Yann Balland

Luc Absil

Franck Pereira

Dos Santos

Quectonewton local force sensor

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Quectonewton local force sensor

Yann Balland, Luc Absil, and Franck Pereira Dos Santos LNE-SYRTE (Dated: October 21, 2023) We report on the realization of a quantum sensor based on trapped atom interferometry in an optical lattice for the measurement of atom-surface interactions, with sub-micrometer-level control of the mean atom-surface separation distance. The force sensor reaches a short-term sensitivity of 3.4 × 10 -28 N at 1 s and a long-term stability of 4 qN (4 × 10 -30 N). We perform force measurements in the 0-300 µm range, and despite significant stray forces caused by adsorbed atoms on the surface, we obtain evidence of the Casimir-Polder force.

Short-range forces are one of the many frontiers of modern physics [1,2]. In the submillimeter scales, quantum electrodynamics (QED) interactions are dominant, and give rise in the case of atom-surface interactions to the Casimir-Polder force [3]. Since the first highlight of this force [4], several different methods [5] have been able to bring out Casimir-Polder forces, notably by measuring the transmission of an atomic beam through a micronsized cavity [6], diffracting matter waves on a surface [7] or performing spectroscopy in vapor cells [8,9]. However these approaches have struggled to achieve the high measurement sensitivity required to detect the very weak forces involved all while maintaining a good understanding of the setup geometry, particularly the distance separating atoms from the surface.

Few experiments have achieved measuring Casimir-Polder forces while controlling directly the atom-surface distance. In the range from tens to hundreds of nanometers, the Casimir-Polder potential was measured directly by reflecting the atoms on an evanescent field [10,11]. In the micrometer range (around 6 µm), the Casimir-Polder effect has been highlighted by tracking the oscillations of a Bose-Einstein Condensate (BEC) inside a harmonic trap [12], a method that does not measure the force though, but rather its gradient. An appealing method that we will use here to perform spatially resolved and sensitive force measurements is to trap atoms during extended measurement times in a shallow vertical optical lattice.

The lattice potential, tilted by gravity g or any other external uniform force F e , leads to a Wannier-Stark Hamiltonian. Its eigenstates |W m ⟩ are localized in every well m of the lattice and separated by the Bloch frequency ν b = (m Rb g +F e )λ l /(2h), where m Rb is the mass of the atom and λ l the wavelength of the lattice beam [13] (Fig. 1). The Wannier-Stark ladder holds even if the F e (z) force is not uniform but remains perturbative relatively to the gravity force m Rb g. Different protocols have been proposed to measure Casimir-Polder forces using such optical lattices, based for instance on performing interferometry with a BEC [14] or measuring the frequency shift δν b of Bloch oscillations [15,16]. In the experiment reported here, which shares common features with these last two methods, we use Raman transitions to couple two eigenstates distant of ∆m wells apart [17], as illustrated in Fig. 1. Using Rabi spectroscopy or Ramsey interferometry, we then probe the corresponding transition energy ∆m ν b with the very high sensitivity associated to frequency measurements with an atomic clock, and thus the external force applied to the atoms.

|𝐹 = 1⟩ |𝐹 = 2⟩ m=1 m=2 m=3 m=4 m=5 𝑧 𝐸 Ω Δ𝑚 ⃗ 𝑔 𝜆 𝑙 /2 ℎ𝜈 𝐵 ℎ𝜈 𝐻𝐹 𝑆 Mirror
We present here the features and performances of such a high sensitivity quantum sensor for local force measurements. We demonstrate a short-term sensitivity of 3.4×10 -28 N at 1 s and a long-term stability of 4×10 -30 N on the force measurements, which allows us to measure interaction forces at the percent level in the range of tens of micrometers. Finally, we compare our measurements with the expected Casimir-Polder force.

87 Rb atoms are first laser cooled in a magneto-optical trap, before being transferred inside a crossed optical dipole trap. A 2 s long evaporative cooling stage leaves us with about 120 000 atoms at a temperature of 300 nK. The vertical size of the cloud is estimated to an rms radius of 10 µm. By adiabatically ramping up the dipolar trap at the end of the evaporation, we reduce this size by a factor of 3, at the cost of heating the atoms up to 1500 nK. This preparation step is done in a first chamber, 30 cm below our surface of interest, a silica dielectric dichroic mirror, highly reflective at 532 nm for our measurement shallow lattice, but transparent at 780 and 1064 nm, the wavelengths of our other lasers. The transport of the atomic cloud to the vicinity of this mirror is performed with a moving lattice described thoroughly in [19]. Two counter-propagating beams (Fig. 2) are detuned with a controllable frequency difference, creating a moving lattice of controllable speed in which the atoms are trapped. This setup allows a very fine control on the transport distance ∆z with a resolution in the tens of nanometers.

This method provides us with a precise knowledge of the travelled distance ∆z from the dipole trap [19]. However, the distance z at between the atomic cloud and the surface is not directly known. This distance depends on the exact distance z tot = z at + ∆z between the dipole trap and the mirror which fluctuates depending on beam alignments and thermal variations. To calibrate this distance z tot we use the mirror surface directly as a position reference by transporting our atomic cloud in its vicin- ity, similarly to [16]. Since atoms kicked into the surface are lost, the exact position of the surface can be determined by measuring the number of remaining atoms, as shown in Fig. 3. Assuming a Gaussian density distribution and fitting the atom number by an erf function, we extract the distance z tot , as well as the 1/e 2 size of the cloud. This lets us to adjust the atom-surface distance z with sub-micrometer uncertainty, limited by the fit uncertainty. The distance z tot remains stable at the 0.1 µm level during a few hours but fluctuates by a few micrometers over days, therefore we perform this distance calibration step twice an hour along the force measurements to track any fluctuation.

As for the vertical size σ z of our atom cloud, our measurements show that it remains unchanged during transport with a 1/e 2 size σ z of 3.5 µm.

We end up with 20 000 atoms at most after the transport, in a mixture of |F = 1⟩ and |F = 2⟩. Atoms in |F = 2⟩ are depumped in |F = 1⟩ before the transfer of the atoms in the green static shallow lattice used for the force measurement. Being blue-detuned, the light potential confines only in the vertical direction, so we superimpose (Fig. 2) an additional IR laser beam, progressive and red-detuned to transversally confines the atoms at the center of the lattice. After 500 ms trapping time, we are left with at most around 1000 trapped atoms in |F = 1, m F = 0⟩ to perform the measurements.

Two counterpropagating Raman lasers, detuned from resonance by 300 GHz, couple the |W m , F = 1⟩ and |W m+∆m , F = 2⟩ states for every initial well number m. These beams both enter the chamber at the bottom. At the top exit of the chamber, one of the two Raman beams (Raman 2) is reflected out by a cube while the other (Raman 1) is transmitted and retroreflected on a Raman mirror. This common implementation of the Raman lasers ensures the stability of the Raman phase difference, whose equiphases are tied to the position of the Raman mirror. Using Ramsey interferometry, one can thus probe the energy difference between the states |W m , F = 1⟩ and |W m+∆m , F = 2⟩ [18]. The two output ports of the interferometer being labeled by their hyperfine levels, their populations are measured with in situ state selective fluorescence imaging, using crossing pairs of retroreflected laser beams with an intensity of 8.5 mW/cm 2 slightly red-detuned from resonance to enhance the fluorescence collection time, up to 1 ms. An imaging system with a numerical aperture of 0.32 collects about 40 photons per atoms onto an Electron Multiplying CCD (EMCCD) camera.

To measure the Bloch frequency, we perform a Ramsey interferometer, with two Raman π/2 pulses separated by the interrogation time T . The output phase Φ for this interferometer is given by Φ = 2π(ν R -ν HF S -∆m ν B ) T where ν R is the frequency difference between the two Raman beams and ν HF S is the hyperfine splitting frequency. By tuning the Raman frequency difference to keep the output phase null, we get ν R = ν HF S + ∆m ν B .

To remove the dependence on the hyperfine frequency, which can slowly fluctuate with laser light shifts, we alternately perform measurements for ±∆m. The difference between the two measurements gives the Bloch frequency free from the hyperfine frequency ν

B = (ν +∆m R - ν -∆m R )/2∆m.
Note that all six vertical beams (Fig. 2) are overlapped, in order to avoid parasitic couplings to transverse states [20]. The uncertainty in the overall vertical alignment has been reduced down to 1.5 mrad using a liquid mirror as a reference, while all beams are superimposed on one another to better than 0.1 mrad.

We now quickly discuss the limitations of the sensitivity of the force measurement. The contrast of the interferometer is typically of 40 % for a coherence time of about 150 ms. It is mainly limited by inhomogeneities in the light shift. Additionally, at less than 20 µm of the mirror, inhomogeneities of the atom-surface force decrease further the interferometer contrast.

As for the measurement noise, we identified three main contributions. The first is Raman phase noise due to vibrations of the mirror of interest relatively to the Raman retroreflecting mirror. The support of the mirror of interest is attached to the end of a 3-axis in-vacuum manipulator, clamped to the overall structure with sorbothane pads used for isolation in order to suppress undesired vibration induced tunneling between neighbouring wells of the lattice [21]. Still, residual vibrations are estimated to induce fluctuations on ν B at the level of 480 mHz rms per shot. By implementing an optical Michelson interferometer that combines the incoming and retroreflected lattice laser beams, we are able to track vibrations of the mirror of interest relatively to the optical bench. By correcting a posteriori the Bloch frequency measurement from the effect of measured vibrations, similar to [22], we reduce the contributions of vibration noise by about 30%, down to the level of 300 mHz per shot.

The second noise source is the quantum projection noise, corresponding to 200 mHz for the typical atom number we had in the measurements, of the order of 500. The third is a background detection noise, equivalent to 20 atoms, which adds 190 mHz. These three contributions sum up (quadratically) to 410 mHz, which is in agreement with the measured noise of 430 mHz. At the end, since we use ∆m = 6, we reach a sensitivity on the measurement of the Bloch frequency of 71 mHz/shot.

Given our cycle time of 3.6 s, this corresponds to a sensitivity of 3.4 × 10 -28 N at 1 s. For an averaging time of 5.5 h, the noise averages down as white noise and a long-term stability of 4 quectonewtons is reached. This lies above the best sensitivity of 7 × 10 -30 N at 1 s reported both with this method [23] and an alternative one based on the amplitude modulation of the lattice [24], but far from any surface. But it outperforms any other local force measurements, such as based on monitoring the dynamics of trapped atoms [25] or ions [26][27][28], or of microscopic objects [29]. It is to the best of our knowledge state-of-the-art for surface force measurements.

This high sensitivity associated with the control of the distance to the mirror allows us to perform sensitive force measurements in the vicinity of the mirror. By performing interleaved differential measurements close to the mirror and further away at a distance of 300 µm, we suppress the gravity force and any long-range bias on the force, such a parasitic vertical dipole force from the IR transverse confinement beam. This dipole force has been evaluated by varying the IR power as an offset of the order of 3 × 10 -29 N with no resolved dependence on the distance to the mirror. These differential force measurements, displayed Fig. 4, exhibit a clearly attractive behaviour. As a reference, the effect expected from the Casimir-Polder force is also displayed as a green curve. It was calculated using the energy shifts of the Wannier Stark states due to the Casimir-Polder interaction, as computed in [30,[START_REF] Maury | Effet Casimir-Polder sur des atomes piégés[END_REF] for our exact experimental configuration. From a simulation of the fringe patterns averaged over the finite size of the cloud, we extract the frequency shift of the central fringe, and the corresponding expected force.

The observed force, which has a longer range and greater magnitude, cannot be explained by the Casimir-Polder effect alone. Another interaction is at play, likely due to electrostatic interactions between our atomic sample and electric dipoles by atoms adsorbed on the dielectric surface, as already put into evidence in [START_REF] Mcguirk | [END_REF].

Rb atoms adsorbed on neutral surfaces create dipole electric moments whose value has been evaluated to µ = 3.2 D in [33] for a silica substrate. An atom brought near the surface thus experiences an attractive potential

U elec = α0 2 | ⃗ E| 2
where α 0 is the static polarizability of the atom ground state and ⃗ E is the electrostatic field produced by these moments. So a force ⃗ F elec = -α0 2 ⃗ ∇| ⃗ E| 2 is exerted on our atoms, to the vertical component of which our sensor is sensitive.

We observed that we were able to reduce by half the force measured for distances above 20 µm, where the Casimir-Polder effect is weak, by simply turning off the experiment for a month. It returned to a new steady state after a week of operation. We interpreted this as a first evidence supporting the idea of stray electric fields from adsorbed atoms, which slowly desorb or diffuse, as already demonstrated in [33]. Furthermore, by applying uniform vertical external electric fields E z ext using electrodes surrounding the mirror as shown Fig. 2, we have been able to get a direct evidence of these electric fields. This amplifies the force F z elec by -α 0 E z ext d z E z and allows for measuring vertical gradients of the parasitic electric field d z E z . First measurements show a measured vertical electric gradient of the order of 2.0(1) × 10 7 V/m 2 at a distance of 20 µm, in marginal agreement with the gradient expected from a model described below (3.3(4) × 10 7 V/m 2 ).

To model these parasitic electric fields, we consider a Gaussian distribution of N dipole moments over an rms radius σ r : n(r) = N/(2πσ r 2 ) exp(-r 2 /(2σ r 2 )). The resulting electrostatic force along the vertical direction depends only on the two parameters N and σ r which we estimate by fitting our force measurements in the range z > 20 µm where the Casimir-Polder force is negligible, see Fig. 4. This gives us an estimate for σ r around 90 µm and N around 2.0×10 10 , corresponding to a radius twice as large as the radial width of the atomic cloud at the end of the transport, and a number of adsorbed atoms compatible with a continuous launch of atoms over several weeks of the running experiment, during which most of the atoms are not stopped at the end of the transport.

Remarkably, by using this estimated bias electrostatic force to correct our force measurements over the full range, we obtain good agreement between these corrected points, depicted with diamond points in Fig. 4, and the expected Casimir-Polder force in the range z < 20 µm.

In conclusion, we demonstrated a state-of-the art short-range force sensor using atom interferometry inside an optical lattice. In the regime of distance explored, the electrostatic force from adsorbed atoms predominates, but we achieved an evidence of Casimir-Polder force from a mirror in the micrometer range. A thorough characterization of stray electric fields would allow for a better comparison between expected and measured Casimir Polder forces, opening the way for tests of gravity at short range. The removal, or at least the reduction of the number of adsorbed atoms, thanks to the heating of the surface [33] would be an asset for such studies. We expect to get better spatial resolutions in the future, starting with even smaller clouds or implementing spatial selection. This selection could be realized by lifting the degeneracy between Wannier-Stark transitions, for instance by using magnetic field gradients [34], or by shaping the trapping potential as a superlattice [START_REF] Pereira Dos Santos | Frequency Standards and Metrology[END_REF]. The resulting loss in the number of atoms could be mitigated by improving the efficiency of the atom transport to increase the number of atoms. This would allow for even more sensitive and spatially resolved measurements of atom-surface interactions.

Figure 1 .

 1 Figure 1. Wannier-Stark ladder potential created by the optical lattice retroreflected on a mirror. Two neighbouring states |Wm, F = 1⟩ and |Wm+∆m, F = 2⟩ separated by the energy h(νHF S + ∆m νB) can be coupled by Raman transitions with a Rabi frequency Ω∆m. Close to the surface, Casimir-Polder interactions modify the external potential (dashed lines) and perturb the value of νB.

Figure 2 .

 2 Figure2. Scheme of the upper chamber setup, with the mirror of interest at its center. Two counter-propagating beams at 780 nm, red detuned from the Rb87 D2 line by 250 GHz, ensure the transport of the atoms, while two other counter-propagating Raman beams, detuned from the D2 line by 300 GHz, are used to drive the interferometer. The Raman 1 beam is actually retroreflected on a Raman mirror, not depicted here, located above the mirror of interest[18]. Atoms end up trapped vertically in a lattice resulting from the retroreflection of a green laser beam at λ l = 532 nm onto the surface of the mirror, and transversally with a progressive IR beam at 1064 nm. Four linear electrodes (displayed as grey circles) are used to produce external electrostatic fields.

Figure 3 .

 3 Figure 3. Number of remaining atoms detected by fluorescence near the mirror surface as a function of the transport distance ∆z. We extract the position of the surface ztot and the width σz of the atomic cloud from a fit to the data (dashed line).

5 0

 5 Selected zone for fitting long range interactionsDistance to mirror 𝑧 at[µm]

Figure 4 .

 4 Figure 4. Force measurements close to the mirror surface. Blue circles: force measurements. Dashed red: fit to these above 20 µm by a model of the interactions with a Gaussian distribution of electric dipoles. Purple diamonds: difference between this model and the measurements. Green line: expected Casimir-Polder force for our setup. The error bars denote the 1-sigma statistical errors.

We thank Xiaobing Deng for early contributions, and Axel Maury and his colleagues for providing us with the calculations of the energy levels in our system. This research has been carried out in the frame of the Quan-tERA project TAIOL, funded by the European Union's Horizon 2020 Research and Innovation Programme and the Agence Nationale de la Recherche (ANR-18-QUAN-0015-01).

Y.B. and L.A. contributed equally to this work.


Jamal Assaad
email: jamal.assaad@univ-valenciennes.fr

Emmanuel Moulin

Najib Abou Leyla

Sébastien Grondel

Frederic Jenot

Mustapha Baouahi

Application of a pseudo-3D modeling to Lamb waves generation by a surface-bonded apodized transducer: Experimental results

Keywords:

The objective of this paper is to predict the ultrasonic field radiated by piezoelectric transducers with finite dimensions in an isotropic plate. This prediction allows a good understanding of physical phenomenon of Lamb wave propagation used on the Integrated Structural Health Monitoring System. In order to describe correctly the radiated field, the stress field in the length direction of the transducer has been supposed to have a parabolic shape. The displacement field generated at the surface of an aluminum plate by a surface-mounted PZT transducer excited by a 5-cycle, Hanning-windowed sinusoid signal, has been measured for different directions, using a laser interferometer. Experimental and numerical results are successfully compared.

INTRODUCTION

Built-in ultrasound interrogation systems are seen as a promising solution for structural health-monitoring of civil and aeronautical structures. In this context, integration of small piezoelectric transducers into plates is of particular interest. The generated Lamb waves can be used to monitor large areas of the plate. The modeling of a full Lamb-wave emission and reception system, including a piezoelectric emitter and a piezoelectric receiver, can be done in a relatively simple and flexible way [START_REF] Moulin | Modeling surfacemounted Lmb wave emission-reception system : Application to strcutural health monitoring[END_REF]. First, the emitter behavior and the Lamb wave propagation are predicted using a two-dimensional (2D) hybrid finite element-normal mode expansion model. The receiver electrical response can be obtained from a finite element computation with prescribed displacements [START_REF] Moulin | Modeling surfacemounted Lmb wave emission-reception system : Application to strcutural health monitoring[END_REF]. Then, a numerical correction is applied to the 2D results to account for the in-plane radiation divergence caused by the finite length of the emitter. In order to describe correctly the radiated field, the stress field in the length direction of the transducer has been supposed to have a parabolic shape. This choice, which is a common way of taking into account apodization of a transducer, allows ensuring stress continuity at the transducer edges. This work is a continuation of the paper published about pseudo-3D modeling technique [START_REF] Moulin | Pseudo-3d modeling of a surface-bonded lamb wave source[END_REF][START_REF] Baouahi | Pseudo-3d modeling of the lamb wave field generated by a surface-bonded, finite-sized piezoelectric transducer[END_REF]. In the following the modeling of the receiver will not be shown and only the displacement field for different directions (at point P, see Fig. 1) will be shown.

A complete summary of guided Lamb waves for identification of damage in composite structures can be found in reference [START_REF] Zu | Guided Lamb waves for identification of damage in composite structures : a review[END_REF].

The displacement field generated at the surface of an aluminum plate by a surface-mounted PZT transducer excited by a 5-cycle, Hanning-windowed sinusoid signal, has been measured for different directions, using a laser interferometer. The predicted displacement fields obtained by pseudo-3D modeling technique are successfully compared the measurement one. σ applied on its upper surface. The excitation area S is supposed to be rectangular shaped, with length L and width W as indicated in Fig. 1. As discussed in previous papers [START_REF] Moulin | Pseudo-3d modeling of a surface-bonded lamb wave source[END_REF], some conditions are required for the application of the pseudo-3D modeling. They are reminded below:

III. APPLICATION TO A BONDED TRANSDUCER

In the case where the excitation is caused by a piezoelectric transducer bonded on the plate surface, the amplitude n A associated to each propagating mode (Eq. 3) can be computed by using a two-dimensional, plane strain hybrid modeling technique [START_REF] Moulin | Modeling of lamb waves generated by integrated transducers in composite plates using a coupled finite element -normal modes expansion method[END_REF], which will not to be reminded here.

The stress profile oy σ , necessary for the computation of n α (Eq. 4), is zero outside the interval / 2/ 2 Ly L -≤ ≤ and unknown under the transducer. Its exact determination would require the full 3D modeling that we precisely intend to avoid. If L is significantly larger than W , the simplest hypothesis is to consider oy σ as uniform. Then, the directivity n α is easily determined from Eq. 4 and the signal received at point P with polar coordinates (, ) r θ can be obtained by adding the results

given for each mode by Eq. 3.

This modeling principle is illustrated in the example presented below. It corresponds to a PZT transducer with T = 1 mm, W = 4 mm and L = 20 mm stuck at the surface of a 1-mm thick aluminum plate and excited by a 5-cycle, Hanningwindowed sinusoid signal with a central frequency f = 700 kHz. The reception point P is located at r = 63 mm and θ = 0°, which makes conditions #1 and #3 (see section II) ensured. The normal displacement obtained by the pseudo-3D model (Fig. 2.a) is compared to the result of a full-3D finite element modeling (Fig. A very good agreement can be observed between both results. In particular, the amplitude of each mode is correctly predicted by the pseudo-3D method. The remaining oscillations after 0.028 ms in Fig. 2.b corresponds to the wave reflection at the edge of the plate (which is necessarily limited in the FEM model) and should not be taken into account. This example, along with other cases presented previously [START_REF] Zu | Guided Lamb waves for identification of damage in composite structures : a review[END_REF], constitute a clear validation of the method, at least when the measurement point belongs to the transducer axis. In a second example, the same configuration has been considered, except a slightly off-axis reception point is considered with θ = 5°. In that case, 0.03 g ≤ (Eq. 2) and thus condition #3 is respected. Here again, the pseudo-3D result has been compared to a 3D-FEM result (Fig. 3.a and 3.b, respectively).

In that case, although all applicability conditions #1, #2 and #3 are respected, a clear amplitude mismatch is observed (especially for the A 0 mode)). Thus, obviously, application of the pseudo-3D model, such as presented above, leads to erroneous results as soon as 0 θ ≠°.

However, the results presented in a previous paper [START_REF] Moulin | Pseudo-3d modeling of a surface-bonded lamb wave source[END_REF] demonstrate that this problem does not occur when a known stress distribution is applied at the transducer location. Consequently, this is the estimation of 0 σ that should be questioned here. Since ox σ is numerically computed and then incorporated in A n (see Eq. 3) using a fully-validated technique [START_REF] Moulin | Modeling of lamb waves generated by integrated transducers in composite plates using a coupled finite element -normal modes expansion method[END_REF][START_REF] Duquenne | Transient modeling of lamb waves generated in viscoelastic materials by surface bonded piezoelectric transducers[END_REF], the source of the problem is likely to be oy σ .

Thus, instead of a basic uniform profile, a parabolic shape will be tried for oy σ . This choice is justified by two simple arguments. First, since it allows to ensure stress continuity at the transducer edges ( o σ = 0 at y = ±L/2), it is likely to be a more physically acceptable solution. And second, a parabolic function is a quite common way of taking into account apodization of a transducer. Though quite convincing, the very good matching observed in Fig. 4.b should be confirmed for higher angles. Therefore a last example is presented in Fig. 5. Here, the excitation frequency is 200 kHz and the reception point is located at 63 rm m = and 22 θ =°. Here again, the comparison between the 3D-FEM and pseudo-3D results is very satisfying. The fact that the maximum value of function g (Eq. 2) is approximately 0.26, which is less favorable as referred to condition #3, only makes the matching slightly less perfect than in the previous case.

In conclusion, the directivity computed using a parabolic shape for oy σ appears to be quite representative of the actual radiation of the transducer. Then, this is what will be considered in all future pseudo-3D modeling.

IV. EXPERIMENTAL RESULTS

The experimental setup considered in this study consisted of a 7-mm wide, 43-mm long and 1-mm thick PZT transducer stuck on the surface of a 6-mm thick aluminum plate. The excitation signal is a 5-cycle, Hanning-windowed sinusoid waveform at 200 kHz and of 10 V amplitude. The normal displacement at the surface of the plate has been measured using a laser interferometer. The measurement points considered are represented in Fig. 6. They correspond to the following locations: P 1 (r = 135 mm, θ = 0°), P 2 (r = 133 mm, θ = 10°), P 3 (r = 130 mm, θ = 20°), P 4 (r = 125 mm, θ = 30°). For each location, the maximum value (this means for the smallest wavelength) of the function g defined in Eq. 2 has been reported into Fig. 7, corresponding to a level-curve representation of (/ , ) gW λ θ . This is a quick way of knowing to which extent condition #3 (see section II) is satisfied.

Then in Fig. 8, the experimental results have been compared to the corresponding pseudo-3D predictions. Naturally, as discussed in the previous section, a parabolic shape has been imposed in the model to oy σ . Globally, the results appear to be very satisfying both in terms of absolute amplitude and waveform shape. Reception points P3 and P4 are too close from the plate edge to allow a complete time separation of the direct and reflected signals. Still, the significant wave packets in Fig. 8.c and d are not polluted and interpretation of the results is fully valuable. Logically, the amplitude for each mode reach a maximum for θ = 0° and is decreased as θ increased. Since A 0 has a smaller wavelength than S 0 , its radiation is also more directive. Accordingly then, the ratio between the S 0 and A 0 amplitudes is increasing as θ is increased. Additionally, it can be remarked that the theoretical condition #3 [START_REF] Moulin | Pseudo-3d modeling of a surface-bonded lamb wave source[END_REF] about function g is not so restrictive. In practice a value of g less than 0.3 leads to good results.

V. CONCLUSION

The work presented in this paper consisted in an application of a pseudo-3D technique to the modeling of Lamb wave generation by a surface-bonded transducer. First, the excitation by the transducer and the subsequent modal fields are predicted in the sagittal plane using a 2D hybrid method. Then, the directivity in the plane of the plate can be computed separately for each mode. The stress profile along the third dimension (i.e. the transducer length) has been proved satisfyingly described by a parabolic function. Experimental and numerical results are in good agreement. This model has many advantages. One can firstly quote its ease of use, secondly the results can be easily interpreted and it enables to save calculation time. Moreover, its flexibility should make it possible to relatively easily take into account important additional physical effects in practice. One can in particular quote plate anisotropy, attenuation and interaction of these Lamb waves with defects.

Figure 1 .

 1 Figure 1. Description of the 3D problem geometry

  2.b).

Figure 2 .

 2 Figure 2. Theoretical normal displacement at point P for θ =0°. (a) From pseudo-3D modeling. (b) From 3D FEM

Figure 3 .

 3 Figure 3. .Theoretical normal displacement at point P for θ = 5°. (a) From pseudo-3D modeling. (b) From 3D FEM observed (especially for the A 0 mode).

Fig. 4

 4 Fig. 4 below is intended to show clearly the influence of oy σ on the results provided by the pseudo-3D model. For better visibility, only the A 0 contribution has been represented. Fig. 4.a corresponds to the results computed with the uniform profile for 5θ =° (and thus is nothing more than a zoomed version of Fig.3). Fig.4.b corresponds to the results for 5θ =°

Figure 4 .

 4 Figure 4. Theoretical normal displacement at point P for 5 θ =°, computed using the pseudo-3D model (dashed line) and 3D-FEM (solid line). (a) oy σ is

Figure 5 .

 5 Figure 5. Sme ss Fig. 4, but for θ = 22°.

2 P 3 P 4 Figure 6 .

 2346 Figure 6. Description of the experimental setup and locations of the measurement points.

Figure 7 .

 7 Figure 7. Contour plot of the function (/ , ) gW λ θ

Figure 8 .

 8 Figure 8. Comparison between pseudo-3D (dashed line) and measurement (solid line) results. (a) Point P 1 , (b) Point P 2 , (c) Point P 3 , (d) Point P 4 .


Wahiba Lfarh
email: wahibalfarh@aero.obs-mip.fr

Florian Pantillon

Jean-Pierre Chaboureau

Impact of surface turbulent fluxes on the formation of convective rolls in a Mediterranean windstorm

Impact of surface turbulent fluxes on the formation of convective rolls in a Mediterranean windstorm

Wahiba Lfarh, Florian Pantillon, Jean-Pierre

Introduction

In late October 2018, a mid-tropospheric trough formed over the eastern Atlantic, extending from Scandinavian to the Iberian Peninsula. At the surface, a broad cyclonic area developed over the western Mediterranean, to the east of the trough axis. On October 29, the strongly baroclinic environment led to the formation of windstorm Adrian (also known as Vaia) [START_REF] Cavaleri | The October 29, 2018 storm in Northern Italy -An exceptional event and its modeling[END_REF][START_REF] Davolio | Heavy precipitation over Italy from the Mediterranean storm "Vaia" in October 2018: Assessing the role of an atmospheric river[END_REF][START_REF] Giovannini | Multi-model convection-resolving simulations of the October 2018 Vaia storm over Northeastern Italy[END_REF]. Owing to a strong contrast between the cold air and the warm sea surface, the low-pressure system rapidly intensified and reached a minimum pressure of 977 hPa as it moved between Corsica and northwest Italy.

In Corsica, Adrian caused wind gusts of up to 180 km h -1 , resulting in extensive damage to infrastructure and considerable economic losses.

Recent research reveals that the strong winds of windstorm Adrian have their origin from a mesoscale cold conveyor belt, which refers to a cold air flow in the lower levels of the troposphere [START_REF] Lfarh | The downward transport of strong wind by convective rolls in a mediterranean windstorm[END_REF]. The fine-scale processes lead-ing to the downward transport of strong winds were examined using a Large Eddy Simulation (LES), for which the strong winds are located in a convective boundary layer. The phase of maximum intensity of Adrian is marked by a combination of thermal instability resulting from a strong air-sea temperature contrast, and dynamic instability due to strong vertical wind shear. Such thermodynamic conditions foster the formation of convective rolls in the boundary layer.

Convective rolls are defined as quasi-two-dimensional vortices, oriented in the wind direction, creating alternating upward and downward air motions [START_REF] Etling | Roll vortices in the planetary boundary layer: A review[END_REF]. These structures are also observed in different meteorological conditions from those of Mediterranean windstorms, such as cold-air outbreak events [START_REF] Gryschka | Roll convection during a cold air outbreak: A large eddy simulation with stationary model domain[END_REF][START_REF] Chen | A high-resolution simulation of roll convection over the Yellow Sea during a cold air outbreak[END_REF] and hurricanes [START_REF] Foster | Why rolls are prevalent in the hurricane boundary layer[END_REF][START_REF] Li | Effects of roll vortices on the evolution of hurricane harvey during landfall[END_REF]. Convective rolls are known for transporting momentum, heat and moisture through their descending and ascending branches between the top of the boundary layer and the sea surface [START_REF] Weckwerth | An observational study of the evolution of horizontal convective rolls[END_REF]. Previous studies have shown that convective rolls contribute half, or even much more, of the total momentum flux in hurricanes, and can reinforce air-sea interactions [START_REF] Morrison | An observational case for the prevalence of roll vortices in the hurricane boundary layer[END_REF]. In Adrian, the wind is stronger along downward branch of the rolls due to higher momentum air transport from the cold conveyor belt [START_REF] Lfarh | The downward transport of strong wind by convective rolls in a mediterranean windstorm[END_REF].

The transport of momentum, heat and moisture by convective rolls in the boundary layer could influence surface fluxes. This raises the question of how these fluxes in turn affect the characteristics of the rolls. Exchanges at the air-sea interface strongly influence the thermodynamic conditions of the atmosphere. Heat fluxes, in particular, are among the diabatic processes involved in the development of extra-tropical cyclones. Previous studies have shown that the release of latent heat during condensation processes can strengthen vorticity at lower levels, and therefore contribute to the cyclogenesis [START_REF] Booth | Sensitivity of midlatitude storm intensification to perturbations in the sea surface temperature near the gulf stream[END_REF][START_REF] Ludwig | The role of anomalous sst and surface fluxes over the southeastern north atlantic in the explosive development of windstorm xynthia[END_REF]. As well as their mesoscale impact, air-sea interactions play a significant role in modulating surface wind speeds during windstorms. [START_REF] Schultz | Using frontogenesis to identify sting jets in extratropical cyclones[END_REF] suggested that heat fluxes promote shallow convection and facilitate wind mixing from the upper boundary layer towards the surface.

Turbulent fluxes at the air-sea interface occur on spatial and temporal scales too small to be explicitly resolved in numerical weather prediction models, and are therefore parameterized. The importance of air-sea interactions has led to widespread interest in improving their representation in numerical weather prediction systems [START_REF] Lewis | The ukc3 regional coupled environmental prediction system[END_REF]. To date, many uncertainties persist in flux measurements at the air-sea interface, particularly in high wind situations. Flux measurements are limited to a wind speed of about 20 m s -1 . This may explain the underestimation of near-surface wind speed in windstorms compared to in situ and satellite observations [START_REF] Pineau-Guillou | Strong winds in a coupled wave-atmosphere model during a north atlantic storm event: evaluation against observations[END_REF]. Although direct measurements from ocean buoys, GPS dropsondes and airborne radar can be employed to quantify surface fluxes under strong wind conditions, the scarcity of such observations presents major challenges for model validation [START_REF] Richter | Evidence of spray-mediated air-sea enthalpy flux within tropical cyclones[END_REF][START_REF] Zou | Drag coefficients derived from ocean current and temperature profiles at high wind speeds[END_REF]. In addition, the air-sea interface becomes more complex under strong wind conditions, with the presence of foam and sea spray resulting from breaking waves. These factors make measuring and modeling turbulent fluxes all the more difficult [START_REF] Emanuel | A similarity hypothesis for air-sea exchange at extreme wind speeds[END_REF]Sroka & Emanuel, 2021).

Theoretical studies and numerical simulations have shown that the microphysical processes governing the generation and evaporation of sea spray can have a significant impact on air-sea exchanges. In strong winds, breaking waves at the air-sea interface can produce large quantities of spray droplets. As they float in the air, spray droplets carry a large amount of energy due to their speed and abundance, generating more intense momentum fluxes [START_REF] Veron | Sea spray spume droplet production in high wind speeds[END_REF][START_REF] Sroka | Sensitivity of sea-surface enthalpy and momentum fluxes to sea spray microphysics[END_REF]. In addition, sea spray has the effect of cooling the air and transferring latent heat to the atmosphere when it evaporates, creating favorable conditions for more powerful heat exchange between the sea surface and the atmosphere [START_REF] Andreas | An improved bulk air-sea surface flux algorithm, including spray-mediated transfer[END_REF][START_REF] Jeong | Enthalpy transfer across the air-water interface in high winds including spray[END_REF]. Using a coupled ocean-wave-atmosphere model, [START_REF] Zhao | Sensitivity of typhoon modeling to surface waves and rainfall[END_REF] found that sea spray caused an increase in cyclone intensity by strengthening air-sea fluxes. Sroka and Emanuel (2021) concluded that the inclusion of sea spray in turbulent flux parameterizations is crucial in extreme wind situations.

The Mediterranean is an ideal test-bed for investigating these complex surface processes under strong wind situations. Regional winds bring cold, dry continental air over the warm waters of the Mediterranean, leading to intense air-sea interactions [START_REF] Flamant | Alpine lee cyclogenesis influence on air-sea heat exchanges and marine atmospheric boundary layer thermodynamics over the western mediterranean during a tramontane/mistral event[END_REF]. In this cyclogenesis-prone region, low-pressure systems are not only maintained, but also reinforced by exchanges resulting from variations in temperature and humidity between the atmosphere and the sea [START_REF] Jansà | Mediterranean cyclones: Subject of a wmo project[END_REF].

Unlike tropical cyclones and Atlantic windstorms, there are few studies devoted to the role of surface turbulent fluxes in the mesoscale dynamics of Mediterranean windstorms. Furthermore, the impact of these fluxes on fine-scale processes is largely unknown for extra-tropical cyclones. The main objective of this study is to examine how turbulent fluxes at the air-sea interface affect the formation and organization of convective rolls and, hence, surface winds during the Mediterranean windstorm Adrian. To achieve this purpose, sensitivity tests to different parameterizations of turbulent fluxes are carried out using LES.

The paper is structured as follows: Section 2 briefly presents the modeling strategy, the parameterizations used to calculate turbulent fluxes and the analysis tools. Section 3 examines the influence of turbulent fluxes at the mesoscale, particularly on changes in the trajectory and intensity of the windstorm. Section 4 investigates the effects of parameterizations on fine scale convective rolls and surface winds before discussing the impact of each turbulent flux independently of the others. Finally, section 5 summarizes the main results.

Methods

Numerical configuration and sensitivity experiments

Two reference simulations, a mesoscale one and the other a LES, are carried out with the Meso-NH non-hydrostatic atmospheric model [START_REF] Lac | Overview of the Meso-NH model version 5.4 and its applications[END_REF] to study the strong mesoscale winds in windstorm Adrian and the fine-scale processes responsible for their transport to the surface, respectively. Both simulations share the same vertical grid with 72 levels reaching 22 km and a finer grid spacing near the surface (10 m at the first level). They are initialized at 0600 UTC on October 29, 2018 from the operational analysis of the European Centre for Medium-Range Weather Forecasts (ECMWF) and run for 15 h.

The mesoscale simulation covers a domain centered on the western Mediterranean (domain D1 in Figure 1a) with a horizontal grid spacing of 1000 m allowing explicit representation of deep convection. The LES is nested in the mesoscale simulation using a one-way nested method. It has a horizontal grid spacing of 200 m over a large domain covering part of the Mediterranean Sea and the whole of Corsica (domain D2 in Figure 1b). Both simulations are based on the same parameterization schemes, except for three. The mesoscale simulation uses the fifth-order WENO (weighted essentially non-oscillatory) advection scheme for momentum variables [START_REF] Shu | Efficient implementation of essentially nonoscillatory shock-capturing schemes[END_REF], the EDMF (eddy-diffusivity mass flux) parametrization of Pergaud et al. (2009) for shallow convection, and the 1D version of the scheme of [START_REF] Cuxart | A turbulence scheme allowing for mesoscale and large-eddy simulations[END_REF] for turbulence. In the LES, the EDMF parameterization is deactivated, as convection is explicitly represented, and a more accurate fourth-order centered advection scheme is used along with a 3D version of the turbulence scheme. For further details on the simulation configuration, the reader is referred to [START_REF] Lfarh | The downward transport of strong wind by convective rolls in a mediterranean windstorm[END_REF].

Sensitivity experiments to turbulent fluxes at the air-sea interface are carried out at both scales based on the two reference simulations. In order to assess the impact of turbulent flux parameterizations on the evolution of the Adrian storm at mesoscale, a first series of sensitivity experiments are performed with a horizontal grid spacing of 1000 m on the D1 domain. These simulations are initialized at 1200 UTC from the reference simulation and run for 6 h, including the development and intensification phase of windstorm Adrian. To shed light on the impact of turbulent exchanges between the air and the sea on the convective rolls that bring strong winds to the surface, a second series of sensitivity experiments is performed at a horizontal resolution of 200 m on the D2 domain. These experiments are also initialized from the reference simulations and conducted over a short period from 1500 UTC to 1530 UTC, when the windstorm reached its maximum intensity.

Surface-atmosphere interactions are managed by a surface modeling platform called SURFEX (Surface Externalised), which is coupled to Meso-NH [START_REF] Masson | The surfexv7.2 land and ocean surface platform for coupled or offline simulation of earth surface variables and fluxes[END_REF]. This platform integrates several physical models that deal with the different characteristics of the surface of the Earth. In reference simulations, turbulent air-sea exchanges are represented by default using the COARE3 (Coupled Ocean-Atmosphere Response Experiment) parameterization [START_REF] Fairall | Bulk parameterization of air-sea fluxes: Updates and verification for the coare algorithm[END_REF]. Other parameterizations of air-sea exchanges widely used are tested at both 1000 m and 200 m resolutions and are described hereinafter. In addition to the different parameterizations, three other sensitivity experiments to momentum and heat fluxes are performed in this study. The aim is to isolate the impact of each type of fluxes on convective rolls and surface winds. Using the COARE3 reference parameterization, the NoM, NoH and NoLE simulations are performed at a horizontal resolution of 200 m, with momentum, sensible heat and latent heat fluxes set to zero, respectively.

The simulations are labelled according to the name of the parameterization used, followed by "1" and "2" to refer to the horizontal resolutions of 1000 and 200 m.

The model configuration for each simulation is summarized in Table 1. at the air-sea interface that can impact the fluctuations, such as waves, sea spray, etc. Air-sea fluxes are calculated on the basis of the [START_REF] Monin | Basic regularity in turbulent mixing in the surface layer of the atmosphere[END_REF] similarity theory, enabling the estimation of fluxes generated by complex turbulent processes in the boundary layer as a function of mean model parameters. These parameters are based on global or bulk aerodynamic algorithms [START_REF] Liu | Bulk parameterization of air-sea exchanges of heat and water vapor including the molecular constraints at the interface[END_REF]. The parameterizations calculate turbulent fluxes as follows:

τ bulk = ρw ′ u ′ = -ρu 2 * H bulk = ρc p w ′ θ ′ = -ρc p u * θ * LE bulk = ρL v w ′ q ′ = -ρL v u * q * ( 1
)
where τ is the momentum flux, H the sensible heat flux and LE the latent heat flux. ρ is the air surface density, c p the heat capacity of air at constant pressure and L v the latent heat of vaporization. u * , θ * and q * are scaling parameters for momentum, potential temperature and humidity, respectively. Bulk algorithms are based on the determination of constants called exchange coefficients (or aerodynamic coefficients), which establish a relationship between turbulent fluxes and mean parameters. Turbulent fluxes are defined as a function of vertical gradients of wind speed ∆U , potential temperature ∆θ and humidity ∆q, with C D , C H and C E the exchange coefficients for momentum, sensible heat and latent heat, respectively :

τ = -ρC D ∆U 2 = -ρC D (U a -U s ) 2 H = -ρc p C H ∆U ∆θ = -ρc p C H (U a -U s )(θ a -θ s ) LE = -ρL v C E ∆U ∆q = -ρL v C E (U a -U s )(q a -q s ) (2)
where the a index corresponds to the first atmospheric level, and the s index to the sea surface.

Even though all parametrizations are based on the same theory, they differ from one another. Each parametrization uses its own stability function and closure assumption, allowing simplified consideration of complex variations in the nearsurface atmosphere. The formulations of turbulent flux parametrizations differ in their expressions of exchange coefficients and roughness length calculations, and also in the way they do or do not take into account the effects of specific physical processes [START_REF] Brunke | Which bulk aerodynamic algorithms are least problematic in computing ocean surface turbulent fluxes?[END_REF]. In this article, four well-established parameterizations are used: COARE3.0 [START_REF] Fairall | Bulk parameterization of air-sea fluxes: Updates and verification for the coare algorithm[END_REF], ANDREAS [START_REF] Andreas | An improved bulk air-sea surface flux algorithm, including spray-mediated transfer[END_REF], ECUME6, the sixth version of Exchange Coefficient Unified Multi-campaign Experiments used operationally by Météo-France [START_REF] Belamari | Report on uncertainty estimates of an optimal bulk formulation for surface turbulent fluxes[END_REF], and WASP (Wave-Age Stress dependant Parametrization) [START_REF] Sauvage | Characterization of the air-sea exchange mechanisms during a Mediterranean heavy precipitation event using realistic sea state modelling[END_REF]. They are detailed in Appendix A.

Differences between parameterizations are illustrated at 1530 UTC, when wind speeds reaching 40 m s -1 occur in a narrow belt along the south flank of the lowpressure center (white square in Figure 1b). The momentum, sensible and latent heat fluxes are represented in Figure 2 as a function of wind speed at 10 m. For all four parametrizations, momentum fluxes show a similar increase with wind speed for speeds below 30 m s -1 (Figure 2a). At speeds above 30 m s -1 , they diverge depending on the parameterization used. The strongest momentum fluxes are produced by the COARE and ANDREAS parametrizations. In contrast, ECUME and WASP show a relatively similar evolution under strong wind conditions, characterized by lower fluxes of momentum, with a tendency to saturation at speeds above 30 m s -1 .

This contrast shows the impact of accounting or not for the saturation of the drag coefficient when wind speed exceeds 30 m s -1 . The differences between sensible and latent heat flux parameterizations become apparent at speeds of 20 m s -1 and above. The COARE, ECUME and WASP parametrizations show similar heat fluxes at high speeds, varying between 200 and 500 W m -2 for sensible heat fluxes, and between 500 and 1500 W m -2 for latent heat fluxes. ANDREAS reveals large deviations for wind speeds in excess of 20 m s -1 , illustrating the impact of the sea spray included in the parametrization. This translates into fluxes in excess of 2000 W m -2 and 4000 W m -2 for sensible and latent heat respectively, at wind speeds in excess of 45 m s -1 .

Auto-correlation function

As in [START_REF] Lfarh | The downward transport of strong wind by convective rolls in a mediterranean windstorm[END_REF] fine-scale wind speed structures associated with the Adrian storm are characterized using the spatial auto-correlation function (ACF).

The ACF is a statistical approach commonly used to assess the spatial variability of turbulence organization in the atmospheric boundary layer, defining the main shape, direction and characteristic size of structures [START_REF] Lohou | Spatial and temporal characteristics of horizontal rolls and cells in the atmospheric boundary layer based on radar and in situ observations[END_REF][START_REF] Brilouet | Trade wind boundary layer turbulence and shallow precipitating convection: New insights combining SAR images, satellite brightness temperature, and airborne in situ measurements[END_REF]. In practice, ACF measures the correlation of a variable with itself at spatial positions shifted along the x and y directions. In this case study, the 2D ACF R F (δ x , δ y ) is calculated from the vertical wind speed fields F (x, y) at lags δ x and δ y , using the method of Granero Belinchon et al. ( 2022) as follows:

R F (δ x , δ y ) = ⟨F (x, y) × F (x + δ x , y + δ y )⟩ (3)
where F (x, y) is the field value at position (x,y), F (x + δ x , y + δ y ) is the field value at a position lagged by (δ x ,δ y ) from (x,y) and ⟨...⟩ denotes the spatial average.

The first step is to extract the main shape of the structures, which can be either elliptical, suggesting the presence of convective rolls, or circular, as in the case of convective cells. The analysis is based on the integral length L es , defined as the integral of the ACF where the field F (x, y) remains correlated with itself. L es is calculated in different directions to detect possible anisotropy [START_REF] Lohou | Numerical study of the impact of coherent structures on vertical transfers in the atmospheric boundary layer[END_REF].

The geometry of the structures can be determined by fitting the set of points corresponding to L es . An elliptical fit indicates an elongated, elliptical structure, while a circular fit suggests a circular structure. In a second step, the geometric parameters of the structure are determined, including the major axis r a and the minor axis r b .

These two axes are used to quantify the flatness parameter f calculated as follows:

f = r a -r b r a (4)
where r a and r b are the major and minor radii of the structure, respectively. This parameter is used to distinguish the roll structures, where f tends towards 1, from the cellular structures, where the parameter tends towards 0. The third step is to estimate the length scale of the organized structure L os , which represents its characteristic size. This length is defined as the distance between two correlation maxima in the ACF.

3 Impact of surface fluxes on the mesoscale dynamics

The effects of the four parameterizations are assessed here on the evolution of windstorm Adrian at mesoscale. On the morning of October 29, 2018, a large lowpressure system intensified into windstorm Adrian in the western Mediterranean as a result of baroclinic interaction. The observed (best-track) and simulated trajec- Furthermore, the simulated trajectories remain close to each other despite slight positional shifts. This suggests that the track of Adrian over this short period does not depend much on the air-sea flux parameterization used, but is mainly determined by the baroclinic interaction.

Temporal evolutions of mean sea level pressure (MSLP) and 99th percentile wind speeds at 10 m over the sea are calculated from 1200 to 1800 on the D2 domain (Figures 3b and3c). The evolution of pressure indicates that Adrian intensified rapidly, reaching its maximum intensity between 1500 and 1600 UTC, i.e., 1 2 :0 0 1 3 :0 0 1 4 :0 0 1 5 :0 0 1 6 :0 0 1 7 :0 0 1 8 :0 1 2 :0 0 1 3 :0 0 1 4 :0 0 1 5 :0 0 1 6 :0 0 1 7 :0 0 1 8 :0 In the early afternoon, wind speeds exceed 25 m s -1 due to the intensification of the low-pressure system and increase rapidly to reach a peak at around 1500 UTC, with 99th percentile values of 29 m s -1 for Coare1 (Figure 3c). As Adrian weakens, wind speeds gradually decrease until 1800 UTC. Unlike the MSLP, wind speed evolutions show immediate differences at the start of the simulations, becoming more noticeable during the maximum intensification of the depression.

The difference reaches 1 m s -1 in Ecume1 and Wasp1 compared with Coare1, and rises to 2 m s -1 in Andreas1. The intensity differences obtained with the AN-DREAS parametrization are consistent with the results of [START_REF] Perrie | Sea spray impacts on intensifying midlatitude cyclones[END_REF] and Zhang2006, who found that including sea spray in the surface fluxes leads to a decrease in surface pressure during windstorms by a few hPa and an increase in surface wind speed by a few m s -1 . Moreover, as in the case of Adrian, the authors found that sea spray has a minor effect on the windstorm trajectory. As saturation of the drag coefficient is not taken into account in the COARE parameterization, it is not surprising that the Coare1 simulation produces the lowest wind speeds compared with the other simulations. After 1600 UTC, the differences become negligible and reflect an almost identical evolution of wind speeds.

Due to the passage of Adrian near Corsica, stations recorded high wind speeds, particularly in the western and northern coastal stations, where measurements exceeded 30 m s -1 (Figure 4a). To assess the impact of the parameterizations on In conclusion, the four simulations reproduce station observations in a similar way, demonstrating that turbulent fluxes at the air-sea interface have little impact on wind speed over land. Over the short 6-h period, the trajectory of the windstorm is unaffected by the choice of surface flux parameterization. This confirms that the trajectory is mainly determined by large-scale flow. Regarding the intensity of Adrian, the ANDREAS parameterization generates a more pronounced low-pressure system, resulting in slightly higher wind speeds compared to the reference simulation Coare1, due to the inclusion of sea spray. Consequently, LES sensitivity tests can be carried out over short periods to assess the impact of turbulent exchanges at the air-sea interface on convective rolls, while preserving mesoscale characteristics of the windstorm.

4 Impact of turbulent fluxes on the finescale

Morphology of convective rolls

In the following, the study focuses on a 20 km by 20 km region, representative of the narrow band of strong winds generated on the southern flank of the low-pressure center (white square in Figure 1b). Wind speeds lie above 20 m s -1 , a threshold at which turbulent flux parameterizations begin to diverge (Figure 2).

The analysis is based on LES simulations, inspecting horizontal and vertical crosssections of wind speeds (Figure 5). In the reference simulation Coare2, the horizontal section (Figure 5a The boundary layer organization in Ecume2 and Wasp2 simulations is similar to the reference simulation (Figures 5e-5h). The Andreas2 simulation stands out the most. The horizontal section shows coherent structures that are narrower than in Coare2 (Figure 5c). The upward and downward motions are more pronounced and extend over a greater vertical distance (Figure 5d). Accordingly, the boundary layer height is higher, showing marked fluctuations between 1000 and 1500 m. ntation and characteristic size of convective rolls.

The spatial characteristics of the convective rolls are examined in the four simulations using the ACF calculated from the vertical wind speed field at 1530 UTC over the zoomed area shown in Figure 5. chosen to distinguish roll structures with f > 0.7 from disorganized structures with f < 0.7. In Coare2, the flatness parameter reveals the presence of rolls up to z=400 m and a transition into disorganized structures above (Figure 6a). The mean wind direction varies slightly with height, from 60°to 50°clockwise, while the roll direction evolves counter-clockwise from 50°to 70°. Below z=400 m, this results in rolls mainly oriented in the wind direction, with a minor difference of around 10°377 (Figure 6b). Accordingly, observational studies have shown that convective rolls are often aligned within 10 to 20°from the mean wind direction [START_REF] Atkinson | Mesoscale shallow convection in the atmosphere[END_REF][START_REF] Foster | Why rolls are prevalent in the hurricane boundary layer[END_REF]. Finally, the size of rolls can be determined up to 400 m height and reaches 2000 m mostly, giving an aspect ratio λ/z i of 2.73. Altogether, this implies that elliptical roll structures are present below 400 m in Coare2. The morphological characteristics of the convective rolls at 1530 UTC are comparable to those found at 1515 UTC in [START_REF] Lfarh | The downward transport of strong wind by convective rolls in a mediterranean windstorm[END_REF], except that the size of the rolls was slightly larger, reaching 2400 m.

In the Ecume2 and Wasp2 simulations, the size of the convective rolls vary slightly, between 2200 and 2400 m, but remain comparable to that in the reference simulation (Figure 6c). The other morphological characteristics are similar (Figure 6a,b). For Andreas2, the flatness parameter remains more constant with height, the direction of the rolls is closer to the mean wind direction and their smaller size of 1400 m is uniform over a longer vertical extent of 500 m. The aspect ratio in An-

Vertical wind transport by convective rolls

Over the same region shown in Figure 5, the wind speed increases progressively from z=1200 m downward and reaches a maximum near z=400 m, then decreases as approaching the surface (Figure 7a). Compared to the other simulations, An-dreas2 shows slower winds near z=400 m and above. At the surface, the wind speed ranges from 30 m s -1 in Coare2 to 33 m s -1 in Andreas2, while values in Ecume2 and Wasp2 fall between these two extremes. In the Coare2 reference simulation, the virtual potential temperature profile shows a decrease from the surface up to an altitude of 200 m, indicating instability characterized by a warm, moist surface layer favorable to convection (Figure 7b). Ecume2 and Wasp2 show a similar profile, with slightly higher temperatures in Ecume2. The surface temperature is higher and the vertical gradient is more pronounced in Andreas2, showing a rapid decrease with altitude. This is due to the enhanced surface fluxes of sensible and latent heat, which increase air temperature and humidity.

The boundary layer stability can be assessed with the instability parameter ζ = -z i /L, where L is the length of [START_REF] Monin | Basic regularity in turbulent mixing in the surface layer of the atmosphere[END_REF] and represents the distance above the surface where the production of buoyancy turbulence exceeds that of shear turbulence. Previous studies revealed that in a fully convective boundary, i.e. ζ ≫ 1, buoyancy dominates turbulence generation throughout the boundary layer [START_REF] Khanna | Three-dimensional buoyancy-and shear-induced local structure of the atmospheric boundary layer[END_REF][START_REF] Salesky | On the Nature of the Transition Between Roll and Cellular Organization in the Convective Boundary Layer[END_REF]. The ζ values are 0.5, 1.56, 0.71 and 0.52 for Coare2, Andreas2, Ecume2 and Wasp2 respectively (Table 2). and with the recent study of [START_REF] Stopa | Automated global classification of surface layer stratification using high-resolution sea surface roughness measurements by satellite synthetic aperture radar[END_REF], which reveals that coherent roll structures occur in a slightly unstable but nearly neutral atmosphere.

While convective rolls pilot vertical motion in the boundary layer, turbulence consisting of irregular eddies is one of the most important transport processes near the surface. This raises the question of the respective roles of convective rolls and turbulent motion in vertical momentum transport. The distinction is estimated here by the separation between resolved motion, which has been shown to consist in organized roll structures, and subgrid-scale motion represented by the turbulence scheme. This assumption likely overestimates the turbulent contribution, because convective rolls are not fully resolved with 200 m horizontal grid spacing [START_REF] Lfarh | The downward transport of strong wind by convective rolls in a mediterranean windstorm[END_REF].

In Coare2, the sum of resolved momentum fluxes and subgrid-scale momentum fluxes (solid green line in Figure 7c), which corresponds to the total momentum transport, shows a gradual strengthening with decreasing height. Resolved momentum fluxes dominate at altitudes above z≈200 m (dashed lines), while they decrease progressively toward the surface and subgrid-scale motion becomes responsible for most of the momentum transport. Vertical profiles in Ecume2 and Wasp2 remain similar to those observed in Coare2. In contrast, vertical profiles in Andreas2 reveal a clear distinction from the other simulations. The total transport is much greater both near the surface and aloft due to stronger resolved momentum fluxes, which dominate down to z≈100 m. At z=200 m, resolved fluxes are enhanced by 40% compared to the reference simulation. This enhanced downward transport in Andreas2 is consistent with the larger vertical extent of convective rolls and can explain the weaker winds in the middle boundary layer and the stronger winds near the surface compared to the other simulations (Figure 7a). Interestingly, the enhancement of vertical momentum transport by 40% in Andreas2 compared to Coare2 roughly matches the increase from n≈10 to n≈14 in the number of rolls over the narrow band of strong winds (or the decrease in their size from λ=2000 m to λ=1400 m). Thus, the transport of momentum per roll is equivalent between the two simulations and the smaller but more numerous rolls in Andreas2 imply a stronger total transport. In contrast, simulations Ecume2 and Wasp2 show values of resolved vertical momentum transport similar to Coare2 but with less, larger rolls (n≈8 and λ=2400 m), which also increases the transport of momentum per roll. The contrast can be explained by the stronger surface momentum fluxes in ECUME and WASP compared to the COARE and ANDREAS parameterizations (Figure 2a).

To assess the impact of the different parameterizations on horizontal and vertical winds in the middle of the boundary-layer and near the surface, Figure 8 shows probability density functions at 400 m, where convective rolls begin to form, and at 10 m. As in the reference simulation, Ecume2 and Wasp2 show narrow distributions of vertical wind speeds at z=400 m (Figure 8a). The standard deviation values of 0.9 in Coare2 and 0.7 m s -1 in Ecume2 and Wasp2 indicate moderate vertical motion. Andreas2 shows a broader distribution and a standard deviation of 1.5 m s -1 with a peak shifted towards negative velocities. This shows higher variability in vertical velocities, marked by more pronounced upward and downward motions. At z=400 m, the horizontal wind speed is lower in Andreas2 compared to the other parameterizations, with a mean value of 46 instead of 47 m s -1 (Figure 8b). The slower wind is explained by the enhanced downward transport of momentum at this altitude. Near the surface, the distribution of wind speeds differs between the simulations. For Coare2, wind speeds are relatively low and this is consistent with the strong surface fluxes of momentum with the COARE parameterization. In contrast, they are significantly higher in Andreas2, averaging 3 m s -1 more than in Coare2.

This difference can be explained by the more intense transport in Andreas2 (Fig- ure 8c), which thus largely compensates for the strong surface momentum fluxes with the ANDREAS parameterization (Figure 2).

In summary, the increase in heat fluxes due to sea spray led to a more important thermal instability at the air-sea interface. This instability generates more intense vertical circulations in the boundary layer, leading to the formation of more and smaller convective rolls, over a larger vertical extent than in the reference simulation. As a result, momentum transport to the surface increases significantly, leading to stronger surface wind speeds than those found with other parameterizations.

Individual contribution of surface fluxes

The findings presented in the previous section suggest that surface heat fluxes contribute significantly to the organization of convective rolls and surface winds in combination with surface momentum fluxes. This section focuses on the NoH, NoLE and NoM simulations to determine the role of each kind of turbulent flux separately from the others. Without latent heat flux (NoLE), the boundary layer organization into convective rolls (Figures 9c and9d) is similar to that of Coare2 (Figures 9a and9b). In the absence of sensible heat flux (NoH), strong wind speeds in the form of a large packet are confined between z=250 m and a lower boundary layer height of around z=500 m (Figures 9e and9f). The boundary layer is devoid of alternating upward and downward motions. This is due to reduced heating of the surface layer, which is typically responsible for convection and thus affect vertical motion.

Without momentum flux (NoM), the wind speed is strongly enhanced (higher locally than 55 m s -1 ) and disorganized, reaching lower levels because friction forces are not taken into account (Figures 9g and9h).

The evolution of roll morphological parameters in Coare2, NoH, NoLE and NoM is evaluated every minute for the full 30-min duration of the simulations to assess their persistence in response to surface fluxes. The elliptical shape of the coherent structures is retained over time in Coare2 with a flatness parameter f superior to 0.7 (Figure 10a). The difference between wind and structure directions does not exceed 20°(Figure 10b). Overall, characteristic roll sizes fluctuate between 2000 and 3000 m, with an average of around 2100 m over 30 min (Figure 10c). Note that roll sizes are calculated with an accuracy of 200 m, i.e., the resolution of the simulations. The NoLE simulation shows a similar evolution for all parameters, albeit with a modest deviation from Coare2, particularly at 1530 UTC. Convective rolls in NoLE are on average slightly larger than in Coare2 with a size of around 2300 m.

The divergences from the reference simulation are immediately apparent in NoM, whereas they appear after about 5 min in NoH. In both simulations, the flatness parameter tends towards very low values, and the structures become misaligned with the wind direction. Given that characteristic sizes are defined for structures with a flatness parameter greater than 0.7, the ACF indicates that there is no spatial periodicity in the vertical velocity field. Consequently no roll structures are present after a few minutes of simulation.

Vertical wind speed profiles in the NoH and NoLE simulations are comparable to the reference simulation Coare2, with a slightly sharper decrease toward the surface in NoH (Figure 11a). In contrast, the NoM profile remains clearly distinct, with high wind speeds remaining almost uniform in the lower boundary layer. Unlike NoLE, which shows a decrease with height in virtual potential temperature in the first levels similar to Coare2, NoH is distinguished by a profile that maintains an almost constant virtual potential temperature, reflecting a nearly neutral atmosphere (Figure 11b). The instability parameter ζ tends to zero in that case, implying that shear is the main source of turbulence in the boundary layer (Table 2). ζ is no longer relevant in NoM. A value of ζ=0.50 in NoLE, as in Coare2, suggests that latent heat fluxes have weak impact on atmospheric instability and therefore on the formation of convective rolls. Although stability conditions are comparable, the momentum transport is somewhat weaker in NoLE (Figure 11c). This is attributed to the slightly higher roll size in NoLE, which results in fewer rolls in the region. In NoH, resolved fluxes remain limited compared to the Coare2 simulation and result in lower total transport, particularly at altitudes below 400 m where convective roll structures are more pronounced in the reference simulation. Moreover, turbulence becomes dominant in transport at higher altitude than in Coare2. Finally, the vertical transport profile in NoM is completely different from other simulations and vanishes near the surface due to the absence of momentum fluxes.

In short, the absence of sensible heat flux results in nearly neutral atmospheric conditions, inhibiting convection and hence the formation of convective rolls even in the presence of strong winds and intense shear. Consequently, momentum transport is considerably reduced, and is mainly controlled by subgrid-scale fluxes. Likewise, without momentum fluxes, wind shear is impacted, rapidly slowing down the formation of rolls. However, in the absence of latent heat fluxes, the convective rolls characteristics remain similar to the reference simulation.

Conclusions

Ensuring an accurate representation of air-sea fluxes is crucial to a better understanding of the marine atmospheric boundary layer dynamics. However, uncertainties persist in the parameterization of fluxes during windstorms, due to the lack of observations for wind speeds above 20 m s -1 . Such strong winds occurred during the passage of the Mediterranean cyclone Adrian studied here. The main objective of this study is to investigate the role of air-sea fluxes in the formation of fine-scale convective rolls using large eddy simulations (LES) run with the Meso-NH atmospheric model. Convective rolls, rarely studied in mid-latitude windstorms, have been shown to be involved in the downward transport of strong winds and the formation of maximum surface wind gusts associated with Adrian.

First, the impact of four air-sea flux parameterizations on mesoscale windstorm development is examined, using convection-permitting simulations over a period of 6-hours. The parameterizations used differ in the formulation of surface fluxes of latent heat, sensible heat and momentum, and also in the way they take into account the effects of physical processes such as sea spray. The choice of surface flux parameterization does not affect the windstorm trajectory. This shows that, on this relatively short time scale covering the mature phase of cyclogenesis, the windstorm trajectory is mostly driven by the large-scale flow. However, the windstorm intensity shows a slight sensitivity to the inclusion of sea spray effects in heat fluxes, which reinforce heat fluxes. As a result, surface pressure drops by a few hPa more than in the other simulations, leading to a slight increase in surface wind speed. As shown in previous studies [START_REF] Li | Effects of roll vortices on the evolution of hurricane harvey during landfall[END_REF][START_REF] Lfarh | The downward transport of strong wind by convective rolls in a mediterranean windstorm[END_REF], a large-scale simulation approach is indispensable for understanding the fine-scale processes responsible for transporting strong winds to the surface, such as the convective rolls in this study. The findings here highlight that, despite sufficient resolution to resolve such processes, an incomplete representation of the physical mechanisms at the air-sea interface under strong wind conditions can lead to significant discrepancies, which can have an impact on wind forecast and, consequently, on the safety of people and property.

The contribution of additional surface processes other than sea spray might be considered in future work. Although foam is regularly observed at the air-sea interface in strong winds, experiments to measure its impact on surface fluxes are relatively recent (Sroka & Emanuel, 2021). The role of foam in convective rolls and surface winds should be assessed and probably a contribution to consider in improving surface flux parameterizations. Apart from the turbulent exchange of air-sea fluxes, wind-wave interactions modify atmospheric and oceanic conditions, which can impact winds during windstorms. Considering the dynamic effects of waves can lead to improved forecasting of wind speeds in storms [START_REF] Wahle | An atmosphere-wave regional coupled model: improving predictions of wave heights and surface winds in the southern north sea[END_REF]. A first task for further studies is to couple the atmospheric model with a wave model. While surface latent heat exchange does not appear to be crucial for the development of WASP (Wave-Age Stress dependant Parametrization) [START_REF] Sauvage | Characterization of the air-sea exchange mechanisms during a Mediterranean heavy precipitation event using realistic sea state modelling[END_REF] is based on the Coare3.0 and Coare3.5 parameterizations and allows wave growth to be explicitly taken into account in the calculation of roughness length in a wind range between 5 and 20 m s -1 . Above 20 m s -1 , the contribution of wave breaking is dominant, so wave age is no longer a sufficient parameter to represent the effect of sea state on surface roughness. Since different mechanisms are involved at low, moderate and high wind speeds, a piecewise continuous description is adopted to describe the Charnock parameter that relates waves and wind stress as a function of sea state. This approach makes it possible to represent the observed decrease in drag coefficient under intense wind conditions.

Figure 1 :

 1 Figure 1: Wind speeds at 10 m height at 1530 UTC, from reference simulations at 1000 m (a) and 200 m (b) horizontal resolution. The black squares represent the D1 and D2 domains in(a). The wind direction at 10 m is represented by the green arrows. In (b), the white square indicates the zoomed-in area of 20 by 20 km shown in Figure 5.

2. 2

 2 Parameterizations of turbulent fluxes at the air-sea interface Parameterizing turbulent fluxes consists in defining a formulation of turbulent fluctuations close to the surface. The formulation takes into account all processes

Figure 2 :

 2 Figure 2: Turbulent fluxes of (a) momentum, (b) sensible and (c) latent heat as a function of wind speed at 10 m from the Coare2 (green), Andreas2 (blue), Ecume2 (pink) and Wasp2 (orange) simulations at 1530 UTC. Turbulent fluxes are calculated at each grid point on the zoom represented by the white square on Figure 1b.

  tories from the Coare1, Andreas1, Ecume1 and Wasp1 simulations are shown over the D2 domain from 1200 to 1700 UTC (Figure3a), until the cyclone moves out of the domain and arrives on land. At 1200 UTC, Adrian is located to the west of Sardinia, then moves northwards along a meridional trajectory. It approaches Corsica between 1500 and 1600 UTC. All simulated trajectories remain close to the best track, demonstrating the ability of the model to capture the trajectory of Adrian.

Figure 3 :

 3 Figure 3: (a) Trajectories of Adrian as observed (black line) and simulated by Coare1, Andreas1, Ecume1 and Wasp1 (colored lines) every 15 min between 1200 and 1700 UTC. The black circular markers indicate the position observed every hour. Terrain height is shown in color. The best observed track is estimated from the center of the cyclonic cloud roll-up on the Spinning Enhanced Visible and InfraRed Imager (SEVIRI) High Resolution Visible (HRV) satellite image, while simulated trajectories are estimated from the position of the lowest mean sea-level pressure. (b) Temporal evolution of minimum pressure at sea level and (c) 99th percentile wind speed at 10 m over sea. The evolutions are calculated over the sea part of the D2 domain between 1200 and 1800 UTC.

Figure 4 :

 4 Figure 4: (a) Maximum wind speed at 10 m observed between 1200 and 1800 UTC at stations in Corsica. Markers with large circles correspond to coastal stations recording wind speeds exceeding 20 m s -1 . (b) Box plots of the bias and root-mean-square error (RMSE) between simulated and observed wind speeds at the 11 coastal stations, calculated over a period from 1200 to 1800 UTC every 6 min. Boxes represent simulations Coare1, Andreas1, Ecume1 and Wasp1 in green, blue, pink and orange colors, respectively.

  ) illustrates band-like wind structures where speeds exceed 48 m s -1 . The structures are approximately oriented in the wind direction from the southwest (red arrows). Horizontal sections are shown at z = 200 m, because the structures are well marked at this altitude. The vertical extent of the structures is roughly equivalent to the height of the boundary layer z i , which reaches around 700 m (Figure 5b). A regular alternation of upward and downward velocities can be seen on vertical winds (black contours), which is a signature of rolls in the boundary layer. This result is in line with the findings of Lfarh et al. (2023), who showed that the boundary layer in strong wind regions is organized into convective rolls at 1515 UTC in windstorm Adrian. Such structures are observed throughout the narrow belt of strong winds and persist until the weakening phase of the windstorm.

  Figure 6 presents vertical profiles of the flatness parameter, orie According to Brilouet et al. (2023) a threshold on the flatness parameter is

1 )

 1 Figure 5: (left) Horizontal sections at z=200 m and (right) vertical sections of horizontal wind speed at 1530 UTC from (a, b) Coare2, (c, d) Andreas2, (e, f) Ecume2 and (g, f) Wasp2. The black lines in the horizontal sections shows the location of the vertical sections. The wind direction is shown by the red arrows. The thick red line in the vertical sections corresponds to the height of the boundary layer. Black contours indicate vertical velocity, solid lines correspond to 1 m s -1 and dotted lines to -1 m s -1 .

Figure 6 :

 6 Figure 6: Vertical profiles of (a) flatness parameter f , (b) mean wind direction dotted curves) and structure direction (solide curves), (c) convective roll size identified from the auto-correlation function at 1530 UTC. Dashed lines in (a) indicate the boundary layer height averaged over the zoomed domain in Figure 5 for each simulation.

Figure 7 :

 7 Figure 7: Vertical profile of (a) horizontal wind speed, (b) virtual potential temperature and (c) vertical fluxes of zonal momentum at 1530 UTC. The dotted horizontal lines indicate the boundary-layer height in (a). The dashed, dotted and solid lines show the resolved, subgrid and total contributions in (c). The profiles are averaged over the area shown in Figure 5.

Figure 8 :

 8 Figure 8: Probability density functions (PDFs) of (a) vertical wind speed at 400 m height and horizontal wind speed at heights (b) 400 m and (c) 10 m. The vertical lines indicate the mean values in (b, c). The PDFs are calculated over the area shown in Figure 5.

  Second, the impact of the four parameterizations on the development and organization of the convective rolls is examined through LES performed at a horizontal grid spacing of 200 m. Convective rolls are systematically formed in the strong wind region due to dynamic and thermal instabilities, whatever the air-sea flux parameterization used. A schematic (Figure12) summarizes the main results relative to the sensitivity of convective rolls to surface fluxes. When heat fluxes are moderate, rolls are characterized by sizes of around 2000 m and extend vertically to the middle of the boundary layer (Figure12, left panel). However, the morphological characteristics of the rolls are significantly influenced by surface fluxes, in particular the increased heat fluxes resulting from sea spray. Stronger heat fluxes lead to more important thermal instability at the air-sea interface. Hence, convection-induced vertical motions become more pronounced in the boundary layer. The convective rolls formed in such conditions are of reduced size, greater in number, and exhibit a higher vertical extent (Figure12, middle panel). With this organization, the vertical transport of momentum by the convective rolls is increased by 40% and persists closer to the surface, before turbulence takes over. This results in higher surface wind speeds, although surface momentum fluxes are also strengthened by sea spray. This increase in vertical momentum transport approximately matches the increase in roll number, which leaves the transport per roll unaltered. Hence, it is assumed that the transport in the presence of sea spray is enhanced by the larger number of rolls. Furthermore, the stronger atmospheric instability results in a deeper boundary layer. This allows the rolls to form over a wider vertical extent and, consequently, to transport additional momentum from higher altitudes. Third, the respective role of surface latent heat, sensible heat and moment fluxes is disentangled by performing LES sensitivity tests. Even in the absence of latent heat flux, convective rolls continue to form and retain similar characteristics, although their size is slightly affected. This indicates that latent heat fluxes have little impact on roll organization. In contrast, convective rolls quickly disappear without sensible heat fluxes (Figure 12, right panel). They also disorganize within a few minutes without surface momentum fluxes. Both of these fluxes are directly responsible for the combination of thermal and dynamic instability required to form and maintain convective rolls.

Figure 12 :

 12 Figure 12: Summary of the main results of the paper. Red and blue vertical arrows represent surface sensible and latent heat fluxes, respectively, while green horizontal arrows represent the background wind shear and circling arrows represents convective rolls. See text for details.

Table 1 :

 1 Configuration of air-sea turbulent flux sensitivity experiments including the main parameterization schemes used.

	Simulation Grid mesh (m) Advection Shallow convection Turbulence	Air-Sea fluxes
	Coare1	1000	WENO	Yes	1D	COARE
	Andreas1	1000	WENO	Yes	1D	ANDREAS
	Ecume1	1000	WENO	Yes	1D	ECUME6
	Wasp1	1000	WENO	Yes	1D	WASP
	Coare2	200	CEN4TH	No	3D	COARE
	Andreas2	200	CEN4TH	No	3D	ANDREAS
	Ecume2	200	CEN4TH	No	3D	ECUME6
	Wasp2	200	CEN4TH	No	3D	WASP
	NoM	200	CEN4TH	No	3D	No momentum fluxes
	NoH	200	CEN4TH	No	3D	No sensible heat fluxes
	NoLE	200	CEN4TH	No	3D	No latent heat fluxes

Table 2 :

 2 Descriptive parameters of the boundary layer and of convective rolls in the LES sensitivity tests.

	Simulation	L (m)	z i (m) ζ =-z i /L	λ	λ/z i
	Coare2	-1453	732	0.50	2000 2.73
	Andreas2	-701	1098	1.56	1400 1.27
	Ecume2	-1089	780	0.71	2400 3.07
	Wasp2	-1331	703	0.52	2400 3.41
	NoLE	-1372	692	0.50	1600 2.31
	NoH	-1938793	509	0.0002	-	-
	NoM	0	854	-	-	-
	Therefore, as opposed to other simulations where shear dominates turbulent gen-
	eration, the enhanced thermal instability by a factor 2-3 in Andreas2 means that
	turbulence is mostly generated by buoyancy. Furthermore, the instability param-

eter is widely used to predict roll convection and quantify the effects of shear and buoyancy on roll formation. The obtained values of ζ are consistent with results from

[START_REF] Weckwerth | An observational study of the evolution of horizontal convective rolls[END_REF]

, which indicate that roll structures appear for ζ < 10,

Acknowledgments

Computer resources for running Meso-NH were allocated by CALMIP through project P20024 and GENCI through project 0111437. This work was supported by the French National Research Agency under grant agreement ANR-21-CE01-0002 and by COST Action CA19109. The authors thank Carlos Granero Belinchon for providing the auto-correlation tools and for the discussions that contributed to the elaboration of a part of the paper and Météo France for making the in-situ observations available online through the MeteoNet application https://meteonet.umr-cnrm.fr/. The authors also thank Juan Escobar and Joris Pianezze of the Meso-NH support team for their assistance during this study.

manuscript submitted to JGR: Atmospheres

convective rolls, another approach would be to investigate the impact of cloud microphysics, focusing on the effect of evaporative cooling on roll formation and wind transport. It has been shown that evaporative cooling can reduce the buoyancy of air parcels in windstorms, facilitating their descent to the surface and contributing to the amplification of surface wind gusts [START_REF] Browning | The role of evaporating showers in the transfer of sting-jet momentum to the surface[END_REF][START_REF] Ludwig | Secondary Cyclogenesis along an Occluded Front Leading to Damaging Wind Gusts: Windstorm Kyrill[END_REF].

In this study, all available data, including in-situ observations and satellite images, are exploited to evaluate the different simulations. However, when it comes to winds at sea, these data proved insufficient. Measurements from buoys at sea are lacking in the region of strong winds, and would be extremely useful for completing the validation of the simulations. Although large eddy simulations provide detailed information on fine-scale processes, their evaluation is hampered by the lack of high-resolution observations. In this context, synthetic aperture radar (SAR) satellite images have potential to reveal the details of convective rolls, as demonstrated by [START_REF] Brilouet | Trade wind boundary layer turbulence and shallow precipitating convection: New insights combining SAR images, satellite brightness temperature, and airborne in situ measurements[END_REF]; [START_REF] Stopa | Automated global classification of surface layer stratification using high-resolution sea surface roughness measurements by satellite synthetic aperture radar[END_REF]. Such observation may constrain parameterizations of surface fluxes via their control on roll characteristics. Thus, the methodology of LES over large domain developed here should be extended to other case studies. This would be essential to reach more general conclusions about the impact of air-sea fluxes on the transport of strong winds to the surface.

Appendix A Description of the turbulent flux parameterizations

The Coupled Ocean-Atmosphere Response Experiment (COARE3.0) [START_REF] Fairall | Bulk parameterization of air-sea fluxes: Updates and verification for the coare algorithm[END_REF] used in the reference simulations, is one of the most commonly used parameterizations for interactions at the air-sea interface. COARE3.0 is derived from the COARE2.6 algorithm [START_REF] Fairall | Bulk parameterization of air-sea fluxes for tropical ocean-global atmosphere coupled-ocean atmosphere response experiment[END_REF] initially developed from observations made during the TOGA COARE experiment in the North Pacific [START_REF] Webster | Toga coare: The coupled ocean-atmosphere response experiment[END_REF]. COARE3.0 uses a new formulation for roughness length which slightly increases the fluxes for wind speeds above 10 m s -1 . The wave effect is taken into account through the roughness length. However, COARE3.0 is mainly valid for wind speeds up to 20 m s -1 due to the lack of observations for strong wind conditions. ANDREAS [START_REF] Andreas | An improved bulk air-sea surface flux algorithm, including spray-mediated transfer[END_REF] is the only parameterization that distinguishes two different contributions to turbulent heat fluxes: standard air-sea interfacial fluxes and spray fluxes controlled by microphysical processes around the spray droplets. The small droplets are ejected into the atmosphere by bubble bursting at the air-sea interface during wave breaking or by the wave clipping mechanism of strong winds [START_REF] Veron | Ocean spray[END_REF]. The contribution to heat exchange becomes significant for wind speeds above 13 m s -1 . The total flux in the presence of sea spray is therefore the sum of the interfacial flux calculated by the bulk method and the flux related to sea spray calculated using the microphysical algorithm described in [START_REF] Andreas | Approximation formulas for the microphysical properties of saline droplets[END_REF].The ANDREAS parameterization was developed with observations including wind speeds up to 25 m s -1 .

ECUME6 is the new version of Exchange Coefficient Unified Multi-campaign

Experiments. This parameterization is developed and used operationally by Météo-France [START_REF] Belamari | Report on uncertainty estimates of an optimal bulk formulation for surface turbulent fluxes[END_REF]. ECUME6 is based on in-situ measurements of oceanatmosphere fluxes from different field campaigns, considering strong wind conditions. Contrary to the COARE algorithms, the turbulent exchange coefficients are computed directly from the observations, which makes it impossible to take into account the effect of waves explicitly in the roughness length calculation. In ECUME6, measurements realized by [START_REF] Powell | Reduced drag coefficient for high wind speeds in tropical cyclones[END_REF], [START_REF] Donelan | On the limiting aerodynamic roughness of the ocean in very strong winds[END_REF] and [START_REF] French | Turbulent fluxes in the hurricane boundary layer. part i: Momentum flux[END_REF] allowed to consider the saturation and decrease of the exchange coefficients for wind speeds higher than 30 m s -1 .