The development of bedforms under unidirectional, oscillatory and combined-flows results from temporal changes in sediment transport, flow and morphological response. In such flows, the bedform characteristics (for example, height, wavelength and shape) change over time, from their initiation to equilibrium with the imposed conditions, even if the flow conditions remain unchanged. These variations in bedform morphology during development are reflected in the sedimentary structures preserved in the rock record. Hence, understanding the time and morphological development in which bedforms evolve to an equilibrium stage is critical for informed reconstruction of the ancient sedimentary record. This article presents results from a laboratory flume study on bedform development and equilibrium development time conducted under purely unidirectional, purely oscillatory and combined-flow conditions, which aimed to test and extend an empirical model developed in past work solely for unidirectional ripples. The present results yield a unified model for bedform development and equilibrium under unidirectional, oscillatory and combined-flows. The experimental results show that the processes of bedform genesis and growth are common to all types of flows, and can be characterized into four stages: (i) incipient bedforms; (ii) growing bedforms; (iii) stabilizing bedforms; and (iv) fully developed bedforms. Furthermore, the development path of bedform; growth exhibits the same general trend for different flow types (for example, unidirectional, oscillatory and combined-flows), bedform size (for example, small versus large ripples), bedform shape (for example, symmetrical or rounded), bedform planform geometry (for example, two-dimensional versus three-dimensional), flow velocities and sediment grain sizes. The equilibrium time for a wide range of bed configurations was determined and found to be inversely proportional to the sediment transport flux occurring for that flow condition.
Over the past few decades, several studies have analysed the development of bedforms under unidirectional and oscillatory flows (e.g. Oost & Baas, 1994; Baas, 1999; Faraci & Foti, 2002; Admiraal et al., 2006; Cataño-Lopera & García, 2006; Landry & Garcia, 2007). This research has provided a much improved understanding of temporally varying bedforms, with particular emphasis on the importance of initial conditions and the time required to reach equilibrium between the bed and flow. The experiments of Doucette & O'Donoghue (2006) under pure oscillatory flows showed that the final geometry of bedforms is independent of the initial conditions and that the time to reach equilibrium depends strongly on the mobility number (dimensionless velocity). Similar observations were made by Baas (1994, 1999) in purely unidirectional flows, which showed that the equilibrium time for current ripple development was proportional to the mean unidirectional velocity. Moreover, the experimental work of Baas (1994, 1999) has shown that there are four stages in the development of ripples under purely unidirectional flows, commencing from an early stage on a flat bed where a few grains are transported in patches, a few grains in height and a few centimetres long, which develop into incipient ripples. These incipient ripples gradually grow in height and wavelength, forming straight and sinuous ripples in the second development stage. As ripple height and wavelength continue increasing in the third stage of development, the bedforms become more three-dimensional (3D) and attain a more linguoid shape, eventually reaching the final stage of development when their equilibrium height and wavelength are reached. Although this bedform development sequence has been well-detailed for unidirectional flows, and despite recent research documenting the development of oscillatory-flow bedforms (e.g. Admiraal et al., 2006; Doucette & O'Donoghue, 2006; Calantoni et al., 2013), no study has yet outlined and characterized the stages of bedform development within purely oscillatory regimes. Furthermore, the current state of knowledge concerning bedforms developed under combined-flows is even sparser, with few studies having investigated the morphology and stability of these bedforms (Arnott & Southard, 1990; Dumas et al., 2005). Thus, to date, the development stages of bedforms under such combined-flows, and how similar they may be to unidirectional or oscillatory flows, have not been investigated.
In addition, in many natural environments, prolonged steady flow conditions may often be infrequent, and thus the bed morphology has to readjust continuously to the flow conditions, which results in non-equilibrium bedforms predominating in many situations (Yalin, 1975; Wijbenga, 1990). However, under some conditions the bed morphology can reach an equilibrium stage before any significant change in the flow field. For example, under relatively high flow velocities, ripples can reach their equilibrium size in tens of minutes to a few hours (Baas, 1994). Thus, considering the range of times needed to achieve an equilibrium bed morphology, and the fact that bedforms undergo different stages of development (Baas, 1994), it is crucial to understand and quantify the geometry of bedform development, and how this may vary between differing flow types. This is particularly important because most relations that relate bedform dimensions with sediment and flow parameters only exist for equilibrium bed-flow conditions (Garcia, 2008). Additionally, interpretation of bedform geometries within the ancient sedimentary record must account for the presence of equilibrium and non-equilibrium bedforms. In this case, it thus becomes important to expand the current knowledge of bedform development, and temporal changes in the bedform characteristics en route to an equilibrium state, to facilitate better recognition of such bedforms in the sedimentary record.
In order to address the above issues, the present study presents results from a series of laboratory experiments that sought to quantify bedform development under a range of unidirectional, oscillatory and combined-flow conditions, to gain insights on the equilibrium development time for different bed states, the temporal changes of bedform height and wavelength, and their relations with flow velocity and sediment transport rates. This study proposes a unified model of bedform development across all flow types and illustrates how the equilibrium bedform development time across all these flows can be collapsed as a function of the dimensionless bedload transport.
Experimental set-up and methodology
A series of experiments was conducted in the Large Oscillatory Water Sediment Tunnel (LOWST) at the Ven Te Chow Hydrosystems Laboratory, University of Illinois, which enables a combination of oscillatory and unidirectional flow velocities of up to 2 m sec−1 and 0·55 m sec−1, respectively. The tunnel is U-shaped, with one leg containing three pistons to generate the water oscillation, whilst the other leg is an open reservoir (Fig. 1). The unidirectional component of flow is generated by two centrifugal pumps. The tunnel has a working test-section 0·8 m wide, 1·2 m deep and 12·5 m in length. Uniform quartz sand (ρs/ρf = 2·65, where ρs and ρf are the sediment and fluid densities, respectively; D50 = 250 μm, D10 = 185 μm and D90 = 373 μm) was used as a bed and had a thickness of half the tunnel height (0·60 m). Sediment traps located at both ends of the test-section collected the sediment that was transported as bedload (Fig. 1), and no additional sand was introduced to the system during the experiments. For a typical experiment, the average amount of sand transported to the ends of the flume and retained in the sand traps was ca 150 kg, but for high sediment transport experiments the traps held up to 300 kg. In addition, for the case of combined-flows, the downstream trap contained up to five times more sand than the upstream trap. To create flatbed conditions at the start of the experiments, a cart composed of two blades that covered the entire width of the tunnel was pulled back and forth along the tunnel to redistribute the wet sediment and leave a smooth and flat surface. However, even after smoothing the bed, small defects on the order of a few grain diameters (ca 0·5 to 1 mm) were present. Full details of the experimental set-up and facilities are given in Perillo (2013).
Bedform development was studied under unidirectional, purely oscillatory and combined-flow conditions with oscillation periods (T) of 4, 5 and 6 sec (Table 1). The maximum orbital velocity (Uo) was varied from 0·10 to 0·70 m sec−1, while the unidirectional component (Uu) was varied from 0 to 0·50 m sec−1. The two pure unidirectional experiments had unidirectional velocities (Uu) of 0·40 m sec−1 and 0·50 m sec−1. Experiments started from a smooth flat bed and during each run a high-resolution webcam (Logitech Webcam Pro 9000, Video Capture Resolution = 1600 × 1200; Logitech Europe S.A., Lausanne, Switzerland) recorded the bed through one of two large windows (0·6 m high by 0·7 m wide) located in each side of the central region (Fig. 1). During all experiments, velocity profiles were measured with an Ultrasonic Doppler Velocity Profiler produced by Met-Flow S.A. (Lausanne, Switzerland) with 1 MHz transducer sensors (Best et al., 2001).
Table 1. Summary of flow and bedform development characteristics. The table is divided into pure unidirectional, pure oscillatory and combined-flow experiments
Uo (m sec−1)
Uu (m sec−1)
qb (cm2 sec−1)
T, oscillatory period; Uo, maximum orbital velocity; Uu, unidirectional velocity; ηe, bedform height at equilibrium; cη, bedform height adaptation constants; R2(η) and R2(λ) are the square of the sample correlation coefficient for Eqs (2) and (3), respectively; λe, bedform wavelength at equilibrium; cλ, bedform wavelength adaptation constants; qb, bedload transport per unit width (Eq. (8)); te, time until flow-bedform equilibrium obtained from Eq. (7). SR, symmetrical small ripples; AR, asymmetrical small ripples; ALR, asymmetrical large ripples; CR, current ripples.
Stable planform geometry is 2D, otherwise it is 3D.
In addition to the velocity data, a custom-designed sonar system was used to measure the bed evolution from flat bed to fully developed equilibrium conditions. An L-shaped Imagenex 881L Digital Multi-Frequency Profiling Sonar (Imagenex Technology Corp., Port Coquitlam, BC, Canada) was installed in a Velmex B4800TS Motorized Rotatory Table with a Velmex VXM controller (Velmex Inc., Bloomfield, NY, USA). The combination of the pencil beam sonar and the custom-designed positioning system allowed the sonar to be rotated around two axes: (i) a horizontal axis, which allowed the sonar to cover a fan-shaped region as the single beam rotated to cover a length of the bed; and (ii) a vertical axis, which by rotating along this axis, allowed many crossing lines to be acquired that thus allowed a complete survey of the bed below the sonar. In this study, the single beam sonar was set in the longitudinal direction along the centre of the tunnel to measure ‘continuous’ streamwise bed profiles during the experiments. This longitudinal sonar data were obtained every 3 sec over a distance of 1 m along the centre of the tunnel to measure the spatio-temporal development of the bed morphology. To eliminate errors in reconstructing the bed elevation, due to sonar reflections from suspended sediment, several profiles at different phases of the oscillation cycle were averaged over ca 30 sec (Pedocchi & García, 2009; Pedocchi, 2009; Perillo, 2013). The bedform height (η) was calculated for each bedform by measuring the distance between the local maximum (bedform crest) and local minimum (bedform trough) from the longitudinal sonar data, whereas the bedform wavelength (λ) was obtained from the distance between local minima (i.e. the bedform troughs). A spatial average was also used to determine the mean bedform height and wavelength for a particular time, t:
where N is the number of bedforms present in the 1 m centre-span of the tunnel surveyed by the sonar. It is important to note that the η and λ data sets include multiple measurements of the same bedform as it migrated in the survey area. However, since the bedforms evolved over time, each bedform does not contribute to the same height or wavelength values for every measurement, and hence no oversampling occurred. The equilibrium conditions were established following the methodology of Prins & de Vries (1971), Gee (1974) and Baas (1993), where the development was estimated by:
where t is time in minutes, and cη and cλ are the bedform height and wavelength adaptation constants. The subscripts indicate the actual (t) and equilibrium (e) values of height and wavelength. A non-linear fit, using a least absolute residual method, was used to find the best solution for Eqs (2) and (3) to fit the spatially averaged height and wavelength data. From these solutions, the values of [ηe, cη ] and [λe, cλ] were obtained. The equilibrium time te is defined as the time when the actual bedform height (ηt) and wavelength (λt) reach 90% of their equilibrium height (ηe) and wavelength (λe):
Therefore, by applying Eq. (4) to Eqs (2) and (3), the equilibrium time for bedform height was computed as:
and for bedform wavelength as:
The final equilibrium time was computed as the average of Eqs (5) and (6):
The equilibrium time (te), equilibrium bedform height (ηe) and equilibrium bedform wavelength (λe), as well as the adaptation constants for the height (cη) and wavelength (cλ), are shown in Table 1. In addition, once the bed was at equilibrium (t > te), the bedload transport per unit width qb, was determined by:
(Simons et al., 1965), where λp is the bed porosity (λp ≈ 0·3), βbf is the bedform shape factor defined as the ratio between the bedform cross-sectional area Abf and ηλ (Rubin & Hunter, 1982):
and is the mean bedform celerity. The bedform porosity, λp, was estimated for three experiments (Table 2) by measuring the difference between the dry and wet weight of sand collected from an end of experiment bedform, instantaneously after the flume water was drained but whilst the bedform was still wet. The bedform porosity, λp, for all three cases was close to 0·3, and this value was assumed to be true for all experiments. The mean bedform celerity, , was computed by time-averaging the instantaneous bedform migration velocity Cbf(t) for t > te:
Table 2. Bedform porosity measurements
Uo (m sec−1)
Uu (m sec−1)
T, oscillatory period; Uo, maximum orbital velocity; Uu, unidirectional velocity; ηe, bedform height at equilibrium; λe, bedform wavelength at equilibrium; λp, bedform porosity.
Cbf(t) was recorded by tracking the displacement of bedform crests ∆x = x2−x1 over the time interval ∆t = t2−t1 (e.g. Coleman, 1969; Dinehart, 2002; Claude et al., 2012).
Bedform classification has always been a topic of discussion that has created debate within different disciplines (e.g. Ashley, 1990). Herein, the terminology adopted by Dumas et al. (2005) is used, with the modification that instead of dividing small and large ripples at a wavelength of 1 m, the division was lowered to 0·5 m. Four types of bedforms were recognized herein:
Symmetrical small ripples (SSR) were characterized by a symmetrical streamwise profile with sharp and narrow crests. Their wavelengths varied between 70 mm and 110 mm, their heights between 5 mm and 20 mm, and the average lee side angle was 14°.
Asymmetrical small ripples (ASR) were characterized by an asymmetrical profile, with a lee side angle in the range of 10° to 20°. The ripple wavelength ranged between 0·11 m and 0·21 m, and the ripple height between 12 mm and 30 mm.
Symmetrical large ripples (SLR) were generally formed by increasing the oscillatory velocity beyond that associated with SSRs. The ripple wavelengths varied between 1·11 m and 2·24 m, and the ripple height between 0·06 m and 0·27 m. The average lee slope angle was 16°.
Asymmetrical large ripples (ALR) were characterized by sharp crests, deep troughs and round stoss sides (RRI > 0·4). The angle of the lee sides varied between 10° and 30° but, in general, they were close to the angle of repose (ca 32°). These large bedforms possessed wavelengths ranging from 1 m to almost 5 m, and ripple heights between 0·1 m and 0·4 m.
In addition to the classification of combined-flow bedforms, the distinction between current-dominated or wave-dominated flows (Yokokawa, 1995; Perillo, 2013) is based on the definition that wave-dominated combined-flows are those in which the maximum shear stress at flow reversal is larger than the critical shear stress for sediment entrainment, whereas current-dominated combined-flows are those in which the maximum shear stress at flow reversal is less than the critical shear stress for sediment entrainment.
The evolution of bedforms from an initially flat bed was studied for different types of flow conditions: purely unidirectional (Fig. 2A), purely oscillatory (Fig. 2B) and combined-flows (Fig. 2C). The development of these bedforms exhibited the same general trend described by Baas (1993) under purely unidirectional flows. The evolution of the bed showed a characteristic asymptotic growth (Fig. 2; Sutherland & Hwang, 1965; Baas, 1994, 1999; Coleman et al., 2003) and once the equilibrium stage was reached (Eq. (7)), bedform wavelength and height fluctuated around their equilibrium values as long as the flow conditions remained unchanged (i.e. to the right of the vertical line in Fig. 2). This development behaviour was found for all experiments (i.e. purely unidirectional, purely oscillatory and combined-flows) regardless of their final stable planform geometry (i.e. 2D = continuous and straight crestlines and 3D = discontinuous and curved crestlines) or size (small ripples or large ripples).
The development of the bedforms is successfully estimated by Eqs (2) and (3) whose fits (red lines in Fig. 2) and their 95% confidence intervals (grey-shaded areas) are plotted for three characteristic experiments. Baas (1994, 1999) also reports successful application of Eqs (2) and (3) to characterize the development of bedforms under purely unidirectional flows. The strength of the prediction of these equations is shown by the relatively high R2 values obtained from the fits (Table 1), despite the large scatter of the data. The mean R2 value for all the fits was 0·66, similar to that reported in the experimental study of Baas (1993) where the mean R2 was 0·70. In addition, most of the experimental data lie within the 95% confidence intervals, which denotes that the fit produces a reliable estimate of the development history of the bedforms. The mean R2 value obtained by the fits for all the purely oscillatory-flow bedform data sets is 0·78 (Table 1) which is, in general, a better overall fit than the mean R2 value for purely unidirectional (mean R2 = 0·72) and combined-flow bedforms (mean R2 = 0·63). The better performance of Eqs (2) and (3) to explain the overall development of the purely oscillatory-flow bedforms is mainly driven by the high R2 values obtained for the 2D bed configurations. The mean R2 value obtained for all the 2D bed morphologies (only under purely oscillatory flows, Experiments 1, 6 and 19) was 0·96, whereas for the 3D bedforms the R2 values ranged from 0·35 to 0·97, with a mean R2 of 0·56. Equations (2) and (3) provide a better representation of the 2D bed morphologies due to the low variability (only 18·3% of the data lie outside of the 95% confidence intervals) that occurs once the bed reaches equilibrium conditions (i.e. to the right of the vertical line in Fig. 2B). However, for the case of 3D bedforms, the scatter was much larger, with almost 50% of the data lying outside of the 95% confidence intervals.
The presence or absence of scatter in this data reflects the temporal and spatial characteristics of the 2D or 3D bedforms at equilibrium conditions (Fig. 3 and Videos S1 and S2). Once the 2D bedforms reached equilibrium conditions, they retained their continuous and straight crestlines with zero to minimal lateral migration (Fig. 3A). This static stage where the bedform wavelength and height remained unchanged with a complete lack of crestline motion is referred to as a frozen or static-equilibrium. On the other hand, once the bed reached equilibrium conditions, the 3D bedforms exhibited a dynamic-equilibrium (e.g. Wilson, 1972), where migration, separation and amalgamation of the bed features occurred constantly (Fig. 3). In addition to the dynamic change in the 3D bedforms, the other principal factor driving the observed fluctuations around the mean values is the inherent three-dimensionality of the bed configuration. The experimental results of Baas (1994) showed that the dynamic evolution of bedforms introduces intrinsic scatter into the data. The planform geometry of the equilibrium 3D bedforms is best described as linguoid and lunate ripples (Fig. 3B). For example, a linguoid or lunate ripple possesses its maximum height if measured in a cross-sectional plane through the centre of the ripple and parallel to the mean flow direction, whereas the height measurements through all other planes will record a smaller value. Hence, the transverse three-dimensionality of these bedforms generates fluctuations around the mean bedform height.
Similar to Baas (1994), four major stages of bedform development were distinguished for all types of flows; unidirectional (Fig. 4), oscillatory (Fig. 5) and combined-flows (Fig. 6): (i) incipient bedforms; (ii) growing bedforms; (iii) stabilizing bedforms; and (iv) fully developed bedforms. The characteristics and the processes governing the development of the bedforms at each development stage shared similarities and differences for the different flow types.
Stage 1 – Incipient bedforms
Stage 1 starts with the initiation of sediment transport from an initially flat bed, until the first signs of bedform growth (Figs 4 to 6). Once the threshold for sediment entrainment is exceeded, a series of small incipient bedforms starts to propagate. For the case of unidirectional flows (Fig. 7A to D), the bed deformation started as longitudinal streaks (Fig. 7B) that rapidly transformed into very small, 3D, bedforms (Fig. 7C) which then merged into straight and sinuous small bedform trains (Fig. 7D). The presence of 2D ripples in the early stages of development under purely unidirectional flows has been observed in previous studies (e.g. Harms et al. 1975; Baas 1994). Under purely oscillatory flows (Fig. 7E to H), the predominant morphological feature during this early stage was 2D rolling-grain ripples, similar to previous studies on purely oscillatory flows (e.g. Bagnold, 1946; Lofquist, 1978; Pedocchi, 2009). These rolling-grain ripples exhibited a characteristic alternation of crestline orientation from upstream to downstream during every cycle (Fig. 7E to H). In addition, their planform geometry changed within this first stage (Fig. 7E to H), with development from an early, very short wavelength, 2D stage (dashed line in Fig. 7E) to a more stable 2D set of rolling-grain ripples (Fig. 7F to H). For the combined-flow conditions, a clear distinction existed between the predominant morphological features present under wave-dominated (Fig. 7I to L) or current-dominated (Fig. 7M to P) combined-flow conditions. For the case of wave-dominated combined-flow bedforms (Fig. 7I to L), the incipient bedforms resembled those for purely oscillatory bedforms: rolling-grain ripples (Fig. 7E to H). However, for the case of current-dominated combined-flows, the incipient bedforms behaved as those described for the purely unidirectional flows (Fig. 7A to D). The diagnostic features used to differentiate the incipient bedforms generated under wave-dominated combined-flow versus current-dominated combined-flow were the presence (Fig. 7I to L) or absence (Fig. 7M to P) of upstream crestal movement during the wave cycle. It is important to note that crestal movement or migration in the upstream direction was not necessarily associated with a change in bedform asymmetry. If the upstream migration of the crest was significant, this might produce an actual change in asymmetry and thus a crestal reversal. During the first stages of bedform development under a wave-dominated combined-flow (Fig. 7I to L), there was a clear back and forth movement of the crestal direction (white arrows in Fig. 7I to L). On the other hand, during the genesis of bedforms for current-dominated combined-flows, the bedform crestline did not move in the upstream direction (Fig. 7M to P). Nevertheless, despite the different behaviour in crestline movement, the final planform configuration of the incipient bedforms for both wave-dominated and current-dominated combined-flows shared the same 2D planform geometry to that described for purely unidirectional and purely oscillatory-flow conditions (Fig. 7D, H, L and P).
The morphological difference between wave-dominated and current-dominated combined-flow bedforms (i.e. the presence and absence of crestal movement in the upstream direction for the incipient bedforms during each wave cycle) is a clear representation of the interactions between fluid flow, sediment transport, and bedform size and geometry in the flow—sediment-transport—bedform ‘trinity’ (Leeder, 1983; Best, 1993, 1996; Perillo, 2013). Figure 8A and C shows the streamwise phase-averaged velocity profiles u(z), and Fig. 8B and D shows a point measurement at z = 0·28 m during the oscillation cycle for a wave-dominated combined-flow (Fig. 8A and B) and a current-dominated combined-flow (Fig. 8C and D). The interactions of the wave and current boundary layers play a significant role in the dynamics of wave-dominated and current-dominated combined-flow bedforms (Fig. 8). For example, for the case of experiments 4 and 14, the maximum combined-flow velocity for both the wave-dominated (Fig. 8A) and current-dominated (Fig. 8C) combined-flow conditions was roughly the same (ca 0·5 m sec−1), which produced a relatively similar velocity profile (especially away from the bed) when the direction of the oscillatory velocity coincided with the unidirectional flow (phase ca 90° in Fig. 8A and C). However, for the case where the oscillatory velocity direction opposed the unidirectional flow, the velocity profiles appear very different (phase ca 270° in Fig. 8A and C). These differences between wave and current-dominated combined-flows not only reflect the mean or upper flow conditions but also the dynamics occurring very close to the bed. Near the bed, two boundary layers are present due to oscillatory and unidirectional flow (Davies et al., 1988; Mathisen & Madsen, 1996a,b, 1999; Fredsøe et al., 1999). The boundary layer formed by the wave is relatively thin compared to that generated by the unidirectional flow, and thus the effects of the oscillatory conditions are clearer near the bed (for example, flow reversal for z < 0·05 m, Fig. 8C) than in the upper parts of the flow (for example, u > 0 for all wave phases for z > 0·1 m, Fig. 8C). This explains why current-dominated combined-flows, as classified herein, can exhibit flow reversal (z < 0·05 m, Fig. 8C), yet the magnitudes of such velocities are insufficient to entrain sediment (θ = −0·01 < θc = −0·04) in the upstream direction. On the other hand, in the case of the wave-dominated combined-flows, the unidirectional velocity was not strong enough to prevent the oscillatory velocity fully reversing (z < 0·05 m, Fig. 8A). Hence, a net sediment transport in the upstream direction occurred, which can be clearly observed in the upstream migration of the crestlines (Fig. 7E to H and I to L).
Stage 2 – Growing bedforms
No difference was found between purely unidirectional, oscillatory and combined-flow conditions regarding the mechanisms of bedform growth. The bedforms grew in size by a combination of sediment being captured by the bedforms and amalgamation between adjacent bedforms (Figs 4B, 5B and 6B). For the case of current-dominated combined-flows and purely unidirectional flows, sediment incorporation into the bedforms occurred as it was transported as bedload over the stoss side and partly deposited in the lee side of the bedform. On the other hand, for the case of wave-dominated combined-flows and purely oscillatory flows there was a net transport of sediment from the troughs to the crests, and in every wave cycle there was transport both upstream and downstream direction. Bedform growth during this stage accounts for ca 85% of the total bedform enlargement under purely unidirectional flows, 93% under pure oscillatory flows and 95% under combined-flows. This result is consistent with the experiments reported by Baas (1994) that showed that the large majority of overall growth occurs in this stage, with an average overall growth of 75%.
During this stage, the bed configuration transitioned from 2D bedforms to 2·5D (where the bedform crestline is either continuous or straight) or 3D bedforms. Furthermore, for the case of purely oscillatory-flow conditions, the alteration of the planform geometry (i.e. 2D to 3D) takes place as the rolling-grain ripples transition to small 3D vortex ripples (Video S1). Once this transition occurred, the bed remained populated with 3D bedforms unless the stable configuration was 2D (under some pure oscillatory-flow conditions; Table 1), where the bed rearranged its planform geometry to become 2D during the last stage of development.
Stage 3 – Stabilizing bedforms
Stage 3 is defined as a transitional stage between the moment when asymptotic bedform growth ends and the time when the bed reaches equilibrium conditions (Figs 4 to 6). On the one hand, the bed is not fully developed, since the criterion defined in Eq. (4) is not fulfilled and, in addition, there are still signs of net bedform growth (mainly by merging of adjacent bedforms, Figs 4 to 6). On the other hand, the bed morphology exhibits all of the ‘visual’ characteristics that are displayed by fully developed bedforms (for example, planform geometry and shapes), but they are just smaller. This observation is also consistent with that of Baas (1994).
The rate at which the bedforms grow during Stage 3 was significantly lower than for Stage 2. The mean overall growth during Stage 3 for purely unidirectional flows was ca 10% of the overall growth. For unidirectional flow ripples, Baas (1994) reported that during the stabilizing bedform state, the mean height and wavelength values increased from 10 to 13 mm and from 100 to 116 mm, respectively, accounting for an estimated ca 14% of the total growth. Similar to unidirectional flows, the growth rate for oscillatory (ca 5%; Fig. 5) and combined-flow (<3%, Fig. 6) bedforms during Stage 3 was also significantly lower than for Stage 2. However, similar to purely unidirectional flows, this result is similar to Stage 2, in which there was a greater percentage growth under oscillatory and combined-flow conditions than under unidirectional flows (ca 93% and 95% versus ca 85%).
Stage 4 – Fully developed bedforms
Once the bed reached a morphodynamic equilibrium (i.e. fully developed bedforms under the flow conditions acting on the bed), the average bedform height and wavelength converged to singular values, ceasing to change with time. However, the height and wavelength of individual bedforms continued to fluctuate around the equilibrium average values (Figs 2 and 4 to 6). Therefore, due to this characteristic variability, and despite the good predictive power of Eqs (2) and (3), the use of a unique value (ηe or λe) to describe the full range of geometries is insufficient for some conditions (for example, Fig. 2). This characteristic has been discussed previously within the context of unidirectional bedforms, where probability density functions have been employed instead of a single value to describe the distribution of bedform dimensions under equilibrium conditions (e.g. Paola & Borgman, 1991; Leclair & Bridge, 2001). A Gaussian density function was thus applied to characterize the bedform size distributions:
where μ is the mean (peak location) and σ is the standard deviation.
Unlike the unidirectional and combined-flow bed morphologies, oscillatory bedforms exhibited two different stable planform bed configurations: 2D and 3D (Table 1). In the previous development stages (Stages 1 to 3), there was no difference in the development history between bedforms that displayed a stable 2D configuration or a 3D configuration. However, for the case of the ‘fully developed’ bedform stage (Stage 4), there is a dependence of development history on the planform geometry (i.e. 2D versus 3D).
It is important to note that the 2D stable planform bed configurations occurred only under oscillatory-flow conditions. The three conditions in which the final stage presented a stable 2D configuration obeyed the empirical relation proposed by Pedocchi & García (2009):
where D∗ is a dimensionless grain size:
and Rew is the wave Reynolds number:
There are two principal reasons why 2D oscillatory-flow bedforms were different than 3D bedforms, or behaved differently to 3D bedforms, during the fourth stage of development. Firstly, there was a temporal lag in the development history of 2D oscillatory bedforms between the time required for these bedforms to reach equilibrium sizes (i.e. ηtca 0·9 ηe and λtca 0·9 λe) and the time to reach equilibrium planform geometry (i.e. the time for the bed to be fully 2D; Fig. 5D and E). For example, during the fourth development stage for Experiment 19 (dark-grey box; Fig. 5), the bed reached size equilibrium after 253·8 min from the initiation of the experiment, although the bed did not achieve a fully 2D planform configuration until 738·6 min (Table 3). Similarly, the bedforms in Experiment 1 reached size equilibrium (t > te) at 132 min, but did not exhibit a fully 2D planform configuration until 141 min (bottom two photographs in Fig. 9). Secondly, the behaviour of 2D oscillatory-flow bedforms exhibited very little variability once the bed reached size-equilibrium conditions (Video S1), regardless of the planform geometry (t ≥ te; to the right of the vertical line, Fig. 2B; and vertical line 4, Fig. 5). For example, during Experiment 1 (t > te = 132 min) the overall wavelength and height remained the same for 30 min (Fig. 9), despite the difference in planform geometry through time. The histogram of the recorded wavelengths (Fig. 10A) and heights (Fig. 10B) for all equilibrium bedforms for a two-dimensional symmetrical small ripple (Experiment 1) includes all the observed bedforms for each time (the N bedforms from Eq. (1)) from the time t = te to the end of the experiment (tt, Table 1). Both the distribution of bedform wavelengths (Fig. 10A) and heights (Fig. 10B) show a narrow distribution once the bed reached an equilibrium state (for example, ση/μη × 100 ca 7·3% and σλ/μλ × 100 ca 9·3%). Assuming a normal distribution for both the bedform height and wavelength, it can be found that 80·9% of the variability in bedform height lies within one standard deviation (ση) and 98·5% lies within 2ση; whereas for bedform wavelength, 77·1% lies within one σλ and 97·1% within 2σλ. Moreover, 100% of the variability lies within ca 7·5 σ; ∆η(t > te) ≈ 15·0mm ≈ 5·6 ση and ∆λ(t > te) ≈ 0·22 m ≈ 7·5 σλ.
Table 3. Summary of equilibrium times of size and planform geometry for the bedform experiments with stable two-dimensional planform geometries
T, oscillatory period; Uo, maximum orbital velocity; Uu, unidirectional velocity; ηe, bedform height at equilibrium; λe, bedform wavelength at equilibrium; D*, dimensionless grain size; Rew, wave Reynolds number; qb, bedload transport per unit width (Eq. (8)); te, time until flow-bedform equilibrium obtained from Eq. (7).
Empirical relation by Pedocchi & García (2009) = 2D > 0·15 > 3D (Eq. (13)).
SSR, 2D symmetrical small ripples.
The narrow spectrum of bed morphologies for 2D oscillatory-flow bedforms can be explained by the relatively fixed spatial position of the bedforms once the bed reached equilibrium conditions (for example, Figs 3A and 9). During the time interval between size equilibrium and planform-equilibrium, a few fluctuations (for example, Fig. 5) were recorded since the bed was still moving (for example, Fig. 9). However, this migration culminated when the bed reached a fully 2D planform geometry, where no further changes in height and wavelength were recorded, producing a ‘static-equilibrium’ (for example, Fig. 3A). The bedform wavelength and height present at the bed during the ‘static-equilibrium’ stage will persist until the flow conditions are changed, at which point the bed will break its two-dimensionality to rearrange itself to a new equilibrium stage that is stable with the new flow condition. This ‘static-equilibrium’ behaviour is a fundamental characteristic of 2D oscillatory-flow ripples, and is not seen in any other type of flow or bedform.
During the fourth stage of development, the wavelengths and heights of 3D bedforms fluctuated around their equilibrium average values (for example, Figs 4 and 6), providing a wide spectrum of wavelengths and heights (Fig. 10C to F). The histogram of bedform height and wavelength from Experiment 12 (pure oscillatory-flow condition: three-dimensional symmetrical small ripples; Fig. 10C and D) and Experiment 3 (current-dominated combined-flow condition: three-dimensional asymmetrical small ripples; Fig. 10E and F) are used to illustrate the relatively large dispersion of sizes during the equilibrium stage for 3D bedforms. The distribution of bedform wavelengths (Fig. 10C and E) and heights (Fig. 10D and F) shows significant variations of the spatially averaged wavelength and height once the bed reached an equilibrium state. The average dispersion in sizes during the fourth stage of development for 3D bedforms is ca 40% of the mean height and wavelength; 3D pure unidirectional flow bedforms (ση/μη × 100 ca 43·52% and σλ/μλ × 100 ca 33·3%), pure oscillatory-flow bedforms (ση/μη × 100 ca 54·1% and σλ/μλ × 100 ca 38·5%), wave-dominated combined-flow bedforms (ση/μη × 100 ca 47·9% and σλ/μλ × 100 ca 33·9%) and current-dominated combined-flow bedforms (ση/μη × 100 ca 45·2% and σλ/μλ × 100 ca 28·9%). Moreover, if the distribution of equilibrium bedforms is represented by a normal distribution, ca 70% of the variability in bedform size lies within one standard deviation (σ) and more than 95% lies within 2σ.
The wider ranges of 3D bedform sizes (ca 40% from the mean values) compared with the 2D oscillatory bedforms (ca 8% from the mean values) can be explained by the dynamic spatio-temporal behaviour of these bedforms once the bed reached equilibrium conditions (static versus dynamic-equilibrium, Fig. 3). During this fourth stage of development, the bedforms were constantly moving and generating new bed features that promoted separation of existing forms or the merger between nearby bedforms.
Equilibrium time as a function of the flow and sediment transport conditions
The empirical model proposed by Baas (1999) was used to predict the equilibrium time as a function of the flow conditions. This model uses an inverse power of the flow velocity to find the best-fit for the measured equilibrium time (Baas, 1999). However, Baas (1999) reported an individual fit for the time required for the height and wavelength to reach equilibrium conditions. Hence, Eq. (16) was used to re-evaluate the experimental data from Baas (1999) (D50 = 0·238 mm) and obtain a unified bed state equilibrium time as a function of flow velocity:
(see solid red line in Fig. 11). In addition, since Eq. (16) does not account for the oscillatory component of the combined-flow, a peak combined-flow velocity value was used instead of Uu. Therefore:
(see solid blue line in Fig. 11), where Ucf is the unidirectional velocity linearly added to the maximum orbital velocity:
The regression statistics for both equations were similar. The R2 for Eq. (16) (Baas, 1999) being 0·55, whereas for Eq. (17) the R2 = 0·63. Equation 17 provides a slightly better prediction, but both equations are unable to predict the large scatter. This observation can be seen by the multiple points that lie outside the grey-shaded area in Fig. 11, which represent the 95% confidence interval for Eq. (17). In addition, even though the mean absolute percentage deviation (MAPD column in Table 4) is relatively large, the true difference from the fitted coefficient is not significant (∆ column in Table 4).
Table 4. Comparison between the coefficients of Baas (1999) and Eq. (17). Equation for the equilibrium time as a function of flow velocity: U = A tB + C
Based on the distribution and relatively large overlap of the different equilibrium times for different bedforms (Fig. 11B), it can be concluded that te is not a good variable to differentiate bed states. This result is consistent with the observations of Baas et al. (1993). However, based on the overall distribution of time scales (Fig. 11B), in general large ripples required less time (only a few tens of minutes) to reach equilibrium conditions, whereas small ripples generally required several hours. As expected from bedform size alone, with large ripples being larger than small ripples, large ripples would require more time to reach equilibrium than small ripples. Yet, since the flow conditions for large ripples mobilized significantly more sediment than small ripples, large ripples reached equilibrium faster. This result illustrates that the sediment transport dynamics play a key role in controlling the equilibrium time of bedforms. Moreover, the similarities between Eqs (16) and (17) can be explained because the time required for a bed to reach equilibrium is related to the total amount of sediment transported, regardless of what type of flow is producing the transport. Therefore, estimations of the equilibrium time of bedforms should place more emphasis on the sediment transport characteristics, rather than the flow conditions in the upper flow (i.e. Ucf in Eq. (17)). Hence, using the data extracted from the sonar, the bedload transport was estimated for each experiment, and used to reformulate Eq. (17) as:
is the dimensionless expression of qb, which is bedload transport per unit width (m2 sec−1).
Equilibrium times were calculated for different bedforms (Fig. 12) generated under the range of wave periods utilized in this study, as a function of the dimensionless bedload transport per unit width. Despite the large dispersion, the fitted Eq. (19) is better than Eqs (16) and (17) at predicting the time at which the bed configuration reaches equilibrium conditions (R2 = 0·92). In addition, for the case of relatively low sediment transport rates (q* < 10) and long equilibrium times (te > 1 h), all of the experimental data lie within the 95% confidence interval.
The evolution of bedforms from a flatbed was studied for a range of unidirectional, oscillatory and combined-flow conditions over a 250 μm diameter sand bed. Based on the bedform growth path described for the different types of flow, it can be concluded that the overall development of bedforms is clearly independent of the flow type (unidirectional, oscillatory and combined). Combining the oscillatory-flow data of Doucette & O'Donoghue (2006) with those of unidirectional flows (Sutherland & Hwang, 1965; Baas, 1999) produces the same development curves as the present study. Furthermore, despite the difference in bedform planform shape (for example, 2D versus 3D), the ‘asymptotic growth’ behaviour is also independent of the final bedform stage. In other words, the general shape of the development curve is independent of the bedform size (i.e. small versus large ripples), shape (i.e. symmetrical or asymmetrical) and planform geometry (i.e. 2D versus 3D). In addition to the independence on the flow types (for example, unidirectional versus combined-flow) and the final bedform geometry (for example, 2D versus 3D or current ripples versus symmetrical small ripples), the development growth path is also found to be grain-size independent. The experiments of Faraci & Foti (2002) and those herein (Table 1) were all conducted using a 0·25 mm diameter sand bed, whereas Lofquist (1978), Baas (1993) and Doucette & O'Donoghue (2006) conducted experiments over a range of grain sizes: D50 = 0·18 mm, D50 = 0·21 mm and D50 = 0·55 mm (Lofquist, 1978); D50 = 0·095 mm and D50 = 0·238 mm (Baas, 1993); and D50 = 0.44 mm (Doucette & O'Donoghue, 2006). All results describe the same behaviour. Therefore, it can be concluded that the development path described previously is ubiquitous for all types of subaqueous bedforms.
Because of this invariance of bedform development to flow type (i.e. unidirectional, oscillatory or combined-flow), the analysis of the equilibrium time presents the same conclusion (see symbol scheme in Fig. 12). The different flow conditions (i.e. different flow velocities or oscillation periods) alter the sediment transport stages, but no differences were found between different periods (i.e. 4, 5 and 6 sec) or flow types. However, the equilibrium time was not independent of the final equilibrium stage for bedforms generated in this study, where the larger bedform morphologies (i.e. large ripples) took the least amount of time to reach equilibrium conditions, since they were generated under the most active sediment transport regimes. On the other hand, the smaller bedforms (i.e. ripples) formed under lower sediment transport rates, and hence required more time to reach their stable configuration. This result depends on the fact that, for this particular grain size (D50 = 250 μm), the larger bedforms only formed under higher flow velocities and sediment transport rates and the smaller bedforms only under low transport stages. Both sediment transport and bedform states have a strong correlation with grain size, and hence the relation between equilibrium time and bed stage presented in Fig. 12 should not be generalized for other grain sizes. For example, for coarser grain sizes, gravel dunes can be generated under flow conditions that are just above the threshold for sediment transport (Carling, 1999), and hence require a long time to reach their stable configuration. It is important to note that all of the results described in this study do not take into account any bedform hysteresis to changing flow conditions. Since the bedforms in this study were generated from an initial flat bed by a specific flow condition, the results should not be extrapolated to bed morphologies that are generated by temporally varying unsteady flows.
In addition, despite the limited data from only three experiments, a clear lag was found in the development history of 2D oscillatory-flow bedforms. The time required for the bedforms to reach equilibrium sizes (i.e. Eq. (4)) was significantly less than the time that the bed required to reach the 2D planform geometry. After detailed examination of the development curves (for example, Fig. 5) and the photographs taken through the side window (for example, Fig. 10), it was observed that by the time the bed reached size-equilibrium (t = te) the crests of the bedforms still possessed numerous defects (for example, Fig. 5D). The presence of these defects prevented the bed from being fully 2D, since lateral discontinuities in the crestline produced different flow velocities, which then induced differential sediment transport (Venditti et al., 2005). Thus, additional time is necessary to fully reshape the bed from the 3D configuration (multiple defects on the bedform crest) to the final 2D configuration (very few defects on the bedform crest; Fig. 9). Resent work by Lamb et al. (2012) has shown the significance of grain size in understanding the planform geometry of pure oscillatory bedforms. Furthermore, Pedocchi & García (2009) argue that the stability of 2D bedforms lies in the ratio between the grain size, D*, and wave Reynolds number, Rew (Eq. (13)). Equation (13) implies that the larger the grain size becomes (D*) and/or the smaller the level of turbulence (Rew), the more likely it is that the bedform will be 2D. Since larger grain sizes (Ds) will be less susceptible to turbulent fluctuations (the flow is 3D in nature; e.g. Carstensen et al., 2010), they may be likely to be more responsive to the mean purely oscillatory flow (2D motion) or require more time to achieve any three-dimensionality at a given sediment transport rate, and thus may tend to be more 2D in planform. This grain-size behaviour is consistent with past work that has documented that 2D bedforms are predominately formed in coarser sand (e.g. Leckie, 1988; Southard, 1991; O'Donoghue & Clubb, 2001; O'Donoghue et al., 2006). However, the larger the grain size, the higher the threshold of sediment transport, and thus the longer it will take to reshape the bedforms into a 2D form. This result can even be seen within the limited data available herein (Table 2).
The development of bedforms was studied under a range of unidirectional, oscillatory and combined-flow conditions over an initially flat bed. Based on the results of previous work and the new results presented herein, eight conclusions can be reached:
The development path of bedform growth exhibited the same general trend regardless of the flow type (for example, unidirectional versus combined-flow), bedform size (for example, ripples versus dunes), bedform shape (for example, symmetrical or rounded), bedform planform geometry (for example, two-dimensional versus three-dimensional) and sediment grain size.
Bedform development can be divided into four main stages regardless of the flow conditions: (i) incipient bedforms; (ii) growing bedforms; (iii) stabilizing bedforms; and (iv) fully developed bedforms. This division is consistent with the scheme proposed by Baas (1994, 1999) for purely unidirectional flows.
Stage 1 is defined from the initiation of sediment transport until the first signs of bedform growth. The dominant morphological feature for purely unidirectional and current-dominated combined-flows started as longitudinal streaks that developed into very small three-dimensional ripples. On the other hand, for purely oscillatory and wave-dominated combined-flows, the main juvenile bedform was transient rolling-grain ripples.
Once the bed was fully covered with bedforms, the bedform transitioned to Stage 2 where the bedforms started to grow exponentially with time. This growth was proportional to ≈1−e−ct and was successfully estimated by Eqs (2) and (3). The large majority of bedform enlargement occurred during this stage. The main mechanisms of bedform growth were sediment capture and amalgamation with nearby bedforms.
Stage 3 was defined as the temporal gap between the cessation of asymptotic growth and the time when the bed reached equilibrium conditions. Compared with the growth in Stage 2, bedform enlargement during Stage 3 was minimal.
Once the bedform reached its equilibrium stage (Stage 4), the bed morphology remained in that regime until the flow conditions were changed. The equilibrium time, te, was defined by applying Eq. (15) to the fitted data. The time of development was found to be inversely proportional to the amount of sediment transport occurring for that flow condition (Eq. (18)).
During the fourth stage of bedform development, the wavelength and height fluctuated from their equilibrium mean wavelength and height. This variation was more marked for the case of three-dimensional bedforms, since the bed was constantly moving, with bedforms merging and splitting and producing a relatively large range of active sizes.
The behaviour of bedforms during the fourth stage can be divided based on the planform geometry. For the case of two-dimensional bedforms occurring only under purely oscillatory flows, there is a temporal separation between the time when the bed morphology reaches its equilibrium size and the planform geometry becomes fully two-dimensional. In addition, three-dimensional bedforms generated under purely unidirectional, purely oscillatory and combined-flow, exhibited a wider range of bedform sizes (ca 40% from the mean values) compared with two-dimensional oscillatory bedforms (ca 8% from the mean values). These results can be explained by the clear differences in the final bedform state, with two-dimensional bedforms at planform-equilibrium being in a ‘static-equilibrium’, whereas all three-dimensional bedforms are in a ‘dynamic-equilibrium’. Nevertheless, almost all of the variation (ca 70%) in bedform height and wavelength lies within one standard deviation.
This study has reported on the first experimental data on bedform development under combined-flows, and outlined a unified model for bedform development and equilibrium under unidirectional, oscillatory and combined-flows. Based on these experimental results, it was observed that the genesis and growth processes are common for all types of flow independent of bedform size (for example, small versus large ripples), bedform shape (for example, symmetrical or rounded), bedform planform geometry (for example, two-dimensional versus three-dimensional), flow velocities and sediment grain size.
The authors would like to acknowledge the support of Erik van Dusen, Scott David and Yoshitsugu Hasegawa for their assistance in the laboratory. We are very grateful to Francisco Pedocchi, Jaco Baas, Eric Prokocki and Frank Engel for many useful discussions during the course of this work. The Large Oscillating Water Sediment Tunnel was built with support from the DURIP Program ONR grant N00014-01-1-0540. In addition, the completion of this work was made possible by the Leighton Graduate Student Research Award to MMP, established and supported by Dr. Morris W. Leighton. We also acknowledge the valuable comments by Editor Stephen Rice, Associate Editor David Mohrig, Paul Myrow and Brandon McElroy.