The limits to growth: how large can subaqueous, flow-transverse bedforms ultimately become?

Abstract

Based on field and experimental evidence, the average initial spacing (seed wavelength) of flow-transverse bedforms (ripples and dunes) appears to lie between 80 and 130 grain diameters (L = 80–130D_mm). Starting with an average initial spacing of L = 100D_mm, subsequent bedform growth proceeds by amalgamation of two successive bedforms, which results in a doubling of the spacing in each step. Geometric principles dictate that the combined volume of two smaller bedforms lacks about 40% of the volume required for a fully developed amalgamated bedform. The missing volume is gained by excavation of the troughs, i.e., by lowering the base level. Where base level lowering is prevented by the presence of a coarse-grained armor layer or hard ground pavement, the larger amalgamated bedform remains sediment starved. In its simplest form, bedform growth proceeds by continuous doubling of the spacing in response to increases in flow velocity, the process being reversible in response to flow decelerations. Bedform growth terminates when the shear velocity (u_*) at the crest reaches the mean settling velocity (w_s) of the sediment. At this point, 40% of the bed material is in suspension, at which point the missing volume can no longer be compensated by trough excavation. In shallow water, maximum bedform size is dictated by the water depth, whereas in deep water, bedforms can potentially grow to their ultimate size. Evaluation of bedform data from deep water settings suggests that the largest two-dimensional, flow-transverse bedforms in terms of grain size (phi) can be approximated by the equations: lnL_max = 13.72–4.03D_phi and lnH_max = 9.95–3.47D_phi for grain sizes <  ~ 0.2 mm (> ~ 2.32 phi), with L and H representing bedform spacing and height in meters and D the grain size in phi. For grain sizes >  ~ 0.2 mm (< ~ 3.23 phi), the corresponding relationships are lnL_max = 6.215–0.69 D_phi and lnH_max = 3.18–0.56D_phi, with notations as before, or in terms of grain diameters in mm: L_max = 5 × 10⁵D_mm.

1 Introduction

Phase diagrams of flow-transverse bedforms observed in flumes and shallow natural flows are commonly distinguished between ripples and dunes. However, both follow the same spacing (L) versus height (H) relationship (Fig. 1; cf. Flemming 1988, 2000a; Ashley et al. 1990). The conventional morphological criteria for their distinction are that ripples are overall steeper than dunes (e.g., Lin and Venditti 2013) and that ripples are restricted to sizes of L <  ~ 0.6 m (Allen 1984; Sumer and Bakioglu 1984; Lapotre et al. 2017), whereas dunes have spacings of L >  ~ 0.6 m (e.g., Simons and Richardson 1966; Costello and Southard 1981; Allen 1984; Sumer and Bakioglu 1984; Boguchwal and Southard 1989; Ashley et al. 1990; Chiew 1991; van den Berg and van Gelder 1993; Kleinhans et al. 2002). For convenience, bedform steepness values have been added to Fig. 1 (right ordinate). They show that bedform steepness decreases systematically with increasing bedform size. The apparent data gap between the spacing of 0.5 and 1 m apparent in Fig. 1 was subsequently cited as observational evidence in support of the distinction between ripples and dunes (e.g., Ashley et al. 1990; Southard 1991; Leeder 1999).

Fig. 1

Global mean and maximum trends of bedform height versus bedform spacing. Note that the global mean trend was determined by non-linear regression, whereas the global maximum trend was fitted by eye (Flemming 1988; cf. also Ashley et al. 1990; Flemming 2000a). Also note the apparent gap between ripples and dunes in this pre-1985 dataset. Since then, the gap is increasingly being filled, notably by bedforms in coarse-grained sediments

In addition, Yalin (1977) defined dunes as bedforms that inherently interacted with the water surface, whereas ripples did not. He did not, however, provide any boundary conditions for which this definition was supposed to be valid. This criterion has recently been challenged by Bartholdy et al. (2015) for very shallow flows, and there is overwhelming evidence that it does, in any case, not apply to deep flows where the frictional boundary layer either equals the water depth or remains fully submerged (cf. Grant and Madsen 1986; Flemming and Bartholomä 2012, and citations therein). Because water depth is an important limiting factor in bedform growth (e.g., Rubin and McCulloch 1980; Flemming 2000a; Latrubesse 2015; Bradley and Venditti 2016), a careful distinction must be made between depth-dependent and depth-independent bedforms (Bartholdy et al. 2004; Gómez et al. 2010), the dynamics of the former being controlled by much more complicated hydraulic conditions than the latter. Reviews of currently accepted definitions and delimitations can be found in Ashley et al. (1990), Carling (1999), Best (2005), and Lin and Venditti (2013).

In the context of the above considerations, the main purpose of this paper is to present a data-based conceptual model in which (a) the growth of the smallest and largest bedforms are not limited by water depth, and (b) where the dimensions defining the smallest and largest bedforms are expressed by empirical regression equations in terms of spacing (L) and height (H) as functions of mean grain size or, alternatively, as functions of the number of mean particle diameters required in each case.

2 The grain-scaling concept in expressing bedform spacing

Yalin (1977) was the first to apply the grain-scaling concept, suggesting that flow-transverse bedforms had minimum average spacings of about 1000 grain diameters (L_min =  ~ 1000D_mm) which, if correct, would be more or less in agreement with the conventional dimensional distinction between ripples and dunes. However, a decade or so later, Führböter (1991) suggested a relation of L_min =  ~ 450D_mm. More recently, Coleman and Eling (2000; cf. also Coleman and Nikora 2009, 2011) proposed the relation of L_min =  ~ 130D_mm for so-called seed waves on the basis of observational data, while Coleman and Melville (1996) recorded minimum lengths of ~ 100D and Raudkivi (1997) of ~ 80D. Furthermore, an L_min =  ~ 80D_mm for a mean grain size of ~ 4 mm can be inferred from data of Gomez et al. (1989; cf. also Carling 1999). Flemming (2019), in turn, recently documented small flow-transverse bedforms of L = 8–9 cm in coarse sand (D₅₀ = 0.87 mm) and L = 20 cm in very coarse sand to fine gravel (D₅₀ =  ~ 2 mm), both approximating an L_min of ~ 100D_mm. In the former case, the small-scale bedforms were exposed at low tide on an intertidal channel bar located along the southwestern macrotidal shore of the San José Gulf, Valdéz Peninsula, Argentina (Fig. 2), and in the latter case in an ephemeral stream in Namibia a few days after a heavy downpour (Fig. 3). Note that the dunes in Fig. 3 are covered by a ca. 1-mm thin drape of late-stage suspension fall-out composed of silt-dominated mud (light gray color). Important here is the fact that in both cases, the smallest bedforms, which incidentally display fully developed bodies, evolved by splitting processes in the course of decelerating flows.

Fig. 2

Ripples in coarse sand (D₅₀ = 0.87 mm) observed by the author in August 2015 on an intertidal channel bar at low tide (southwestern San José Gulf, Valdez Peninsula, Argentina). Note that the smallest examples have spacings of 8–9 cm, which more or less correspond to 100 grain diameters

Fig. 3

Bedforms in an ephemeral stream with dimensions falling into the apparent gap of Fig. 1. The bedform bodies are composed of material having a mean grain size of D_mean =  ~ 2 mm. Also note the small bedforms at top left with a spacing of about 20 cm, which correspond to hundred grain diameters (100 D)

The main point of interest here is that the concept expressing the smallest spacings of bedforms, variously called seed waves, wavelets, ripplets, or embryonic ripples, in terms of grain diameters has become progressively downscaled over the past decades and today ranges from 80 to 130D_mm, a size interval that can be construed to represent the initiation or seeding region of flow-transverse bedforms (Fig. 4). This concept has recently received strong support by the experimental and numerical studies of Kidanemariam and Uhlmann (2017) who proposed the smallest bedform spacing in the range of 75–100D_mm, and Scherer et al. (2020) who identified two scaling relations, a “shorter domain” around 80D_mm and a “longer domain” from 150 to 180D_mm, the two domains relating to different (shallow) fluid depths. The grain-scaling criterion for initial bedform spacing not only implies that, at their initiation, there is considerable dimensional overlap between conventionally defined ripples and dunes, but that ripples should also occur in non-cohesive, silt-sized sediments (region outlined by the ellipse in Fig. 4), as indeed documented by Rees (1966), Mantz (1978), Jopling and Forbes (1979), van den Berg and van Gelder (1993), and Culp et al. (2021).

Fig. 4

Diagram of ripple/dune spacing vs grain size showing various suggestions for initial ripple/dune spacings (L_min) in terms of grain diameters: 1000 D (Yalin 1977); 430 D (Führböter 1991); 130 D (Coleman and Nikora 2011); 100 D (Flemming, this paper); and 80 D (Gomez et al. 1989; cf. also Carling 1999). The size interval of 80–130 D (sloping gray bar) is considered to represent the region of bedform initiation. Also shown are the L_max trend lines defined by multiples of the 100 D criterion, and the L_max trend (thin gray line) proposed by Flemming (1988). Also shown are the data points of large dunes (red dots) on the basis of which the trend lines of L_max have been defined. The numbers refer to the data sources as follows: (1) (Flemming and Bartholomä 2012); (2) (Rubin and McCulloch 1980); (3) (Zhou et al. 2020); (4) (van Landeghem et al. 2009); (5) (Mosher and Thomson (2000); and (6) (Franzetti et al. 2013). The ellipse at lower left highlights the L_max region of ripples known to occur in coarse silt

3 Materials and methods

The present study is based on partly published and partly unpublished data that were mostly acquired in the course of geological and geophysical surveys of the continental shelf off the South African east coast, which is swept by the Agulhas Current (Flemming 1978, 1980, 1988, 2000a, 2019; Flemming and Hay 1988; Ashley et al. 1990; Flemming and Bartholomä 2012). Additional data were obtained from the international literature (cf. corresponding citations in the caption of Fig. 4 and in the text below) and by coincidental, not previously published, observations in the field, and the grain-size analysis of simultaneously collected sediment samples. When sampling bedforms exposed sub-aerially at low tide, care was taken to avoid contamination by late-stage suspension fall-out composed of fine-grained sediments that commonly drape such bedforms, and also by material from underlying, coarser-grained basal strata over which the bedforms migrated.

Grain-size analyses were performed by sieving and/or settling tube, settling velocities being subsequently expressed in terms of equivalent settling diameters (for details, see Flemming and Thum 1976; Brezina 1979). The cumulative grain-size frequency distribution presented in Fig. 2c represents the material shown in inset b, which is a blow-up of the sampling area indicated by the arrow in Fig. 2a. It demonstrates that the overall contamination from the two potential sources outlined above amounted to less than 1% (0.4% late-stage fall-out and 0.5% basal material). The bulk of the sample (99.1%) collected from the bedform core was found to be log-normally distributed and moderately well sorted with a median diameter of D₅₀ = 0.87 mm. The same basically applies to the bed material shown in the inset of Fig. 3 which had a D₅₀ of ~ 2 mm. The sample was recovered immediately below the fine-grained drape covering the dunes. The grain-size parameters were in all cases calculated on the basis of the inclusive percentile statistics of Folk and Ward (1957).

While the trend line in Fig. 4 defining maximum bedform size (L_max) for grain sizes < 0.2 mm is well constrained, the trend line for grain sizes > 0.2 mm is based on much fewer data (numbered red dots) provided by (1) Flemming (1980), Flemming (2000a), and Flemming and Bartholomä (2012); (2) Rubin and McCulloch (1980); (3) Mosher and Thomson (2000); (4) van Landeghem et al. (2009); (5) Franzetti et al. (2013); and (6) Zhou et al. (2020). The dearth of data in this context is rather unfortunate, only very few publications actually providing specific grain-size data for individual large dunes. In many cases, only grain-size ranges are cited for a variety of bedform size ranges (e.g., van Landeghem et al. 2009; Franzetti et al. 2013), which complicates the issue. However, to date, no data were found that plotted above the trend lines. In cases where the measurement of dune lengths is doubtful but a reliable height is available, the associated minimal dune length can be calculated by the equation L = 6.456H_max^0.84, which represents the regression equation when the axes in Fig. 1 are reversed.

In analogy, values for both H_mean and H_max are calculated for L values at 0.5-phi intervals extracted from Fig. 4 using the equations H_mean = 0.0677L^0.8098 and H_max = 0.16L^0.84 (cf. Figure 1). These were then used to construct the corresponding model of H vs grain size. The conceptual models derived in this way can be regarded as first-order approximations that may be refined as more data become available.

The regression equations describing the relationships between grain size and maximum bedform spacing and height were determined by the application of the software package TableCurve 2D v501.02 (Systat Software, Inc., San José, California). Of interest here is that good fits were only achieved when grain sizes were expressed in the binary phi notation (phi = –log₂D_mm).

4 Results

A first controversy to be resolved is whether the apparent size gap between 0.5 and 1.0 m in the H vs L diagram of Flemming (1988) displayed in Fig. 1 (cf. also Ashley et al. 1990) is real or whether it is merely an observational gap, as argued by Flemming (2000a). In this context, it needs to be appreciated that the data presented in Fig. 1 represent the pre-1985 state of knowledge where grain-size data for dunes of known size, especially in coarse-grained sediment, was still rather sparse. In the meantime more such data have become available (e.g., Gomez et al. 1989; Leclair et al. 1997; Carling 1999; Mosher and Thomson 2000; Flemming and Bartholomä 2012), some of which either fall into the ominous gap, e.g., Leclair et al. (1997: L = 0.83–0.94 m, D = 0.42 mm) and Bradley and Venditti (2016, their Fig. 4: at least 40 dunes within the gap), or would do so when applying the L_min = 80–130D criterion (Gomez et al. 1989; Carling 1999: L_min = 0.3–0.4 m; D =  ~ 4 mm). Another example are the dunes shown in Fig. 3 (Flemming, hitherto unpublished), which had spacings of 0.7–0.85 m in sediment of D₅₀ =  ~ 2 mm. Individuals about one-third that size (L =  ~ 0.2 m) are visible in the upper left of Fig. 3, which precisely corresponds to 100D_mm and hence represents the smallest possible forms in that grain size. Of interest here is the fact that the mean or median grain sizes of dunes that fall into the 0.5–1.0 m gap are without exception coarser than ~ 0.4 mm. These examples demonstrate that the apparent gap in the pre-1985 data shown in Fig. 1 was fortuitous, being solely due to a lack of data from coarse-grained sediments.

Besides being fortuitous, the gap in the pre-1985 data also highlights two aspects that could easily have gone unnoticed. First, for any given grain size, dune growth by amalgamation inherently results in dimensional gaps between successively larger dunes, the gap progressively widening as the dunes grow in size, and second, the gaps also increase in width with increasing grain size. This is illustrated in Fig. 4 where the distance between the sloping parallel lines representing L_min for 80, 100, and 130D_mm progressively increases with increasing grain size as implied by the log-scaling of the ordinate (bedform spacing).

In its simplest form, amalgamation (growth) or splitting of flow-transverse bedforms can be construed to proceed in steps where two successive bedforms either merge to form the next larger category or split up to form the next lower category. In this process, the spacing doubles or halves in the course of each amalgamation or splitting step. Examples of this are illustrated in Fig. 5 for ripple-sized and in Fig. 6 for dune-sized bedforms. Thus, in Fig. 5a, large ripples (L ~ 30 cm) can be seen to split up into progressively smaller forms by exactly halving their spacing in each step as the flow expands into a landward runnel after passing through a gap in a beach ridge. Figure 5b shows several sequences of large ripples that are laterally interrupted by narrow bands of smaller ripples, suggesting corresponding lateral changes in flow speed that cause the large ripples to split up into several smaller ones. Finally, Fig. 5c shows two sets of differently spaced ripples produced by laterally variable wind-driven flows in very shallow water on an intertidal flat. The larger ripples have a spacing that is about twice that of the smaller ones. As indicated by the shadow width along the lee slopes, the larger ripples are also clearly higher than the smaller ones.

Fig. 5

Images illustrating dune amalgamation/splitting in response to changes in either the streamwise (a) or lateral (b, c) flow velocity. Further details are provided in the text

Fig. 6

a Sand ribbon on the south-east African continental shelf (~ 80 m water depth) showing progressive and quasi-symmetrical splitting of dunes towards both margins. b Dunes in a tidal channel of the Wadden Sea at spring tide (upper panel) and at neap tide (lower panel). Note the splitting into 3–4 smaller dunes towards neap tide

The same response to changes in flow speed can be observed in the behavior of dunes. Figure 6a shows a several hundred-meter-wide sand ribbon with superimposed dunes located in 80 m water depth on the SE African continental shelf (Flemming 1980, 2000b). The ~ 30-m long dunes at the center of the ribbon can be seen to split up laterally (towards both margins) into progressively smaller dunes by halving their spacing in each step, suggesting the existence of parallel-aligned corridors above the ribbon where the water flowed at different speeds. Such a flow structure is quite compatible with helical flow models commonly associated with sand ribbons (e.g., McLean 1981; Viekman et al. 1992). A second example from a tidal channel of the Wadden Sea (Fig. 6b) documents the splitting up of dunes spaced about 40 m apart formed by strong spring tide flow (upper bathymetric profile) into three to four smaller dunes in response to an about 20% lower flow velocity at neap tide (lower bathymetric profile). The process reversed towards the next spring tide. Clearly evident in Fig. 6b is the raised base level (or trough level) due to accretion in the troughs of the larger dunes. The higher base level is again lowered by excavation during renewed amalgamation towards the next spring tide.

While straightforward doubling/halving of bedform spacing results in regular size sequences (Figs. 5a and 5c and 6a), the images in Figs. 5b and 6b reveal that bedform amalgamation/splitting does not necessarily follow the simplest form but frequently involves more than two bedforms, both in the course of amalgamation and splitting. Where three bedforms are involved, the growth or splitting step will produce a ripple/dune size that is out of sequence with those produced by the doubling/halving process. Such events result in irregular size sequences, which are frequently observed in nature (e.g., in Fig. 6b).

The potentially largest bedforms (L_max) as a function of grain size are defined by the trend lines in Figs. 4 and 7 which are based on observations in nature. The criterion applies to active two-dimensional (straight-crested to mildly sinuous) bedforms in conditions where water depths are sufficiently large, current velocities are sufficiently strong, and sediment is of sufficient supply (e.g., Flemming 2000a). Three-dimensional (strongly sinuous to lunate) bedforms are explicitly excluded here because these are in a state of transition to upper plane bed conditions where the steepness ratio (H/L) begins to strongly decrease (e.g., Lin and Venditti 2013). If a smaller or larger initial grain-scaling approach was applied, the corresponding L_max trend lines would lie slightly above or below the 100D_mm trend lines. For comparison, the original L_max trend line of Flemming (1988) is also shown (thin gray line). The divergence of the latter for bedform spacings larger than about 100 m is due to the fact that, in pre-1985 times, grain-size data for individual large dunes was still rather scant. As argued by Flemming (2000a), the growth of two-dimensional bedforms ultimately ends when the shear velocity (u_*) at the bedform crest reaches the mean settling velocity (w_s) of the sediment (i.e., u_* = w_s), a point at which 40% of the bed material is in suspension (Graf and Acaroglu 1966). This criterion applies to any water depth and is supported by the fact that the steepness ratio begins to decrease when the ratio of u_*/w_s exceeds unity as documented in Lin and Venditti (2013) and Bradley and Venditti (2016). In Fig. 7, the progressive growth of bedforms towards their maximum size in terms of grain diameters is illustrated by broken parallel lines representing factor 10 multiples of the 100D_mm criterion. Of interest here is the fact that maximum bedform size for grain sizes > 0.2 mm uniformly corresponds to about 500,000 (5 × 10⁵) grain diameters, whereas for grain sizes < 0.2 mm, the number of grain sizes defining the maximum size rapidly decreases with decreasing grain size. The sharp change in the slope of L_max at a grain size of about 0.2 mm (~ 2.32 phi) is a feature that obviously requires an explanation (cf. Discussion).

Fig. 7

Diagram of ripple/dune spacing vs grain size showing bedform growth in terms of equivalent numbers of grain diameters, here as factor ten multiples of the L_min = 100 D criterion (sloping broken lines). Note the steeply sloping regression trend of L_max for grain sizes < 0.2 mm, and the more gently sloping trend for grain sizes > 0.2 mm, which runs parallel to the L_min trend and corresponds to about 500,000 grain diameters. Also shown are the critical shear velocities at which maximum growth is postulated to be reached

As pointed out before, the potentially largest bedform spacing (L_max) for any given grain size can only be achieved if water depth is large enough and flow velocity is strong enough. In the diagram of Fig. 7, the maximum spacing is represented by two trend lines, one with a steeper slope applying to grain sizes < 0.2 mm, and another with a gentler slope applying to grain sizes > 0.2 mm. The steeper trend line, which defines maximum spacings up to about 85 m, can be mathematically approximated, with the grain size in phi, by the regression equation:

$$\begin{array}{cc}\ln L_{max}=13.72-4.03D_{\mathrm{phi}}&[m] \end{array}$$

(1)

The more gently sloping trend line defining dune spacing larger ~ 85 m can be mathematically approximated by two approaches. As can be seen in Fig. 7, the dune spacing (expressed in meters) can alternatively also be approximated in terms of grain diameters:

$$\begin{array}{cc}{L}_{\mathrm{max}}=500{D}_{\mathrm{mm}}& \left[\mathrm{m}\right]\end{array}$$

(2a)

or in terms of the grain size in phi, by the regression equation:

$$\begin{array}{cc}\mathrm{ln}{L}_{\mathrm{max}}=6.215-0.69{D}_{\mathrm{phi}}& \left[\mathrm{m}\right]\end{array}$$

(2b)

In the above equations, L_max refers to the maximum bedform spacing, D_mm refers to grain diameters expressed in millimeters, and D_phi refers to grain size expressed in phi. In all cases, the values obtained when resolving the equations represent spacing in meters. The convergence point between the more steeply and more gently sloping trend lines occurs at a grain size of approximately 0.2 mm (200 µm). It separates a domain to the right (grain sizes > 0.2 mm), where the bed material is exclusively transported in bed load, from a domain to the left (grain sizes < 0.2 mm), where part of the bed material is transported in suspension. It can be construed that the 0.2-mm separation line corresponds to the situation where w_s = 1.6u_*, with w_s representing the settling velocity and u_* the shear velocity. The proportion of material transported in suspension increases towards the left of the separation line and continues until w_s = u_* when ~ 40% of the bed material is in suspension and further bedform growth is terminated.

Using the initial and maximum bedform spacing determined by the 100D_mm criterion and the regression equations above, it is possible to calculate the corresponding mean and maximum bedform heights as a function of grain size on the basis of the equations of Flemming (1988), i.e., H_mean = 0.0677L^0.8098 and H_max = 0.16L^0.84 (cf. Figure 1). The trend lines obtained by this exercise are illustrated in Fig. 8. Minimum and maximum initial bedform heights can be approximated by the power relationships:

Fig. 8

Diagram of bedform height vs grain size. Note the similarity of the trends as in Figs. 4, and 7 for dune spacing. Also shown are the shear velocities (at a water temperature of 10 °C) for selected grain sizes (full phi intervals) at which 40% of the bed material is in suspension (w_s = u_*), as a consequence of which the growth in height (H_max) is terminated

$$\begin{array}{cc}{H}_{\mathrm{mingd}}={7.138D}^{-0.247}+3.92& \left[\mathrm{mm}\right]\end{array}$$

(3a)

and

$$\begin{array}{cc}{H}_{\mathrm{maxgd}}={22.406D}^{-0.171}-0.171& \left[\mathrm{mm}\right]\end{array}$$

(3b)

where H_mingd and H_maxgd are the ripple heights expressed in grain diameters (subscripts gd) of a given grain size expressed in mm. Worth noting here is the fact that the maximum ripple height is approximately twice that of the mean ripple height for any given grain size.

As in the case of spacing, the mean and maximum bedform heights are also represented by more steeply and more gently sloping trend lines separated at a mean or median grain size of 0.2 mm. The more steeply sloping trend lines, which apply to grain sizes < 0.2 mm (H_mean <  ~ 3 m; H_max <  ~ 8 m), can be mathematically approximated, in terms of the grain size in phi, by the respective regression equations of:

$$\begin{array}{cc}\mathrm{ln}{H}_{\mathrm{mean}}=8.98-3.4{D}_{\mathrm{phi}}& \left[\mathrm{m}\right]\end{array}$$

(4a)

and

$$\begin{array}{cc}\mathrm{ln}{H}_{\mathrm{max}}=9.95-3.4{D}_{\mathrm{phi}}& \left[\mathrm{m}\right]\end{array}$$

(4b)

The more gently sloping trend lines, which apply to grain sizes > 0.2 mm (H_mean >  ~ 3 m; H_max >  ~ 8 m), can be mathematically approximated, in terms of the grain size in phi, by the respective regression equations of:

$$\begin{array}{cc}\mathrm{ln}{H}_{\mathrm{mean}}=2.28-0.55{D}_{\mathrm{phi}}& \left[\mathrm{m}\right]\end{array}$$

(5a)

and

$$\begin{array}{cc}\mathrm{ln}{H}_{\mathrm{max}}=3.18-0.55{D}_{\mathrm{phi}}& \left[\mathrm{m}\right]\end{array}$$

(5b)

As before, D_phi refers to grain sizes expressed in phi and the values obtained when resolving the equations represent heights in meters. Also shown in Fig. 8 is the criterion at which further vertical growth of flow-transverse bedforms comes to a halt, i.e., when ~ 40% of the bed material is in suspension (for explanation, see Flemming 2000b). As pointed out before, this condition is reached when the shear velocity equals the settling velocity (u_* = w_s), here expressed in cm/s at a water temperature of 10 °C for grain sizes at full phi intervals. As in Fig. 7, the separation of the bed load domain from the combined suspension and bed load domain occurs at a grain size of about 0.2 mm.

An interesting feature is documented by the sloping lines representing the heights at the initiation of bedform formation in relation to grain size. Here, the mean and maximum absolute heights as well as minimum and maximum initial heights can be distinguished, all of which can be expressed in either metric measures or number of grain diameters. For example, at a mean grain size of 4 phi (0.063 mm), the minimum initial height corresponds to ~ 0.001 m, which amounts to about 18 grain diameters, from where it progressively increases to 0.036 m at a grain size of − 2 phi (4 mm), which corresponds to about 9 grain diameters. The progressively smaller grain number expressing the bedform height with increasing grain size is in agreement with the general observation that the height/spacing ratio decreases (or, inversely, the ripple index L/H increases) as grain size increases; i.e., bedform height increases less rapidly than spacing (cf. Figure 1).

With the above relationships between bedform spacing and height as a function of grain size at hand, the size range occupied by flow-transverse bedforms for any particular grain size can be displayed in the L vs H diagram (Fig. 9). For convenience, selected size ranges corresponding to the mm values at full phi intervals are shown together with the H_mean and H_max trend lines of Flemming (1988). Furthermore, to avoid confusion between minimum (or embryonic) and maximum bedform sizes for given grain sizes, the two have been plotted into separate diagrams, maximum sizes in Fig. 9a (downward pointing arrows) and minimum (or initial) sizes in Fig. 9b (upward pointing arrows). The corresponding shear velocities at which bedform growth for the selected grain sizes comes to a halt (here at a water temperature of 10 °C) are shown at the top of Fig. 9a.

Fig. 9

Size ranges (H vs L) of bedforms as a function of selected mean grain sizes. a Maximum bedform sizes. b Initial bedform sizes. Note that, for all grain sizes shown here (full phi steps from − 2 to + 4 phi or 4.0–0.063 mm), the initial forms have spacings < 0.5 m. Also shown (at the top of diagram a) are the corresponding shear velocities at which 40% of the bed material is in suspension, i.e., where w_s = u_*, here valid for a water temperature of 20 °C

5 Discussion

The narrow size interval defined by the smallest dune spacings in terms of the 80–130D_mm criterion based on data of Coleman and Eling (2000; cf. also Coleman and Nikora 2009, 2011), Gomez et al. (1989), Carling (1999), and Flemming (this article) can be viewed as the seeding zone within which flow-transverse bedforms begin to develop (Figs. 7 and 8). It should also be noted that the 1000D_mm and 450D_mm proposals of Yalin (1977) and Führböter (1991), respectively, are in conflict with the size of ripples known to occur in silts (ellipse in Fig. 4), which therefore disqualifies those proposals.

A specific aim of this study was to address the question of how large flow-transverse bedforms can ultimately become (L_max) under ideal conditions, i.e., where bedform growth is not impaired by water depth. In contradiction to Ashley et al. (1990), the field evidence available today clearly shows that bedform growth is ultimately limited by a grain-size-dependent critical shear velocity, provided that water depth and flow velocity are large enough. This is illustrated in Figs. 7 and 8 where L and H are plotted against grain size. The more steeply sloping trend lines represent grain sizes up to about 0.2 mm. This region is well constrained by observational data (e.g., Francken et al. 2004; Flemming and Bartholomä 2012). For coarser grain sizes, the trend line follows a substantially lower gradient, which appears to run parallel to the trend line defining the spacings and heights at bedform initiation. Although both trends are supported by field data, the data base for the lower gradient trend line (valid for grain sizes > 0.2 mm) is much poorer because dunes more rarely reach their potential maximum size the coarser the sediment becomes, and which frequently results in sediment-starved individuals. Nevertheless, the spacings and heights of the very large dunes of Mosher and Thomson (2000), van Landeghem et al. (2009), Franzetti et al. (2013), and Zhou et al. (2020) are in good agreement with the fitted trend lines, the dunes having almost reached their maximum sizes (Fig. 4). In the case of Mosher and Thomson (2000), it should be noted that the dune body itself is incomplete or sediment starved; i.e., its high steepness ratio is simply due to the very strong bipolar tidal currents in the Strait of Juan-de-Fuca located between Vancouver Island (British Columbia, Canada) and the Olympic Peninsula (WA, USA). In such environments, dunes frequently have trochoidal shapes resulting in unusual heights (e.g., van Landeghem et al. 2009; Ma et al. 2019; Zhou et al. 2020). In this context, it should also be noted that particle shape and grain-size dispersion appear to have profound effects on bedform scaling parameters (Jain et al. 2019), a phenomenon that requires further investigation.

The break in slope gradient between grain sizes smaller and larger than 0.2 mm may be explained by the observation that it separates a domain of pure bedload transport from a domain of combined bedload and suspension transport. In Fig. 10, the Shields diagram as modified by Van Rijn (1993), the grain size of 0.2 mm represents the situation where w_s = 1.6u_*, which represents the criterion for beginning suspension (e.g., Graf 1971). Bedform evolution can thus be said to develop under stable transport conditions at grain sizes > 0.2 mm (only bedload transport), whereas it becomes progressively more unstable as grain sizes become finer (increasing suspension transport). The proportion of suspended material then gradually increases until the condition of w_s = u_* is reached, where, according to Graf and Acaroglu (1966), approximately 40% of the bed material is in suspension. In Fig. 10, this point is reached at a grain size of about 0.15 mm. Flemming (2000a) showed that, when two dunes (or ripples) of similar size amalgamate, the geometric relationships dictate that about 40% of the larger dune body is initially missing. With little or no suspension transport, the missing volume is gradually regained in the course of amalgamation by lowering the base level through trough scouring (Flemming 2000b). However, when 40% of the bed material is in suspension, the missing volume cannot be regained, and further, bedform growth becomes impossible. It marks the onset of gradual dune destruction by transformation into three-dimensional forms and eventually upper plane bed conditions. The time required for this transformation increases with increasing bedform size. Because of the increasing proportion of suspension transport at grain sizes < 0.2 mm, bedform evolution reacts very sensitively to even small changes in bed shear stress. As a consequence, bedform development proceeds under increasingly more unstable conditions the finer the sediment is, a fact which may explain why the bedforms are still comparatively small when their growth comes to a halt.

Fig. 10

Shields diagram (Shields 1936, as modified by Van Rijn 1993) illustrating the transition from stable bedforms to increasingly more unstable bedforms in response to increasing suspension transport beginning when w_s = 1.6u_* and reaching 40% when w_s = u_*. The corresponding trend lines are valid for a water temperature of 10 °C. With increasing water temperature, these would gradually shift towards finer grain sizes (i.e., towards the left) because of corresponding increases in w_s

A final comment concerns the difference between depth-independent and depth-dependent bedforms. This is illustrated in Fig. 11 by the two dune sequences shown in box A and box B, respectively. The data points represent the average trends of dune parameters published by Bartholdy et al. (2002, 2004, 2005) and Ernstsen et al. (2005). The dunes plotted in box A represent a depth-independent sequence, and those in box B represent a depth-dependent sequence. Both sequences occurred in similar water depths (~ 12 m) of the same tidal channel (Danish Wadden Sea). Sequence A represents superimposed dunes on larger dunes exposed to maximum depth-averaged flow velocities of ~ 1.0 m/s, and sequence B represents the larger dunes exposed to the same depth-averaged flow. In the case of the depth-independent dunes (sequence A), the maximum shear velocity is smaller than the settling velocity and is practically identical for all the dunes. As a consequence, the size of the dunes decreases with increasing grain size. The exact opposite trend is observed in the depth-dependent, larger dunes (sequence B) which have all reached their maximum size for the given water depth and flow velocity, and where the critical shear velocity equals (or even slightly exceeds) the settling velocity of the bed material. As a consequence, the size of the dunes increases with increasing grain size, the reason being that the critical shear velocity at the crest of the finer-grained dunes is reached before that of the coarser-grained ones. The clear relationship between maximum dune height and grain size in the depth-dependent dunes illustrated in Fig. 11 suggests that corresponding relations proposed by Allen (1968) and Yalin (1977) (cf. Flemming 2000a), which have ignored grain size as a functional parameter in this context, represent averages for the coincidental, but unknown, grain size ranges characterizing the sediment in their studies. The same argument would apply to depth-dependent dune spacing as described by Jackson (1976).

Fig. 11

Diagram illustrating the H vs D trends described by sets of depth-dependent (group A) and depth-independent (group B) bedforms recorded in the Grådyb tidal channel, Danish Wadden Sea (Bartholdy et al. 2002, 2004, 2005). Note the opposing trends defined by the depth-independent (group A) and depth-dependent (group B) dunes, respectively. Also note the different d/H ratios from larger to smaller with increasing grain size in the depth-dependent (group B) dunes. For both groups the average water depth was identical (~ 12 m)

6 Conclusions and recommendations

The main conclusions and recommendations of this investigation are as follows:

• The initial spacing of flow-transverse bedforms in terms of grain diameters follows the rule of L_min being ≥ 80D_mm to ≤ 130D_mm, which appears to be valid for all grain sizes.

• Maximum spacing and height are a function of grain size, water depth, and a critical shear velocity approximated by the relation of u_* = w_s, where ~ 40% of the bed material is transported in suspension.

• By implication, maximum potential bedform size increases with increasing grain size provided that water depth and flow velocity are large enough and sufficient sediment is available to allow the full body of the bedform to develop.

• Bedform experimentalists are encouraged to also study bedform evolution from larger to smaller forms (splitting) in the course of progressive flow reduction.

• Bedform researchers are encouraged to sample the bed material of specific large dunes to increase the data base for optimizing the conceptual models presented in Figs. 7 and 8.