Bedforms and roughness
Definition of Bedforms:
Bedforms are relief features generated by fluid flow over a bed consisting of movable sediments.
This is the common definition for Bedforms, other definitions can be discussed in the article

This article deals with smallmedium scale bedforms (ripples and dunes) that influence hydrodynamics by generating bed roughness. Largescale bedforms are dealt with in other articles: Nearshore sandbars, Rhythmic shoreline features, Sand ridges in shelf seas.
Contents
 1 Introduction
 2 Bedforms in unidirectional and oscillating flow
 3 Currentgenerated ripples
 4 Dunes
 5 Sand bars
 6 Transitional regime
 7 Antidunes
 8 Wavegenerated bedforms
 9 Formation of bedforms
 10 Bed roughness
 11 Related articles
 12 Appendix Empirical estimates of bedform characteristics and associated bed roughness
 13 References
Introduction
Bedforms are the most important source of flow resistance at the local scale in sandbed river channels and estuaries. Bedforms generate turbulent eddies and flow separation that are responsible for the transfer of mean flow momentum to turbulence, leading to momentum dissipation by viscous forces as heat^{[3]}.
Many types of bed forms can be observed in nature. The bed form regimes for steady flow over a sand bed can be classified into (see Figs. 1, 2):
 lower transport regime with flat bed, ribbons and ridges, ripples, dunes and bars,
 transitional regime with washedout dunes and sand waves,
 upper transport regime with flat mobile bed and sand waves (antidunes).
When the bed form crest is perpendicular (transverse) to the main flow direction, the bed forms are called transverse bed forms, such as ripples, dunes and antidunes. Ripples have a length scale much smaller than the water depth, whereas dunes have a length scale much larger than the water depth. The crest lines of the bed forms may be straight, sinuous, linguoid or lunate. Ripples and dunes travel downstream by erosion at the upstream face (stossside) and deposition at the downstream face (leeside). Antidunes travel upstream by leeside erosion and stossside deposition. Bed forms with their crest parallel to the flow are called longitudinal bed forms such as ribbons and ridges.
In the literature, various bedform classification methods for sand beds are presented. The types of bed forms are described in terms of basic parameters (Froude number, suspension parameter, particle mobility parameter; dimensionless particle diameter).
A flat immobile bed may be observed just before the onset of particle motion, while a flat mobile bed will be present just beyond the onset of motion. The bed surface before the onset of motion may also be covered with relict bed forms generated during stages with larger velocities.
Bedforms in unidirectional and oscillating flow
Many of the processes that generate bedforms in unidirectional flow also generate bedforms in oscillatory flows. This holds for dunes and bars which have similar characteristics in rivers and estuaries. Bed ripples are an ubiquitous feature in rivers as well as in coastal environments. However, wavegenerated ripples have specific characteristics and are therefore discussed in more detail in two separate articles, Wave ripples and Wave ripple formation. Empirical formulas of bedform characteristics and bed roughness are given in the Appendix, including wavegenerated ripples. Formulas for estuaries are given in the article Bed roughness and friction factors in estuaries.
Currentgenerated ripples
Smallscale ribbon and ridge type bed forms parallel to the main flow direction have been observed in laboratory flumes and small natural channels, especially in case of fine sediments (grainsize [math]d_{50}[/math] typically smaller than 0.1 mm). They are probably generated by secondary flow phenomena and nearbed turbulence effects (burstsweep cycle) in the lower and transitional flow regime. These bed forms are also known as parting lineations because of the streamwise ridges and hollows with a vertical scale equal to about 10 grain diameters. These bed forms are mostly found in fine sediments ([math]0.05 \lt d_{50} \lt 0.25 \; mm[/math]).
When the velocities are somewhat larger (10%20%) than the critical velocity for initiation of motion and the median particle size is smaller than about 0.5 mm, small (mini) ripples are generated at the bed surface. Ripples that are developed during this stage remain small with crest is perpendicular (transverse) to the main flow direction and a ripple length much smaller than the water depth (Fig. 1A). The characteristics of mini ripples are commonly assumed to be related to the turbulence characteristics near the bed (burstsweep cycle). Current ripples have an asymmetric profile with a relatively steep downstream face (leeside) and a relatively gentle upstream face (stossside). As the velocities near the bed become larger, the ripples become more irregular in shape, height and spacing yielding strongly threedimensional ripples. In this case the variance of the ripple length and height becomes rather large. These ripples are known as lunate ripples when the ripple front has a concave shape in the current direction (crest is moving slower than wing tips) and are called linguoid ripples when the ripple front has a convex shape (crest is moving faster than wing tips, Fig. 3). The largest ripples may have a length up to the water depth and are commonly called megaripples.
Dunes
Another typical bed form type of the lower regime is the dunetype bed form (Figs. 1B, 1C). Dunes have an asymmetrical (triangular) profile with a rather steep leeside and a gentle stossside. A general feature of dune type bed forms is leeside flow separation resulting in strong eddy motions downstream of the dune crest. The length of the dunes is strongly related to the water depth ([math]h[/math]) with values in the range of [math](3  15) \, h[/math], with [math](6  7) \, h[/math] as most usual value. Extremely large dunes with heights ([math]\Delta[/math]) of the order of 7 m and lengths ([math]\lambda[/math]) of the order of 500 m have been observed in the Rio Parana River (Argentina) at water depths of about 25 m, velocities of about 2 m/s and bed material sizes of about 0.3 mm. In shallow flows, highangle dunes develop that are asymmetric with an upstream (stoss) slope averaging 2–6 degrees and a maximum leeside slip face slope > 24 degrees. The lee slopes are controlled by granular avalanches (sediment flows dominated by graingrain collisions). In deeper rivers, lowangle dunes develop that are more symmetrical and have slip face slope < 24 degrees, i.e. less than the theoretical limit for granular avalanches dominated by graintograin contacts^{[3]}.
Sand bars
The largest bed forms in the lower regime are sand bars (such as alternate bars, side bars, point bars, braid bars and transverse bars), which usually are generated in areas with relatively large transverse flow components (bends, confluences, expansions). Alternate bars are features with their crests near alternate river bends. Braid bars actually are alluvial "islands" which separate the anabranches of braided streams. Numerous bars can be observed distributed over the crosssections. These bars have a marked streamwise elongation.
Transitional regime
It is a wellknown phenomenon that the bed forms generated at low velocities are washed out at high velocities (Figs. 1D, 1E). It is not clear, however, whether the disappearance of the bed forms is accomplished by a decrease of the bed form height, by an increase of the bed form length or both. Flume experiments with sediment material of about 0.45 mm show that the transition from the lower to the upper regime is effectuated by an increase of the bed form length and a simultaneous decrease of the bed form height. Ultimately, relatively long and smooth sand waves with a roughness equal to the grain roughness are generated ^{[4]}.
In the transition regime the sediment particles will be transported mainly in suspension. This will have a strong effect on the bed form shape. The bed forms will become more symmetrical with relatively gentle leeside slopes. Flow separation will occur less frequently and the effective bed roughness will approach to that of a plane bed. Largescale bed forms with a relative height ([math]\Delta / h[/math]) of 0.1 to 0.2 and a relative length ([math]\lambda / h[/math]) of 5 to 15 were present in the Mississippi river at high velocities in the upper regime.
Antidunes
In the supercritical upper regime the bed form types will be plane bed and/or antidunes. The latter type of bed forms are sand waves with a nearly symmetrical shape in phase with the water surface waves (Fig. 1F, Fig .4). Due to the supercritical flow regime the water surface elevation mimics the bed elevation. Antidunes are bed forms with a length scale of less than 10 times the water depth. They move upstream due to strong leeside erosion and stossside deposition. Antidune bedforms evolve rapidly, growing in amplitude as they migrate upstream and collapse when the resultant wave at the water's surface breaks (Fig. 1G). Trains of antidunes waves are highly unstable, following a quasistochastic process of formation and destruction. When the flow velocity further increases, finally a stage with chute and pools may be generated (Fig. 1H).
Wavegenerated bedforms
Bedforms generated by waves do not grow as large as currentinduced bedforms. Bagnold and Taylor (1946^{[5]}) distinguished two types of wavegenerated bedforms, namely the 'rolling grain ripples' and the 'vortex ripples'. Rolling grain ripples arise when the waveinduced shear stress on the seabed is just above the critical value for setting sediment grains in motion. The height is only a few grain diameters, and the ripple steepness is therefore small. When the orbital velocity amplitude or wave period are increased, the ripples grow longer and taller (Fig. 5) and vortices start to form due to flow separation. Vortex ripples are usually categorized into three regimes: orbital, suborbital or anorbital, see Wave ripples. Orbital bedforms are typically generated when the wave period [math]T[/math] is small. They are characterized by a wavelength [math]\lambda_r[/math] proportional to the amplitude [math]a[/math] of the horizontal wave orbital motion, [math]\lambda_r = (1.31.4) \, a [/math] ^{[6]}. Suborbital bedforms arise for longerperiod waves and commonly have a wavelength inversely proportional to the orbital diameter of waves, [math]\lambda \propto a^{1} [/math]. Anorbital bedforms typically show a wavelength proportional to the mean grain size [math]d_{50}[/math], [math]\quad \lambda_r \approx (400600) \; d_{50} [/math] and a small ripple height [math]\eta_r[/math] on the order of [math]\eta_r \approx (0.10.12) \; \lambda_r[/math]. They occur for large wave orbital excursions [math]a \gt 10^3 \, d_{50}[/math] ^{[7]}.
When ripple steepness [math]\eta_r / \lambda_r[/math] is greater than about 0.1, vortices are generated due to flow separation. Vortex dynamic is crucial for sediment transport and kinetic energy dissipation over vortex ripples. At the beginning of each half flow cycle, the leeside vortex starts to grow over the ripple flank and reaches its maximum strength at about the time of maximum freestream flow. Later, during the flow deceleration phase, the vortex starts to move away from the bed toward the ripple crest. At the time of flow reversal, the vortex is ejected and carried over the ripple crest toward the opposite side until it is fully dissipated. The vortices change the boundary layer structure by their organized convective motion and suspend a significant amount of sediment into the water column^{[8]}. Laboratory experiments by van der Werf et al. (2007^{[9]} showed that ripple geometry, vortex generation and ripple migration depend on the skewness and asymmetry of the horizontal wave orbital velocity (the asymmetry in velocity magnitude between positive and negative wave phases and the asymmetry in acceleration between onshore and offshore flow reversal, respectively). This implies that boundary layer flow, sediment transport, ripple geometry, and migration are all interrelated processes, with important implications for the shoreface profile of sedimentary coasts. See also Wave ripple formation.
Formation of bedforms
The formation of bedforms is related to the inherent instability of the sediment bed. It can be qualitatively understood as follows^{[10]}. A small perturbation of the sediment bed generates a perturbation of the flow pattern and associated zones of sediment transport convergence and divergence. The perturbation will grow if the sediment transport convergence zone comprises the crest region of the initial perturbation. The underlying causes are the compression of the streamlines on the upstream side of the bedform (inertial effect) and the short range of the additional turbulent friction and momentum dissipation produced by the bed perturbation. Consider, for example, a bed perturbation that restricts the flow depth (or the flow width) of a channel. While passing the perturbation crest, the flow accelerates. This will be the case for the flow zone most remote from the bed perturbation (near the surface or opposite the width restriction), which is hardly influenced by the additional frictional momentum dissipation generated by the perturbation. In contrast, the flow in the zone closest to the bed perturbation is strongly perturbed. After an initial acceleration on the upstream ramp, the flow slows down before passing the perturbation crest due to the increasing shear stress. This deceleration increases the internal velocity shear and thus counteracts the acceleration of the remote flow downstream of the bed perturbation. The flow deceleration close to the bed perturbation locally diminishes the bedload sediment transport capacity. This implies sediment deposition, that leads to growth of the perturbation if it is assumed that most of the sediment load is transported close to the bed. The process is schematically illustrated in Fig. 6, for unidirectional flow and for oscillating flow.
Different types of bedforms can be generated through this mechanism, depending on their impact on the structure and intensity of turbulent fluid motions and momentum dissipation in steady flow, tidal flow or wave dominated flow. The type of perturbations with the highest initial growth rate will become the dominant bedform. Perturbations with small length scales produce the strongest turbulence and will thus be favoured. However, bedforms with small length scales are steep and therefore subject to avalanching. Due to this antagonistic mechanism, the fastest growing bedforms are not the steepest ones, but bedforms with intermediate length scales. These length scales can be estimated by performing a linear stability analysis, as explained in the articles Stability models and Wave ripple formation. In this way it is possible to understand the formation of almost all bedforms that are observed in sandy coastal environments^{[11]}. See also the articles Rhythmic shoreline features and Sand ridges in shelf seas.
Field observations show that dunes can develop from an initial ripple field by a nonlinear process of pattern coarsening, i.e. successive stages of ripple merging and growth^{[12]}. However, stable bedforms with wavelengths intermediate between ripples and dunes do not exist. This can be explained by the smoothness of these bedforms, which causes a shift of the maximum sediment bedload flux from a position upstream of the crest to a downstream position. These bedforms decay unless they become covered with ripples and thus become sufficiently rough to shift the location of maximum bedload flux back to the upstream side of the bed (Fig. 6), allowing further growth into mature dunes^{[13]}.
Bed roughness
Nikuradse^{[14]} introduced the concept of an equivalent or effective sand roughness height, [math]k_s[/math], to simulate the roughness of arbitrary roughness elements of the bottom boundary. In case of a movable bed consisting of sediments the effective bed roughness [math]k_s[/math] mainly consists of grain roughness ([math]k'_s[/math]) generated by skin friction forces and of form roughness ([math]k''_s[/math]) generated by pressure forces acting on the bed forms. Similarly, a grainrelated bedshear stress ([math]\tau'_b[/math]) and a formrelated bedshear stress ([math]\tau''_b[/math]) can be defined. The effective bed roughness for a given bed material size is not constant but depends on the flow conditions. Analysis results of [math]k_s[/math]values computed from Mississippi River data (USA) show that [math]k_s[/math] strongly decreases from about 0.5 m at low velocities (0.5 m/s) to about 0.001 m at high velocities (2 m/s), probably because the bed forms become more rounded or are washed out at high velocities.
The fundamental problem of bed roughness prediction is that the bed characteristics (bed forms) and hence the bed roughness depend on the main flow variables (depth, velocity) and sediment transport rate (sediment size). These hydraulic variables are, however, in turn strongly dependent on the bed configuration and its roughness.
A second problem is the almost continuous variation of the discharge during rising and falling stages. Under these conditions the bed form dimensions and hence the friction coefficient are not constant but vary with the flow conditions. Bedforms observed in the field are generally in a transient state.
A third problem is the influence of biota on bed friction. Microorganisms living in and on the bed can stick sediments together and reduce bed friction by forming surface films (algae mats) and by excreting large organic molecules (EPS). Other organisms can enhance sediment mobility and bed friction through bioturbation. Bed friction can be strongly reduced by seagrass and seaweed. For a discussion of these various biotic factors, see Biogeomorphology of coastal systems.
Related articles
 Bed roughness and friction factors in estuaries
 Wave ripples
 Stability models
 Wave ripple formation
 Rhythmic shoreline features
 Sand ridges in shelf seas
 Biogeomorphology of coastal systems
Appendix Empirical estimates of bedform characteristics and associated bed roughness
Several formulas have been derived from laboratory and field experiments for estimating bedform characteristics (especially height and wavelength). Bedforms influence currents and waves by enhancing shear stresses and associated frictional momentum dissipation in the turbulent boundary layer. Conversely, currents and waves are themselves the primary agents responsible for the emergence and development of these bedforms by inducing sediment transport. Spatial patterns in sediment transport are both cause of and caused by the development of bedforms due to the inherent instability of the flat sediment bed. Bedform characteristics depend on flow properties and flow properties depend on bedform characteristics. The empirical formulas presented in this appendix establish relationships between flow properties and bedform characteristics.
The case of currents only
Roughness and friction
Steady or lowperiodic water motions in coastal waters are turbulent over the whole water column. The seabed is subjected to a shear stress [math]\tau_b[/math] which is related to the depthaveraged current velocity [math]U[/math] through the friction coefficient [math]c_D[/math]:
[math]c_D = \Large\frac{\tau_b}{\rho U^2}\normalsize ,[/math]
where [math]\rho[/math] is the density of water. Sometimes other friction factors are used, such as the Chezy coefficient, the DarcyWeisbach friction factor or the Manning coefficient, see Bed roughness and friction factors in estuaries. The friction coefficient [math]c_D[/math] is related to the thickness [math]k_s[/math] of the nearbed roughness layer (the Nikuradse length) ^{[15]},
[math]k_s = 12 h \exp(\large\frac{0.4}{\sqrt{c_D}}\normalsize) , [/math]
where [math]h[/math] is the total water depth.
For a flat sediment bed [math]k_s \approx (23) \, d_{50} ,[/math] where [math]d_{50}[/math] is the median sediment grainsize.
The roughness height [math]k_s[/math] according to van Rijn (2007^{[16]}) can be estimated from
[math]k_s = d_{50} \, \big[85 – 65 \, \tanh(0.015(\Psi150)) \big] ,[/math]
where [math]\Psi = \Large\frac{U^2}{g (s1)d_{50}}\normalsize[/math] is the mobility parameter.
Ripple height and wavelength
Several empirical formulas have been established for the mean ripple height and ripple wavelength from field and laboratory experiments. These formulas involve the following variables:
 [math]d_* = d_{50} \, \big( \large\frac{g (s1)}{\nu^2}\normalsize \big)^{1/3} \approx 25,000 \, d_{50}[/math] is the dimensionless mean grainsize
 [math]s = \rho_{sediment}/\rho \approx 2.65[/math] is the effective sediment density
 [math]\nu \approx 10^{6} m^2/s[/math] is the kinematic viscosity of water
 [math]\Theta = \Large\frac{\tau_{b}}{g \rho (s1) d_{50}}\normalsize[/math] is the Shields parameter
 [math]\Theta_{cr} \approx 0.05 [/math] is the critical value of the Shields parameter for the onset of particle motion
 [math]\tau_b = \rho u_*^2[/math] is the bed shear stress
 [math]u_* = \sqrt{c_D} U[/math] is the friction velocity
A few popular empirical formulas for the mean ripple height and wavelength are indicated in the table below.
Reference  Data  Ripple height  Ripple wavelength 

Soulsby et al. (2012^{[17]})  Fully developed field of bottom ripples with flowperpendicular straight crestlines the ripple height  [math]\eta_r = 202 \, d_{50} \, d_*^{0.554}[/math]  [math]\lambda_r = d_{50} \, (500 + 1881 \, d_*^{1.5}) [/math] 
Flemming (1988^{[18]})  Field observations in rivers and estuaries  [math]\eta_r = 0.677 \; \lambda_r^{0.809} [/math]  
Lapotre et al. (2017^{[19]})  Field data and dimensional analysis  [math] \lambda_r = 2504 \, d_{50}^{1/3} \Theta^{1/6} (\nu/u_*)^{2/3} [/math]  
Venditti and Bradley (2022^{[3]})  Field observations of river dunes  [math]\log_{10}(\Large\frac{\eta_r}{h}\normalsize) =  0.397\, \big[ \log_{10}(\Large\frac{\Theta}{\Theta_{cr}}\normalsize) 1.14 \big]^2  0.503 [/math]  [math] \log_{10}(\Large\frac{\lambda_r}{h}\normalsize) = 0.098\, \big[ \log_{10}(\Large\frac{\Theta}{\Theta_{cr}}\normalsize)  1.09 \big]^2 + 0.791[/math] 
The case of waves only
A more detailed discussion of wave ripples is given in the article Wave ripples and Wave ripple formation.
Roughness and friction
The frequent reversal of the wave orbital motion along the seabed hampers the development of the turbulent boundary layer. The height of the wave boundary layer [math]h_w[/math] is generally small, of the order of one to ten centimeters. It depends on the wave phase and is largest shortly before reversal of the orbital motion^{[20]}. For a rough sediment bed the height of the wave boundary layer depends primarily on the height [math]k_s[/math] of the roughness elements, the median sediment grainsize [math]d_{50}[/math] and the horizontal wave orbital excursion [math]2a[/math]. The wave orbital excursion is related to the maximum wave orbital velocity [math]U_b[/math] at the top of the wave boundary layer, [math]a= U_b / \omega [/math], where [math]\omega=2 \pi / T[/math] is the wave angular frequency and [math]T[/math] the wave period. For a rippled seabed, the wave boundary layer also depends on the ripple characteristics (height [math]\eta_r[/math] and wavelength [math]\lambda_r[/math]). The following empirical orderofmagnitude estimates have been derived based on laboratory and field experiments:
 For a smooth flat sediment bed, the height of the wave boundary layer [math]h_w \approx \sqrt {\nu T /\pi }[/math]
 For flat rough bed^{[21]} [math]k_s \approx (23) \, d_{50}[/math] and [math]\; h_w \approx 0.09 \, k_s (\large\frac{a}{k_s}\normalsize)^{0.82}[/math]
 For a stonecovered bed^{[22]} [math]\; h_w \approx 0.08 \, k_s \big[ 1 + (\large\frac{a}{k_s}\normalsize)^{0.82} \big] [/math]
 For a rippled bed^{[23]} [math]\; k_s \approx 8 \Large\frac{\eta_r^2}{\lambda_r}\normalsize + 170 \, d_{50} \sqrt{\Theta  \Theta_{cr}}[/math]
The symbols are defined as follows:
 [math]\Theta = \Large\frac{\tau_{bw}}{g \rho (s1) d_{50}}\normalsize[/math] is the Shields parameter
 [math]\Theta_{cr} \approx 0.165\, (R_f+0.6)^{0.8} + 0.045 \, \exp[40 R_f^{1.3}] [/math] is the critical value of the Shields parameter for the onset of particle motion^{[24]}
 [math]\tau_{bw}[/math] is the maximum waveinduced shear stress
 [math]R_f=\sqrt{\large\frac{f_w}{2}}\large\frac{d_{50} U_b}{\nu}\normalsize[/math] is the grain Reynolds number
Orderofmagnitude estimates are [math]\Theta_{cr} \approx 0.05 , \; k_s \approx 25 \Large\frac{\eta_r^2}{\lambda_r}\normalsize [/math].
The wave friction coefficient is defined as [math]f_w = \Large\frac{2 \, \tau_{bw}}{\rho U_b^2}\normalsize .[/math] Several empirical formulas have been established for the wave friction coefficient. Most of the formulas, valid for [math]a\gt 1.6 k_s[/math], are of the form [math]f_w = c_1 \, \exp\big[c_2 \, (\large\frac{a}{k_s}\normalsize)^{c_3} \big] [/math] with the following values: ([math]c_1 = 0.00251, c_2 = 4.57, c_3 = 0.19[/math]) ^{[25]}, ([math]c_1=0.00123, c_2 = 5.5, c_3 = 0.16[/math]) ^{[26]}, ([math]c_1 = 0.00184, c_2 = 5.5, c_3 = 0.2[/math])^{[23]}. For smooth sediment beds the friction coefficient is related to the Reynolds number [math]Re=\Large\frac{a U_b}{\nu}\normalsize[/math] ^{[21]}: [math]f_w \approx 2 \; Re^{ 0.5} [/math] in case of laminar flow and otherwise [math]f_w \approx 0.035 \, Re^{0.16}[/math]. For beds covered with very large roughness elements ([math]a/k_s \sim O[1][/math]) the friction coefficient can be approximated by^{[22]} [math]\; f_w \approx 0.32 \, (a/k_s)^{0.8}[/math] and for the opposite case [math]a \gt \gt k_s[/math] ^{[21]}, [math]\; f_w \approx 0.04 \, (a/k_s)^{ 0.25}[/math].
Ripple height and wavelength
A large number of field and laboratory studies have been dedicated to determining the height and wavelength of seabed ripples generated by wave action. Empirical formulas for the ripple height and wavelength (orbital ripples) of a fully developed field are indicated in the table below. More empirical formulas are presented in the review article by Nelson et al. (2013^{[27]}). Key variables appearing in these formulas are the wave orbital excursion amplitude [math]a[/math] and the sediment mobility parameter [math]\Psi = \Large\frac{U_b^2}{g (s1) d_{50}}\normalsize[/math].
Reference  Ripple height  Ripple wavelength 

Nielsen (1981^{[28]}), O'Donoghue et al. (2006^{[29]})  [math]\eta_r = a \, \Big(0.275  0.022 \, \Psi^{0.5}\Big)[/math]  [math] \lambda_r = a\, \Big(1.97  0.44 \, \Psi^{0.21}\Big)[/math] 
Soulsby and Whitehouse (2005^{[30]})  [math] \eta_r = 0.15 \, \lambda_r \Big(1\exp((5000 \large\frac{d_{50}}{a}\normalsize)^{3.5}) \Big) [/math]  [math] \lambda_r = a \, \Big(1+0.00187 \large\frac{a}{d_{50}}\normalsize \big[1\exp((5000 \large\frac{d_{50}}{a}\normalsize\Big)^{1.5}) \big] \Big)^{1} [/math] 
Goldstein et al. (2013^{[31]})  [math] \eta_r = 313 \, d_{50} \, \lambda_r [/math]  [math] \lambda_r = 2a \, \Big(1.12 + 2180 \, d_{50}\Big)^{1} [/math] 
Ruessink et al. (2015^{[32]}) for [math]d_{50}\gt 0.3 mm[/math]  [math] \eta_r = 0.164 \, \lambda_r \Big(1\tanh(0.63 \Theta)\Big) [/math]  [math] \lambda_r = 0.676 \, a \, \Theta^{0.163} \quad[/math] 
Grasmeijer and Kleinhans (2004^{[33]})  [math]\eta_r = 2 a \Psi^{1}[/math]  [math]\lambda_r = \eta_r \, \Big(0.078 + 0.355 \Psi^{0.221} \Big)^{1}[/math] 
van Rijn (2007^{[16]})  [math]\eta_r = a \Big( 2.8 \; 10^{13} \, (250  \Psi)^5 \Big)[/math]  [math]\lambda_r = 5 \; 10^6 \eta_r \, (250  \Psi)^{2.5}[/math] 
Chen et al. (2023^{[34]})  [math]\eta_r = a \, \Big(0.2 + 0.6 \, \Psi^{0.2} \Big)[/math]  [math]\lambda_r = \eta_r \, \Big(0.1  3.9 \; 10^{4} \, \Psi \Big)^{1}[/math] 
thumb rule orbital ripples  [math]\eta_r \approx 0.1 \, \lambda_r[/math]  [math]\lambda_r \approx 1.35 \, a[/math] 
The predictions of the empirical formulas span a wide range (Fig. 7). This wide range partly reflects the strong scatter in the experimental data on which the formulas are based. However, it also suggests that ripple dimensions do not only depend on the wave orbital excursion [math]2a[/math], the sediment mobility parameter [math]\Psi[/math] and the mean grainsize [math]d_{50}[/math]. Other possibly relevant variables proposed in the literature include the wave period [math]T[/math], the Shields parameter [math]\Theta[/math], the grain Reynolds number [math]R_f[/math] and the sediment fall velocity [math]w_s[/math]. ^{[27]}
References
 ↑ Simons, D.B. and Richardson, E.V. 1966. Resistance to flow in alluvial channels. U.S. Geological Survey Professional Paper 422J. Washington, DC: U.S. Government Printing Office
 ↑ Southard, J.B. and Boguchwal, L.A. 1990. Bed configurations in steady unidirectional water flow part 2. Synthesis of flume data. Journal of Sedimentary Petrology 60: 658–679
 ↑ ^{3.0} ^{3.1} ^{3.2} Venditti, J.G. and Bradley, R.W. 2022. Bedforms in sand bed rivers. Treatise on Geomorphology 2nd edition, Ch. 6.13. Elsevier
 ↑ Van Rijn, L.C., 1993, 2012. Principles of sediment transport in rivers, estuaries and coastal seas. Aqua Publications, Amsterdam, The Netherlands (WWW.AQUAPUBLICATIONS.NL)
 ↑ Bagnold, R. A. and Taylor, G. 1946. Motion of waves in shallow water. Interaction between waves and sand bottoms. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 187(1008): 1–18
 ↑ Komar, P.D. 1974. Oscillatory ripple marks and the evaluation of ancient wave conditions and environments. Journal of Sedimentary Research 44: 169–180
 ↑ Vaucher, R. and Dashtgard, S.E. 2022. Nearshore Bedforms. Treatise on Geomorphology 2nd edition Elsevier. Ch. 8.09, pp. 230246
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 ↑ van der Werf, J. J., Doucette, J. S., O'Donoghue, T. and Ribberink, J. S. 2007. Detailed measurements of velocities and suspended sand concentrations over fullscale ripples in regular oscillatory flow. Journal of Geophysical Research 112(2): 1–18
 ↑ Dronkers, J. 2017. Dynamics of Coastal Systems. Advanced Series on Ocean Engineering Vol. 41. World Scientific Publ. Co. Singapore, 753 pp.
 ↑ Vittori, G. and Blondeaux, P. 2022. Predicting offshore tidal bedforms using stability methods. EarthScience Reviews 235, 104234
 ↑ Fourrière, A., Claudin, P. and Andreotti, B. 2010. Bedforms in a turbulent stream: formation of ripples by primary linear instability and of dunes by nonlinear pattern coarsening. J. Fluid Mech. 649: 287328
 ↑ Vinent, O.D., Andreotti, B., Claudin, P. and Winter, C. 2019. A unified model of ripples and dunes in water and planetary environments. Nature Geoscience 12: 345–350
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<ref>
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