. We therefore use the ratio of settling to horizontal sediment velocity as the scale factor to find the horizontal sediment flux associated with D. This horizontal sediment flux associated with deformation, the deformation flux, q_{sD}, can then be found as

Settling velocities for the sand in both the field and laboratory cases were calculated using the formulation of Dietrich [1982] for the settling of natural sediments. At a minimum the sediment that is in transit to deform the bed must be traveling slightly faster than the propagation speed of the bed forms themselves. Maximally, it cannot be traveling faster than the fluid. Horizontal sediment velocity can be calculated from empirical relations [Fernandez-Luque and van Beek, 1976] or with a model based on flow and sediment characteristics [Wiberg and Smith, 1985]. The Wiberg and Smith [1985] model is calibrated to the empirical data of Fernandez-Luque and van Beek [1976], and the results of these calculations naturally agree. Horizontal sediment velocities in this work were calculated from the model relation.

[29] In summary, the translation flux is calculated with the traditional bed form-bed load equation [Simons et al., 1965] that accounts for all of the material that moves from the stoss to lee slope of a migrating bed form. This is the flux responsible for migrating the bed profile in a mean sense. When flow conditions are steady (i.e., constant discharge), the rate of bed form translation is constant, the characteristic velocity. Therefore, the translation flux must be constant for a topographic profile composed of equilibrium bed forms. Additionally, because the total bed material flux should be constant under the same conditions, the deformation flux, the difference between total and translation flux, should also be constant. In section 4.6 we will evaluate the magnitude of the deformation flux.

4.6. Deformation Flux Results

[30] For the North Loup River case, basal shear stress is estimated from the product of mean depth, h = 0.25 m, and slope, S = 1.3 × 10^{−3}, [Mohrig and Smith, 1996; Mohrig, 1994] to have a value of 3.2 Pa. The modal grain size of the well-sorted bed material is 0.3 mm and has a critical shear stress of 0.16 Pa. Critical shear stress is estimated using the Shields Diagram. Using the Wiberg and Smith [1985] model this produces a sediment velocity of 0.40 m/s. This horizontal grain velocity is intermediate between the characteristic velocity of the bed forms of 6.2 × 10^{−4} m/s and 0.70 m/s for the fluid flow. With a settling velocity of 0.039 m/s, the q_{sD} is estimated to be 1.4 × 10^{−5} m^{2}/s. This flux value is approximately 65% the size of the translation flux, q_{sT} = 2.2 × 10^{−5} m^{2}/s, found using (9) with a bed form height is 0.11 m.

[31] For the B2 experimental case, the median bed material is 0.68 mm and has a critical shear stress of 0.42 Pa. The mean water depth is 0.24 m and the surface slope is 2.2 × 10^{−3} giving a basal shear stress of 5.2 Pa. The Wiberg and Smith [1985] model predicts a sediment velocity of 0.62 m/s which is intermediate between a mean fluid velocity of 0.69 m/s and a bed form translation velocity of 8.7 × 10^{−4} m/s. With a settling velocity of 0.105 m/s, a profile length of 28.31 m and a sample spacing of 0.01 m, the deformation flux is found to be 4.0 × 10^{−7} cm^{2}/s. This flux value is approximately 1% of the 3.7 × 10^{−5} m^{2}/s translation flux, estimated using (9) and a mean bed form height of 0.13 m.

[32] For both the field and laboratory cases, the ratio of horizontal sediment velocity, V_{s}, to settling velocity, w_{s}, is of order 10^{1}. This must be generally true because ratios nearer to10^{2} or greater would represent sediments in the wash load component of sediment flux that do not participate in bed evolution. In contrast ratios nearer to 10^{0} or lower represent sediments under conditions much closer to the threshold of motion, and that sediment is not likely to be exchanged between the bed load and suspended load sediment reservoirs. Therefore it is not likely to participate in bed deformation. We speculate that this is generally true in nature. Specifically, the bulk of the sediment that can travel in intermittent suspension and is responsible for instigating bed deformation has a horizontal velocity that is approximately an order of magnitude greater than its settling velocity.

5. Discussion

[33] The first major difference between the two cases is in the magnitude of the deformation flux that is calculated for them. This is shown by the ratio of the deformation flux to the total bed material flux. It is termed the deformation fraction, F, given by

where q_{sT} is the translation flux from the bed form-bed load equation, i.e., q_{sT} = 〈q〉 from (9). The field data show that approximately 40% of the flux of bed material is not captured by the bed form-bed load equation while the deformation fraction for the lab experiments is only 1%.

[34] The discrepancy in the deformation fraction as well as the characteristic deformation for these two data sets raises the question, “What controls rates of deformation?” Although the total flux of bed material is greater in the flume compared to the field, the laboratory case has a lower flux representing bed deformation. We do not think these cases define a general difference between all lab and field data. Because deformation is a consequence of the exchange of material between the suspended and bed load sediment reservoirs, we hypothesize that a primary control on the deformation rate is the ability of the flow to suspend the sediment. This is best summarized by the Rouse parameter, the ratio of the sediment fall velocity to the skin friction shear velocity, P = w_{s}/κu_{sf}*, where κ is von Karman's constant. Skin friction shear velocities were calculated with the method of Wright and Parker [2004], and the resulting Rouse parameters for the North Loup and B2 cases are 2.3 and 4.3, respectively. Sediment sizes with P > 2.5 are normally considered to be bed load while those with values less than 2.5 considered to be part of the suspended load. For the North Loup case, the modal sediment size is barely suspended, and in contrast the modal sediment size of the B2 experiment should be traveling as bed load. This supports our hypothesis that the ability to suspend the bed material has an effect on the deformation fraction. However, with only two points, more work will need to be done to further support or refute this hypothesis.

[35] Another hypothesis is that the observed differences are related to the range of bed roughness scales present in these systems. Even though the experimental bed forms are considerably steeper h/L_{c} = 0.13/1.80 compared to 0.11/2.90 the field bed forms, the variability in the river flux could likely be due to a greater abundance of roughness elements at finer scales below ∼0.20 m (Figure 7). It has been found that the width to depth ratio controls both the mean geometry of bed forms [Crickmore, 1970; Williams, 1970] and the variability in bed form geometries [van der Mark et al., 2008]. While the size and variability of bed form geometry should not control the characteristic deformation (if it is indeed an independent scale of the bed dynamics), the deformation could definitely be limited to expression within the range of possible bed geometries. If a narrower flume only allows a small range of bed form heights and lengths to exist than a wider flume or a natural channel would, then the dynamics in the narrower flume can only explore the topographies in that smaller range. However, the data here suggest that the B2 experimental dunes were actually able to explore a larger range of geometries than the North Loup data. Undoubtedly, this question will form a basis for future work.

[36] The second major difference is that the histogram of deformation rates, Π, generated for the field data is nearly symmetric while the lab data render a Π distribution that is positively skewed toward high values of excess elevation change (Figure 6). For the laboratory case there is a large fraction of the bed that is losing elevation slightly faster than expected from translation alone, and a smaller bed fraction is gaining elevation much faster than expected. This behavior can be observed both in Figures 2b and 6b. In Figure 2b the bed is largely moving in a coherent fashion except for a small segment, i.e., approximately two dunes located near 25 m, that are moving at a somewhat greater velocity. Because the lee slopes are steeper than the stoss slopes, the bed at the positions of the lee faces for these two dunes has undergone larger excess gains in elevation while the bed covered by the stoss sides of these dunes has undergone more modest excess decreases in elevation. Furthermore, the stoss sides are longer than the lee sides. This means that in the frame of reference of the bed there are more positions along its surface that have seen modest elevation decreases and fewer that have seen large increases in elevation. The resulting distribution shows a larger, less populated range of deformation rates, Π, above zero and a smaller, more populated range of deformation rates below zero. This is the origin of the skewness of the distribution.

[37] In contrast, deformation rates from the field data closely approximate a normal distribution. This reflects a greater randomness in topographic changes due to the coexistence and interactions of topographic elements with many length scales, ripples through dunes, on the river bottom [Jerolmack and Mohrig, 2005]. The three main styles of bed deformation can be seen and are labeled in the caption of Figure 2; these are changes in bed form size, shape, and spacing. The experimental data seem to be dominated by changes in bed form spacing. For the North Loup data, small bed forms are more abundant (Figure 7) and therefore can add to the diversity of measureable deformation rates. As a result the rates of deformation, Π, appear to be more equably distributed for the field data.

5.1. Apparent Time Dependence of Deformation Flux

[38] We have argued that bed deformation is a stochastic process in the context of bed topography in a dynamic equilibrium, and this is made by equations (10) and (11). The assessment of deformation (equations (18) and (19)) explicitly depends on the elapsed time between bed profile surveys. This is because deformation is perpetually reversing, undoing and redoing, previous bed changes. In short, it is a process that imparts no net elevation change to the bed profile through time. As typical examples, troughs deepen and then refill or two bed forms partially merge and then split. The end result is very little net change in the topographic profile while there is significant movement of sediment beyond that associated with the translating dunes. Because of this the magnitude of the deformation flux appears to depend on the time step used in its calculation even though we have already shown that for steady flow conditions and constant bed form migration rates the real deformation flux must be temporally invariant.

[39] The explanation for the apparent inconsistency is found in stochastic theory wherein the time history of deformation is viewed as a random walk. The classical central limit theory [e.g., Feller, 1966] predicts that the variability of the rate of stochastic bed changes, Π, should decrease as the square root of the dt, Π ∼ dt^{−0.5}. The deformation flux is proportional to the deformation rate, and it should therefore scale in the same manner, q_{sD} ∼ dt^{−0.5}. This scaling prediction is based on the assumption that bed deformation at each step in bed evolution is independent and identically distributed in the family of distributions whose tails decay approximately exponentially. If on the other hand, the transport of sediment could allow for rare events of rapid erosion or deposition in the bed's frame of reference, then the scaling of the deformation rate would be described by fractional dispersion (please see R. Schumer et al. (What are fractional advection-dispersion equations?, submitted to Journal of Geophysical Research, 2009) for a discussion of the differences between normal and fractional advection-dispersion).

[40] The North Loup and B2 experimental data show that the deformation flux scales as q_{sD} ∼ dt^{−[0.75, 0.63]} (Figure 8). This suggests that classical theory does not describe the rate of deformation of sandy beds. However, Sayre and Hubbell [1965] used radioactively tagged tracer particles to track the downstream motion of an assemblage of grains in the North Loup River over two weeks. Their findings were consistent with classical central limit theory. D. N. Bradley (Fractional dispersion in a sand bed river, submitted to Journal of Geophysical Research, 2009) and G. E. Tucker (Trouble with diffusion: Reassessing hillslope erosion laws with a particle-based model, submitted to Journal of Geophysical Research, 2009) offer an alternative to the analysis of Sayre and Hubbell [1965]. Although, we currently do not know which theory is a better descriptor of the system, we surmise that they both partly fail because they are based on an assumption that consecutive events are independent. It is quite likely that deformation rates are correlated in both space and time. Although it is not possible to uniquely evaluate the effects of correlation in general, we can still be certain that the decrease in deformation rate as a function of timescale is a result of the summation of a series of individual (likely correlated), stochastic deformation events bounded to a maximal range of elevation changes set by the height scale of the largest dunes.

[41] In section 4.6, we calculated the deformation fluxes for both data sets from the nontranslational changes in bed topography over the minimum time step between profiles. These estimates of deformation flux are minima. Figure 8 shows that for all time steps measured, the deformation flux is a power law function of time step. However, there must exist a rollover point at small timescales that is associated with the individual motions at the grain scale. Because this was not realized in our data, the calculated deformation fluxes must be considered a minimum. Future work will elucidate the fundamental time and length scales of deformation.

5.2. Suggestions for Designing a Survey Campaign to Assess Bed Material Flux

[42] One significant implication of the apparent time dependence of the deformation flux is that the survey repeat interval must be minimized for the most accurate assessment. Greater durations between repeat surveys will result in lower estimates for the apparent deformation flux and greater differences between the apparent flux and the real, constant deformation flux. Longer intervals between repeat surveys increase the error associated with estimates of deformation flux. This means that larger, systematic measurement errors will be accrued in estimates from surveys with longer repeat intervals.

[43] While it was shown above that the deformation of a sandy bed can be calculated rather straightforwardly, there are a few points to consider in optimizing the tradeoffs between accurate assessments and total survey time. First, deformation can be calculated with the methods above using only two surveys. The bed form-bed load equation (9), will yield the translation flux, and the deformation flux can be measured from the profile differences after removing the mean translation. Because the deformation flux explicitly depends on the timescale over which it is estimated, any measure generated this way will be a minimum value for the deformation flux. A series that contains at least three surveys is needed to define how the deformation flux varies with timescale and to produce a trend line similar to that shown in Figure 8 that yields a first-order extrapolation to the short-time deformation flux. We therefore recommend that all repeat surveys designed to determine deformation flux as well as translation flux should use a minimum of two repeat surveys after an original survey. Results generated using these surveys can be further improved by choosing time intervals between the survey repeats that are different. For instance, surveying at time 0, t, and 2t will yield only two sampling intervals, dt = t and 2t. In contrast, surveying at time 0, t, and 3t will give three distinct observation intervals, dt = t, 2t, and 3t. Even though these three increments will not be completely independent, they will provide a better estimate of apparent deformation flux reductions than surveys at time 0, t, and 2t only.

[44] These suggestions with regard to the best practices for determining deformation flux should be taken relative to estimates of the importance of suspended load transport. In conditions where the transport stage is low or where the suspendibility of bed material is low, deformation flux can probably be ignored without accruing large errors in estimates of bed material flux using the bed form-bed load equation with dune tracking. In contrast, whenever suspension of bed material is significant, surveys that are not designed to account for deformation or separately measure suspended load will underestimate bed material flux. Examples presented here indicate that this error could be 65% or greater.

6. Conclusion

[45] We have developed a method for incorporating a stochastic component of sediment transport into theory of sandy bed form evolution. In a manner analogous to general methods of separating mean and fluctuating parts of variables, bed topographic evolution can be broken up into translation and deformation. Translation is associated with a flux of bed material that can be calculated by the bed form-bed load equation [Simons et al., 1965]. Deformation is found after removing from the total bed elevation changes that part which is expected from the mean translation of the original topography. The deformation rate can be seen as a source/sink term in a kinematic wave equation, thus providing a unifying framework for simultaneously investigating the deterministic and stochastic components of sandy bed evolution. Deformation can be interpreted as the exchange of sediment between the suspended load and bed load reservoirs, and as such represents a vertical flux of bed material. The horizontal flux of bed material associated with that deformation can then be determined by the vertical flux of bed material scaled by the ratio of horizontal sediment velocity to settling velocity. Sediment that participates in deformation will have a characteristic velocity scaling ratio that is order 10^{1}. For significantly larger velocity ratios the sediment will be a part of the wash load and not participate in the bed evolution. For significantly smaller ratios, the sediment will rarely be exchanged between the bed load and suspended load sediment reservoirs.

[46] For the steady flow conditions investigated, total bed material fluxes were approximately constant as were the characteristic velocities of the topographic profiles. This condition requires that deformation flux also be constant and can be represented as a fraction of the total sediment flux. Under steady flow conditions where bed profiles are in dynamic equilibrium, deformation results in the decoupling of the bed from previous states as it evolves. This decoupling is expressed as an exponential decay in correlation of topography. The field profile decorrelated much more rapidly than the laboratory profile, with decay constants separated by an order of magnitude. This corresponds to a laboratory deformation flux that was only ∼1% of the total flux of bed material while the deformation flux in the field setting was nearly 40% of the total bed material flux. Although laboratory data have consistently shown that the bed form-bed load equation does an adequate job of determining bed material flux from translating bed forms alone, we surmise that this will not be the general case in the field. Instead, in cases where bed material can be somewhat suspended, it is more likely that deformation will play a larger role in bed evolution. Similarly then, less of the total bed material flux will be captured by dune tracking methods.

[47] Deformation fluxes have an apparent dependence on the interval between the repeated profile surveys because, over the long-term, deformation produces no net change in the topographic profile. However, deformation flux must be constant in steady flow conditions, so longer intervals between repeated bed surveys result in greater underestimates of actual deformation flux. A reliable estimate of the deformation flux can be made by employing a surveying strategy with at least two survey repeats. This will allow for making plots analogous to Figure 8 and for extrapolating to apparent deformation fluxes at short timescales, the most accurate estimate of deformation flux.

Notation

α

attenuation coefficient of light through water;

c

characteristic deformation, exponential decay rate of R_{max}(L_{T}*);

C

mean sediment concentration in water column;

d

bed profile length;

D

average bed elevation change in bed's migrating reference frame;

F

deformation fraction;

h

bed form height;

H

depth of flow;

I

light intensity;

I_{0}

incident light intensity;

κ

von Karman's constant;

l

shift length for cross correlation;

λ

interfacial half-life;

L_{c}

mean bed form length;

L_{T}

bed translation length;

L_{T}*

dimensionless bed translation length;

m

bed state memory;

η

bed elevation;

N

number of measurements in bed profile;

P

Rouse parameter;

p

bed porosity;

Π

deformation rate;

q

bed sediment flux per unit width;

q_{0}

bed sediment flux past dune trough, constant of integration;

〈q〉

bed sediment flux averaged over a bed form length;

q_{sD}

deformation flux;

q_{sT}

translation flux, equivalent to 〈q〉;

R

cross-correlation function;

R_{max}

cross-correlation maximum;

σ

standard deviation of bed profile elevations;

t

time;

Δt

time step between profiles;

ΔT

arbitrary time step;

u_{sf}*

skin friction shear velocity;

V_{c}

characteristic bed velocity;

V_{s}

horizontal sediment velocity;

w_{s}

sediment settling velocity;

x

streamwise direction;

Δx

bed profile measurement spacing;

y

cross-stream direction;

z

vertical direction.

Acknowledgments

[48] This work was partially supported by the STC program of the National Science Foundation via the National Center for Earth-Surface Dynamics under the agreement EAR-0120914. Additional support came from the STC International Research Experience Program of the NSF through the Nanobiotechnology Center under agreement ECS-9876771. A great thanks to A. Blom and C. F. van der Mark for excellent discussion and help with the laboratory data while B. McElroy was visiting the Environmental Fluid Mechanics group at Delft University.