Benoît Camenen
email: benoit.camenen@cemagref.fr

Estimation of the wave-related ripple characteristics and induced bed shear stress

Keywords: wave ripples, ripple height, ripple steepness, roughness height, bed shear stress, predictive formula

A large data set on ripples was collected and examined. A set of new formulas for the prediction of the ripple characteristics is proposed with an emphasis on the disappearance of the ripples. The ripple wavelength was observed to be proportional to the bottom wave excursion but also to be a function of the grain-related Shields parameter and wave period parameter introduced by Mogridge et al. (1994). The ripple steepness was found to be nearly constant for orbital ripples, and with a sharp decrease for suborbital ripples. Two empirical functions are added including the effects of the critical Shields parameters (inception of transport and inception of sheet flow), i.e. giving the boundaries for the ripple existence's domain. The proposed formulas yield better prediction capabilities compared to the previously published formulas, especially when ripples are washed out. The effect of the ripple characteristics on the roughness height and the calculation of the bed shear stress is also discussed. It appeared that the bed shear stress calculation is more sensitive to the empirical coefficient a r introduced in the estimation of the ripple-induced roughness height or to the limits of existence of the ripples than the ripple characteristics themselves.

Introduction

An important issue for the prediction of the suspended load in coastal waters is to estimate the bottom shear stress due the oscillatory movement of the waves. Close to the shore, the wave-related bed shear stress often prevails on the current-related bed shear stress, and thus significantly influences the estimation of the bottom reference concentration and sediment diffusivity, which are the main parameters for the suspended load [START_REF] Camenen | A suspended load sediment transport formula for the nearshore[END_REF]. When ripples appear, their effects on the roughness height and bottom shear stress are very significant [START_REF] Van Rijn | Principles of sediment transport in rivers, estuaries and coastal seas[END_REF]. [START_REF] Van Der Werf | Modelling and measurement of sand transport processes over full-scale ripples in oscillatory flow[END_REF] also showed the strong influence of ripples on the direction of the net suspended load, as they induce some phase-lag in the suspension concentration. It is then extremely important to predict correctly the characteristics of the ripples and induced roughness height.

Unlike current-related bed forms, wave ripples are generally symmetric with sharper crests due to the to and fro movement of wave orbital velocity. [START_REF] Bagnold | Motion of waves in shallow water: interaction between waves and sand bottom[END_REF] called rolling-grain ripples the ripples first appearing on a flat bed exposed to wave-action. For this regime close to the inception of motion, the grains roll back and forth and form small triangular ridges. According to [START_REF] Bagnold | Motion of waves in shallow water: interaction between waves and sand bottom[END_REF], these ripples are stable only if the shear stress stays close to its critical value for the inception of movement. If the flow magnitude increases, the lee-side vortex becomes strong enough to initiate grain motion in the space between the two ripples. The ripples grow until an equilibrium geometry corresponding to the so-called vortex ripples.

The ripple wave length L r is often assumed to be proportional to the semiexcursion amplitude of orbital motion A w : 1 < L r /A w < 2. And the ripple wave steepness H r /L r (where H r is the ripple height) varies from 0.1 (vortex formation) to 0.25 (slope stability). [START_REF] Wiberg | Ripple geometry in wave-dominated environments[END_REF] made a very careful description of ripple characteristics. In their classification, ripples are subdivided into three groups: orbital, suborbital and anorbital. The basic scaling of orbital ripple wavelength is on A w , whereas the basic scaling of anorbital ripples is on the median grain size d 50 ; suborbital ripples represent the transitional region between these two limits. Orbital ripples have an almost constant steepness of H r /L r = 0.17, while anorbital ripples are less steep, with a steepness decreasing as orbital diameter increases for a given grain size.

Recently, [START_REF] Hanes | Wave-formed ripples at Duck, North Carolina[END_REF] observed in the field long-wave ripples (LWR) with relatively small heights (ranging from 3 mm to 6 cm) and large lengths (ranging from 35 to 200 cm). LWR were observed to be not as dynamics as the small-wave ripples (SWR) commonly observed. And they also observed some coexistence for both types of ripples. One explanation for LWR is that the peak frequency near the seabed may be different (smaller) than the peak frequency of the surface elevation because of the frequency-dependent reduction of fluid orbital amplitude with depth. Other assumptions for their origin may be a larger suspension within the boundary layer and a regime close to the sheet-flow regime [START_REF] Hanes | Wave-formed ripples at Duck, North Carolina[END_REF]. [START_REF] Kleinhans | Phase diagram of bed states in steady, unsteady , oscillatory and mixed flows[END_REF] described them as skewed hummocks with much sediment suspension but no strong vortex shedding, associated with large orbital flows possibly combined with small currents.

For strong conditions (typically storm conditions), ripples are washed out. It corresponds to the inception of the sheet-flow regime where most of the sediment transport occurs in a thin layer close to the bed. This regime occurs when the Shields parameter, mobility parameter, orbital Reynolds number or the ratio A / d 50 reach some critical value: θ w > 0.8, Ψ w > 100, ℜ w > 2 × 10 6 or A / d 50 = 6000 approximately (cf. [START_REF] Van Rijn | Principles of sediment transport in rivers, estuaries and coastal seas[END_REF][START_REF] Camenen | Phase-lag effects in sheet flow transport[END_REF] where the wave-related Shields parameter, mobility parameters, wave period parameter, and orbital Reynolds number are defined as follows: (3)

ℜ w = U w 2 ων (4)
with U w the wave orbital velocity, s the relative sediment density, g the acceleration of gravity, f w the wave-related friction factor, A w = U w T w /(2π) the wave orbital semi excursion, T w the wave period, ω = 2π/T w , and ν the kinematic viscosity of water.

Many authors proposed empirical relationships to describe the ripple characteristics, based on the mobility parameter Ψ w : [START_REF] Nielsen | Dynamics and geometry of wave generated ripples[END_REF], [START_REF] Wikramanayake | Calculation of movable bed friction factors[END_REF][START_REF] Van Rijn | Principles of sediment transport in rivers, estuaries and coastal seas[END_REF], or [START_REF] Grasmeijer | Observed and predicted bed forms and their effect on suspended sand concentrations[END_REF]; on the Shields parameter θ w : [START_REF] Grant | Movable bed roughness in unsteady oscillatory flow[END_REF]; on the wave period parameter χ: [START_REF] Mogridge | Geometry prediction for wavegenerated bedforms[END_REF]; on the ratio A w /d 50 : [START_REF] Wiberg | Ripple geometry in wave-dominated environments[END_REF] or [START_REF] Soulsby | Dynamics of marine sands, a manual for practical applications[END_REF][START_REF] Soulsby | Prediction of ripple properties in shelf seas: Mark 1, predictor[END_REF][START_REF] Van Rijn | Unified view of sediment transport by currents and waves. I: Initiation of motion, bed roughness, and bed-load transport[END_REF]; or on the orbital Reynolds number: [START_REF] Williams | Unifying large and small wave-generated ripples[END_REF]. These equations are presented in detail in appendix. Most of these formulas are however only based on experimental data where ripples exist. In order to properly estimate the total shear stress, it is also fundamental to validate these formulas over plane beds when the ripples are washed out. For random waves, authors usually suggested to use the significant wave characteristics for the prediction. Then, most of the formulas are assumed valid for both monochromatic and random waves.

The main purpose of this study is to provide a formula which correctly predicts ripple characteristics but also ripple disappearance in order to estimate properly the total bed shear stress for any wave condition. This paper is organized as follows:

The experimental data used for this study are described and compared with the studied formulas. New formulas for the prediction of the ripple characteristics and their validation are presented next. Then, a presentation of the computation of the total shear stress is made as well as a sensitivity analysis on the main parameters. Finally, conclusions concerning the validity of the formulas and a discussion on the calculation of the total shear stress in the nearshore are also provided.

2 On the prediction of ripple characteristics

Experimental data

In order to compare the results obtained using empirical formulas with experimental data, several data sets were compiled. In Tab. 1, the main hydrodynamic and sedimentologic parameters are presented, as well as the type of the experiment. For all the experiments presented in Tab. 1, sand with a relative density s = 2.65 was used. Several of these data sets were obtained from the data compilation provided by the Sedmoc (2001) European Union research project.

Comparison between experimental data and empirical relationships for ripple characteristics

A comparison between the experimental data and the empirical relationships presented previously was undertaken. Tab. 2 summarizes the results obtained for the prediction of the ripple height and length where P 2 and P 5 indicate the percentage of data correctly predicted within a factor 2 or 5, respectively. The mean value (µ f ) and standard deviation (σ f ) of the function f (X) = log |X pred /X meas | is also presented, where X pred and X meas are the predicted and measured values (H r or L r ), respectively. To avoid zero-values in the experimental data or predicted data, a minimum value for H r and L r was assumed (equal to 2d 50 and 10d 50 , respectively).

The [START_REF] Williams | Unifying large and small wave-generated ripples[END_REF] formulas were proposed for suborbital ripples only and thus are valid for large values of the ratio A w /d 50 only. Because of its obvious limits, this formula will not be used in the following study. for the ripple height prediction compared to the Grant & Madsen, Soulsby & Whitehouse and Van Rijn formulas. This may be partly explained as former formulas do not take into account the critical Shields parameters for the presence of ripples. As the comparison is made with data where ripples exist, a model that predicts the absence of ripple would yield significant underestimation. It is particularly significant for the effect of θ cr as several data indicate the occurrence of ripples whereas the Shields parameter is smaller than θ cr (estimated in this paper using Eq. 5). Actually, as noted by Soulsby & Whitehouse (2007?), when θ w ≤ θ cr , ripples should take pre-existing values of H r and L r , which are not necessarily null. However, in nature, there are many cases where the bed is flat. Even if some uncertainties exist on the prediction on the critical Shields parameters (inception of sediment movement or sheet flow), it is very important to be able to predict the appearance and disappearance of ripples and not to overestimate the roughness height when the bed is flat. The Nielsen, Van Rijn and Grasmeijer & Kleinhans formulas give a sheet flow threshold for (θ w ) cr,sf = 1 or (Ψ w ) cr,sf = 250 or 156, respectively. Another remark is that the Wiberg & Harris and Soulsby & Whitehouse formulas predict a nearly constant value for the ratio H r /L r ≈ 0.15 when A w /d 50 < 1000 and a rapid decrease in steepness when A w /d 50 > 3000. For these two formulas, the threshold of sheet-flow is described in term of A w /d 50 ((A w /d 50 ) cr,sf ≈ 6000, which seems to be too large, see section 3.1). As a conclusion, the Soulsby & Whitehouse formula yields the best prediction for the estimation of the ripple characteristics using the collected data; the Van Rijn formula yields a lower score for the prediction but it seems to better take into account the effect of the threshold for the inception of sheet flow.

Capabilities of the various formulas to predict the absence of ripples

To confirm these observations, a comparison is provided using data for the inception of sheet flow (cf. Sec. 3.1 and [START_REF] Camenen | Phase-lag effects in sheet flow transport[END_REF]. When the sheet flow regime just appears, the bed is flat and the roughness height may be assumed to be equal to the grain-related roughness, i.e. k s = k sg = 2d 50 . In Tab. 3, the "measured" (assuming k s = k sg ) and "estimated" (based on the estimated ripple characteristics and using Eq. 17) roughness heights are compared.

The results from Tab. 3 clearly indicates the weakness of most of the predictive formulas. As they are based and fitted on experimental data where ripples exist, they do not forecast properly the absence of ripple. Only the Nielsen, Van Rijn and the Grasmeijer & Kleinhans formulas predict a disappearance of the ripples even if some large uncertainties are encountered to predict this limit. The Nielsen formulas yield the best results just because it tends to underestimate θ cr,sf . 3 A new model for the calculation of the ripple characteristics

Limits for the ripple existence

A need clearly comes out from the above study for the estimation of the ripple characteristics, which is to estimate carefully their limits of existence, i.e. the critical Shields parameters corresponding to their appearance and disappearance. As concerns the appearance of ripples, it may be assumed that they exist as soon as sediments move. The critical Shields parameter for the inception of movement may be estimated using the equation proposed by [START_REF] Soulsby | Threshold of sediment motion in coastal environment[END_REF],

θ cr = 0.30 1 + 1.2d * + 0.055 [1 -exp(-0.02d * )] (5)
where [START_REF] Camenen | Phase-lag effects in sheet flow transport[END_REF] proposed a maximum value of the wave orbital velocity U w,cr,sf for the inception of the sheet flow regime or wash-out of the wave ripples,

d * = [g(s -1)/ν 2 ] 1/3 d 50 is the dimensionless grain size.
U w,cr,sf = 8.35 (s -1)g d 50 δ w (1 + r w ) (6)
where δ w = νT w /π is the thickness of the Stokes boundary layer and r w the wave asymmetry (r w = (u w,max -u w,min /(u w,max + u w,min ) with u w,max and u w,min are the maximum onshore and minimum offshore velocity, respectively). A simple empirical equation has been suggested by [START_REF] Hanson | Closed form solution for threshold velocity for initiation of sediment motion under waves[END_REF] for the wave induced friction factor (validated with plane bed data), which may be written as follows:

f w = 0.5 √ π d * -3/4 d 50 δ w (7)
Thus, an expression for the critical Shields parameter θ cr,sf for the inception of sheet-flow may be obtained from Eqs. 1, 6 and 7:

θ cr,sf = 10 d * -3/4 d 50 δ w (1 + r w ) 2 (8)
Using the same experimental data as [START_REF] Camenen | Phase-lag effects in sheet flow transport[END_REF], Eq. 8 shows relatively good results (cf. Fig. 1). It can also be observed from the data that the critical Shields parameter for the inception of sheet flow may vary from 0.2 to 2.0. It should be noted that the threshold of sheet-flow in term of (A w /d 50 ) cr,sf (as used by [START_REF] Wiberg | Ripple geometry in wave-dominated environments[END_REF][START_REF] Soulsby | Prediction of ripple properties in shelf seas: Mark 1, predictor[END_REF] varies from 60 to 6000 using the same experimental data as [START_REF] Camenen | Phase-lag effects in sheet flow transport[END_REF]. Thus, the Wiberg & Harris and Soulsby & Whitehouse formulas seem to overestimate this threshold as they assume (A w /d 50 ) cr,sf ≈ 6000.

New equations for the description of the ripple characteristics

Following the previous observations, two simple equations function of the skin Shields parameter and its critical values only (inception of movement and inception of sheet flow) are proposed to predict the ripple length and steepness, respectively:

L r A w = L r A w eq f θ cr θ wg f θ wg θ cr,sf (9)
H r L r = H r L r eq f θ cr θ wg f θ wg θ cr,sf (10)
where the subscript eq corresponds to the equilibrium value of the ratio L r /A w and H r /L r , respectively, θ wg is the grain-related Shields parameter (computed assuming k s = 2d 50 ) and f is a function defining the existence of the ripples or not:

f (x) = exp -α cr x 4 (11)
with α cr a coefficient. [START_REF] Grant | Movable bed roughness in unsteady oscillatory flow[END_REF], [START_REF] Van Rijn | Principles of sediment transport in rivers, estuaries and coastal seas[END_REF] or [START_REF] Grasmeijer | Observed and predicted bed forms and their effect on suspended sand concentrations[END_REF] observed that (L r /A w ) eq is a decreasing function of Ψ w or θ w . On the other hand, [START_REF] Mogridge | Geometry prediction for wavegenerated bedforms[END_REF] found a relationship between the ratio L r /d 50 and the wave period parameter χ. In Fig. 2, the ripple wavelength normalised by the wave orbital semi-excursion L r /A w was plotted against the grain-related Shields parameter θ wg with the wave period parameter χ emphasised (for practical purpose, flat beds are represented in this figure by the value L r /A w = 10 -2 ). It appears that the ratio L r /A w does decrease with the grain-related Shields parameter θ wg but with a varying slope that depends on χ. A relationship for the equilibrium value of the ratio L r /A w is suggested as a function of θ wg and χ:

L r A w eq = 1.6 exp(-5 × 10 -3 χ -0.4 ) θ wg -0.025χ -0.2 (12)
Some uncertainties obviously exist in the determination of the critical Shields parameter for the inception of sheet flow. Indeed, many data from Hanes et al. indicate a disappearance of ripples for much smaller Shields parameter than expected. Unlike the observations by [START_REF] Wiberg | Ripple geometry in wave-dominated environments[END_REF], the collected data do not show any clear relationship between the ripple length and the grain size (when A w > 3000) corresponding to the anorbital ripples. Only orbital and suborbital ripples were observed. It should also be noted that there is a relationship between χ and Eq. 8 for θ cr,sf , and χ = 10 -8 π -2 θ cr,sf 8 d * 3 . The ripple steepness was often observed to be nearly constant, and H r /L r ≈ 0.16 [START_REF] Wiberg | Ripple geometry in wave-dominated environments[END_REF]. As observed by [START_REF] Soulsby | Prediction of ripple properties in shelf seas: Mark 1, predictor[END_REF], the ripple steepness appears to be strongly dependent on the ratio A w /d 50 with a smaller steepness for suborbital ripples (A w /d 50 > 1000). In Fig. 3, the ripple steepness H r /L r was plotted against wave orbital semi-excursion normalised by the median grain size A w /d 50 with the wave period parameter emphasised. A clear decrease of the ripple steepness with the ratio A w /d 50 may be observed. The following equation for the equilibrium ripple steepness is then suggested,

H r L r eq = 0.16 exp -1.0 × 10 -7 A w d 50 2 (13)
The introduction of the function f in Eqs. 9 and 10 was proposed to properly evaluate the non-existence of the ripples. As some uncertainties exist on the prediction of θ cr or θ cr,sf , this exponential function allows positive values for H r and L r , but much smaller than their equilibrium values.

In Fig. 4, Eqs. 9 and 10 are plotted versus the grain-related Shields parameter for four values of the critical Shields parameter for the inception of sheet flow 10 -7 < χ ≤ 5x10 -7 5x10 -7 < χ ≤ 2x10 -6 2x10 -6 < χ ≤ 10 -5 10 -5 < χ (θ cr,sf =0.75, 1.0, 1.25 and 1.5) for a given grain size (d 50 = 0.3 mm). The wave period is estimated using the relationship between χ and Eq. 8 for θ cr,sf . As expected, the proposed formulas yield a ripple steepness which depends mainly on the ratio θ wg /θ cr,sf and a slightly more complex function for the ratio H r /A w . Compar-ing with the experimental data (cf. Fig. 2), it appears that Eqs. 9, 10, 11 (with α cr = 1), 12, and 13, provide a good overview of the experimental data as soon as the critical Shields parameter for the inception of sheet flow is properly estimated. When θ cr,sf < 0.8 (or χ < 10 -7 ), ripples present much smaller height and steepness or even may not occur whatever the Shields parameter. Using Eqs. 5 and 8 for the estimation of the different critical Shields parameter, Eqs. 9 and 10 yield good results for the prediction of the ripple characteristics (see Tab. 2). Of course, even if a large data set has been used, these good results (compared to the other formulas) are slightly biased as the newly proposed formulas were optimized using this data set. As shown in Tab. 3, the correct prediction of θ cr,sf and its integration in the ripple formulas are fundamental not to overestimate largely the roughness (and thus the Shields parameter) when the ripples are washed out. The influence of the coefficient α cr in Eq. 11 appears to be significant when sheet flow regime is reached. Indeed, if the choice of α cr = 0.01 or 1 does not affect significantly the results for the prediction of the ripple characteristics (results slightly poorer when α cr = 1, see Tab. 2), the choice of α cr = 1 seems much more appropriate to predict the absence of ripples (cf. Tab. 3).

Large wave ripples

For large wave ripples (LWR), all the proposed formulas overestimate the ripple height and steepness. In Fig. 5, the wavelength and steepness with LWR data from [START_REF] Hume | What happens at the seabed of a headland during tropical cyclone?[END_REF] and [START_REF] Hanes | Wave-formed ripples at Duck, North Carolina[END_REF] have been plotted against the grain-related Shields parameter. Compared to Eqs. 9 and 10, the behaviour against θ wg seems to be properly reproduced (decrease of L r /A w with an increasing θ wg ). However, the ripple wavelength is underestimated by a factor 4 to 8. Apart from the data by Hume et al., the wave steepness is also underestimated by a factor 4 to 8, which means that the ripple height is properly estimated. Thus, some LWR could correspond to a more complex energetic system at the boundary layer where several frequencies coexist due to the frequency-dependent reduction of fluid orbital amplitude with depth. A higher frequency may control the ripple height whereas the smaller frequency may control the ripple length. Another hypothesis for the occurrence of the LWR is the relict (or fossil) ripples built during a previous event that may affect the ripple characteristics during the following period.

Effects of a steady current

When a current was superimposed on the waves, Van Rijn et al. (1993) observed that ripples become asymmetrical and rapidly three-dimensional with increasing current. [START_REF] Khelifa | Prediction of sand ripple geometry under waves and current[END_REF] showed that positive and negative longitudinal current velocities produce a similar effect on the ripple characteristics, i.e. an increase of the ripple length and height with increasing current. Based on kinetic considerations, they suggested to use the effective fluid orbital semi-excursion instead of the wave orbital semi-excursion,

A cw = A w 2 + T w U c 2 2 + A w T w |U c || cos φ| = A w 1 + πU c U w 2 + 2π |U c | U w | cos φ| (14)
where U c is the steady current and φ the angle between the wave and current directions. Following this idea, the Shields parameter in a wave and current interaction may be estimated such as θ cw = θ w (A cw /A w ) 2 . It is thus possible to use Eqs. 9 to 13 in case of a wave and current interaction by replacing A w and θ w by A cw and θ cw , respectively. In Fig. 6, the ratio between the predicted (using Eqs. 9 to 13) and measured ripple length and height is plotted versus the ratio |U c |/U w . It appears that, if both ripple length an height are properly estimated when U c = 0, a significant over-estimation exists as soon as |U c |/U w > 10 -2 (where L r,pred /L r,meas ≈ 2). No explanation was found to explain this reduction in the ripple dimensions due to a relatively low current. The change from 2D symmetrical ripples to 3D asymmetrical ripples may affect the average ripple length and height. When |U c |/U w > 10 -2 , ripple characteristics are increasing as predicted by Eq. 14. As observed by [START_REF] Van Rijn | Transport of fine sands by currents and waves (II)[END_REF], the angle between the wave and current directions has no significant influence on the ripple dimensions. Figure 6: Prediction of the ripple wavelength and steepness using Eqs. 9 and 10 as a function of the ratio between the steady current amplitude of the wave orbital velocity.

As the steady current appears to induce two opposite effects on the ripple dimensions (decreasing when |U c |/U w < 3 × 10 -2 and increasing |U c |/U w > 3 × 10 -2 ), Eqs. 9 to 13 for waves alone still yield correct results (cf. Tab. 4). Using Eq. 14 improves the behaviour of the prediction against the ratio |U c |/U w . However, it tends to overestimate ripple dimensions; and the critical Shields parameter for inception of sheet flow is reached more easily (as θ cw > θ w ), which yields a too fast disappearance of the ripples. An empirical coefficient for Eqs. 9 and 10 is suggested to improve the prediction of the ripple characteristics under waves and current:

r wc = 1 -0.6 tanh 50 |U c | U w 1 + 3 tanh |U c | U w (15)
As observed in Tab. 4, Eq. 15 significantly improves the results for the ripple prediction in case of a wave and current interaction.

Table 4: Prediction of the ripple characteristic under waves and current.

ripple length L r ripple height H r Equation P 2 P 5 µ f σ f P 2 P 5 µ f σ f Eqs. 9 to 13 69% 100% +0.16 0.24 59% 98% +0.21 0.24 Eqs. 9 to 13 using Eq. 14 45% 95% +0.16 0.52 50% 91% -0.08 1.15 Eqs. 9 to 13 using Eq. 15 81% 99% +0.08 0.23 73% 98% +0.14 0.24 4 Estimation of the wave-related Shields parameter in case of ripples

The calculation of the bed shear stress in the nearshore is not trivial. The main difficulty lies in the determination of the wave-related friction factor f w . f w was calculated using the Swart formula (1974):

f w = min exp -6.0 + 5.2 A w k s -0.19 , 0.3 (16)
where the roughness height k s needs to be estimated. Many uncertainties are then combined: they stem from the prediction of ripple characteristics but also their effects on the bed shear stress and the limits for the presence of ripples.

Calculation of the roughness height for a rippled bed

In the off-shore region where the ripples are generally observed, the shear stress due to the current is often much smaller than the shear stress due to the waves, and then, may be neglected in this study. The total roughness height k s was estimated using the method proposed by [START_REF] Soulsby | Dynamics of marine sands, a manual for practical applications[END_REF] by summing the grain-related k sg ≈ 2d 50 , form-drag k sf , and sediment transport k ss components, respectively: k s = k sg + k sf + k ss . When sheet flow regime occurs, the [START_REF] Wilson | Friction of wave induced sheet flow[END_REF] formula was used, i.e. k ss = 5θ w d 50 . Thereafter, the grain related Shields parameter θ wg based on k sg will be distinguished from the total Shields parameter θ w based on k s . The ripple-related roughness height k sf is generally calculated using the following formula,

k sf = a r H r 2 L r (17)
where a r is a constant (5 < a r < 40). [START_REF] Nielsen | Coastal bottom boundary layers and sediment transport[END_REF] suggested a r = 8 whereas Van Rijn (1993) proposed a r = 20.

More recently, [START_REF] Kim | Effective form roughness of ripples for waves[END_REF] investigated the effective form roughness of the wave ripples using a numerical model with a mixing length hypothesis to simulate the wave boundary layer flow over ripples. He obtained the ripple-length average bed shear stress (or form drag friction) from integration of the computed pressure field. An interesting conclusion from his study is that the effective roughness for sharpcrested ripples becomes about three times larger than that for the sinusoidal shape. It means that the presence of a steady current may considerably modify (reduce) the effective roughness as it will smooth the ripple crests. Nevertheless, this effect will be neglected in this study but may partly explain the different values suggested for a r . Another explanation is the choice of the formula for f w , which may influence afterwards the choice for a r .

The ratio H r 2 /L r appeared to be a fundamental parameter for the estimation of the roughness height in the ripple regime, and thus for the estimation of the total Shields parameter. In Fig. 7(a), the ratio H r 2 /L r is plotted versus the ratio A w /d 50 using all the different studied formulas (assigned with initials of the authors or with the three first letters of the first author): The Nielsen (1981, Nie), Grant & Madsen (1982, GM), Wikramanayake & Madsen (1991, WM), Van Rijn (1993, VR), Mogridge et al. (1994, Mog), Wiberg & Harris (1994, WH), Grasmeijer & Kleinhans (2004, GK), Soulsby & Whitehouse (2005, SW), and Williams et al. (2005, Wil) formulas. As most beaches consist of sand with a grain size ranging from d 50 = 0.15 to 0.35 mm, and because the wave period does not vary on a cross-shore profile (if the energy transfer to nearly harmonic wave components is neglected), this sensitivity analysis is presented for a varying wave excursion only, assuming a constant median grain size d 50 = 0.25 mm and a constant wave period T w = 6 s. It should be noted that the calculated values for H r 2 /L r do vary significantly depending on the fixed parameters (d 50 or T w ). For the Soulsby & Whitehouse formula, U 1/10 was used instead of U 1/3 to represent the wave orbital velocity in case of irregular waves. When applied to regular waves, the curve should then be shifted rightward relative to the other curves.

Even if large discrepancies are observed between the different formulas, a similar trend for the coefficient H r 2 /L r may be observed:

-for A w /d 50 < 1000 (orbital ripples), H r 2 /L r is an increasing function of A w /d 50 .

-when 1000 < A w /d 50 < 3000 (suborbital ripples), H r 2 /L r reaches a maximum which is similar whatever the formula, and then decreases. (H r 2 /L r ) eq ≈ 6 × 10 -3 . This value appeared to be an increasing function of d 50 or T w .

-for A w /d 50 > 3000 (anorbital ripples), H r 2 /L r decreases rapidly toward zero.

It is also in accordance with [START_REF] Van Rijn | Unified view of sediment transport by currents and waves. I: Initiation of motion, bed roughness, and bed-load transport[END_REF], who suggested a constant value for the ripple induced roughness height k sf = 150d 50 when Ψ w < 50 with a quick decrease afterwards. Then, the main differences observed between the formulas are not in the magnitude of the coefficient H r 2 /L r but in the limits for the ripple presence (critical Shields parameters for the inception of movement and inception of sheetflow, respectively). Figure 7: Ratio H r 2 /L r versus the ratio A w /d 50 (a) and the induced Shields parameter θ w versus the ratio A w /d 50 (b) for a constant medium grain size d 50 = 0.25 mm and a constant wave period T w = 6 s using the different studied formulas for the ripple characteristics (the roughness height k s was calculated using Eq. 17 with a r = 10).

Influence of the ripple characteristics on the total Shields parameter

Using Eq. 17 with a r = 10 for the calculation of the ripple-induced roughness height for the eight studied formulas, the total shear stress was estimated for the same conditions as in Fig. 7(a); i.e. with fixed median grain size and wave period. This condition is quite coherent with what is observed on a beach for a specific time: a large range of bed shear stresses with nearly constant median grain size and wave period. Fig. 7(b) shows a typical evolution of the ratio between the total and skin wave-related Shields parameters as a function of the skin wave-related Shields parameter using the different studied formulas.

For very small skin shear stress (θ wg < θ cr ), as only the "GM" and "SW" formulas takes into account θ cr , only these formulas yield realistic results. The other formulas predict a Shields parameter one order of magnitude larger than the skin Shields parameter. Even if there are some uncertainties in the prediction of the critical Shields parameter for the inception of movement (cf. Eq. 5), it is necessary to take into account this critical value.

For relatively small skin shear stresses (θ cr < θ wg < 0.2), most of the formulas present similar results which indicate a total Shields parameter approximately 5 to 10 times larger than the grain-related Shields parameter. It is possible to define an equilibrium value for the ratio between the total and the skin-related Shields parameters (θ w /θ wg ) eq .

Then, for larger skin shear stress (0.2 < θ wg < θ cr,sf ), the ratio θ w /θ wg is decreasing with an increasing skin-related Shields parameter. The ratio also appeared to be a decreasing function of d 50 or T w (suborbital regime). The uncertainties in the prediction of the ripples wash-out are very large. Even if [START_REF] Wiberg | Ripple geometry in wave-dominated environments[END_REF] observed that ripples cannot exist for A w /d 50 > 6000, most of the formulas, including the "WH" formula, yield a wash-out of the ripples (θ w = θ wg ) for larger values of A w /d 50 . As no limit is provided, the "Mog" equation yields a presence of ripples whatever the shear stress, which is unrealistic. On the contrary, the "Nie" and "VR" formulas yield a wash-out of the ripples for θ wg ≈ 0.7 -0.9. This is more in agreement with the observations by Wiberg & Harris even if bed forms have been observed for higher shear stresses. An important parameter for the estimation the roughness height due to ripple is therefore the critical Shields parameter for the inception of sheet flow. It may also be observed that these formulas induce a nearly constant (even decreasing for the "VR" formula) total Shields parameter for an increasing wave orbital velocity when suborbital ripples are present (1000 < A w /d 50 < 5000).

It should be noted that f w and then (θ w /θ wg ) eq are functions of the coefficient a r (cf. Eq. 17) and strongly depend on the chosen value for a r . Based on [START_REF] Kim | Effective form roughness of ripples for waves[END_REF] numerical results, very different roughnesses are observed depending on the shape of the ripple. For sharp-crested ripples (pure oscillatory flow), a r = 13.4 is obtained whereas using a sinusoidal profile with the same characteristics, a r ≈ 6.3 is obtained. In case of a wave and current interaction, where the shape of the ripples is much smoother, a relatively low coefficient should thus be used. Soulsby & Whitehouse (2007?) proposed an alternative approach to compute the friction coefficient f w using the drag coefficient approach on strip roughness elements :

f w = M cd A w H r 2 L r (18)
with M cd a coefficient (Soulsby & Whitehouse suggested to used M cd = 5). In Fig. 8, the friction coefficient is plotted versus the ratio (H r 2 /L r )/A w with observed values from experiments of [START_REF] Jonsson | Experimental and theoretical investigations in a oscillatory turbulent boundary layer[END_REF], and [START_REF] Mathisen | Waves and currents over a fixed ripples bed. 1. bottom roughness experienced by waves in the presence and absence of currents[END_REF] and numerical model results of [START_REF] Andersen | The dissipation of waves over a rippled bed[END_REF] and [START_REF] Kim | Effective form roughness of ripples for waves[END_REF]. Eq. 16 with Eq. 17 and Eq. 18 yield similar results when f w > 0.05 and M cd ≈ a r /2. From the observed values, it may be noted that a r (M cd ) may vary from 5 to 30 (2.5 to 15). In the same way, (θ w /θ wg ) eq may vary by a factor 6 depending on the choice of a r (or M cd ).

Estimation of the total Shields parameter directly from the skin Shields parameter

Following the observations in section 4.2 (and the trends of the empirical formulas in Fig. 7, as well as the study of ripple characteristics proposed in section 3.2) and using θ cr (Eq. 5) and θ cr,sf (Eq. 8), a simple formula is proposed to estimate the total Shields parameter due to the ripples directly from the grain-related Shields parameter: 2 /L r /A w (Eq. 16 and Eq. 18 are plotted, using Eq. 17 with a r = 10 and 20, and M cd = 5 and 10, respectively, together with experimental data from [START_REF] Jonsson | Experimental and theoretical investigations in a oscillatory turbulent boundary layer[END_REF][START_REF] Mathisen | Waves and currents over a fixed ripples bed. 1. bottom roughness experienced by waves in the presence and absence of currents[END_REF], and numerical data from Andersen, 2001[START_REF] Kim | Effective form roughness of ripples for waves[END_REF].

θ w θ wg = 1 + θ w θ wg eq -1 f 2 θ cr θ wg f 2 θ wg θ cr,sf (19)
where the function f 2 was found to be f 2 (x) = f 2 (x) = exp(-2α cr x 4 ), and an estimation of the equilibrium value of the ratio θ w /θ wg may be: θ w θ wg eq = 1 + 1.0 a r exp -1.5 × 10 -6 θ wg χ (20)

Using Eqs. 17 (with a r = 10), 5, 8, 9 and 10, it is possible to estimate the total Shields parameter as a function of the skin Shields parameter. Fig. 9 presents a comparison between the method based on the ripple characteristics (1) and the empirical equation directly based on the skin Shields parameter (2: Eqs. 19 and 20). One can observe that both methods present very similar results. A truncation may appear using the first method for large grain size or wave period values combine with a large a r -value as f w reaches its maximum (max(f w ) = 0.3, cf. Eq. 16). For fine sediments or short wave periods, the behaviour is similar to the Van Rijn formula, i.e. for 0.5 < θ wg < 1 (cf. Fig. 7(b)), the total Shields parameter is a fast decreasing function of the grain-related Shields parameter, thus a decreasing function of the wave height, which needs to be validated. For coarser sediments or longer wave periods (i.e. for larger values of θ cr,sf ), this effect is not as strong.

Comparison with data

Tab. 5 presents the statistical results on the prediction of the total Shields parameter using the different formulas for the prediction of ripple characteristics. As for the previous calculations, Eq. 17 with a r = 10 was used to estimate the rippleinduced roughness height for both measured and estimated data. It appeared logically that the formulas which predict the best ripple characteristics yield the most accurate predictions of the total Shields parameter. Thereby, the Van Rijn, Soulsby & Whitehouse formulas as well as the suggested formula (Eqs. 9 and 10) yield the best results. Using α cr = 1 does influence the results as the Shields parameter is strongly underestimated when θ wg < θ cr whereas ripples were observed. In the same way, Eq. 19, which estimated the total Shields parameter directly from the skin Shields parameter, yields satisfactory results although it may underestimate results for θ wg < θ cr or θ wg > θ cr,sf . On the other hand, much better results are observed for the data set where ripples are washed out. In Fig. 10, it clearly appears that the underestimation is mainly due to the error on the estimation of the critical Shields parameters. Indeed, the formulas induce an underestimation of the results when θ wg < θ cr or θ wg > θ cr,sf , where θ cr and θ cr,sf are estimated values. Figure 10: Prediction of the Shields parameter using Eq. 17 with a r = 20 for the estimation of the roughness and the observed and estimated (Eqs. 9 and 10 with α cr = 1) characteristics of the ripples.

Conclusion

From this sensitivity analysis, it appears that large scatters exist in the prediction of ripple characteristics and so on the ripple-related roughness depending on the formulas applied. One important point to be considered is the critical Shields parameters for the inception of sediment movement θ cr and for the inception of sheet flow θ cr,sf , which should border the limits for the existence of ripples. Only the Grant & Madsen and Soulsby & Whitehouse formulas take into account θ cr and only the Nielsen and Van Rijn formulas include the sheet flow limit, even if both formulas seem to underestimate the wash-out of the ripples. Thus, even if the Soulsby & Whitehouse formula yields the best prediction for the ripple characteristics and induced Shields parameter, it does not take into account properly the inception of sheet flow and largely overestimates the results when ripples are washed out.

A new set of formulas (Eqs. 9 to 13) for ripple characteristics was proposed based on these two critical Shields parameters. The ripple steepness H r /L r and dimensionless ripple length L r /A w were found to reach an equilibrium value as soon as θ c < θ wg < θ cr,sf where θ cr,sf is estimated based on [START_REF] Camenen | Phase-lag effects in sheet flow transport[END_REF] study. For the ripple steepness H r /L r , this equilibrium value appeared to be nearly constant and equal to 0.16. For the dimensionless ripple length L r /A w , it has been found to be a decreasing function of χ and θ wg .

It should be noted that this set of formulas does not take into account relict ripples. As suggested by Soulsby & Whitehouse (2007?), it could be partly taken into account assuming ripples take pre-existing values of H r and L r when θ w < θ cr . However, the hysteresis effect due to relict ripples as observed by [START_REF] Traykovski | Geometry, migration, and evolution of wave orbital ripples at leo-15[END_REF] appears to be much more difficult to model, especially when there is a reorganisation from 2D to 3D wave ripples.

A discussion is also proposed to explain the differences observed for the case of large wave ripples (LWR). The effect of an additional current appeared not to be so significant for the prediction of ripple characteristics. However, it should strongly affect the induced roughness by smoothing the ripples and making the ripple system 3D.

The importance of the critical Shields parameter appeared to be even more significant when calculating the total Shields parameter. The roughness height for a rippled bed was also found to be as sensitive to the parameter a r (which varies from 8 to 37.5 depending on the authors) as to the ripple height and length. One future challenge would be to properly estimate the coefficient a r depending on the shape of the ripples, and so on the ratio |U c |/U w , using both experimental data and numerical tests.

All the uncertainties to estimate these parameters might explain the large scattering observed in many results for the estimation of the sediment transport. A simple empirical formula (Eq. 19) was then suggested to estimate the total Shields parameter in the ripple regime directly from the grain-related Shields parameter. The excess shear stress is proportional to a r and a decreasing function of the ratio θ wg /χ. The proposed formula yields satisfactory results compared to the classical methods based on ripple characteristics.

Using the data by [START_REF] Inman | Wave generated ripples in nearshore sands[END_REF] and [START_REF] Dingler | Wave formed ripples in nearshore sands[END_REF], [START_REF] Nielsen | Dynamics and geometry of wave generated ripples[END_REF] observed a washout of the ripples for a skin Shields number θ ws ≈ 0.8 -1. The maximum ripple steepness also appeared larger for irregular waves (max(H r /L r ) ≈ 0.35) than regular waves (max(H r /L r ) ≈ 0.20). Nielsen proposed an equation for the ripple steepness in case of irregular waves:

H r L r = 0.342 -0.34 θ wg 0.25 (22)
Grant & Madsen formula (1982) [START_REF] Grant | Movable bed roughness in unsteady oscillatory flow[END_REF] presented a set of empirical equations for the prediction of ripple characteristics. They distinguished a break-off point (critical Shields parameter) below/above which the ripple steepness is increasing/decreasing:

θ B = 1.8θ cr d * 1.5 4 0.6 (23)
They proposed the following relationships functions of the skin wave-related Shields parameter: Van Rijn formula (1993) Van Rijn (1993) proposed the following relationships for the wave ripples in case of irregular waves and as functions of the mobility parameter Ψ w :

H r A w =          0
H r A w = 0.22 for Ψ w ≤ 10 2.8 × 10 -13 (250 -Ψ w ) 5 for 10 < Ψ w ≤ 250 (26)

H r L r = 0.18 for Ψ w ≤ 10 2.0 × 10 -7 (250 -Ψ w ) 2.5 for 10 < Ψ w ≤ 250 (27)

Wikramanayake & Madsen formula (1991) [START_REF] Wikramanayake | Calculation of movable bed friction factors[END_REF] proposed a relationship using that the ratio of the mobility number Ψ w to the non dimensional sediment parameter S * :

S * = d 50 4ν (s -1)gd 50 (28)

For irregular waves, they chose U rms to represent the wave orbital velocity U w (Assuming a Rayleigh distribution, U rms = 1/ √ 2U 1/3 ). They found a good correlation with the field ripple data of [START_REF] Inman | Wave generated ripples in nearshore sands[END_REF], [START_REF] Dingler | Wave formed ripples in nearshore sands[END_REF] and [START_REF] Nielsen | Field measurements of time-averaged suspended sediment concentrations under waves[END_REF] using the following equations: It should be noted that the coefficients have been modified as [START_REF] Wikramanayake | Calculation of movable bed friction factors[END_REF] based their results on the root-mean-square value of the wave excursion. Moreover, when Ψ w /S * ≤ 4.2, the ripple height and wave length are only functions of the sediment characteristics.

Mogridge et al. formula (1994) [START_REF] Mogridge | Geometry prediction for wavegenerated bedforms[END_REF] conducted an extensive study on previously published experimental data in both laboratory and field. They found some complex empirical relationships based on the wave period parameter χ. The complexity of the formula makes it however difficult to use. A simplified version is as following:

L r d 50 = min 10 3.373-13.772χ 0.02054 , 1394

H r d 50 = 10 8.542-10.822χ 0.03967 (32)

On a real case, the wave period and median grain size do not vary significantly along a cross-shore beach profile whereas ripple characteristics do vary a lot (from well formed ripples offshore to a flat bed in the surf zone as ripples are washed-out).

Wiberg & Harris formula (1994) [START_REF] Wiberg | Ripple geometry in wave-dominated environments[END_REF] proposed iterative formulas for the prediction of the ripple characteristics which is also difficult to use. However, [START_REF] Malarkey | A non-iterative procedure for the wiberg and harris (1994) oscillatory sand ripple predictor[END_REF] proposed a simplified procedure to avoid the iterative calculation. The ripple characteristics read: with AH = A w /H r,ano where H r,ano is the anorbital ripple height calculated using the following equation for the ripple height with L r = 535d 50 :

A w H r = exp C 1 -C 2 -C 3 ln A w L r (34)
with C 1 = 7.59, C 2 = 33.60 and C 3 = 10.53.

Grasmeijer & Kleinhans formula (2004)

More recently, [START_REF] Grasmeijer | Observed and predicted bed forms and their effect on suspended sand concentrations[END_REF] proposed some equations based on the [START_REF] Nielsen | Dynamics and geometry of wave generated ripples[END_REF] study. They fitted an empirical relationship using laboratory data by Van Rijn et al. (1993[START_REF] Van Rijn | Transport of fine sands by currents and waves (II)[END_REF] and [START_REF] Grasmeijer | Transport of fine sands by currents and waves (iii)[END_REF] and field data by [START_REF] Inman | Wave generated ripples in nearshore sands[END_REF], [START_REF] Hanes | Wave-formed ripples at Duck, North Carolina[END_REF] as well as their own data set measured near the coast of Egmond aan Zee, The Netherlands:

An additional constraint is that ripples can only form and evolve if the wave-induced stress exceeds the threshold of motion of the sediment, i.e. θ wg > θ cr .

Figure 1 :

 1 Figure1: Estimation of the critical Shields parameter for the inception of waverelated sheet flow using Eq. 8.

Figure 2 :

 2 Figure 2: Wave-generated ripple wavelength normalised by the wave orbital semiexcursion as a function of the grain-related Shields parameter with the wave period parameter emphasised.

Figure 3 :

 3 Figure3: Wave-generated ripple steepness as a function of the wave orbital semiexcursion normalised by the median grain size with the wave period parameter emphasised (the solid line corresponds to Eq. 13).

Figure 4 :

 4 Figure 4: Ripple steepness H r /L r (a) and ratio L r /A w (b) versus the skin Shields parameter for several values of the critical Shields parameter for the inception of sheet flow θ cr,sf using Eqs. 9 and 10 with α cr = 1 and d 50 = 0.3 mm.

Figure 5 :

 5 Figure 5: Large wave ripple (LWR) wavelength (normalised by the wave orbital semi-excursion) and steepness as a function of the grain-related Shields parameter.

Figure 8 :

 8 Figure8: Wave related friction coefficient f w as a function of the ratio H r 2 /L r /A w (Eq. 16 and Eq. 18 are plotted, using Eq. 17 with a r = 10 and 20, and M cd = 5 and 10, respectively, together with experimental data from[START_REF] Jonsson | Experimental and theoretical investigations in a oscillatory turbulent boundary layer[END_REF][START_REF] Mathisen | Waves and currents over a fixed ripples bed. 1. bottom roughness experienced by waves in the presence and absence of currents[END_REF], and numerical data from Andersen, 2001[START_REF] Kim | Effective form roughness of ripples for waves[END_REF].

Figure 9 :

 9 Figure 9: Total Shields parameter versus the skin Shields for several median grain sizes (a) and wave periods (b) using the classical method based on the prediction of the ripple characteristics (1: black lines) or Eq. 19 (2: red lines with crosses).

  w /θ cr,sf ≤1.5 1.5<θ w /θ cr,sf

  (2005) proposed a new equation based on an extensive study andWiberg & Hanes (1994) observations. Following findings of O'Donoghue et al. (2005), for irregular waves, they chose U 1/10 , the mean of the highest onetenth velocities to represent the wave orbital velocity U w (Assuming a Rayleigh distribution, U 1/10 = 1.80U rms = 1.27U 1/3 ). In irregular waves, they assumed the peak-period T p gives the best representation of T w .

Table 1 :

 1 Data summary for experiments on ripple characteristics.

	Author(s)			Name	Location	Flow	nbr.	d50	h	Uc	Uw	Tw	Hr	Lr
						Type	exp.	[mm]	[m]	[m/s]	[m/s]	[s]	[cm]	[cm]
	Inman (1957)			Inm57	US Corps of Engineers, USA	field	41	0.1-0.9	0.5-2.0	-	0.06-0.57	5.0-13	0.5-19	6-105
	Lofquist (1978)			Lof78	US Corps of Engineers, USA	oscillatory	13	0.18-0.26	0.4	0	0.41-1.22	2.7-15	0-19	11-120
						flow tunnel
	Sleath (1982)			Sle02	Cambridge University, United	oscillatory	13	0.20, 0.41	0.15	0	0.16-0.44	2.9-5.1	2-5	12-32
					Kingdom	flow tunnel
	Bosman (1982) and Steet-	DLL80s	DHL, Delft, The Netherlands	wave flume	70	0.10	0.1-0.65	0.10-0.32	0.13-0.30	1.4-2.0	1-3	8
	zel (1985)
	Sato (1987)			Sat87	Tokyo University, Japan	oscillatory	66	0.18	0.21	0	0.11-0.65	0.5-7.0	0-3	0-18
						flow tunnel
	Ribberink & Al Salem	Rib90s	DHL, Delft, The Netherlands	large water	71	0.21	0.8	0	0.2-1.5	2.0-12.0	0-35	0-300
	(1994)					tunnel
	Van Rijn et al. (1993)	VR93	Delft Univ., The Netherlands	wave flume	45	0.1-0.22	0.48-0.52	-0.45-0.45	0.14-0.39	2.2-2.7	0.6-2.9	6-20
	Van Rijn & Havinga (1995)	VRH95	DHL, Delft, The Netherlands	wave basin	28	0.1	0.40-0.43	0-0.32	0.14-0.30	2.1-2.3	0.6-1.4	6-11
	Hume et al. (1999)		Hum99	NIWA, USA	field	9	0.4	25	-	0.28-1.06	11.0	3-13	40-90
	Khelifa & Ouellet (2000)	KO00	Laval, Canada	wave basin	48	0.15-0.5	0.3	0-0.34	0.08-0.26	0.9-1.4	0.4-1.7	3-12
	Doucette (2000, 2002)	Dou02	Various beaches, Australia	field	84	0.14-0.62	0.18-1.73	-	0.15-1.00	2.2-12.2	0-14	0-91
	Sedmoc data set (2001)	TUF80s	Grote Speurwerk (35m), DUT,	wave flume	125	0.10-0.22	0.29-0.60	0.07-0.45	0.17-0.55	1.2-2.7	0.2-2.9	0.6-20
					Delft, The Netherlands
	O'Donoghue	&	Clubb	oDC01	Aberdeen University, United	flow tunnel	35	0.18-0.44	0.75	0	0.25-0.94	2.0-15	0.9-19	6-121
	(2001)				Kingdom
	Hanes et al. (2001)		Han01	Duck, North Carolina, USA	field	201	0.12-1.7	1.4-7	-	0.12-1.62	3.1-16	0-13	0-270
	Sleath & Wallbridge (2002)	Sle02	Cambridge University, United	oscillatory	29	0.20-0.80	0.15	0	0.12-1.64	2.8-6.8	0-9	0-50
					Kingdom	flow tunnel
	Faraci & Foti (2002)		FF02	Catania University, Italy	wave flume	38	0.25	0.29	-	0.12-0.35	1.2-4.2	0.7-2.1	4-11
	Delgado Blanco et al.	Del04	Deltaflume, Delft, The Nether-	large wave	17	0.18-0.44	7.0	0	0.14-0.74	6	0.6-5	10-35
	(2004)				lands	flume
	Van der Werf & Ribberink	vdWR04 DHL, Delft, The Netherlands	large water	10	0.35	0.8	0	0.42-0.85	5.0-10.0	2.5-13.9	44-113
	(2004)					tunnel
	Grasmeijer & Kleinhans	GK04	Egmond aan Zee, The Nether-	field	45	0.24	1.5-5.0	-0.40-0.10	0.23-0.99	4.0-10.5	0.7-10	19-200
	(2004)				lands
	Williams et al. (2004)	Wil04	Deltaflume, Delft, The Nether-	large wave	65	0.16-0.35	4.0-4.5	0	0.13-1.03	4.0-6.0	0.3-7	20-104
					lands	flume
	Catano-Lopera & Garcia	Cat06	University of Illinois, USA	wave flume	23	0.25	0.36-0.76	0-0.6	0.25-0.42	2.0-6.9	1.5-3.5	9.1-18.8
	(2006a, b)

Table 2 :

 2 Prediction of the ripple characteristics using various empirical equations and the collected experimental data with purely oscillatory flows. Whitehouse formula yield the best results. For the prediction of the ripple height (which is actually dependent on the prediction of the ripple length), the Van Rijn, Wiberg & Harris, and Soulsby & Whitehouse formulas yield the best results. Apart from the Nielsen and Wikramanayake & Madsen formulas, all formulas tend to generally overestimate measured ripple height and length. The Wiberg & Harris and Mogridge et al. formulas yield smaller dispersion

			ripple length L r			ripple height H r
	Equation	P 2	P 5	µ f	σ f	P 2	P 5	µ f	σ f
	Nielsen Grant & Madsen	35% 61% -0.11 0.84 35% 60% -0.34 0.79 59% 76% +0.36 0.99 44% 71% +0.17 0.85
	Wikramanayake & Madsen Van Rijn	45% 75% +0.04 0.81 37% 67% -0.09 0.74 61% 80% +0.10 0.84 56% 78% +0.02 0.79
	Mogridge et al.	62% 82% +0.36 0.78 45% 71% +0.63 0.82
	Wiberg & Harris	54% 82% +0.09 0.76 55% 82% +0.15 0.66
	Grasmeijer & Kleinhans	58% 81% +0.26 0.89 39% 74% +0.02 0.74
	Soulsby & Whitehouse	66% 79% +0.19 0.80 65% 84% +0.06 0.63
	Eqs. 9 and 10 with α cr = 0.01 66% 81% +0.17 0.76 68% 88% -0.05 0.56 Eqs. 9 and 10 with α cr = 1 64% 80% +0.02 0.83 65% 84% -0.14 0.64
	For the prediction of the ripple length, it appears that the Grant & Madsen,
	Mogridge et al., Van Rijn, Wiberg & Harris, Grasmeijer & Kleinhans formulas and
	more especially the Soulsby &

Table 3 :

 3 Prediction of the roughness height when the ripples are washed out and the sheet flow regime reached.

	Equation	P 2	P 5	µ f	σ f
	Nielsen	87% 97% +0.10 0.20
	Grant & Madsen	20% 41% +0.80 0.47
	Wikramanayake & Madsen 13% 31% +1.13 0.62
	Van Rijn	41% 57% +0.56 0.52
	Mogridge et al.	1% 18% +1.10 0.43
	Wiberg & Harris	0%	0% +1.68 0.39
	Grasmeijer & Kleinhans	24% 78% +0.51 0.26
	Soulsby & Whitehouse	0%	0% +1.71 0.33
	Eqs. 9 and 10, α cr = 0.01	4%	4% +1.57 0.40
	Eqs. 9 and 10, α cr = 1	47% 55% +0.63 0.65

Table 5 :

 5 Prediction of the Shields parameter from the estimation of the ripple characteristics using various empirical equations and the collected experimental data.

		ripple data		data for θ w = θ cr,sf
	Equation	P 1.2 P 2	µ f	σ f	P 1.2 P 2	µ f	σ f
	Nielsen Grant & Madsen Wikramanayake & Madsen 15% 52% -0.14 0.40 10% 38% +0.58 0.38 19% 48% -0.24 0.42 85% 96% +0.04 0.10 16% 53% -0.02 0.45 17% 46% +0.36 0.26 Van Rijn 29% 64% +0.005 0.40 39% 59% +0.27 0.28
	Mogridge et al.	13% 43% +0.51 0.48	0% 14% +0.47 0.16
	Wiberg & Harris	18% 62% +0.09 0.35	0%	0% +0.83 0.14
	Grasmeijer & Kleinhans Soulsby & Whitehouse	10% 50% -0.15 0.39 15% 82% +0.20 0.12 25% 73% +0.02 0.33 0% 0% +0.85 0.11
	Eqs. 9 and 10, α cr = 0.01 Eqs. 9 and 10, α cr = 1 Eq. 19	4% 34% 74% -0.11 0.33 46% 58% +0.29 0.31 5% +0.79 0.22 36% 78% -0.05 0.30 32% 71% -0.12 0.36 30% 45% +0.41 0.36

Author-produced version of the article published in Estuarine Coastal and ShelfScience, vol. 84, n° 4, p. 553-564, doi : 10.1016Science, vol. 84, n° 4, p. 553-564, doi : 10. /j.ecss.2009.07.022 .07.022

Acknowledgment

This work was conducted under the Inlet Modeling System Work Unit of the Coastal Inlets Research Program, U.S. Army Corps of Engineers, and partly sponsored by the Japanese Society for the Promotion of Science. I also would like to thanks R.L. Soulsby for for his very helpful comments, suggestions, and corrections.

Appendix: Predictive equations for ripples characteristics

Nielsen formula (1981) [START_REF] Nielsen | Dynamics and geometry of wave generated ripples[END_REF] proposed the following equation for the ripple wave length based on the field data of [START_REF] Inman | Wave generated ripples in nearshore sands[END_REF], [START_REF] Dingler | Wave formed ripples in nearshore sands[END_REF][START_REF] Miller | A field investigation of the relationship between ripple spacing and near bottom water motions[END_REF][START_REF] Miller | A field investigation of the relationship between ripple spacing and near bottom water motions[END_REF]:


Researcher, Cemagref A Recking
email: alain.recking@cemagref.fr

phD Student, Cemagref V Bacchi
email: vito.bacchi@cemagref.fr

Researcher, Cemagref M Naaim

Researcher, Cemagref P Frey

Antidunes on steep slopes

When increasing the rates of subcritical flow on gentle slopes, the bed successively produces ripples, dunes and flat beds. Antidunes (defined here as all bed undulations for which the surface gravity waves are in phase with the bed profile) appear only in high flow rates and may be found in some extreme natural flow events. Inversely, on steep slopes flume experiment (S≥1% approximately) flows are supercritical and antidunes were observed to appear just after the beginning of sediment motion and to disappear for high flow rates. With new experiments, this study aimed to improve the prediction of antidune geometry on steep slopes. An equation for antidune wavelength was deduced from dimensional analysis and fitted to new experimental data. The equation was successfully evaluated using a data set extended to 167 values with data from the literature, obtained on both steep and gentle slopes. This equation gave results very similar to the usual analytical equation from Kennedy when tested on the results from gentle slope experiments, but proved to be better adapted for antidune wavelength on steep slopes.

INTRODUCTION

Flows over natural sediments generally develop bedforms consisting of ripples, dunes and antidunes [START_REF] Engelund | A monograph on sediment transport in alluvial streams[END_REF][START_REF] Gilbert | The Transportation of Debris by Running Water, Professional paper N°86[END_REF][START_REF] Van Rijn | Sediment transport, Part III : Bed forms and alluvial roughness[END_REF]. Ripples and dunes are downstream migrating bedforms that are triangular in shape with a gently sloping upstream face and a downstream face the slope of which nearly equal the sediment's angle of repose. Antidunes have an approximate sinusoidal shape that is usually characterized by a wave length L and an amplitude A (Figure 1). They were first called antidunes by [START_REF] Gilbert | The Transportation of Debris by Running Water, Professional paper N°86[END_REF] because they contrast with dunes in their direction of movement. [START_REF] Kennedy | Stationary waves and antidunes in alluvial channels[END_REF] also observed downstream migrating antidunes and extended this term to all forms for which the surface gravity waves are in phase with the bed profile. This definition (including nonmoving sinusoidal shapes) will be considered here. Occurrence of dunes and antidunes depends on the sediment and flow properties. It is generally admitted that ripples are associated with smooth turbulent subcritical flows and dunes are associated with rough turbulent subcritical flows, whereas antidunes are associated with supercritical flows. A better comprehension of the mechanisms controlling bedforms is very important because they are associated with the complex behaviour of both the flow resistance and bedload transport equations. More particularly, a better prediction of antidune geometry can help to analyse local scouring associated with hydraulic structures [START_REF] Comiti | Dimensions of standing waves at steps in moutain rivers[END_REF] and to interpret paleo-floods [START_REF] Shaw | Paleohydraulic interpretation of antidune bedforms with applications to antidunes in gravel[END_REF]. Antidune models also represent one (of several) possible mechanisms which may explain the formation of step-pool morphologies in gravel-bed streams [START_REF] Chin | On the origin of step-pool sequences in mountain streams[END_REF][START_REF] Curran | Step-pool formation models and associated step spacing[END_REF][START_REF] Lenzi | Step-pool evolution in the rio Cordon, northeastern Italy[END_REF][START_REF] Weichert | Bed morphology and generation of step-pool channels[END_REF][START_REF] Whittaker | Origin of step-pool systems in mountain streams[END_REF].

Several theoretical developments on flow domains pertaining to antidunes were proposed during the last decades [START_REF] Carling | The antidune transition in fine gravel with especial consideration of downstream migrating antidunes[END_REF][START_REF] Colombini | Revisiting the linear theory of sand dune formation[END_REF][START_REF] Colombini | Coupling and decoupling bed and flow dynamics: fast and slow sediment waves at high Froude numbers[END_REF]2008;[START_REF] Deigaard | Breaking antidunes: cyclic behavior due to hysteresis[END_REF][START_REF] Engelund | Instability of erodible beds[END_REF][START_REF] Hayashi | Formation of dunes and antidunes in open channels[END_REF][START_REF] Hayashi | Dominant wave numbers of ripples, dunes and antidunes on alluvial river beds[END_REF][START_REF] Huang | The formation of dunes, antidunes and rapidly damping waves in alluvial channels[END_REF][START_REF] Kennedy | Stationary waves and antidunes in alluvial channels[END_REF][START_REF] Kubo | Theoretical study on breaking of waves on antidunes[END_REF][START_REF] Parker | Sediment inertia as cause of river antidunes[END_REF][START_REF] Raudkivi | Bedforms in alluvial channels[END_REF][START_REF] Reynolds | Waves on the erodible bed of an open channel[END_REF][START_REF] Sammarco | Nonlinear resonance of free surface waves in a current over a sinusoidal bottom: a numerical study[END_REF]. All these developments considered analytical solutions for the equations governing the mean flow velocity and sediment bedload transport rate. However, the observations and equations were essentially based on gentle slope experiments. On gentle slopes with fine sands, the bed successively produces ripples, dunes and the flat beds close to transition when increasing the subcritical flow rates. Antidunes appear only for very high flow rates (in the upper regime as defined by [START_REF] Simons | Resistance to flow in alluvial channels[END_REF]) and may characterise extreme natural events. On steep slopes (S>1% approximately) antidunes were also observed for intense sediment transport [START_REF] Foley | Scour and fill in an ephemeral stream[END_REF][START_REF] Smart | Sediment transport on steep slopes[END_REF] but in these (near) supercritical flows, antidunes (as defined above, i.e., a free surface and upper bed layer in phase) were also observed for relatively low flow conditions both in the field (Figure 2 shows antidunes observed on the Arveyron river in june 2009 whereas the flow discharge was estimated to be half the 2 years RI discharge Q 2 ) and in the flume [Bathurst, et al., 1982a;[START_REF] Cao | Resistance hydraulique d'un lit à gravier mobile à pente raide; étude expérimentale[END_REF][START_REF] Comiti | Dimensions of standing waves at steps in moutain rivers[END_REF]Recking, et al., 2008a;[START_REF] Shaw | Paleohydraulic interpretation of antidune bedforms with applications to antidunes in gravel[END_REF]. Figure 3 presents the images of antidunes obtained with a uniform gravel measuring 4.9 mm in diameter on a 5% slope as part of the steep slope experiments (143 runs) presented in Recking et al [2008a]. This figure shows that for this experiment, antidunes appear very close to the incipient motion flow condition (characterized here by the ratio between the Shields number θ and its critical value for incipient motion θ c ) and that wavelength increased when bed load increased, resulting in a flatter bed as the Shields number increased until the bed become perfectly flat. The shields number is the dimensionless shear stress defined by:

) 1 ( ) ( ) ( - = - = - = s S D R D g S gR D g b s b s ρ ρ ρ ρ ρ τ θ (1)
where τ is the bed shear stress, R b is the bed hydraulic radius, D is the grain diameter, ρ s is the sediment density, ρ is the water density and s=ρ s /ρ is the relative density. The free surface and bed deformation are maximum for approximately θ/θ c = 1.5 (suggesting that on steep slopes, antidunes could arise in many natural floods, at least on the rising and falling limbs of the hydrograph), whereas in [START_REF] Simons | Resistance to flow in alluvial channels[END_REF]'s experiments antidunes were observed for θ/θ c values in the range. In both steep and gentle slope experiments, antidunes were obtained in supercritical flows, but for very different transport rates and flow relative depth. This paper aims to verify wether these changing flow conditions affect the antidune geometry by comparing antidunes produced on gentle and steep slopes. More particularly, we will investigate whether wavelength equations proposed for gentle slope flows remain valid on steep slopes, where the flow depth h is on the order of magnitude of the sediment grain size D.

New flume experiments, data from the literature and a semi-empirical investigation of antidune wavelength will provide the analysis. First, a set of dimensionless parameters are deduced from a dimensional analysis. Second, these parameters are used with new steep slope flume experiments (19 runs) to fit a relation giving the antidune wavelength. Third, the model is tested with data from the literature (148 runs) and is compared to existing models. To finish, the results are discussed.

MODELLING

Dimensional analyses

To identify the relevant scaling parameters, we used the dimensional analysis method, which assumes minimal a priori knowledge of antidune dynamics. For the sake of brevity and clarity, we restricted the analysis to two-dimensional bedforms (also called long-crested antidunes).

The antidune wavelength L can be expressed as a function of nine variables, as represented symbolically by:

) , , , , , , , , ( 1 D g S U R A f L s b ρ ρ ν = (2)
where A is the antidune amplitude, R b is the bed's hydraulic radius (obtained after side wall correction of the measured hydraulic radius R=hW/[2h+W], where h is the flow depth and W is the flow width), U is the mean flow velocity, S is the energy slope, ν is the water viscosity, ρ is the water density, ρ s is the sediment density, g is the acceleration of gravity and D is the grain diameter. Using the Buckinghams's pi theorem, ten variables involving three basic dimensions would give seven dimensionless pi terms and Eq. 2 could be reduced to:

) , , *, Re Re, , ( 2 s S D R F f D L b = (3)
Where F=U/(gh) 0.5 is the Froude number, Re=Uh/ν is the Reynolds number, Re*=u*D/ν is the grain Reynolds number (where u*=(gRS) 0.5 =(τ/ρ) 0.5 is the shear velocity), R b /D is the relative depth and s=ρ s /ρ is the relative sediment density. Since flows on steep slopes are rough (Re*>70) and turbulent (Re>2000), Re and Re* can be neglected (both for flow resistance and bedload transport).

In Figure 3 antidunes appear with sediment transport and wavelength increases as flow conditions increase (characterized by the θ/θ c ratio), with amplitude remaining almost constant (on the order of magnitude of the grain diameter) until the antidunes completely disappears when θ=2.5θ c approximately. For this reason, we suggest looking for a relation linking the antidune geometry to the Shields ratio θ/θ c . The relative depth R b /D, the relative density s and the slope S can be rearranged to form the Shields number (Eq. 1). The method used in this study is original in that instead of using a constant critical Shields stress θ c we propose using a variation of this parameter with slope in reference to [START_REF] Lamb | Is the critical Shields stress for incipient sediment motion dependent on channel-bed slope?[END_REF] and [START_REF] Recking | Theoretical development on the effects of changing flow hydraulics on incipient bedload motion[END_REF] (this choice will be considered further in the discussion). The following slope-dependent equation proposed in Recking et al [2008b] is used:

275 . 0 15 . 0 S c = θ (4)
Finally, considering a natural sediment mixture (ρ s /ρ constant and equal to 2.65), Eq. 3 can be written:

) , , ( 3 F f D L c θ θ = (5)

Experiments

New experiments were conducted at Cemagref to investigate Eq. 5. The experimental setup was a 6-m-long flume. The flume width was varied between 0.05 and 0.102m. We used three uniform sediments -2.3 mm, 9 mm and 23 mm in diameter -and we varied the slope from 3 to 12.6%. The flow rate at the inlet was ensured by a constant head reservoir and measured by an electromagnetic flowmeter. Before each experiment (for a given material and a given slope), we systematically measured the bed infiltration rate (part of the flow discharge flowing within the gravel bed), which was deduced from the total inlet flow discharge. The sediment feeding system consisted of a customized conveyor belt device ensuring constant feeding (see Recking et al [2008a] for further detail). For each run, measurements were made only after attaining a dynamic equilibrium condition, i.e. near-equality of the feed and trap sediment transport rates, and a near constant time-averaged bed slope. Train antidunes were usually instable and observed over finite distances (generally not affecting the entire flume length). The wavelength L was measured by dividing this distance by the number of antidunes. Antidune height was difficult to measure precisely. The flow velocity and the flow depth were deduced from the flow discharge Q using the following friction equation for rough turbulent flows, based on the analysis of 2282 flume and field flow resistance values [Recking, et al., 2008b]:

        + = D R S gR U BR RL b b α α log 75 . 5 25 . 6 (6)
where U is the vertically averaged flow velocity and

43 . 0 4 -       = D R b RL α with 1 ≤ α RL ≤ 3.5 (7) D R S b BR 85 . 0 7 = α with 1 < α BR ≤ 2.6 (8)
where α RL is a roughness layer coefficient taking into account deviation from the logarithmic profile on small relative depth flows (with an increasing influence of the roughness layer) and α BR is a bedload roughness coefficient taking into account additional flow resistance caused by bedload. This flow resistance equation was derived for flat beds but proved to be valid for flows with nonbreaking wave antidunes. The corresponding wavelength and flow conditions are summarized in Table 1. Φ is the dimensionless solid discharge [START_REF] Einstein | The bed-load function for sediment transportation in open channel flows[END_REF] defined by:

( ) 3 1 gD s q sv - = Φ (3)
Where q sv [m 3 /s/m] is the volumetric transport rate per unit width.

Model fitting

A functional relationship between L/D, θ, θ c and F is sought. The model will be investigated in its simplest form, through a power equation:

γ β α θ ξθ F D L c = (9)
where ξ, α, β and γ are constant values to be fitted from our experimental results. The best fit of Eq. 9 (R²=0.99) gave the following equation (Figure 4):

F D L c 3 093 . 0 θ θ = (10)
These coefficients allowed us to minimize to zero the mean error ε (where ε is the difference between the measured and calculated values of L) with a standard deviation σ ε = 0.02.

When rearranged with Eq. 1 and Eq.4, it was possible to eliminate D from both sides, and considering a natural sediment of relative density s=2.65, this equation could be reduced to:

F S R L b 17 . 0 16 = (11)
Similar equations could have been investigated for the antidune amplitude A, but unfortunately, we did not produce enough data to investigate this parameter.

MODEL VALIDATION AND COMPARISON

In this section, we test the models' ability to reproduce measured wavelengths L with available data from the literature.

The data set

We built a data set composed of 148 values from [START_REF] Kennedy | Stationary waves and antidunes in alluvial channels[END_REF], [START_REF] Simons | Resistance to flow in alluvial channels[END_REF], [START_REF] Shaw | Paleohydraulic interpretation of antidune bedforms with applications to antidunes in gravel[END_REF], [START_REF] Cao | Resistance hydraulique d'un lit à gravier mobile à pente raide; étude expérimentale[END_REF] and [START_REF] Recking | An Experimental Study of Grain Sorting Effects on Bedload[END_REF], obtained in flume experiments with near uniform sediments. The data from [START_REF] Recking | An Experimental Study of Grain Sorting Effects on Bedload[END_REF] were produced in a 10-m-long, 0.1-m-wide flume in the Lyon LMFA laboratory (Laboratoire de Mécanique des Fluides et d'Acoustique) and must be distinguished from the new data produced at Cemagref.

For these runs, the flow velocity was not computed but measured with a marker technique similar to the salt dilution technique (see [START_REF] Recking | An Experimental Study of Grain Sorting Effects on Bedload[END_REF] for further detail). Only a few field values are available from [START_REF] Kennedy | Stationary waves and antidunes in alluvial channels[END_REF]. However, the slope values were missing and,

given the fine grain diameter (approximately 0.3 mm), a 0.001 slope was used for the calculations. Actually the exact slope value is not very important because with a slope exponent equal to 0.17, the model distinguishes between gentle and steep slopes. For all runs, R b was calculated with the measured flow depths, after correction for side wall effects using the procedure proposed by [START_REF] Johnson | The importance of side-wall friction in bed-load investigations[END_REF] and modified by [START_REF] Vanoni | Laboratory studies of the roughness and suspended load of alluvial streams[END_REF]. The data set is presented in Table 1 and indicates that antidunes on steep slopes were obtained in the 1.1 < θ/θ c < 2.5 range, whereas on gentle slopes antidunes were obtained in the 10 < θ/θ c < 90 range.

The data set was compared to the commonly admitted domain permitting antidunes. These limits were derived analytically and give the minimum Froude number F m [START_REF] Kennedy | Stationary waves and antidunes in alluvial channels[END_REF] and the maximum Froude number F M [START_REF] Parker | Sediment inertia as cause of river antidunes[END_REF][START_REF] Reynolds | Waves on the erodible bed of an open channel[END_REF] for the formation of antidunes for a given wave number k=2π/L and flow depth h:

kh kh F m tanh ² = (12) kh kh F M tanh 1 ² = (13)
These equations were compared to the data set in Figure 5 and gave a good estimate of the flow domain permitting antidunes, whatever the slope. Note that antidunes can also appear for a Froude number as low as 0.8.

Model validation

Figure 6 presents a comparison between the measured wavelengths and the wavelengths computed with Eq. 11. The widely used wavelength equation proposed by [START_REF] Kennedy | Stationary waves and antidunes in alluvial channels[END_REF] was also used for the comparison (Figure 7):

kh F 1 ² = (14)
Models' efficincy was tested by calculating a relative root mean square error (RRMSE) defined by:

mes L RRMSE 2 ² ε σ ε + = ( 15
)
where ε is the error calculated from the difference between the measured and calculated values of L and σ ε is the standard deviation of ε. The results are presented in Table 3. The new model improves antidune wavelength prediction on steep slopes, but also on gentle slopes. Figure 8 presents the calculated to measured ratios for each model and for increasing slopes. As in a few cases the calculation error associated with Eq. 14 was very large, it was necessary to remove these outliers in order to calculate average values that were truly representative of the data set. The new model gives results to within ±10% for all slope values. The bad score on the 3% slope corresponds to six data from [START_REF] Shaw | Paleohydraulic interpretation of antidune bedforms with applications to antidunes in gravel[END_REF] (antidunes were obtained during a bed erosion experiment with no sediment feeding).

The model from Kennedy (Eqs. 14) gave similar results with an overestimation on gentle slopes and a progressive decrease of the prediction ratio with increasing slope.

Because in field applications the available parameters may not be the flow depth h but the flow discharge Q, Eq. 11 was also tested by replacing measured R b and F with values calculated from the flow discharge Q, using Eq. 6 (the algorithm presented in [START_REF] Recking | An Experimental Study of Grain Sorting Effects on Bedload[END_REF] takes into account a flume side wall effect). Wavelength prediction was unchanged for steep slope experiments (RRMSE =0.25). The results were not as good for gentle slope experiments [START_REF] Kennedy | Stationary waves and antidunes in alluvial channels[END_REF] and [START_REF] Simons | Resistance to flow in alluvial channels[END_REF]) because they were associated with smooth and transitional flows (Re*<70) for which Eq. 6 is no longer appropriate.

DISCUSSION

Physical significance

Kennedy's equation (Eq.14) was based on the celerity equation for small-amplitude waves [START_REF] Milne-Thomson | Theretical hydrodynamics[END_REF] hypothesizing that flows can deform the boundary to conform to a streamline of a wave in a flow of infinite depth. It considers that for a given Froude number F, only one combination is possible between the flow depth h and the bed deformation L. Doing so, it implicitly considers that processes controlling the flow resistance and bed load transport are unchanged whatever the slope. However recent studies have demonstrated that both the mean flow resistance and bedload transport were strongly affected by changing flow hydraulics with changing slope:

(i) First, all the available flume and field data confirmed an increasing critical Shields stress with increasing slope when plotted together [START_REF] Recking | Theoretical development on the effects of changing flow hydraulics on incipient bedload motion[END_REF]. This result was first formulated by Shields himself [START_REF] Shields | Anwendung der Aehnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung[END_REF], pp. 16-17) and was confirmed by several authors after him [START_REF] Aksoy | The influence of relative depth on threshoàld of grain motion[END_REF][START_REF] Armanini | Incipient sediment motion at high slopes in uniform flow condition[END_REF][START_REF] Bathurst | Critical conditions for bed material movement in steep, boulder-bed streams, paper presented at Erosion and Sedimentation in the Pacific Rim[END_REF][START_REF] Bathurst | Initiation of sediment transport in steep channels with coarse bed material[END_REF][START_REF] Bettess | Initiation of sediment transport in gravel streams[END_REF][START_REF] Bogardi | Sediment transportation in alluvial streams (International Post-Graduate Course on Hydrological methods for developping water resources management[END_REF][START_REF] Graf | Sediment transport in steep channels[END_REF][START_REF] Mueller | Variation in the reference Shields stress for bed load transport in gravel-bed streams and rivers[END_REF][START_REF] Shvidchenko | Flume study of the effect of relative depth on the incipient motion of coarse uniform sediments[END_REF][START_REF] Shvidchenko | Critical shear stress for incipient motion of sand/gravel streambeds[END_REF][START_REF] Tabata | An Experimental Study on Critical Tractive Force of Cobble Gravels[END_REF][START_REF] Tsujimoto | Bed-load transport in steep channels, Fluvial Hydraulics of Mountain Regions[END_REF][START_REF] Vollmer | Predicting incipient motion, including the effect of turbulence pressure fluctuations in the bed[END_REF]. This variation with slope is not as would be expected when studying the gravitational effects acting on the grain alone and several explanations were given for this. Shields considered it was the result of a change in crosssection when the relative depth decreased, whereas others believed it was the consequence of the drag shear stress calculation in presence of form drag at low relative depth [START_REF] Buffington | A systematic analysis of eight decades of incipient motion studies, with special reference to gravel-bedded rivers[END_REF]. However this finding was also confirmed for flows over fine gravels and without bedforms [START_REF] Recking | Variation du nombre de Shields critique avec la pente[END_REF] which argues for another explanation. Recent analyses considered that gravitational effects acting on the grains are balanced by near bed flow velocity and turbulence changes with changing slope [START_REF] Lamb | Is the critical Shields stress for incipient sediment motion dependent on channel-bed slope?[END_REF][START_REF] Recking | Theoretical development on the effects of changing flow hydraulics on incipient bedload motion[END_REF].

Even though the origin of this variation could appear controversial, the critical Shield stress increase with increasing slope is considered here, with several consequences observed not only on bedload transport, but also on flow resistance, as recalled hereafter.

(ii)

The resulting critical Shields stress function θ c (S) improved bedload prediction quite significantly in Recking et al [2008b] when the calculations were based on the concept of excess shear stress (i.e. as Φ=f[θ-θ c (S)]).

(iii) We also found that bedload strongly impacts flow resistance [Recking, et al., 2008a].

With the findings reviewed above, this explains why a slope-dependent roughness parameter k s (S) proved to greatly improve mean flow velocity prediction when used in the logarithmic flow resistance equations for U/u* (Eqs. 6 and 8).

These studies produced slope-dependent θ c (S), Φ(S) and U/u*(S) functions as a consequence of the deviation from the law of the wall on steep slopes and because of flow resistance and bedload interactions. Thus, by impacting all parameters controlling antidunes when the slope is increased, these effects explain why antidune prediction was improved by incorporating the slope in the wavelength equation, through θ c (S). In addition, the good score obtained with Eq.

11 for any slope value may result from the fitting of the θ c (S) function (Eq.4) on the full range of slopes (from 0.001 to 0.1) in Recking [2008b].

Because it is not the gravitational effects associated with the changing slope that are responsible for the observed changes in antidune geometry but the changing flow hydraulics associated with changing slope, Eq. 11 can be rewritten as a function of the friction coefficient considering S= (U/u*) -²F² (where U/u*=(8/f) 0.5 is the Darcy Weisbach coefficient):

35 . 0 35 . 1 *) / ( 16 - = u U F R L b (16)
This equation provides information on the wavelengths in the F-kh plane for a given value of U/u* (Figure 9). A complete physically based investigation would consist in deriving new theoretical solutions incorporating these findings, but this was not the purpose of this paper.

Antidune amplitude

In Figure 3 antidunes appear with sediment transport and the bed is flattened when the flow condition increases. Table 4 presents the corresponding flow conditions and associated theoretical wavelengths as predicted with Eq.11. It indicates that when θ=2.5θ c the wavelength should be 26 cm. The flume length captured by images was approximately 20 cm. This suggests that for such high flow conditions antidunes were still presents, but because of the low amplitude (approximately one grain diameter height) and because the short observation window, they could not be observed. Instead, the bed appears flat.

Given that antidunes were produced with a very wide range of flume widths (0.05-3.2m), another question concerns the effect of flume geometry on antidune geometry (characterized by an amplitude:wavelength ratio). Given that antidune amplitude data are rare, we also used newly produced data to analyse the flume width effect. Figure 10 [START_REF] Simons | Resistance to flow in alluvial channels[END_REF] are also coherent with this result despite a wider scatter. This suggests that the flume width had no or little effect on antidune geometry. These aspects should be evaluated through new experiments in a future investigation.

Field implications

The relevance of this research to field problems deserves discussion. Figure 2 presents antidunes observed on a 3% slope gravel bed river (the Arveyon River in Chamonix, France).

Successive trains of antidunes with a wavelength of approximately 3m were regularly produced at the same place and migrated upstream very quickly over a distance of approximately 50 m, before disappearing. However, this observation is rare under field conditions. First, it appears to be rare to find supercritical flow conditions for long time periods and long reaches (e.g. [START_REF] Grant | Critical flow constrains flow hydraulics in mobile-bed streams: A new hypothesis[END_REF]; it may be more typical to observe longitudinal alternation of supercritical and subcritical flows [START_REF] Comiti | Dimensions of standing waves at steps in moutain rivers[END_REF]. Moreover, if on some occasions antidunes can be observed after flooding [START_REF] Foley | Scour and fill in an ephemeral stream[END_REF], antidunes are usually destroyed on falling water stages [START_REF] Carling | Subaqueous gravel dunes[END_REF], which renders field observations very rare and comparisons difficult. The model fitted the few values presented by [START_REF] Kennedy | Stationary waves and antidunes in alluvial channels[END_REF] in Figure 6 fairly well.

Another important aspect concerns the effect of grain size distribution given that all the experiments presented in this paper were conducted with uniform or near uniform sediments, whereas natural sediments are usually poorly sorted. [START_REF] Smart | Sediment transport on steep slopes[END_REF] also obtained flat beds with their uniform sediments at high flow intensities (as was illustrated in Figure 3), but they obtained antidunes with nonuniform bed material in these flows. They did not provide wavelength values but they described "weak" antidunes. We suspect that such weak antidunes did also exist in our experiments (see discussion above), but could not be observed.

We did not succeed in isolating the hydraulic specificities that could explain any differences using [START_REF] Smart | Sediment transport on steep slopes[END_REF]'s data. However, it was demonstrated that in presence of grain sorting, hydraulics alone could not reproduce all phenomena associated with sediment transport [START_REF] Dietrich | Sediment supply and the development of the coarse surface layer in gravel-bedded rivers[END_REF][START_REF] Iseya | Pulsations in bedload transport rates induced by a longitudinal sediment sorting: a flume study using sand and gravel mixture[END_REF][START_REF] Recking | An Experimental Study of Grain Sorting Effects on Bedload[END_REF] and additional sediment mixture properties are likely needed to fully understand antidunes in poorly sorted sediments.

CONCLUSION

This study intended to investigate antidune characteristics on steep slopes and to compare the results with available results previously obtained on gentle slopes. Using a data set comprising 167 values (19 newly produced data and 148 data from the literature) it was shown that Kennedy's theoretical model for the dominant wavelength (Eq. 14) provided a good estimation of measured wavelength, but with decreased efficiency (under prediction) as slope increased.

A wavelength model based on dimensional analysis and new steep slope experiments was proposed. This model proved to reproduce adequately steep slope data from the literature, but also, to improve wavelength prediction on gentle slopes when compared to Kennedy's models. This model incorporates parameters that are similar to previous ones, i.e. the Froude number and the flow depth, but it also takes into account the changes in flow hydraulics through a slope parameter as was demonstrated in [START_REF] Recking | Theoretical development on the effects of changing flow hydraulics on incipient bedload motion[END_REF]. The available data did not permit to make any definitive conclusion on antidunes' amplitude.

Additional steep slope data are needed to confirm these results. The grain sorting effects on antidune geometry should also be investigated further because all the available data were obtained with uniform sediments only. Only long-crested antidunes were considered. Threedimensional antidunes are expected to have shorter wavelengths than those reported in this paper [START_REF] Kennedy | The mechanics of dunes and antidunes in erodible-bed channels[END_REF].

FIGURES CAPTION

Figure 1: Schematic presentation of antidunes characterized by a sediment wave migration in the upstream direction and a bed surface in phase with the form of the water surface.

Figure 2 : Antidunes on the Arveyron river, a 3% slope gravel bed river in Chamonix (France). The wavelength was approximately 2m.

Figure 3: Images of antidunes for increasing flow conditions characterized by the ratio θ/θ c .

Each image presents the free surface and the moving grains.

  presents antidune amplitude as a function of the wavelength. The first observation is that antidune geometries measured in our 0.1-m-wide flume are very coherent with the antidune geometries measured by Cao in a 0.6-m-wide flume. These steep slope antidunes fit the following equation fairly well (R²=0.97): on gentle slopes in a 2.5-m-wide flume experiment by Simons

Figure 4 :Figure 5 :

 45 Figure 4: Wavelength model fitting

Figure 7 :

 7 Figure 7: Comparison between computed (with Eq. 14) and measured wavelengths (51% of the values are within the envelope ±20%)

Figure 8 :

 8 Figure 8: Calculated-to-measured wavelength ratio for each model and different slopes (each

Figure 9 :

 9 Figure 9 : Wavelengths in the F-kh plane with consideration of U/u* values Figure 10: Amplitude versus wavelength

Figure 3

 3 Figure1 : Schematic presentation of antidunes characterized by a sediment wave migration in the upstream direction and a bed surface in phase with the form of the water surface.Figure2: Antidunes on the Arveyron river, a 3% slope gravel bed river in Chamonix (France). The wavelength was approximately 2m. Figure3: Images of antidunes for increasing flow conditions characterized by the ratio θ/θ c .Each image presents the free surface and the moving grains.

Figure 2 :

 2 Figure1 : Schematic presentation of antidunes characterized by a sediment wave migration in the upstream direction and a bed surface in phase with the form of the water surface.Figure2: Antidunes on the Arveyron river, a 3% slope gravel bed river in Chamonix (France). The wavelength was approximately 2m. Figure3: Images of antidunes for increasing flow conditions characterized by the ratio θ/θ c .Each image presents the free surface and the moving grains.

Figure 9 :

 9 Figure 8: Calculated to measured wavelength ratio for each model and different slopes (each points represent a slope range, from left to right: S<0.005 -0.007 -0.01 -0.03 -0.05 -0.07 -0.09 -0.12). Vertical lines represent the standard deviation. Figure 9 : Wavelengths in the F-kh plane with consideration of U/u* values Figure 10: Amplitude versus wavelengths

Table 2 : Data from the literature 547

 2

	535	TABLES
	536
	537
	538
	539
		Run	W	D	S	Q	U	h	Rb/D U/u*c	F	Re	Re*	θ	θ /θc	φ	L	A
			(m)	(mm)		(l/s)	(m/s)	(cm)									(m)	(m)
		1	0.05	2.3 0.0300	0.24	0.40	1.20	4.66	7.12	1.17	4800	129	0.085	1.48 5.28E-03	0.114
		2	0.05	2.3 0.0300	0.28	0.42	1.33	5.14	7.12	1.16	5600	136	0.093	1.63 1.76E-02	0.143
		3	0.05	2.3 0.0500	0.19	0.38	0.97	4.06	5.61	1.23	3700	154	0.120	1.87 6.34E-02	0.100
		4	0.05	2.3 0.0700	0.13	0.31	0.84	3.51	4.16	1.08	2600	171	0.149	2.06 7.75E-02	0.095
		5	0.05	2.3 0.0900	0.10	0.29	0.69	2.91	3.77	1.11	2000	177	0.159	2.05 4.05E-02	0.087
		6	0.10	9 0.0516	1.50	0.61	2.41	2.56	5.65	1.26 14737	993	0.084	1.21 2.70E-03	0.295 0.015
		7	0.10	9 0.0516	2.00	0.67	2.91	3.08	5.66	1.26 19593	1093	0.101	1.45 1.34E-02	0.360 0.014
		8	0.10	9 0.0516	2.50	0.73	3.38	3.50	5.78	1.26 24512	1177	0.117	1.65 3.05E-02	0.399 0.015
		9	0.10	9 0.0516	3.00	0.77	3.82	3.96	5.74	1.26 29448	1252	0.133	1.86 5.24E-02	0.431 0.011
		10	0.10	9 0.0853	1.00	0.51	1.94	2.11	4.05	1.16	9820	1146	0.111	1.43 1.59E-02	0.243 0.014
		11	0.10	9 0.0853	1.30	0.55	2.31	2.53	3.98	1.16 12756	1250	0.132	1.72 4.28E-02	0.315 0.013
		12	0.10	9 0.0853	1.50	0.58	2.54	2.76	4.02	1.16 14709	1311	0.146	1.87 6.59E-02	0.349 0.013
		13	0.10	9 0.0853	1.70	0.60	2.76	3.02	3.98	1.16 16662	1367	0.158	2.05 9.25E-02	0.350 0.013
		14	0.10	9 0.1260	0.87	0.43	1.98	2.21	2.74	0.98	8538	1406	0.168	1.99 1.01E-01	0.243 0.015
		15	0.10	9 0.1260	1.00	0.45	2.17	2.42	2.74	0.98	9813	1475	0.184	2.18 1.46E-01	0.237
		16	0.10	9 0.1260	1.10	0.47	2.27	2.55	2.79	1.01 10780	1510	0.193	2.29 1.77E-01	0.274
		17	0.10	23 0.1260	2.50	0.61	4.00	1.73	2.75	0.98 24528	5111	0.133	1.55 3.22E-02	0.446 0.022
		18	0.10	23 0.1260	2.70	0.63	4.21	1.80	2.78	0.98 26489	5245	0.140	1.62 4.22E-02	0.465 0.017
		19	0.10	23 0.1260	3.00	0.65	4.51	1.94	2.77	0.98 29396	5433	0.150	1.75 6.04E-02	0.555 0.022
	540
	541					Table 1: Flow conditions associated with antidunes
	542
	543
	544

Table 3 : Relative root mean square error for model efficiency comparison

 3

	W	D	S	Q	U	H	R b /D U/u* c	F	θ	θ/θ c	L	A/L
	(m)	(mm)		(l/s)	(m/s)	(cm)					(m)
	0.1	4.9	0.05 0.80 0.49 1.63 3.18 5.61 1.22 0.10	1.53 0.16	0.0309
	0.1	4.9	0.05 0.90 0.51 1.76 3.42 5.62 1.23 0.11	1.69 0.17	0.0287
	0.1	4.9	0.05 1.05 0.53 1.98 3.83 5.52 1.20 0.12	1.85 0.19	0.0256
	0.1	4.9	0.05 1.20 0.57 2.11 4.05 5.78 1.25 0.13	1.94 0.20	0.0242
	0.1	4.9	0.05 1.30 0.57 2.28 4.38 5.55 1.21 0.14	2.08 0.22	0.0224
	0.1	4.9	0.05 1.50 0.60 2.50 4.78 5.60 1.21 0.15	2.29 0.24	0.0205
	0.1	4.9	0.05 1.70 0.63 2.70 5.14 5.67 1.22 0.16	2.46 0.26	0.0191
	0.1	4.9	0.05 2.00 0.69 2.90 5.47 6.02 1.29 0.17	2.60 0.27	0.0179

Table 4 : Increasing flow conditions for the experiment presented in

 4

ACKNOWLEDGMENTS

This study was supported by Cemagref and funding was provided by the ECCO-PNRH program from ANR/INSU N o . ANR-05-ECCO-015, and the PGRN (Pole Grenoblois des Risques Naturels). We are grateful to the TSI laboratory of Saint Etienne (Christophe Ducottet, Nathalie Bochard, Jacques Jay, and Jean-Paul Schon).

The authors would like to thank Dieter Rickenmann and two other anonymous reviewers who greatly contributed to this paper by providing helpful reviews of an earlier version of this manuscript. Our thanks are extended to Rob Ferguson (Associate Editor) who greatly contributed to this paper by providing additional reviews.