3.1. Dimensions of Simple Free Dunes Versus Grain Size and Flow Strength
 As bed form dimensions relate to transport conditions in general, a direct comparison based on grain size alone would be misleading in cases where flow conditions change. In order to compensate for this, the grain sizes in equations (1) and (2) were used to calculate the form-corrected Shields parameter (θ′ = uf′2/[(s − 1) g d50]), based on the dominant friction velocity, where uf is the friction velocity, (s − 1) the dimensionless submerged density of quarts, g the acceleration due to gravity, and d50 the median grain size (m). The Shields parameter expresses the dimensionless bed shear stress and represents the ratio between shear and gravity forces. The superscript (′) indicates that the shear stress and the Shields parameter have been corrected for form drag (see below). By means of these equations it is possible to establish a local relationship between grain size (d50) and the equivalent sand roughness of the bed (kt), following the model suggested by van Rijn [1982, 1993], in which (in a slightly modified version) skin and form roughness are combined, and which add up to
Originally, van Rijn suggested ks = 3 · d90 for the skin roughness. Other results [see, e.g., Yalin, 1992] suggest that ks ≈ 2 · d50, which is in accordance with the Engelund and Hansen  approach (ks = 2.5 · d50) used in equation (3). van Rijn also suggested that the constant of 1.1 should be reduced to 0.77 when, under field conditions in rivers, the lee sides of the dunes did not reach the angle of repose [Ogink, 1988] (cited by van Rijn ). In our experience, most simple dunes in the marine environment develop angles of repose, and there is thus no need to reduce the constant in equation (3). Although the large to very large dunes (on the stoss-side of which the simple dunes occur) will also contribute to the hydraulic roughness, their effect is considered to be relatively small. van Rijn  does not distinguish between bed form types in exactly the same manner as is done in the present paper. Thus if, according to van Rijn, the largest bed features are sand waves with a low hydraulic roughness, then the hydraulic roughness values calculated for the superimposed dunes alone are absolutely realistic. If, on the other hand, the large bed forms are considered to be dunes and the smaller ones ripples, which according to van Rijn  would be the other alternative, then the calculated hydraulic roughness values become unrealistically high, reaching values close to or over 1 m. On the basis of these considerations, L and H measures of the simple superimposed dunes (equations (1) and (2)) have been used to produce a local Grådyb version of equation (3),
It is important to note here that, although d50 and Md represent the same parameter, they each have a different notation. Here d50 is expressed in meters and Md in dimensionless phi-units. Because of the empirical origin of the equation, both types of grain size notations have to be present. The empirical constant, −0.538, is dimensions less and the other, 0.187, has the dimension “meter.” All grain-sizes used in these calculations were determined from the regression line shown in Figure 1. On the basis of equation (3′), the hydraulic roughness varies from 0.08 m in the inner part to 0.34 m in the outer part of the study reach.
 In the following procedure, the dimensionless bed shear stress, θ (the Shields parameter),
is transformed into the form-corrected Shields parameter, θ′, which considers skin friction only. This was done in order to base the newly derived relationships on the single overriding dynamic factor, namely the dimensionless bed shear stress due to skin friction. The transformation was carried out by means of the method used by, for example, Engelund and Hansen  and Fredsøe and Daigaard ,
where τo is the bed shear stress (N/m2), uf is the friction velocity (m s−1), uf′ is the form-corrected friction velocity (m s−1), ρs is the density of quartz, ρ is the density of water (kg m−3), (s − 1) is the dimensionless submerged density of quartz (1.65), g is the acceleration due to gravity (9.82 m s−2), and d50 is the mean grain size (m). Formula (5) was originally suggested by Engelund , who derived it from similarity principles and calibrated the equation on the basis of the experimental work by Guy et al. . As θ′ depends on θ alone, the following relations, if desired, can easily be recalculated to achieve dependence on θ instead of θ′. The form correction is derived from the currently used method originally suggested by Einstein  and later modified as explained by Engelund and Hansen . The bed shear stress is split up into two components: τ′ and τ″, the former being the effective shear stress, the latter that part of the total bed shear stress which is lost due to form drag on the bed forms.
 The value of uf is found by means of the logarithmic velocity distribution [e.g., Yalin, 1977],
where U4 is the velocity 4 m above the bed (1.0 m s−1) as defined earlier, and kt is found from equation (3′).
 The resulting relationship between average dune heights and lengths and the form-corrected Shields parameter is illustrated in Figure 3. A best fit relationship using polynomial relations between θ′ and H and L resulted in equations (7) and (8), both having an R2 > 0.99. A second-order equation was chosen for the bed form height as this is expected to decrease after reaching a maximum value when approaching the plane-bed phase of the upper flow regime at high values of θ′. A third-order equation was chosen for the bed form length which is expected to continue to grow toward the upper-stage plane bed as suggested by Fredsøe  and also implied by the trends in Figure 4 (to follow).
H and L are measured in meters. As θ′ is dimensions less all empirical constants in equations (7) and (8) also have the dimension “meter.” The advantage of using θ′ as the independent variable is that this parameter is directly related to sediment transport [e.g., Engelund and Hansen, 1972; Engelund and Fredsøe, 1976; Fredsøe and Daigaard, 1992] which makes it the most obvious parameter to relate to bed form development. As the Engelund and Hansen  method implies a direct relationship between θ and θ′, H and L in equations (7) and (8) can equally well be expressed as functions of θ,
Again, all empirical constants have the dimension “meter.”
Figure 3. Dune height (H) and length (L) of simple depth-independent dunes as a function of the form-corrected Shields parameter.
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Figure 4. Dune height (H) and length (L) as a function of the form-corrected Shields' parameter (θ′) for all “8-foot flume” data from Guy et al.  with water depth >0.28 m and dunes present on the bed. The average water depth is 0.31 m (±1 standard deviation of 0.02 m). The lines labeled I are the polynomial regression lines based on the data set: L = 18.75 θ′3 − 19.50 θ′2 + 9.92 θ′ + 0.35; R2 = 0.90; H = 0.45 θ′2 − 0.47 θ′; R2 = 0.60. The lines labeled II are the regression lines from the Grådyb inlet (equations (7) and (8)), multiplied by a factor: αL and αH, giving the smallest error compared to the flume data (these factors are found by means of a maximum likelihood procedure). αL and αH were found as 0.19 and 0.27, respectively.
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3.2. Dune Dimensions
 For almost a century following the pioneering work of Gilbert , studies of bed form development and sediment transport have primarily been carried out using flumes. The foremost reason for this was that the required precision of data on flow conditions, form shape and form migration could only be achieved under controlled laboratory conditions. However, the technological advances of the last 2 decades have now reached a point where field studies also produce sufficiently precise data, especially after the development of integrated survey systems in which differential GPS positioning, acoustic Doppler current profiling and multibeam echo sounding are combined [e.g., Ernstsen et al., 2004]. Most of our fundamental knowledge, however, is still based on flume studies. Of these, the work published by Guy et al.  is in a class of its own. This outstanding detailed data report has functioned as the state of the art on which almost all models dealing with bed form development and sediment transport have been calibrated ever since it was published. When, as in the present paper, it is attempted to relate natural bed form conditions to plausible flow conditions at full scale, it is therefore only natural to test the derived relationships against the flume results. To keep the comparison as realistic as possible, only those data of Guy et al.  have been considered which were least constrained by the laboratory conditions. Thus only the data of the widest flume, namely the “8-foot flume” (2.4 m wide) with the largest water depth (>0.28 m) and with reported dunes on the bed, were used. A minimum water depth of 0.28 m was selected as a trade-off between a sufficiently large number of data sets and as great a water depth as possible. The average water depth in this case is 0.31 m (with a standard deviation of 0.02 m), the grain size in this data set ranges from 0.19 mm to 1.00 mm, and θ′ varies from 0.06 to 0.8 (0.07 < θ < 0.14).
 As flume data sets are depth limited, a comparison between free bed form dimensions and the flume bed forms is expected to reveal scale differences, even when only the largest water depths are used. This is indeed the case (Figure 4). However, when the bed form lengths and heights have been adjusted (using a maximum likelihood procedure by means of a Fortran program which tests all possible reductions and chooses the one with the smallest cumulative error), the similarities are striking. A reduction of the dune heights to 27% of that prescribed by equation (7) produces a distribution which is practically identical to that obtained by a regression analysis of the flume data itself. Likewise, although with a somewhat greater variance, the dune lengths are found to be close to those of a direct regression analysis of the flume data, when the length prescribed by equation (8) is reduced to 19% of the prescribed lengths. The relationship thus produced has the same shape as the flume regressions. The reason for using the second- and third-order regression equations for the flume data relationships between θ′ and H and L, respectively, follows the arguments used for the generation of equations (7) and (8) (see above). The peak value of H = f(θ′) and the inflection points of L = f(θ′) can be found by differentiation. It reveals that the largest bed form height is associated with θ′ = 0.5 to 0.6 (θ = 1.1) for both equations, and that the inflection point where L starts to increase again with increasing shear stress is associated with θ′ = 0.4 (θ = 0.9) for both equations. Besides confirming a strong connection between the two relationships, it also implies that bed form development during accelerating flow reaches a general turning point when the form-corrected dimensionless shear stress reaches values close to 0.5 (θ close to 1.0). This is well above the criterion for suspension suggested by van Rijn , but probably coincides with the condition where a naturally graded sediment becomes fully suspended. For the coarser part of the sand fraction, θ-values around 1 represent the suspension criterion suggested by Bagnold . It is conceivable that bed forms continue to grow until the coarsest part of the transported material goes into suspension, and that the bed form crests begins to erode when the flow strength increases further. The close correspondence between the field and the flume results is regarded as an indication for the validity of the equations, and suggests that scale differences for dunes range from about 70% for H and 80% for L in water depths of 0.3 m compared to free stream conditions.
 To the best of our knowledge, only two earlier studies have linked bed form height to flow strength and grain size while acknowledging the fact that bed forms decrease in height when a certain flow strength is exceeded [Fredsøe, 1982; van Rijn, 1982]. Fredsøe  based his method on theoretical considerations relating to the bed load formula of Meyer-Peter and Müller . In his approach, all suspended bed material is removed from bed form migration by means of the Engelund and Fredsøe  sediment transport model. van Rijn , on the other hand, used a purely empirical approach. He concluded that dune dimensions depend on grain size, water depth and a transport stage parameter, T = (uf′2 − ufc2)/ufc2, in which (by means of his method) uf′ and ufc are the form-corrected friction velocity and critical friction velocity, respectively. He formulated an equation for which, when compared with flume and field data, “the best agreement was obtained” (van Rijn [1984, p. 1738], also referring to Report S 487-III from Delft Hydraulic Laboratory). Both approaches determine the dune height in relation to flow depth, and are therefore restricted to depth-dependent dunes. The dune length is regarded by van Rijn  to be equal to 7.3 times the water depth, whereas Fredsøe  suggested a method which accepts observed variations (e.g., those reported by Fredsøe , Raudkivi , Yalin , and Yalin and Karahan ) of increasing dune steepness with increasing bed shear stress for small bed shear stresses, and decreasing steepness with increasing shear stress for high shear stresses. Fredsøe explains the latter with increasing suspension transport, with references to Kennedy  and Engelund and Fredsøe .
 In Figure 5, the two approaches by Fredsøe  and van Rijn  are compared with those shown in Figure 4 (II) using the flume data mentioned above. Clearly, neither of the two approaches is able to reproduce the observed variation in bed form height and length, the regression lines being more or less horizontal. The mean errors of the reduced equations (7) and (8), calculated as the relation between the summed errors and the summed heights and lengths, respectively, are 22% and 19%, whereas the corresponding percentages using the Fredsøe /van Rijn  methods are 62%/35% and 50%/34%, respectively. It could be argued that, as the reduction of equations (7) and (8) is made on the basis of the data set in question, the comparison is not fair. None of the two other approaches, however, can be changed in the same manner to correlate better than shown in Figure 5. This is because their best fit lines are close to horizontal. Both predict the right order of magnitude for relatively small dune heights but substantially underestimate the larger ones. Using the two approaches on the data set from Grådyb produces quite a different picture. In this case the measured heights change from 0.17 m to 0.49 m over the study reach, which is also predicted by equation (7), whereas they produce a change from 2.4 m/1.1 m to 0.84 m/1.0 m in the case of Fredsøe/van Rijn, respectively. Thus the predictions of the latter approaches produce much too large bed form heights and a decrease in bed form height over the study reach, which is incompatible with the observations and also contrary to the increase in height predicted by equation (7). The reason for this, most likely, is that in this environment the two other approaches aim at the large-scale compound bed forms which decrease in height from about 3 m to about 1 m over the study reach. This is matched relatively well by the Fredsøe approach, whereas the van Rijn approach only predicts the right order of magnitude but misses the relatively rapid height decrease of the compound dunes.
Figure 5. Measured and predicted dune heights (H, left) and lengths (L, right) based on the data and relations shown in Figure 4 (II, diamonds). Similar results based on the method proposed by Fredsøe [1982, p. 941, Figure 6 and 8] (triangles) and van Rijn  (squares) are shown with their linear regression lines for comparison. The three solid triangles are extrapolated, and the one in brackets was excluded from the regression.
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 At this stage, it is not clear what is happening in the transition zone from simple dunes (with heights on the order of 10 cm) to meter high compound dunes with superimposed simple dunes. Nevertheless, equation (7) provides a realistic prediction of simple superimposed dune heights, and, if a proper way of determining a depth-dependent reduction for smaller water depths could be developed, it would also seem capable of predicting flow-strength associated changes under depth-restrained conditions. In accordance with the solution of Fredsøe , also discussed by Fredsøe and Daigaard , the reduction in dune height under dimensionless shear stresses higher than θ′ = 0.5–0.6 (θ = 1.1) is interpreted here as the influence of increasing suspension transport. In the work of Fredsøe , the peak in dune height appears as early as θ′ = 0.2, which is probably a result of his interpretation of suspension as a transport mode which is totally separated from bed form migration. As discussed by Bartholdy et al. , increasing suspension transport is in fact regarded as the primary reason for the combined bed form change (simple and compound) in the outer part of the study reach in the Grådyb inlet.
 The physical interpretation of the change in dune length is highly problematical. In the case of the Grådyb inlet, the approach suggested by Fredsøe  predicts dune lengths in the range 46–35 m, whereas the observed range is 8–12 m. The suggestion by van Rijn  of dune length being about 7.3 times the water depth (in accordance with Yalin ) also produces much too large bed form lengths (about 88 m) in Grådyb. Again, the two approaches appear more appropriate for the prediction of the large compound features, although with the restriction of being unable to reproduce the length reduction from about 200 m to about 50 m observed along the study reach. Thus both of these approaches fail to correctly predict the length of the simple depth-independent dunes, while also failing to predict the increase in length with decreasing height for θ′ values over about 0.5. We are not aware of any theory which can explain the empirically based variations described by equations (7) and (8).
3.3. Dune Migration Rate
 An evaluation of the reported sediment transport values based on the 28 flume data sets mentioned earlier reveals a relatively close relationship (Figure 6) between the sediment transport derived from bed form migration and that calculated by a slightly adjusted version of the bed load formula of Meyer-Peter and Müller ,
where qstv (m3s−1m−1) = H Ub is here defined as the volume of loosely packed sand transported per unit time and width, and calculated as the migration of triangular-shaped dunes where H is dune height and Ub is the dune migration rate. The empirical constant, 39.87, has the dimension, m4/3s−1. As evident from Figure 6, the basic term of the Meyer-Peter and Müller formula, ((θ′ − 0.047)d50)3/2 correlates relatively well with qstv (R2 = 0.71, P < 0.01), and a regression forced through the origin produces a calibration constant of 39.87. This corresponds to a 20% reduction of the calibration constant in the original work of Meyer-Peter and Müller if it is assumed that the bulk dry density of newly transported sand is roughly 1700 kg m−3 [e.g., Bartholdy et al., 1991]. The calibration result can be regarded as being satisfactory when compared with results of other studies [e.g., Yalin, 1977]. It is generally accepted that when sediment transport takes place as bed load alone, the formula of Meyer-Peter and Müller  is well suited to describe the sediment flux. Arriving from a different and much more sophisticated theoretical analysis, Engelund and Fredsøe  developed an expression for bed load transport which, as later confirmed by, for example, Fredsøe and Daigaard [1992, p. 214], “becomes close to the widely used semi-empirical formula of Meyer-Peter and Müller .” This formula was also the choice of McLean et al. [1994, 1996] when they tried to relate the local boundary shear stress to local sediment flux on a bed form, and recently of Colombini  when reanalyzing the instability mechanisms [Kennedy, 1963, 1969] leading to the formation of bed forms.
 There is no exact borderline between bed load and suspended load when both transport modes are present. Describing a succession of grain jumps in terms of their trajectories, Yalin [1977, p. 17] writes: “Indeed, there is no natural indicator to point out which among the transitional paths … should be the border line between the paths of the bed load and suspended load particles.” Dealing with bed form behavior, the advantage of using a sediment transport formula based on bed form migration alone is obvious. As the data sets plotted in Figure 6 are associated with θ-values ranging from under 0.1 to over 1, the data series crosses the border between predominantly bed load and what must be regarded as fully developed suspended load [e.g., Bagnold, 1966; van Rijn, 1993]. Thus the relatively successful calibration of equation (9) suggests that bed form migration represents a continuum describing increasing bed form–related sediment transport as the shear stress increases even after the border line between predominantly “true” bed load and predominantly suspended load is exceeded.
 Calibrating a model on the basis of laboratory results is of course not satisfactory when the aim is to evaluate depth-independent bed forms. For the moment, however, this employment of an existing model is the only possible way of interpreting our results, and equation (9) is believed to be the least feeble of available models which are able to describe bed form migration as a function of bed shear stress and grain-size.
3.4. Flow and Transport Parameters as a Function of Dune Length
 Equations (3), (4), (5), (7), (8), and (9) permit the formulation of direct relationships between dune dimensions and parameters describing flow strength and grain size. The easiest bed form parameter to obtain is the bed form length, L (even from echo sounder profiles at large depths where the estimation of bed form heights may be problematical). Since equation (8) relates L to θ′ through a third-order polynomial equation, a solution with L as the entrance parameter requires an indirect iterative solution. In order to implement a more direct and less laborious procedure, regressions have been calculated between paired results of L and θ′ (equations (10a)–(10c)) as shown in Figure 7. In order to obtain an adequate accuracy, it was necessary to split up the data set into different size classes, 4 m < L < 10 m
10 m < L < 12 m
12 m < L < 20 m
The empirical constants have the dimension mx, where x is the reciprocal of the single power of L. Note that the experimentally obtained results from the Grådyb inlet are limited to 7 m < L < 12 m. Equation (10c) has been included because the results presented in Figure 4 imply a somewhat wider range of applicability because for dune lengths up to about 5 m, the depth-limited flume results correspond to depth-independent bed form lengths of about 25 m (compare equation (8)). It must be stressed, however, that the results are currently still restricted to the range 7 m < L < 12 m for which measurements in nature are available.
 The above relationships between the lengths of simple depth-independent dunes and the form-corrected Shields parameter enables the construction of additional relationships between dune length and grain size, and a large number of other relevant flow and transport parameters. Four examples are given in Figure 8. Friction velocity can be calculated from dune length and grain size using equations (10), (5) and (4). As clearly evident from Figure 8a, a constant grain size and an increasing dune length indicates an increasing friction velocity. Furthermore, with a fixed dune length the friction velocity increases with grain size.
Figure 8. Flow and transport parameters as a function of simple depth-independent dune length and grain size. (a) Friction velocity (m s−1); (b) total hydraulic roughness (m); (c) velocity 4 m above the bed (m s−1); and (d) bed load transport based on bed form migration (m3 s−1 m−1).
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 The same patterns emerge for the current velocity (U4m, Figure 8c) which can be found from dune length and grain size by using equations (10), (7) and (3) to calculate kt, and equations (5) and (4) to calculate uf. With these relationships at hand, the velocity at any height above the bed in the logarithmic layer can be calculated using equation (6). In many deeper water situations the free stream velocity, which would be located well above the bed features, may be the only available indicator of flow conditions. Following the above arguments, dunes in deep flows having the same grain-size should adjust solely to the free stream velocity. It should thus be possible to predict the magnitude of free stream velocities from bed form dimensions and associated grain-sizes. According to the procedure suggested by Soulsby , a reduction of the free stream velocity by 10% will transform it into the corresponding velocity at 4 m above the bed within an error of ±7% for water depths greater than 10 m. Thus
The variations in the equivalent sand roughness of the bed associated with dune length are due to the fact that the hydraulic roughness changes only little with grain size, which gives almost parallel horizontal lines for the kt isolines in Figure 8b. Grain size is of course part of kt (equation (3)) but its influence is small compared to the hydraulic resistance of the bed forms. With increasing dune length, the kt-values grow fast (decreasing distance between the isolines) as long as the dune height grows more rapidly than the dune length. At dune lengths greater than about 12 m (on the border and outside the range of θ′ < 0.5 covered by Figure 8b), dune heights, and therefore also kt, level out and decline with increasing flow strength. This is the prediction of equation (7). However, since it lies outside the experimentally observed range, only the striking resemblance with the larger range flume data (Figure 4) supports this trend. Unfortunately, few researchers have reported on the dimensions of superimposed bed forms, and often clear statements about whether a particular bed form is simple or compound are lacking. This makes it difficult to compare the present findings with other published data. Results published by Harbor  from the Mississippi River and Kostaschuk and Ilersich  from the Fraser River suggests that, under increasing flow strength, simple bed forms can grow higher than predicted by equation (7).
 The bed load transport can be obtained from dune length and grain size by calculating θ′ from equation (8), whereas the transport associated with bed form migration can be calculated from equation (9), with θ′ and grain size as the independent parameters. For a fixed grain size, the bed load transport increases with dune length, as it does with grain size for a fixed dune length.
 Obviously, a field-based calibration of equation (9) would improve the credibility of this approach. However, until that is achieved, the results presented here are claimed to represent the best possible way of expressing bed form migration as a function of flow strength and grain size, and to relate bed form dimensions, in form of bed form length, to a plausible bed load transport rate related to bed form migration. The procedure described in this paper for the first time presents a relationship between bed form dimensions and bed load transport which is based on transparent arguments based on accepted theory and which are in agreement with field (and flume) data. It should be kept in mind, however, that the bed load transport modeling in this case relates to quasi-stationary (dominant flow) conditions only.
 It should also be noted that the algorithms presented above are calibrated for water temperatures of about 10°C and grain sizes in the range of 0.3–0.6 mm. Deviations from these conditions may change the calibration constants if other scaling effects than those represented by θ′ (or θ) influence the result.
 As the primary reason for the observed variations is suggested to be the proportion of suspension in the total transport, it would seem logical that temperature can play a role as one of the parameters controlling the sediment fall velocity. Following from the π-theorem analysis of Yalin , four dimensionless parameters are sufficient to describe what he called “the two-phase phenomenon” of the motion of fluid and bed material. Omitting the dimensionless depth (as we are dealing with depth-independent bed forms) and mass (since only quartz and water are considered), only the grain Reynolds number remains in addition to the Shields parameter. The influence of this dimensionless number is important if variations in water viscosity affect the results. This is not important when the water temperature remains almost constant, as in the present study, but large changes in kinematic viscosity associated with seasonally changing water temperatures will probably have an effect [cf. Krögel and Flemming, 1998]. In fact, observations made in the Grådyb tidal channel suggest a tendency toward larger bed form heights during the summer season. The influence of water temperature should be an important concern of future research, but lies beyond the scope of the present paper. Unfortunately, since most studies involving compound bed forms concentrate on the larger features, only little has been published on free simple (superimposed) dunes with corresponding information on grain size and dominant flow conditions. A considerable amount of such data is probably available in the form of unpublished background information from various field studies dealing with bed forms. We are ourselves currently collecting such data and hereby invite interested colleagues to participate in building up a database on free subaqueous dunes in order to improve our knowledge about bed form generation and its relation to flow conditions and grain size. Of particular interest would be the improvement of equations (7) and (8) based on a wider range of data, and an examination of the extend to which shallow water depth affects the dune dimensions until they become depth independent.