domain: nlin phys sde

text:
H Branger
email: branger@irphe.univ-mrs.fr

M A Manna

C Luneau

M Abid

C Kharif

Growth of surface wind-waves in water of finite depth: a laboratory experiment

Keywords: Wind-generated waves, wind-wave growth rates, Miles' theory, finite depth, wind-wave facility

, as well as measurements from other laboratories.

Introduction

Understanding the coupling between the atmosphere and the ocean surface is important for many scientific, environmental and engineering problems. The wind driven momentum exchange and amplification factor of the water waves by the wind at the air-water interface are key drivers of hydrodynamic and ecological processes in open seas. Surface water waves and their generation by wind is a fascinating issue. The task of coupling atmosphere and oceans requires rather detailed knowledge of the wave-field and, in particular, the momentum transfer between boundary layer winds and surface waves. While coastal zones are limited, compared to the global ocean, their impacts on and response to human activity are profound and should be better understood. The vast majority of air-sea interaction studies involved measurements made across the open ocean and, excepting relatively few studies, the role transition in the momentum flux from open ocean to nearshore has not been fully characterized: amplification of water waves by the wind in finite depth environment is still an unsolved problem. Despite differentiated observed physical air-sea behaviours between deep waters and shallow waters, atmospheric models apply the same wind stress formulation regardless of the depth of the waters. Wave growth parametrizations for deep water are commonly applied to shallow water, although the impact that depth-limited processes have on the growtfh of the waves by the wind remains under-explored. Additional well-controlled experiments are essential to gain a better understanding of the growth of waves over water of finite depth. Any theory needs ultimate validation by experimental means.

A few experiments have been conducted to study the influence of the water depth on air-sea interactions at the water surface. Most of them were dedicated to the role of the finite depth h on the modification of prediction laws of non-dimensional energy (g 2 σ 2 /U 4 ref ), non-dimensional frequency (f U ref /g), wave age (c/U ref ), non-dimensional fetch (gX/U 2 ref ) and non-dimensional depth (gh/U ref ), with σ the standard deviation of the water elevation, U ref the wind speed at a reference level, g the acceleration due to gravity, f the dominant wave frequency, c the wave celerity and X the fetch [START_REF] Thijsse | Dimensions of wind-generated waves[END_REF][START_REF] Bretschneider | Generation of wind waves over a shallow bottom[END_REF][START_REF] Bretschneider | Revised wave forecasting relationships[END_REF][START_REF] Ijima | Numerical calculations of wind waves in shallow water[END_REF]CERC, 1977CERC, , 1984;;Young and Verhagen, 1996 a,b].

Other studies were done on the influence of water depth on momentum flux (τ = ρ a u * 2 ), with ρ a the air density, u * the friction velocity [START_REF] Geernaert | Measurements of the wind stress, heat flux, and turbulence intensity during storm conditions over the north sea[END_REF][START_REF] Donelan | Wave-follower field measurements of the wind-input spectral function. part 1 : Measurements and calibrations[END_REF][START_REF] Donelan | Wave-follower field measurements of the wind-input spectral function. part 2 : Parameterization of the wind input[END_REF][START_REF] Zhao | Typhoon air-sea drag coefficient in coastal regions[END_REF][START_REF] Ortiz-Suslow | Observations of air-sea momentum flux variability across the inner shelf[END_REF][START_REF] Chen | Impact of surface waves on wind stress under low to moderate wind conditions[END_REF], on the drag coefficient (C d = u * 2 /U 2 ref ) [START_REF] Anctil | Air water momentum flux observations over shoaling waves[END_REF][START_REF] Donelan | Modeling waves and wind stress[END_REF][START_REF] Bi | Observed drag coefficients in high winds in the near offshore of the south china sea[END_REF][START_REF] Oost | New evidence for a relation between wind stress and wave age from measurements during asgamage[END_REF][START_REF] Toffoli | On the variability of sea drag in finite water depth[END_REF][START_REF] Shabani | Direct measurements of wind stress over the surf zone[END_REF], or on the roughness length z 0 [START_REF] Anctil | Air water momentum flux observations over shoaling waves[END_REF][START_REF] Smith | Wind stress and heat flux over the ocean in gale force winds[END_REF][START_REF] Gao | Wave-dependence of friction velocity, roughness length, and drag coefficient over coastal and openwater surfaces by using three databases[END_REF][START_REF] Macmahan | Increased aerodynamic roughness owing to surfzone foam[END_REF][START_REF] Jimenez | On the need to modify the sea surface roughnessformulation over shallow waters[END_REF]. These papers show the increase of both u * , C d and z 0 in shallow water environment. Authors explain this increase by the modification of wave dynamics and kinematics by the shallow bottom. Finite depth decreases the wave celerity c, thus the wave age c/U ref , and increases wave height and wave steepness by shoaling effect. This modification of celerity and wave geometric profile seems to explain the increase of drag and roughness. Very close to the shore, the slower wave celerity and the sawtooth wave shape in the surf zone are an important contributor to the increased wind stress. For example, [START_REF] Donelan | Wave-follower field measurements of the wind-input spectral function. part 2 : Parameterization of the wind input[END_REF] argued that shallow water measurements exhibit higher drag coefficients for same wind speed, because the waves of the same frequency are slower and shorter and consequently steeper, thus producing higher stress. Based on their air-flow separation model, [START_REF] Makin | Impact of dominant waves on s ea drag[END_REF] have reproduced the increase in C d with decreasing water depth: at moderate and high winds, decreasing water depth induces a steepening of dominant waves and thus more air-flow separation at the crest of the waves, and then more drag. The constant flux model of [START_REF] Chen | Impact of surface waves on wind stress under low to moderate wind conditions[END_REF] has shown how the waves, modified by the bottom in finite depth conditions, influence the wind profile over the waves and thus the stress budget between the water and the air. Other writers like [START_REF] Macmahan | Increased aerodynamic roughness owing to surfzone foam[END_REF] related the increase of roughness near the coasts to surf-zone foam coverage due to wave breaking. Recently [START_REF] Ortiz-Suslow | Observations of air-sea momentum flux variability across the inner shelf[END_REF] clearly demonstrated that the mechanisms that characterize air-sea interaction in deep water may not apply near shore in finite depth waters.

Very few papers [Young and Verhagen, 1996 a,b;[START_REF] Donelan | Wave-follower field measurements of the wind-input spectral function. part 1 : Measurements and calibrations[END_REF][START_REF] Donelan | Wave-follower field measurements of the wind-input spectral function. part 2 : Parameterization of the wind input[END_REF][START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF][START_REF] Young | An integrated system for the study of wind wave source terms in finite depth water[END_REF] were dedicated specifically to the growth rate modification by the water depth. The experi-ment described in [Young and Verhagen, 1996 a] was the first really comprehensive attempt to measure the evolution of fetch limited waves in water of finite depth. The Lake George finite-depth field experiment is well-documented in literature and we refer to [START_REF] Young | An integrated system for the study of wind wave source terms in finite depth water[END_REF] for the most complete and detailed summary on this experiment. A large set of atmospheric and wave data were collected in finite water depth regimes at the Lake George measurement site. The aim of the Lake George project was to simultaneously measure source and sink functions for wave prediction models, as well as fetch-limited wave evolution. In particular, the effects of water depth and wave steepness were addressed. The lake George measurements have confirmed the water depth dependence of the asymptotic limits to wave growth. [START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF], also reproduced in [START_REF] Young | Wind Generated Ocean Waves[END_REF], derived an empirical relation with appropriate non-dimensional parameters able to reproduce the experimental data of Young and Verhagen [1996 a]. In particular the relationship between the variation of fractional energy and the inverse wage age, found by [START_REF] Donelan | On the growth rate of wind-generating waves[END_REF] for deep water, was extended to the finite depth domain. These open field results together with plots of the empirical laws have shown that, contrary to the deep water case, the wave age at which the growth rate becomes zero is wind-dependent and depth-dependent.

In [START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF] the curves of the wave growth versus wave age have been presented for ranges of a non-dimensional water depth parameter δ. Beyond a critical wave age, the growth rate vanishes.

Pioneering theories on surface wind-wave growth have been developed for deep water waves since [START_REF] Jeffreys | On the formation of waves by wind part1[END_REF][START_REF] Jeffreys | On the formation of waves by wind part2[END_REF][START_REF] Phillips | On the generation of waves by turbulent wind[END_REF] and [START_REF] Miles | On the generation of surface waves by shear flows[END_REF], and until more recently, for example by [START_REF] Belcher | Turbulent shear flow over slowly moving waves[END_REF] or other authors reviewed in [START_REF] Janssen | The interaction of ocean waves and wind[END_REF]. These theories are thus restrictive with regard to wind generated near-shore waves, bays, rivers or shallow lake waves. The well known Miles' theory has been extended to the finite depth under breeze to moderate wind conditions by several authors [Montalvo et al., 2013 a,b;[START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF]. They derived a surface wind-wave growth theory in finite depth based on a) the Euler equations in the water domain for finite depth, and b) the linearised governing equation of steady air flow, with a prescribed mean horizontal velocity U(z) depending on the vertical coordinate z together with perturbations to the mean. Developments lead to the well known Rayleigh equation [START_REF] Rayleigh | On the stability, or instability, of certain fluid motions[END_REF]. Then a non-dimensional water depth parameter depending on the depth and on a characteristic wind speed, induced a family of curves representing the wave growth as a function of the wave phase velocity and the wind speed. Their model provides an agreement with the data and empirical relationships obtained from the Lake George experiment and the Australian Shallow Water Experiment. For small wave age the wave growth rates are comparable to those of deep water, but for higher wave age a finite-depth limited growth is reached, with wave growth rates going to zero. The [Montalvo et al., 2013 a] study was focused on the wave growth of monochromatic waves, the work of [Montalvo et al., 2013 b] was extended to the evolution of wave packets growing with the wind, and [START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF] added the effect of shelter mechanism in the theory [START_REF] Jeffreys | On the formation of waves by wind part2[END_REF], and they also added non-linearity effects and studied the evolution of non-linear breathers and solitons in shallow depth environments and windy conditions.

To date, there is no referenced laboratory direct experimental work that confirms or denies the theoretical study of [Montalvo et al., 2013 a,b;[START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF], more particularly the fact that the wave growth rate induced by the wind could drop to zero amplification for some configurations of water depth, wave dominant frequency, wave age and wind speed. Recently amazing new laboratory experiments have been conducted on air-to-water transfers using laser doppler Anemometry (LDA) or particle Image Velocimetry (PIV) to characterize wave amplification by the wind through form drag and viscous drag repartition along the water surface [START_REF] Veron | Measurements of the viscous tangential stress in the airflow above wind waves[END_REF][START_REF] Buckley | Surface viscous stress over wind-driven waves with intermittent airflow separation[END_REF]Yousefi K. and M.P, 2021b,a;[START_REF] Donelan | On the role of the explosive interaction of three surface waves at the initial stage of spray generation in strong winds[END_REF][START_REF] Kabardin | Approbation of the laser doppler anemometer under kinematic flow parameters diagnostics during spray formation in a laboratory wind-wave interaction modelling[END_REF]. Unfortunately, all these experimental works with optical methods have always been conducted on deep-water conditions and not in intermediate or shallow water conditions. Moreover, they were often done in moderate to high wind speed conditions because most of the studies were focused on airflow separation and aerosol production.

Therefore a well-control experiment on wave growth in finite depth environment is lacking. The goal of our study is to conduct a specific water tank experiment with wind blowing over the water surface, and with different combinations of water depth, dominant waves, and wind speeds. Marine air boundary layer characteristics (friction velocities, drag coefficients, surface roughness lengths) will be estimated, and water elevation will be measured at different locations to measure the wave growth in relatively smooth wind and low steepness conditions: we avoided wave breaking dissipation, because in [Montalvo et al., 2013 a,b;[START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF] theories breaking dissipation is disregarded. Wave growth will be compared to theoretical developments. The asymptotic depth limited growth of nondimensional energy, non-dimensional peak frequency and inverse wave age will be tested and validated.

Notations, parameters and theoretical developments are presented in section 2. Material and methods are described in section 3, with the presentation of the wind-wave facility, the experimental arrangement, data processing, experimental conditions, air and water measured values and their precision. Section 4 describes the effects of water depth on air flow, on air-water transfer parameters, on wave growth rate, and this section ends by the comparison with theory developments. The last section provides a short summary and outlook.

2. The air-water interface in finite depth. Theoretical wave growth rates γ and β 2.1. Wave growth rates γ and β : analytical developments

The classical Miles' theory in deep water is based on the dispersion relation of the air-sea interface and the related Rayleigh equation [START_REF] Rayleigh | On the stability, or instability, of certain fluid motions[END_REF][START_REF] Conte | On the numerical integration of the orr-sommerfeld equation[END_REF][START_REF] Drazin | Hydrodynamic Stability[END_REF]. We consider the water and air particles to be located relative to a two-dimensional Cartesian coordinate system with axes x, z, origin x = z = 0, x ∈ [-∞, +∞], z being the upward vertical direction and z ∈ [-h, +∞]. Before perturbations the water lies between the impermeable bottom à z = -h and the still water surface at z = 0. The undisturbed water steady state is incompressible, inviscid and with zero surface tension. It is assumed that the air is also inviscid and incompressible. In absence of surface waves the air can be described by a prescribed mean shear flow U (z) in the x-direction which varies with the height z above the surface i.e.:

⃗ U = U (z)⃗ e x (1)
with ⃗ e x the unitary vector in the x direction.

Let us consider a perturbation to the free surface which propagates, under the wind action, like a progressive plane wave of wave number k, phase velocity c and constant amplitude η 0 , i.e., η(x, t) = η 0 e ik(x-ct) . Miles's mechanism of wave generation by wind assumes that η(x, t) induces small perturbations of the air pressure and velocities. From reference [Montalvo et al., 2013 a] we have that the perturbed air pressure P a (x, z, t) is given by

P a (x, z, t) = P 0 -ρ a gz + ikρ a ∞ z [U (z) -c]w a (x, z, t)dz (2)
where P 0 is the atmospheric pressure, ρ a the air density, w a (x, z, t) = W a (z)e ik(x-ct) the z-component of the perturbed air velocity. The expression for P a is of primary importance in the Miles mechanism of wave-generation by wind since it is supposed that the momentum transfer from the air into the surface waves is due to the air pressure oscillations in quadrature with the surface slope [START_REF] Deardorff | Aerodynamic theory of wave growth with constant wave steepness[END_REF]. In (2) neither w a (x, z, t) nor c are known. The perturbed velocity w a (x, z, t) is calculated using the Rayleigh equation [Montalvo et al., 2013 a] and c is determined by the relation dispersion which reads ρ w η(x, t) (c 2 k coth kh -g) + P 0 = P a (x, η, t).

(3)

In the single-domain problem P a (x, η, t) = P 0 and equation (3) brings to the usual expression c 0 of the phase velocity governing surface waves in finite depth: c = c 0 = (g/k) tanh (kh). In this case in order to calculate c from (3) we use P a evaluated at z = η from (2). We obtain [Montalvo et al., 2013 a] g

(1 -s) + c sk 2 W 0 I 1 -c 2 ( sk 2 W 0 I 2 + k coth(kh)) = 0, (4)
with

I 1 = ∞ z0 U (z)W a (z)dz, I 2 = ∞ z0 W a (z)dz, (5)
with z 0 the aerodynamic sea surface roughness located just above the interface, W 0 = lim z→z0 W a (z), and s the ratio of air density to water density: s = (ρ a /ρ w ) ∼ 10 -3 . The function W a (z) is complex and consequently c too, so c = Re{(c)} + i Im{(c)} where Im{(c)} (Re{(c)}) is the imaginary (real) part of c. The parameter s is small (ρ a /ρ w ∼ 10 -3 ) and the solution of (4) may be approximated as

c = c 0 + sc 1 + O(s 2 ). ( 6
)
c 0 is real, so the imaginary part Im{(c)} is s Im{(c 1 )}, and it gives the growth rate of η(x, t) defined by γ = k Im{(c)} = sk Im{(c 1 )} (7)

The theoretical and numerical results concerning the growth rate γ are studied and computed with three non-dimensional parameters δ, θ dw and θ f d defined by

δ = gh U 2 1 , θ dw = 1 U 1 g k , θ f d = θ dw T kh , with, T kh = tanh (kh) = tanh ( δ θ 2 dw ) (8)
The non-dimensional parameter δ, for constant U 1 , measures the influence of the finite fluid depth on the rate of growth of η(x, t). The parameter θ dw is a theoretical analogous of the deep water wave age. It measures the relative value of the deep water phase speed in relation to the characteristic wind velocity U 1 . Finally θ f d is a finite depth wave age θ f d . We introduce the following non-dimensional variables and scaling, hats meaning non-dimensional quantities

U = U 1 Û , W a = W 0 Ŵa , z = ẑ k , c = U 1 ĉ, t = U 1 g t. (9)
Using ( 6) and ( 9) in equation ( 4) and retaining only the terms of order O(s) we obtain ĉ,

ĉ = ĉ0 + sĉ 1 = θ dw T 1/2 kh - s 2 θ dw T 1/2 kh + s 2 {T kh Î1 -θ dw T 3/2 kh Î2 }, (10)
and with (7), ( 9) and (10 , we derive the non-dimensional growth rate γ = U1 g γ which reads γ = s 2 (

T kh Im Î1 θ 2 dw - T 3/2 kh Im Î2 θ dw ). ( 11
)
From the coupled water/air interface, [START_REF] Miles | On the generation of surface waves by shear flows[END_REF] derived the phase velocity as :

c 2 = c 2 0 + ρ a ρ w (α + iβ) 2 U 2 1 ( 12
)
where α and β are non-dimensional coefficients. Using (6) we obtain

α = 2c 0 Re{(c 1 )} U 2 1 , and
β = 2c 0 Im{(c 1 )} U 2 1 ( 13
)
where β is the β-Miles parameter. Using γ = U1 g γ, and equations ( 8) and (9), we obtain the following transformation rule between non-dimensional parameters β and γ

β = 2γ s θ 3 dw T 1/2 kh , (14)
The deep water limit δ → ∞ (T kh → 1) gives back γ and β for Miles' theory. The existence of a finite depth h transforms the unique curve of wave growth rate γ or β in deep water in a family of curves indexed by δ = gh/U 2 1 , i.e., a curve for each value of δ [Montalvo et al., 2013 a,b;[START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF].

2.2. Wave growth rates γ and β : wind-wave tank environment Experimental results are obtained using appropriate parameters and formulae commonly employed in the laboratory routine. In this section we are looking for the transformation rules between theoretical expressions and their laboratory counterpart.

Let us consider a monochromatic progressive wave moving in a wave tank. The wave frequency is ω, the wavenumber k and the amplitude a(x, t) with x the distance from the wavemaker which is termed the fetch x and t is the time

η(x, t) = a(x, t) sin(kx -ωt) (15)
If we consider the wind action, with wind U ref (the wind speed at the altitude z = z ref ), as the unique process causing the wave growth, and we consider the viscosity and bottom friction the only two existing mechanisms of wave dissipation, (i.e. no wave breaking because we did not see wave breaking during our experiments, and the present theory excludes wave breaking), we may write :

( 1 a ∂a ∂x ) |U ref = γ es -ν (16)
where γ es is the experimental spatial wave growth rate , and ν is the total rate of dissipation by viscosity and friction along the walls and on the bottom of the tank.

For each experiment we did two runs: the first one was done with mono-frequency paddle waves with amplitude a, frequency ω and wind speed U ref and the second one was done with the same mono-frequency waves (amplitude a, and frequency ω) but without wind speed, i .e. with U ref = 0 m/s. When there is no wind speed, the wind notation is U 0 .

We made the strong hypothesis that the dissipation process was the same in presence of wind and without wind. This hypothesis is questionable because there may have many efficient processes that dissipate the waves in a closed basin [START_REF] Miles | Surface-wave damping in closed basins[END_REF]. Molecular viscosity and friction dissipation along the bottom and along the tank walls are always an active process and occur in all conditions [START_REF] Van Dorn | Boundary dissipation of oscillatory waves[END_REF]. Wave breaking is another strong dissipation process but it occurs with steep waves and/or during moderate to strong wind conditions, and/or very close to the shore-line [START_REF] Perlin | Breaking waves in deep and intermediate waters[END_REF]. Droplets and spume generation is also an efficient dissipation process, but it occurs only at relatively high winds [START_REF] Veron | Ocean spray[END_REF]. Another process could be the non-linear interaction between the shear-water current generated by the wind drag at the surface, and the orbital velocities of the waves leading to possible wave attenuation. But this phenomenon of interaction between water vorticity and water waves is not well-known and quite impossible to quantify [START_REF] Hsu | A nonlinear schrodinger equation for gravitycapillary water waves on arbitrary depth with constant vorticity. part 1[END_REF]. We decided to avoid it. Air-flow separation, occurring at the crest of steep, breaking or near breaking waves, generates vortical motions in the air that may contribute to the dissipation of wave energy [START_REF] Iafrati | Modeling of ocean atmosphere interaction phenomena during the breaking of modulated wave trains[END_REF]. But we were always in low steep waves environment, and we did not take this effect into account. In our study, we always generate low steep mechanical paddle waves and conducted runs with low wind speed in order to avoid wave breaking, aerosol production, and air-flow separation above waves. This is the reason why we made the strong hypothesis that the main dissipation process was the molecular viscosity.

So without wind we may write:

( 1 a ∂a ∂x ) |U0 = -ν (17)
Finally, the spatial growth rate may be inferred from ( 16) and ( 17)

( 1 a ∂a ∂x ) |U ref -( 1 a ∂a ∂x ) |U0 = γ es (18)
In many papers, model theories and wave prediction models, the temporal growth rate γ [START_REF] Janssen | The interaction of ocean waves and wind[END_REF]Montalvo et al., 2013 a,b] or the temporal growth rate β M iles [START_REF] Miles | On the generation of surface waves by shear flows[END_REF] are used instead of the spatial growth rate. By analogy we can define an experimental temporal wave growth rate γ et , related to the experimental spatial growth rate γ es by the group velocity C g [START_REF] Gaster | A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability[END_REF] :

γ et = C g γ es (19)
The non-dimensional temporal wave growth γet may be defined as [START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF] :

γet = ( u * gκ )γ et ( 20
)
with u * the friction velocity and κ = 0.41 the von Karman constant. [START_REF] Miles | On the generation of surface waves by shear flows[END_REF] used the other non-dimensional temporal wave growth β :

β = 2 ( c u * ) 2 κ 2 1 ω ( 1 ρa ρw ) γ et (21)
Another widely used non-dimensional temporal growth rate [START_REF] Plant | A relationship between wind stress and wave slope[END_REF][START_REF] Belcher | Linear dynamics of wind waves in coupled turbulent air-water flow. part1 theory[END_REF][START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF]

is γet f = γet2π ω
Experimentally, if the wave height is measured at two different fetches (i.e. different distances from the wind inlet and the wavemaker), namely a 0 and a 1 at fetch x 0 (close to the beginning of the tank) and x 1 (far away from x 0 ), the dimensional spatial growth rate may be estimated by (c.f. equation ( 18)):

γ es = ( 1 a 1 a 1 -a 0 x 1 -x 0 ) |U ref -( 1 a 1 a 1 -a 0 x 1 -x 0 ) |U0 (22)
In this relation, subscripts U ref and U 0 refer to experiments with wind speed U ref and experiments with wind speed U 0 = 0

The non-dimensional temporal growth rates γet , β and γet f could be estimated by (c.f. equations ( 19), ( 20)and ( 21) ):

γet = ( u * gκ ) C g (( 1 a 1 a 1 -a 0 x 1 -x 0 ) |U ref -( 1 a 1 a 1 -a 0 x 1 -x 0 ) |U0 ) (23) β = 2 ( c u * ) 2 κ 2 1 ω ( 1 ρa ρw ) C g (( 1 a 1 a 1 -a 0 x 1 -x 0 ) |U ref -( 1 a 1 a 1 -a 0 x 1 -x 0 ) |U0 ) (24)
and

γ et /f = C g f (( 1 a 1 a 1 -a 0 x 1 -x 0 ) |U ref -( 1 a 1 a 1 -a 0 x 1 -x 0 ) |U0 ) (25)
Usually only ω is measured. k is not directly measured. Wavenumber k is estimated by solving the dispersion relationship between k and pulsation w :

ω 2 = gk tanh(kh) (26)
In some wave tank experiments, c and C g are measured using cross-correlation of two or more wave probes located not far from each other [START_REF] Shemer | Spatially evolving regular water wave under the action of steady wind forcing[END_REF][START_REF] Longo | Wind-generated water waves in a wind tunnel: Free surface statistics, wind friction and mean air flow properties[END_REF]. Unfortunately, during the experiments, the electronic part of the bank of wave gauges usually used in the wind-wave tank was not available and we had only two wave gauges available for the experiments separated by twenty meters. The distance between the two remaining wave gauges was too far to have a good estimation of the wave celerity and wave group by the cross correlation technique. So we decided to use the traditional dispersion relationship to estimate experimentally c and C g from k :

c = c 0 = g k tanh(kh) (27)
C g = C g0 = c 0 2 (1 + 2kh sinh(2kh) ) ( 28
)
3. Material and methods

Wind-wave facility and experimental arrangement

Experiments were conducted in the IRPH É/Pythéas wind-wave tank, in order to estimate the wave-growth of regular monochromatic mechanical waves under the action of the wind for different water depth configurations. The IRPH É/Pythéas wind-wave tank, sketched in Figure 1, consists of a closed-loop tunnel for air-circulation with a 1.5 m-high air cavity above a basin, with a test section 40 m-long and 2.60 m-wide. A complete description of the facility can be found in [START_REF] Coantic | Wind-water tunnel simulation of smallscale ocean-atmosphere interactions[END_REF], and at the following address: https://www.osupytheas.fr/?-Uniqueness-.The wind is generated by an air blower in the upper part of the tunnel, and guided through a settling chamber with a honeycomb and a converging nozzle. This system produces a uniform airflow at the entrance of the water tank. The bottom is flat, and the water depth may be adjusted from 0.1 m up to 0.80 m depending upon the water quantity put in the tank. An adjustable 6 m-long divergent section was added recently between the entrance of the air section and the water section when the tank is not full of water, thus giving a smooth and non-perturbing air transition between the two air and water sections. The height of this air-divergent can be adjusted to any water depth (see Figure ( 2)). Consequently for each run, the wind was always tangent to the mean water surface at the entrance of the water section. There is a slightly divergence of the air section along the wind-wave tank, allowing a zero pressure longitudinal gradient, and a constant momentum transfer flux layer from the air to the water along the tank. The air blower is controlled by a graduated potentiometer.

At the upwind-end of the basin, an immersed paddle wave-maker (piston-type) is used for generating monochromatic waves propagating along the wind direction. The wave maker is controlled by a Labview program. These paddle waves do not hit the air divergent located just above them. At the downwind-end, a permeable absorbing beach, with a 6°slope, prevents wave reflection. The reflection rate ranges between 2% to 5% depending upon the wavelength of the incoming waves. The basin side walls are punctuated with windows to observe the water surface. An inside view of the facility showing the experimental arrangement is shown in Figure 3. A sketch of the experimental arrangment is drawn in Figure 4 showing the location of the measuring devices.

The instantaneous horizontal wind velocity was measured with a Constant Temperature Anemometer (E+E Elektronik, Langwiesen, Austria). This sensor was calibrated against a reference sonic GILL 2D anemometer on several occasions during the experiments.The anemometer was supported by a vertical telescopic pole allowing it to be moved at different heights above the water level. Measurements were done from a point slightly above the highest wave, up to about several vertically distributed points and a duration of 300 s to obtain the vertical profile of the mean horizontal velocity U (z). The precision of the anemometer was about ±4 cm/s For these experiments, we used two DISA capacitance-type wave gauges in order to measure the wave growth between two different locations. One wave gauge was located at fetch x 0 = 2.40 m, the other one at fetch x 1 = 20.10 m. The origin of the fetch (x = 0 m), is by definition, the location where the wind hits the water, i.e. at the downwind end of the air divergent. For each run, wave gauges signals were acquired during 20 minutes at a sampling rate of 256 Hz. Before every acquisition, it was verified that the waves in the wave tank were in stationary fetch-limited conditions by waiting a sufficiently long time before acquiring data. Calibrations of the wave gauges were required every day.

Experimental conditions

Due to: a) the intrinsic mechanical performance of the piston wavemaker in low depth conditions (this wavemaker was designed long time ago for deep waters) , b) the fact that the paddle waves should not hit the rigid air divergent close to the water surface, c) the requirement to have low steep waves to avoid wave amplitude modulation by modulational instabilities and wave breaking, d) the requirement to have low wind speeds to avoid air-flow separation and wave breaking too, the possible combinations of depths, wave frequencies, wave steepnesses, and wind speeds were quite low. We used 4 different water depths, 4 different wave paddle frequencies, and different initial wave amplitudes. Finally we conducted 15 different runs.

Main basic physical parameters characterizing waves and wind are listed in Table 1 . Water depth h ranged from 0.14 m to 0.74 m, mono-frequency waves f 0 from 1.1 Hz to 1.7 Hz, k 0 h from 1.07 to 8.59. Finite depth is usually reached when the ratio of water depth over wavelength: h/L 0 is lower than 1/2, which corresponds, in terms of kh to: kh < π = 3.14. According to the traditional classification of waves with respect to depth [START_REF] Young | Wind Generated Ocean Waves[END_REF] runs number 1 to 7 and run 10 correspond to intermediate depths: π/10 < kh < π, and other runs to deep water conditions.

Initial steepness (ak) 0 ranged from 0.026 to 0.168. Wind speed U 10 from 3.18 m/s to 5.81 m/s (see next paragraph for the definition and computation of U 10 ). It was not possible to use higher wind speeds and/or higher initial wave steepness because we wanted to avoid natural wave breaking. Wave breaking induces additional wave dissipation which is not included in the theory here. With higher wind speeds or higher initial wave steepnesses we effectively obtained wave breaking inside the wave tank, either by wave amplification by the wind, or by natural sub-harmonic Benjamin-Feir instability leading to strong wave groupiness with the highest wave breaking inside the group [START_REF] Bliven | Experimental study of the influence of wind on benjamin-feir sideband instability[END_REF][START_REF] Benjamin | The disintegration of wave trains on deep water[END_REF][START_REF] Hwung | Threshold model on the evolution of stokes wave side-band instability[END_REF].

Measurements 3.3.1. Wind parameters

In the facility, air and water temperature were roughly at the same temperature, which means that the marine atmospheric conditions were "neutral". The friction velocity u * is considered constant along the test section because of the slightly divergence of the air section along the wind-wave tank allowing a zero pressure longitudinal gradient [START_REF] Coantic | Wind-water tunnel simulation of smallscale ocean-atmosphere interactions[END_REF]. Assuming a logarithmic wind profile and neutral conditions, [START_REF] Monin | Basic laws of turbulent mixing in the surface layer of the atmosphere[END_REF] scaling allows evaluation of friction velocity u * , roughness length z 0 , and U 10 , from the measured U (z) profiles values with:

U (z) = u * κ ln z z 0 ( 29
)
where κ = 0.41 is the von Kármán constant, z is the measurement height above mean water level. The logarithmic profile is the solution of the horizontal momentum equations for the near-surface boundary Run sublayer dominated by friction forces, both viscous and turbulent [START_REF] Komen | Sea surface winds and atmospheric circulation[END_REF] ∂τ /∂z = 0, with τ the momentum flux from air to water. A linear fit between ln z and U (z) gives easily an estimation of u * and z 0 . Once u * and z 0 are evaluated, U 10 is taken as :

h f 0 L 0 k 0 h (
U 10 = u * κ ln 10 z 0 (30)
This method, called the "profile method", is routinely used for wind parameter determination in this facility (see for example recent papers of [START_REF] Villefer | Influence of following, regular and irregular long waves on wind-wave growth with fetch: an experimental study[END_REF][START_REF] Bruch | Seaspray-generation dependence on wind and wave combinations: A laboratory study[END_REF] where vertical wind profiles are displayed with u * , z 0 , and U 10 determinations). Two examples of vertical wind profiles are plotted in Figure ( 5). We did several times the same runs to estimate the precision on u * and z 0 . The relative precision was estimated to be 15% for u * and U 10 and 30% for z 0 : du * u * = 15%, dU10 U10 = 15%, dz0 z0 = 30%. The estimated values of u * , U 10 , and z 0 for the 15 different runs are listed in Table 2.

The drag coefficient C d10 = ρau * 2 ρaU 2 10 = ( u * U10 ) 2 , with ρ a the air density, is a commonly used parameter to characterise the shear air-flow above the waves and the non-dimensional momentum transfer from the air to the water. τ = ρ a u * 2 is the wind stress on the water surface. Values of C d10 for the 15 different runs are listed in Table 2.

In open seas, it is very difficult to measure z 0 . A simple parametrization like : z 0 = α Ch u * 2 /g, with g the acceleration due to gravity, and α Ch the Charnock parameter could be used to estimate z 0 from u * [START_REF] Charnock | Wind stress on a water surface[END_REF]. Values of α Ch for the 15 runs are listed in Table 2

Wave parameters

The goal of the experiments was to measure the spatial and temporal growth rates of mechanical monochromatic waves forced by wind for different water depth conditions and to compare these growth rates with the theoretical values of [Montalvo et al., 2013 Table 2: Experimental conditions: u * and z 0 are the friction velocity and roughness length estimated by the wind profile method, C d10 is the drag coefficient

C d10 = (u * /U 10 ) 2 , α Ch is the Charnock constant α Ch = gz 0 /u * 2 .
wave growth instead of a spatial wave growth. But it does not work for a few reasons: a) turning the wind "on" (from 0.0 m/s to, say, 5.1 m/s) takes time: it is not possible to command the wind turbine motor with a voltage step, it needs to be done progressively, and consequently the temporal growth will depend also on the wind gradient used to turn the wind on for the experiment at the beginning of the run; b) in a wave tank, in each location, the equilibrium state is reached quite rapidly, and in presence of low frequency mechanical waves, there are not enough waves before the equilibrium state to have a statistical good estimation of the temporal growth rate; c) It is difficult to determine the quantity da dt (with a the amplitude of the waves ) in a wavy surface environment, more particularly if the wave spectrum is not narrow, i.e. with wind on.

Wave amplification factors with the respective effect of wind and depth will be discussed in section 4.2 .

An example of the energy density spectra of the waves recorded during run 5 is displayed in Figure ( 7). The spectra are plotted at fetches x 0 = 2.4 m and x 1 = 20.1 m, when there is no wind and when the wind is on. Wind waves appear clearly on spectra (d) at high frequencies, and only the second harmonics of the dominant waves are visible at fetch x 1 when wind is blowing.

In equations ( 23), ( 24) and ( 25), a 0 and a 1 are not the amplitude of each individual waves, but they must be considered as the average amplitude of waves with frequency close to the frequency f 0 , the initial frequency of the wavemaker.

The determination of the amplitude a of the paddle waves, at the location of the wave gauges, was computed from the recorded time series of the wave gauges. We did a standard up-zero crossing analysis to separate each individual wave. Only the waves with a frequency f ranging between 0.8f 0 < f < 1.2f 0 were considered to avoid noisy short wave or long wave perturbations and to focus on the monofrequency generated waves. For each wave number i, the amplitude a i was determined as half the crest-to-trough height of the wave number i :

a i = (H max -H min ) i 2 (31)
The mean amplitude ā was computed as :

ā = 1 N N i=1 a i (32)
with N the total number of waves of the run.

The experimental non-dimensional temporal wave growths were estimated using equations ( 23), ( 24) and ( 25), with ā1 and ā0 instead of a 1 and a 0 , and with x 1 = 20.1 m, x 0 = 2.4 m.

Measured values and precision of the measurements

Main parameters of the 15 runs were listed in Table 1. Other environmental parameters defined in sections 3.3.1 and 3.3.2 which were computed from wind and wave measurements are listed in Table 2. Friction velocity u * ranged from 0.108 m/s to 0.231 m/s, roughness length z 0 from 0.030 mm to 0.402 mm. The drag coefficient C d10 ranged from 1.0210 -3 to 1.6410 -3 , and the Charnock constant α Ch from 0.0096 to 0.107.

In the following paragraphs, we will consider non-dimensional variables which may be active in fetch or duration limited growth. Like in [START_REF] Sverdrup | Wind, sea and swell: theorie of relation for forecasting[END_REF][START_REF] Kitaigorodskii | The physics of air-sea interaction, English Edition[END_REF][START_REF] Young | Wind Generated Ocean Waves[END_REF] these quantities are the non-dimensional energy g 2 σ 2 U 4 10 , with σ the standard deviation of the water elevation, the non-dimensional frequency f0U10 g , the non-dimensional depth gh U10 , the inverse wave age U10 c .

It is quite important to evaluate the precision of the measurements and the errors made on the environmental parameters computed from the raw data acquired during the experiments. A good comparison with theory may be done only if the uncertainties on measured data are clearly given. By doing the same experiment several times, we have estimated the relative errors on the measurements we conducted in the facility. All the estimations and error computations are detailed in Appendix A. Relative errors on the different parameters are summarized in Table 3. 4, as well as the estimated wave celerity c 0 (relation ( 27)), wave group velocity C g0 (relation ( 28)), wave age c 0 /u * , and non-dimensional parameters δ, θ dw , θ f d (relation ( 8)), γet (relation ( 23)), β (relation ( 24)) and γ et /f 0 (relation ( 25)). Parameter δ ranges between 4.7 to 76.0 and parameter θ f d ranges between 1.6 to 4.9. Amplification factor β ranges between 0.10 (quite no amplification) to 17.0 (large amplification) seas, when U 10 is known from satellite or models, a model of the drag like : C d10 = a + b U 10 , a and b being constant, is widely used to estimate u * from U 10 : u * = U 10 (C d10 ) 0.5 . A quite large range of a and b values have been reported from measurements like, for example : a = 0.75 10 -3 , b = 0.067 10 -3 [START_REF] Garratt | Review of drag coefficients over oceans and continents[END_REF]; a = 0.61 10 -3 , b = 0.063 10 -3 [START_REF] Smith | Wind stress and heat flux over the ocean in gale force winds[END_REF]; a = 0.80 10 -3 , b = 0.065 10 -3 [START_REF] Wu | Wind-stress coefficients over sea surface from breeze to hurricane[END_REF]; a = 0.96 10 -3 , b = 0.041 10 -3 [START_REF] Donelan | The dependence of the aerodynamic drag coefficient on wave parameters[END_REF]; a = 0.58 10 -3 , b = 0.084 10 -3 [START_REF] Geernaert | Measurements of the wind stress, heat flux, and turbulence intensity during storm conditions over the north sea[END_REF]; a = 0.49 10 -3 , b = 0.071 10 -3 [START_REF] Anderson | A study of wind stress and heat flux over the open ocean by the inertial-dissipation[END_REF]; a = 0.50 10 -3 , b = 0.071 10 -3 [START_REF] Yelland | Wind stress measurement of the open ocean drag coefficient corrected for air flow disturbance by the ship[END_REF]]; a = 0.96 10 -3 , b = 0.060 10 -3 [START_REF] Toffoli | On the variability of sea drag in finite water depth[END_REF]. Our measured trend of C d10 with U 10 is equivalent to the trend observed in literature (i.e. the same as [START_REF] Smith | Wind stress and heat flux over the ocean in gale force winds[END_REF]: b = 0.063 10 -3 ) for large depth configuration. But for low depth, C d10 values are significantly higher and the trend is much higher : b = 0.112 10 -3 .

x h f 0 c 0 C g0 L 0 k 0 h ā σ (

Discussion

( ā0 ) |U0 ( ā0 ) |U ref ( ā1 ) |U0 ( ā1 ) |U ref c 0 C g0 c 0 /u * δ θ dw θ f d γet β γ et /f 0 cm cm cm cm m/s m/
[ [START_REF] Bi | Observed drag coefficients in high winds in the near offshore of the south china sea[END_REF] investigated the drag coefficient C d in the South China Sea from 2008 to 2014, and one of their towers to collect data was located in 15 m of water. Some of their data were considered to be in finite depth environment (i.e. depth lower than half the dominant wavelength, i.e. kh < π). [START_REF] Bi | Observed drag coefficients in high winds in the near offshore of the south china sea[END_REF] found that the drag coefficient was higher in shallow water by 40% for U 10 < 10 m/s. [START_REF] Jimenez | On the need to modify the sea surface roughnessformulation over shallow waters[END_REF] showed that the drag at the location of FINO1 offshore platform in shallow waters was about 30% higher than the drag in deep waters. Very close to the shore, the drag coefficients over the surf zone during onshore winds were determined by [START_REF] Shabani | Direct measurements of wind stress over the surf zone[END_REF] to be almost 200% the values expected at the same wind speed and open ocean conditions ! [START_REF] Anctil | Air water momentum flux observations over shoaling waves[END_REF] found that the wind dependence of C d derived for shallow water differs from that of deep water. sometimes, their C d values were 40% to 80% higher for shallow water. They explain the modification of C d by the modification of wave age c/U 10 and modification of the root mean square wave slope by the bottom effect on waves in shallow water. The geometric shape of the wave and the kinematics are modified by shoaling effects, thus modifying air-water transfers. [START_REF] Toffoli | On the variability of sea drag in finite water depth[END_REF] argued that water depth effects together with large wind result in a steeper and more vertically asymmetric profile for dominant waves, which enhances breaking probability, air flow separation and sea drag. They proposed a parametrisation of C d with an inverse quadratic function of a non-linear parameter µ depending on both wave steepness and water depth. In our figure (8) we see that our Drag coefficient is 45% higher at low water depth (kh < 1.8) if compared to higher water depth (kh > 2.0). We notice also that the higher wind speed, the higher relative difference between C d values at low depth and C d values at large depth. [START_REF] Zhao | Typhoon air-sea drag coefficient in coastal regions[END_REF] explored also C d in the South China Sea from wind propeller anemometers stationed on a coastal observation tower (water depth 14 m), to observe the effect of water depth on C d . But their study was more focused on typhoon measurements. [START_REF] Zhao | Typhoon air-sea drag coefficient in coastal regions[END_REF] concluded that water depth unequivocally affects the drag coefficient when compared to deep open ocean results, but more particularly at high winds (U 10 > 15 m/s), which is beyond our scope here. They found that in shallow water depth environment, the C d maximum occurs approximately at 24 m/s, which is 5 to 15 m/s lower than of the open sea.This is prospectively due to wave shoaling and wave breaking conditions pertaining to the shallow water in high wind conditions. [START_REF] Zhao | Typhoon air-sea drag coefficient in coastal regions[END_REF] proposed a C d formulation as a function of wind speed and water depth, but in their formulation, the water depth plays a role only in the U 10 > 15 m/s part of the C d curves. Before them, [START_REF] Geernaert | Variation of the drag coefficient and its dependance on sea state[END_REF] have also shown that the shallow water C d values for wind speeds above 15 m/s are higher than those over the open ocean and they attributed the difference to the variations in the surface wave energy spectrum. Based on their air-flow separation model, [START_REF] Makin | Impact of dominant waves on s ea drag[END_REF] and their WOWC (Wind Over Waves Coupling) model have reproduced the increase in C d with decreasing water depth: at moderate and high winds, decreasing water depth induces a steepening of dominant waves and thus more air-flow separation at the crest of the waves, and more drag. [START_REF] Makin | Drag of the water surface at limited fetch: laboratory measurements and modelling[END_REF] explained the wind speed, wave age and finite depth dependences of the sea drag in a wave tank environment with this model.

What is new in our measurements, is the fact that we find a significant increase of C d , even at relatively low wind speed (i.e. U 10 < 6 m/s). [START_REF] Donelan | The dependence of the aerodynamic drag coefficient on wave parameters[END_REF][START_REF] Oost | New evidence for a relation between wind stress and wave age from measurements during asgamage[END_REF][START_REF] Gao | Wave-dependence of friction velocity, roughness length, and drag coefficient over coastal and openwater surfaces by using three databases[END_REF][START_REF] Toffoli | On the variability of sea drag in finite water depth[END_REF] have also shown that the C d over shallow water is higher than that over deep water at the same wind speed, but they have related this phenomenon to wave state parameters, e.g., wave age and wave steepness. They do not include water depth h or non-dimensional parameter kh in their drag model.

Depth influence on aerodynamic roughness length z 0

The evolution of z 0 with the wind U 10 is shown on figure (9). z 0 values are much higher for low water depths (i.e kh < 1.8) than for greater depths (kh < 2.0), and the variation trend of z 0 with U 10 is six times higher for low depth than deep water. Recently [START_REF] Jimenez | On the need to modify the sea surface roughnessformulation over shallow waters[END_REF] pointed out the need to modify the sea surface roughness formulation over shallow waters taking into account the water depth. [START_REF] Anctil | Air water momentum flux observations over shoaling waves[END_REF] explained the increase of z 0 by the increase of the "inverse wave age" U 10 /c, the increase of root mean square wave slope η X , and the increase of significant wave height H s due to shoaling effect near the coast in intermediate and shallow water conditions. They proposed a relationship like : z 0 = a 1 H s (U 10 /c) b1 (η X ) c1 . [START_REF] Smith | Sea surface wind stress and drag coefficients: the hexos results[END_REF] have suggested that celerity and steepness are the wave properties more likely to influence the surface roughness z 0 in finite depth environment because celerity is restricted by water depth and wave age is shorter at short fetch. [START_REF] Gao | Wave-dependence of friction velocity, roughness length, and drag coefficient over coastal and openwater surfaces by using three databases[END_REF] using three different databases, illustrated the increase of z 0 with low depth, because the finite depth change the wave celerity c, and by the way the inverse wave age (U 10 /c). They also proposed a model of z 0 with the inverse wave age U 10 /c. [START_REF] Oost | New evidence for a relation between wind stress and wave age from measurements during asgamage[END_REF]] also took into account the wave steepness in their model of z 0 , the waves being steeper when the depth is low, but they do not include parameter kh in the z 0 estimation. [START_REF] Macmahan | Increased aerodynamic roughness owing to surfzone foam[END_REF] related the increase of roughness length z 0 near the coasts in finite depth to surfzone foam coverage due to wave breaking. But in our experiment we did not observe wave breaking although roughness parameter z 0 was higher in low depth conditions.

Depth influence on Charnock parameter α Ch

In open seas, the Charnock parameter α Ch is generally assumed constant, although a range of values have been reported from measurements: 0.014 [START_REF] Garratt | Review of drag coefficients over oceans and continents[END_REF], 0.018 [START_REF] Wu | Wind stress coefficient over sea surface near neutral conditions[END_REF], 0.019 [START_REF] Geernaert | Variation of the drag coefficient and its dependance on sea state[END_REF], 0.035 [START_REF] Kitaigorodskii | On the roughness parameter of the sea surface and the calculation of momentum flux in the near water layer of the atmosphere[END_REF] among others. [START_REF] Vickers | Fetch limited drag coefficients[END_REF]] conducted measurements by fetch length and they measured α Ch = 0.018 for onshore and α Ch = 0.073 for offshore wind conditions during the RASEX campaign [START_REF] Lüko | Observation of wave-driven air-water turbulent momentum exchange in a large but fetch-limited shallow lake[END_REF], i.e. at very short fetches and intermediate water depths. Above the water surface in the intertidal area and in the inner-surf and swash zones, which means in very shallow water environment and even sometimes with no water but along the shoreline, during the Stradbroke 2012 campaign, [START_REF] Shabani | Direct measurements of wind stress over the surf zone[END_REF] observed Charnock coefficients an order of magnitude greater than open ocean values : the average Charnock coefficient of their data was α Ch = 0.110 ! To our knowledge, these are the highest observed α Ch values so far. They explained this large increase of Charnock coefficients by the slower celerity in the surf zones, the breaking of the waves when they reach the shore and more particularly the sawtooth wave shape in the inner surf zone and swash zone, with asymmetric and very steep waves. From Lake George measurements, [START_REF] Toffoli | On the variability of sea drag in finite water depth[END_REF] have proposed a parametrisation of α Ch using an inverse quadratic function of a complex non-linear parameter µ depending on wave steepness and water depth.

Figure (10) shows the evolution of our estimated α Ch data with the non-dimensional depth parameter kh. For high depth, our mean α Ch value is 0.018, which is the same value reported by [START_REF] Wu | Wind stress coefficient over sea surface near neutral conditions[END_REF] and [START_REF] Vickers | Fetch limited drag coefficients[END_REF]. This value of 0.018 was also proposed by [START_REF] Charnock | Wind stress on a water surface[END_REF]. But for low depth configuration (i.e. kh < 1.8), α Ch is 5 times higher: 0.086 . This value is slightly greater than the RASEX campaign Charnock value for offshore winds [START_REF] Lüko | Observation of wave-driven air-water turbulent momentum exchange in a large but fetch-limited shallow lake[END_REF], but slightly lower than the Stradbroke campaign value recorded along the shore line [START_REF] Shabani | Direct measurements of wind stress over the surf zone[END_REF]. However, one of our measurements (run 1, α Ch = 0.107) is close to equal to their mean recorded value: α Ch = 0.110. The depth effect on Charnock parameter appears clearly on our figure (10). 4.2. Depth influence on wave growth [Montalvo et al., 2013 a,b] and [START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF] have shown that the limit for which the growth rate γet goes to zero was defined by the equation Figure (11) shows the location of the 15 experiments in the (θ f d , δ) plane, together with the [Montalvo et al., 2013 a] limit and the data of [START_REF] Donelan | Wave-follower field measurements of the wind-input spectral function. part 1 : Measurements and calibrations[END_REF][START_REF] Donelan | Wave-follower field measurements of the wind-input spectral function. part 2 : Parameterization of the wind input[END_REF], from the Australian Shallow Water Experiment. This figure shows the wide range of δ and θ f d values covered by our experiments. Even if the relative errors are quite high, values remains below the physical limit for which sea state evolution (wave amplification) is not possible. Figure (12) displays the evolution of the non-dimensional frequency f 0 U 10 /g with non-dimensional depth gh/U 2 10 . This graph shows that are data are inside the asymptotic non-dimensional limit-law proposed by [Young and Verhagen, 1996 a;[START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF]]: f0U10 g = 1.25( gh U 2 10 ) -0.45 . Our data are closer to the [Young and Verhagen, 1996 a] limit than the [START_REF] Bretschneider | Revised wave forecasting relationships[END_REF] Figure (13) displays the evolution of the inverse wave age U 10 /c 0 with non-dimensional depth gh/U 2 10 . This graph shows that the low depth limits the wave age: the wave age cannot be beyond the non-dimensional depth asymptotic limit proposed by [START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF]

θ c f d = √ δ.
: U10 c0 = 1.25( gh U 2 10 ) -0.45 .
Figure ( 14) displays the evolution of the non-dimensional energy g 2 σ 2 /U 4 10 with non-dimensional depth gh/U 2 10 . This graph shows that our data are inside the asymptotic limit-laws proposed by [Young and Verhagen, 1996 a [START_REF] Bretschneider | Revised wave forecasting relationships[END_REF]CERC, 1977] :

] : g 2 σ 2 U 4 10 = 1.06 10 -3 ( gh U 2 10 ) 1.3 and
g 2 σ 2 U 4 10 = 1.40 10 -3 ( gh U 2 10
) 1.5 . The low depth limits the total energy of the sea state. Waves cannot be amplified beyond a threshold value depending on the non-dimensional depth.

The non-dimensional growth rate γet is now compared in figure (15) with the theoretical curves of [Montalvo et al., 2013 a]. The existence of a finite depth h transforms the unique Miles curve of wave growth rate in deep water in a family of curves indexed by δ = gh/U 2 1 , i.e., a curve for each value of δ. [START_REF] Bretschneider | Revised wave forecasting relationships[END_REF]CERC, 1977] The experimental measurements follow more or less the general tendencies of the theoretical curves: a) the grow rate is lower for low values of δ and large values of θ f d ; b) the growth rate with low δ can be sometimes two orders of magnitude lower than run with higher δ. c) usually, but not always, experimental values of γet are higher than the theoretical values. If we draw linear regression lines for several bins of δ values, we find however a good general trend of the experimental measurements with a decrease towards no amplification at high θ f d and low δ. These regression lines are drawn in figure ( 16). The intersection of the regression lines with quite zero non-dimensional growth-rate (here γet = 10 -6 ) gives the critical θ c f d experimental value for the bin of δ data considered, and thus the corresponding δ c = (θ c f d ) 2 value. We note that for each bin of δ, the critical δ c belongs to that bin : 3.42 ∈ [0 6.5], 12.7 ∈ [6.5 21], 34.3 ∈ [21 36], 68.6 ∈ [36 ∞]. This decrease towards zero amplification at θ c f d was the key point developed by the [Montalvo et al., 2013 a,b;[START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF] theory. It is the first time that such a theory is illustrated by wave tank experiments.

In order to compare the measured wave growth with previous well known experiments, we have plotted our computed values of γ et /f 0 versus inverse wave age u * /c 0 in figure ( 17), together with data compiled by [START_REF] Plant | A relationship between wind stress and wave slope[END_REF], as they are shown in figure 4.2, page 52 of [START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF]. Data came from [START_REF] Larson | Wind-generated gravity-capillary waves: Laboratory masurements of temporal growth rates using microwave backscatter[END_REF][START_REF] Wu | The energy transfer due to air-input, non-linear wave-wave interaction and white-cap dissipation associated with wind-generated waves[END_REF][START_REF] Wu | Experimental study of nonlinear wave-wave interaction and white-cap dissipation of wind-generated waves[END_REF][START_REF] Snyder | Array measurements of atmospheric pressure fluctuations above surface gravity waves[END_REF][START_REF] Shemdin | Direct measurement of aerodynamic pressure above a simple progressive gravity wave[END_REF]. We see clearly that some of measured growth rates are right in the [START_REF] Plant | A relationship between wind stress and wave slope[END_REF] tendency, lying along Miles and [Snyder et al., 1981] curves, but others are completely below the proposed range, close to zero amplification. As we saw before, according to [Montalvo et al., 2013 a] theory, this happens because δ is not high and θ f d is not small. In shallow water environment, amplification factor by the wind may be sometimes not efficient.

The comparison between the non-dimensional experimental β Miles parameter and the theoretical ones of [Montalvo et al., 2013 a] is shown in figure (18). Some measured β values seem much higher than theoretical β values. We have done those runs several times and we always found these large β values. In fact, those values seem high in comparison with the [Montalvo et al., 2013 a] theory, but they are not so high if we compare them to previous experimental studies done in the past on the wave growth of wind waves, as shown on figure (19). Our β values are more or less on the lower part of the figure, and we see clearly that some values are very close to zero, if compared with previous published data. In another paper [START_REF] Troitskaya | Quasi-linear approximation for description of turbulent boundary layer and wind wave growth[END_REF], figure (2-c), all the β values computed from a precise DNS model, and also by a quasi-linear model of waves amplified by wind, are all above 20, i.e. above our β measured data.

Regression lines between β and θ f d , for several bins of δ values, are shown in figure (20). The intersection of these lines with the β = 0 axis gives the critical θ c f d experimental value for the bin of δ data considered, and thus the corresponding δ c = (θ c f d ) 2 value. Like in figure( 16), we note again that for each bin of δ, the critical δ c belongs to that bin : 3.3 ∈ [0 6.5], 8.6 ∈ [6.5 21], 26.0 ∈ [21 36], 50.4 ∈ [36 ∞]. This experiment shows that, following [Montalvo et al., 2013 a,b;[START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF] development, wind-wave growth rate by Miles's mechanism in finite water depth environment could be very low.

Conclusions

Experiments were conducted in a wind-wave facility to study momentum transfer from air to water and the growth rate of the waves by the wind for different configurations of water depth. Fifteen different runs were done. Some runs were done many times to check the uncertainties on the measurements. All the error bars on the different graphs were plotted. Wind speeds and wave steepnesses [START_REF] Plant | A relationship between wind stress and wave slope[END_REF] and displayed in [START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF] page 52 (see also [START_REF] Snyder | Array measurements of atmospheric pressure fluctuations above surface gravity waves[END_REF][START_REF] Larson | Wind-generated gravity-capillary waves: Laboratory masurements of temporal growth rates using microwave backscatter[END_REF][START_REF] Wu | The energy transfer due to air-input, non-linear wave-wave interaction and white-cap dissipation associated with wind-generated waves[END_REF][START_REF] Wu | Experimental study of nonlinear wave-wave interaction and white-cap dissipation of wind-generated waves[END_REF][START_REF] Shemdin | Direct measurement of aerodynamic pressure above a simple progressive gravity wave[END_REF]). Solid line: Miles Theory, Dashed line : [START_REF] Snyder | Array measurements of atmospheric pressure fluctuations above surface gravity waves[END_REF] interpolation. other experiments (see legend), data gathered by [START_REF] Plant | A relationship between wind stress and wave slope[END_REF] and displayed in [START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF] page 53 (see also [START_REF] Snyder | Array measurements of atmospheric pressure fluctuations above surface gravity waves[END_REF][START_REF] Larson | Wind-generated gravity-capillary waves: Laboratory masurements of temporal growth rates using microwave backscatter[END_REF][START_REF] Wu | The energy transfer due to air-input, non-linear wave-wave interaction and white-cap dissipation associated with wind-generated waves[END_REF][START_REF] Wu | Experimental study of nonlinear wave-wave interaction and white-cap dissipation of wind-generated waves[END_REF][START_REF] Shemdin | Direct measurement of aerodynamic pressure above a simple progressive gravity wave[END_REF]). were relatively low to avoid wave breaking, aerodynamics effect of high steepnesses on the air-flow. Several studies have been conducted in the past on the effect of shallow water on drag coefficients and momentum transfers, but until this one, no specific work has been done on water tank experimental evidence of wave growth by the wind when the water depth is low without breaking events. Although still limited, the results derived from this unique wind wave tank experiment shed new light on the relationship between drag, U 10 , friction velocity, roughness length, wave growth (γ-types or β-types), and other non-dimensional parameter (depth, energy, frequency)

We confirm that the dependence of C d on wind speed is modified by water depth. Low depth increases Drag coefficient C d , roughness length z 0 , Charnock parameter α Ch . This implies that for a given wind speed, momentum transfer from air to water is higher when the water depth is low.

All data are within the [START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF]Young and Verhagen, 1996 a;[START_REF] Bretschneider | Revised wave forecasting relationships[END_REF] depth asymptotic limits. So, even if our measurements are not open field measurements, they follow well the natural limits observed in real open seas.

When the water depth is very low, the growth rate decreases zero amplification for some combinations of non-dimensional parameters δ = gh U 2 1 = ghκ 2 u * 2 and θ f d = κ u * g k tanh kh. The non-dimensional water depth parameter δ depending on depth and on a characteristic wind speed, induces theoretically a family of curves representing the wave growth as a function of the wave phase velocity and the wind speed Montalvo et al. [2013 a,b]; [START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF]. Our experimental data follow those theoretical curves, with a decrease towards null amplification for some low depth values in conjugation with particular wind speeds. To date, no fully consistent and conclusive theory of wave amplification by wind exists. We are focused here on the Miles' mechanism, because we used quite low wind speed and low wave steepnesses conditions to avoid breaking and/or air flow separation that could be generated by steep waves, but probably the Miles' mechanism is not the only mechanism involved in the amplification of waves in finite depth conditions.

At the water surface the momentum transfer τ = ρ a u * 2 is split into two distinct parts: the form drag τ f orm , and the viscous drag τ viscous [START_REF] Makin | Impact of dominant waves on s ea drag[END_REF][START_REF] Donelan | Modeling waves and wind stress[END_REF]: τ = τ f orm + τ viscous . The form drag τ f orm is the part of the momentum transfer that goes to the amplification of the waves, i.e. it induces directly the wave growth by the wind [START_REF] Grare | Growth and dissipation of wind-forced, deep-water waves[END_REF]. The form drag arises from the form of the waves and becomes higher as the steepness of the waves increases. The steeper the waves the greater the contribution of the wave drag to the total drag. The viscous drag τ viscous , also called surface tangential viscous stress or skin friction drag [START_REF] Veron | Measurements of the viscous tangential stress in the airflow above wind waves[END_REF][START_REF] Buckley | Surface viscous stress over wind-driven waves with intermittent airflow separation[END_REF], comes about through the direct molecular interaction at the interface. This viscous drag is the part of the momentum transfer that goes to the generation and amplification of the surface current, i.e. the surface drift. in this study, we have shown that, in low depth conditions, a) momentum flux τ increases and b) wave growth decreases, i.e. τ f orm decreases. Then, by simple deduction from τ viscous = τ -τ f orm , because τ increases and τ f orm decreases, the viscous drag τ viscous should increase significantly when the water depth is low.

From a theoretical point of view Miles theory in finite depth [Montalvo et al., 2013 a] was recently extended into the case with vorticity [START_REF] Kern | Miles' mechanism for generating surface water waves by wind, in finite water depth and subject to constant vorticity flow[END_REF]. In the future, it would be interesting to conduct experiments in this wave tank to measure the influence of water depth on the surface wind drift current and water vorticity. This kind of experimental study, to our knowledge, has never previously been done. More data and analysis are needed to confirm these findings. Another point is that all the theories and developments on wave growth over finite depth involve water of constant depth, which is very restrictive. Even in the most ideal of field basins such a situation never exists. Variable depth effect on air-sea interactions is still an open issue.

Figure 1 :

 1 Figure 1: External view of the IRPH É/Pythéas wind-wave facility

Figure 2 :

 2 Figure 2: Internal view of the IRPH É/Pythéas wind-wave facility: the height of the downwind end of the air-divergent is adjustable, depending upon the water depth. The wavemaker is under the air-divergent. Mechanical waves do not hit the air divergent.

Figure 3 :

 3 Figure 3: View of the experimental arangment inside the IRPH É/Pythéas wind-wave facility. We used one DISA capacitance wave gauge at fetch x 0 = 2.40 m, another one at fetch x 1 = 21.10 m, a DANTEC Constant Temperature Anemometer (CTA) located on a telescopic arm that can move vertically from the water surface up to 1 m above the waves, thus allowing us to have vertical profiles of horizontal wind speed U (z).

Figure 4 :

 4 Figure 4: Sketch of the IRPH É/Pythéas wind-wave tank facility showing the location of the measuring devices

  a]. An example of time series of the water elevation recorded by wave gauges during run 5 at fetch x 0 = 20.1 m, without wind and with wind on (U 10 = 5.10 m/s is shown on Figure (6). The amplification by the wind of the dominant waves appears clearly on the plot. Looking at Figure (6) it seems possible to compute directly a temporal

Figure 5 :

 5 Figure 5: Two examples of logarithmic vertical wind profiles, with the estimated u * and z 0 values (runs 6 and 9). z is the altitude above the mean water level.

Figure 6 :

 6 Figure 6: Example of time series of water elevation recorded by the wave gauges during run 5 at fetch x 1 = 20.1 m, without wind and with wind on.

Figure 7 :

 7 Figure 7: Density energy spectra of recorded waves during run 5. (a) fetch x 0 = 2.4 m no wind; (b) x 1 = 20.1 m no wind , with axis frequency multiplied by 10; (c) x 0 = 2.4 m U 10 = 5.10 m/s , with axis frequency multiplied by 10 2 ; (d) x 1 = 20.1 m U 10 = 5.10 m , with axis frequency multiplied by 10 3 .

4. 1 .

 1 Depth influence on air-to-water transfers 4.1.1. Depth influence on drag coefficient C d Figure (8) shows the evolution of measured C d10 with estimated U 10 . Drag coefficients in low water depth environment (kh < 1.8) are clearly higher than those in large depth configurations. In open Run

Figure 8 :

 8 Figure 8: Drag coefficient C d10 as function of U 10 . Drag coefficients are higher for low water depth (i.e kh < 1.8)

Figure 9 :

 9 Figure9: Roughness parameter z 0 as function of U 10 . z 0 values are higher for low water depth (i.e kh < 1.8), and the trend is six times higher.

Figure 10 :

 10 Figure 10: Charnock parameter α Ch = gz 0 u * 2 as function of non-dimensional depth parameter kh. α Ch values are higher for low water depth (i.e kh < 1.8)

  Figure (15) shows a theoretical family of γet for four values of δ: δ th = 4, 9, 25, 49 against θ f d parameter. In the same figure are plotted the experimental values of γet . These data are partitioned into four ranges of δ i.e., δ ∈ [0 6.5], [6.5 21], [21 36], [36 ∞]. Small finite values of θ f d correspond to short surface waves characterizing young wind waves. For time increasing the wave age θ f d increases which

Figure 11 :

 11 Figure 11: Location in the (θ f d , δ) plane of the 15 runs. Each run number is printed at its location with the corresponding error bars. The black line is the [Montalvo et al., 2013 a] limit (location of the critical θ c f d values)where there is no amplification by the wind. Black dots are the points from the Australian Shallow Water Experiment[START_REF] Donelan | Wave-follower field measurements of the wind-input spectral function. part 2 : Parameterization of the wind input[END_REF].

Figure 12 :

 12 Figure 12: Non-dimensional frequency f 0 U 10 /g evolution with the non-dimensional depth gh/U 2 10 . Run numbers are written on the plot. Full symbols with error bars : measured values for δ ∈ [0 6.5] (▼), δ ∈ [6.5 21] (•), δ ∈ [21 36] (▲), δ ∈ [36 ∞](■). Straight black line: [Young and Verhagen, 1996 a] depth asymptotic limit : f 0 U 10 g = 0.20( gh U 2 10

Figure 13 :

 13 Figure 13: Inverse wave age U 10 /c 0 evolution with the non-dimensional depth gh/(U 2 10 ). Run numbers are written on the plot. Full symbols with error bars : measured values for δ ∈ [0 6.5] (▼), δ ∈ [6.5 21] (•), δ ∈ [21 36] (▲), δ ∈ [36 ∞](■). Straight black line: [Young, 1997] limit : U 10 c 0 = 1.25( gd U 2 10

Figure 14 :

 14 Figure 14: Non-dimensional energy g 2 σ 2 /U 4 10 evolution with the non-dimensional depth gh/(U 2 10 ). Run numbers are written on the plot. Full symbols with error bars : measured values for δ ∈ [0 6.5] (▼), δ ∈ [6.5 21] (•), δ ∈ [21 36] (▲), δ ∈ [36 ∞](■). Straight black line: [Young and Verhagen, 1996 a] limit : g 2 σ 2 U 4 10

Figure 15 :

 15 Figure 15: Comparison of the non-dimensional growth rate γet values measured in the facility with the theoretical curves of [Montalvo et al., 2013 a]. Empty symbols: theoretical values for δ th = 4 (∇), δ th = 9 (•), δ th = 25 (△), δ th = 49 (2). Full symbols with error bars : measured values for δ ∈ [0 6.5] (▼), δ ∈ [6.5 21] (•), δ ∈ [21 36] (▲), δ ∈ [36 ∞](■). The value of δ for each experimental run is written inside the figure.

  corresponds to mild or moderate wavelengths. The waves develop and the growth rate γet gradually decreases due to the effects of finite depth. As γet goes to zero, each δ-curve approaches the theoretical θ f d -limited growth, θ c f d , given by θ c f d = δ 0.5 . Hence for δ th = 4, 9, 25, 49 the values of θ c f d for γet are: θ c f d = 2, 3, 5, 7. The wave reaches a final state of linear progressive wave with zero growth. In other words, for a given δ the surface wave does not grow old any more beyond a determined θ c f d .

Figure 16 :

 16 Figure 16: Same as figure (15), but we have added linear regression lines between log 10 γet and θ f d . Empty symbols: theoretical values for δ th = 4(∇), δ th = 9(•), δ th = 25(△), δ th = 49(2). Full symbols and regression lines: experimental values: dotted line and (▼) : δ ∈ [0 6.5] , dashed line and (•), δ ∈ [21 36] (▲), δ ∈ [36 ∞](■) : δ ∈ [36 ∞]. The critical experimental values of θ c f d and δ c for which there is no amplification (γet = 10 -6 ) by bins of δ are written in the figure.

Figure 17 :

 17 Figure17: Non-dimensional growth rate γet/f 0 versus inverse wave age u * /c 0 . Solid symbols : our experiment. Open symbols : other experiments, data gathered by[START_REF] Plant | A relationship between wind stress and wave slope[END_REF] and displayed in[START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF] page 52 (see also[START_REF] Snyder | Array measurements of atmospheric pressure fluctuations above surface gravity waves[END_REF][START_REF] Larson | Wind-generated gravity-capillary waves: Laboratory masurements of temporal growth rates using microwave backscatter[END_REF][START_REF] Wu | The energy transfer due to air-input, non-linear wave-wave interaction and white-cap dissipation associated with wind-generated waves[END_REF][START_REF] Wu | Experimental study of nonlinear wave-wave interaction and white-cap dissipation of wind-generated waves[END_REF][START_REF] Shemdin | Direct measurement of aerodynamic pressure above a simple progressive gravity wave[END_REF]). Solid line: Miles Theory, Dashed line :[START_REF] Snyder | Array measurements of atmospheric pressure fluctuations above surface gravity waves[END_REF] interpolation.

Figure 18 :

 18 Figure 18: Comparison of the non-dimensional growth rate β measured in the facility with the theoretical curves of [Montalvo et al., 2013 a]. Empty symbols: β theoretical values for δ th = 4 (∇), δ th = 9 (•), δ th = 25 (△), δ th = 49 (2). Full symbols with error bars : β measured values for δ ∈ [0 6.5] (▼), δ ∈ [6.5 21] (•), δ ∈ [21 36] (▲), δ ∈ [36 ∞](■). The value of δ for each experimental run is written in the figure.

Figure 19 :

 19 Figure 19: Non-dimensional growth rate β versus wave age c 0 /u * . Solid symbols : our experiment. Open symbols :other experiments (see legend), data gathered by[START_REF] Plant | A relationship between wind stress and wave slope[END_REF] and displayed in[START_REF] Young | The growth rate of finite depth wind-generated waves[END_REF] page 53 (see also[START_REF] Snyder | Array measurements of atmospheric pressure fluctuations above surface gravity waves[END_REF][START_REF] Larson | Wind-generated gravity-capillary waves: Laboratory masurements of temporal growth rates using microwave backscatter[END_REF][START_REF] Wu | The energy transfer due to air-input, non-linear wave-wave interaction and white-cap dissipation associated with wind-generated waves[END_REF][START_REF] Wu | Experimental study of nonlinear wave-wave interaction and white-cap dissipation of wind-generated waves[END_REF][START_REF] Shemdin | Direct measurement of aerodynamic pressure above a simple progressive gravity wave[END_REF]).

Figure 20 :

 20 Figure 20: Same as figure (18), but we have added linear regression lines between β and θ f d : Empty symbols: theoretical values for δ th = 4 (∇), δ th = 9 (•), δ th = 25 (△), δ th = 49 (2). Full symbols and regression lines: experimental values: dotted line and (▼) : δ ∈ [0 6.5] , dashed line and (•), δ ∈ [21 36] (▲), δ ∈ [36 ∞](■) : δ ∈ [36 ∞]. The critical experimental values of θ c f d and δ c for which there is no amplification (β = 0) by bins of δ are written in the figure.

Table 1 :

 1 Experimental conditions. h is the water depth, f 0 is the paddle wave frequency, L 0 is the wavelength of the paddle waves, (ak) 0 is the steepness of the paddle waves measured without wind at fetch x 0 , U 10 is the wind speed extrapolated to the standard altitude of 10m.

					ak) 0 U 10
		m	Hz	m	m/s
	1	0.14 1.7 0.51 1.78 0.120 3.18
	2	0.14 1.7 0.51 1.78 0.121 5.58
	3	0.14 1.2 0.85 1.07 0.100 5.70
	4	0.26 1.7 0.54 3.03 0.124 3.39
	5	0.26 1.7 0.54 3.03 0.124 5.10
	6	0.26 1.4 0.77 2.10 0.166 3.46
	7	0.26 1.4 0.77 2.10 0.168 5.43
	8	0.48 1.7 0.54 5.58 0.091 5.70
	9	0.48 1.4 0.80 3.79 0.126 5.71
	10	0.48 1.1 1.27 2.38 0.119 3.60
	11	0.74 1.7 0.54 8.59 0.026 3.96
	12	0.74 1.7 0.54 8.59 0.026 5.79
	13	0.74 1.7 0.54 8.59 0.034 3.92
	14	0.74 1.7 0.54 8.59 0.034 5.79
	15	0.74 1.4 0.80 5.83 0.051 5.81

Table 3 :

 3 Relative errors on estimated values. The measured values of ( ā0 ) |U0 , ( ā0 ) |U ref , ( ā1 ) |U0 , ( ā1 ) |U ref are displayed in Table

Table 4 :

 4 Experimental measured values: mean wave amplitudes at fetch x 0 without wind ( ā0 ) |U 0 and with wind ( ā0 ) |U ref , and at fetch x 1 without wind ( ā1 ) |U 0 and with wind ( ā1 ) |U ref . Estimated values: celerity c 0 , group velocity C g0 , wave age c 0 /u * , non-dimensional parameters δ, θ dw , θ f d , γet, β and γet/f 0 .

Acknowledgements: H.B thanks IRPHE "appel d'offres interne" for the financial support of wind sensors. M.A.M. thanks the KIM Sea and Coast project for their funding.

Credit authorship contribution statement. H. Branger: experiments, software and data processing, original draft preparation, writing, editing, final manuscript. M.A. Manna: conceptualization, supervision, writing, editing, funding. C. Luneau: experiments. M. Abid: expertise, methodology, review. Ch. Kharif: supervision, investigation.

Data statements: Experimental data collected and analysed in this article can be obtained upon request from author Hubert Branger at branger@irphe.univ-mrs.fr

Conflicts of Interest:

The authors declare no conflict of interest.

Appendix A. Relative errors

By making several times the same experiment, the relative precision on ( ā0 ) U0 , ( ā0 ) U ref , ( ā1 ) U0 , and ( ā1 ) U ref was estimated to be 8%: dā ā = 8%. The precision on u * and on U 10 is 15%, the precision on z 0 is 30%. du * u * = 15%, dU10 U10 = 15%, dz0 z0 = 30%. These error values may appear quite important but they are in the range values that can be found in literature for wave tank measurements [START_REF] Zavadsky | Characterization of turbulent airflow over evolving water-waves in a wind-wave tank[END_REF].

Since C d10 = ( u * U10 ) 2 , the relative precision on C d10 is:

u * 2 , the relative precision on α Ch is : dα Ch α Ch = dz0 z0 + 2 du * u * = 60%. The precision on f 0 is 2%: df0 f0 = 2% . The precision on h is 1%: dh h = 1%, the precision on x is 0.5% : dx x = 0.5% . Since L 0 ≈ g 2πf 2 0 , we have dL0 L0 = 2 df0 f0 = 4% . The precision on non-dimensional depth is: d(k0h) k0h = dk0 k0 + dh h = dL0 L0 + dh h = 5% . The precision on steepness is: d(ak)0 (ak)0 = da a + dk0 k0 = 12% . The precision on celerity is: since c 0 ≈ g ω , dc0 c0 = dω ω = df0 f0 = 2% . C g0 = dω dk . We estimated the precision on C g0 as :

Cg0 ≈ dω ω + dk k = 6% . The reference wind speed speed U 1 was taken as U 1 = u * /κ [Montalvo et al., 2013 a,b;[START_REF] Latifi | Linear and weakly nonlinear models of wind generated surface waves in finite depth[END_REF].

From equation ( 23) the precision on γet is:

Cg0 + 4 dā ā + 2 dx x = 15% + 6% + 32% + 1% = 54%.

From equation ( 24) the precision on β is:

Cg0 + 4 dā ā + 2 dx x = 30% + 4% + 2% + 6% + 32% + 1% = 75%.

From equation ( 25) the precision on γ et /f 0 is:

Cg0 + df0 f0 + 4 dā ā + 2 dx x = 6% + 2% + 32% + 1% = 41%.

By making several times the same experiment, the relative precision on σ, the standard deviation of the water elevation, was found to be the same as the precision on the mean amplitude ā : dσ σ = dā ā = 8%. The precision on U 10 /c 0 is : (dU10/c0) U10/c0 = dU10 U10 + dc0 c0 = 15% + 2% = 17% The precision on gh/U 2 10 is :

h + 2 dU10 U10 = 1% + 2(15)% = 31% The precision on f 0 U 10 /g is : d(f0U10/g) f0U10/g = df0 f0 + dU10 U10 = 2% + 15% = 17% The precision on g 2 σ 2 /U 4 10 is :

d(g 2 σ 2 /U 4 10 ) g 2 σ 2 /U 4 10 = 2 dσ σ + 4 dU10 U10 = 2(8)% + 4(15)% = 76%.