Benoît Nicolas Aubin

Benoît Augier

Patrick Bot

Frédéric Hauville

Matthieu Sacher

Richard Flay

Wind tunnel investigation of dynamic trimming on upwind sail aerodynamics

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1 nicolas.aubin@ecole-navale.fr Mainsail sheet length (mm) M Smax Mainsail with maximum camber for the design shape M Sstd

Mainsail with standard camber for the design shape M Sf lat Mainsail with zero camber for the design shape P Mechanical power from the sheet (mW) q Dynamic pressure (Pa) S Sail mould area (m 2 ) T Time period of oscillation (s) U ref

Reference wind velocity (m s -1 ) ρ Density of air (kg m -3 )

INTRODUCTION

A challenging task in yacht design modeling and simulation is the analysis of dynamic effects in the Fluid Structure Interaction (FSI) of the yacht sails and rig. The dynamic behavior can be caused by the sea state or the wind, but can also be caused by the action of the crew while trimming. Literature has pointed out the difficulty of considering the realistic sailing environment of a yacht [START_REF] Charvet | Numerical of the flow over sails in real sailing conditions[END_REF][START_REF] Marchaj | Sail performance: techniques to maximize sail power[END_REF][START_REF] Garrett | The symmetry of sailing: the physics of sailing for yachtsmen[END_REF]. Recent studies have underlined the importance of considering the dynamic behavior: forced pitching motion in the wind tunnel [START_REF] Fossati | An experimental investigation of unsteady sail aerodynamics including sail flex[END_REF], 2D simplified pitching [START_REF] Gerhardt | Unsteady aerodynamics of two interacting yacht sails in twodimensional potential flow[END_REF], interaction of yacht sails in unsteady conditions [START_REF] Gerhardt | Unsteady Aerodynamics of Upwind-Sailing and Tacking[END_REF], full scale experiments and simulations [START_REF] Augier | Experimental validation of unsteady models for fluid structure interaction: Application to yacht sails and rigs[END_REF][START_REF] Augier | Dynamic Behaviour of a Flexible Yacht Sail Plan[END_REF][START_REF] Augier | Numerical study of a flexible sail plan submitted to pitching: Hysteresis phenomenon and effect of rig adjustments[END_REF], and downwind sails [START_REF] Gerritsen | The challenging turbulent flows past downwind yacht sails and practical application of CFD to them[END_REF]Gerritsen, 2006, Deparday et al., 2014). Downwind sail design is where the gain from a dynamic aero-elastic analysis seems to be potentially the greatest due to the large motion and the induced large load variation. The main findings of these different studies are the same, i.e. the aerodynamics can be predicted more accurately with an unsteady approach.

To account for this dynamic behavior, several Dynamic Velocity Prediction Programs (DVPPs) have been developed [START_REF] Masuyama | Dynamic performance of sailing cruiser by a full scale sea reality[END_REF][START_REF] Masuyama | Full scale measurement of sail force and the validation of numerical calculation method[END_REF][START_REF] Richardt | Maneuvering simulations for ships and sailing yachts using FRIENDSHIP-Equilibrium as an open modular workbench[END_REF][START_REF] Keuning | A generic mathematical model for the manoeuvring and tacking of a sailing yacht[END_REF] which need models of dynamic aerodynamic and hydrodynamic forces. While the dynamic effects on hydrodynamic forces have been studied extensively, the unsteady aerodynamic behavior of sails has received much less attention. [START_REF] Schoop | Instationary aeroelastic computation of yacht sails[END_REF] first developed an unsteady aeroelastic model in potential flow dedicated to flexible membranes but neglected the inertia. In a quasi-static approach, a first step is to add the velocity induced by the yacht's motion to the steady apparent wind to build an instantaneous apparent wind [START_REF] Richardt | Maneuvering simulations for ships and sailing yachts using FRIENDSHIP-Equilibrium as an open modular workbench[END_REF][START_REF] Keuning | A generic mathematical model for the manoeuvring and tacking of a sailing yacht[END_REF] and to consider the aerodynamic forces corresponding to this instantaneous apparent wind using force models obtained in the steady state.

Recently, advanced computational resources have enhanced numerical simulations and have allowed coupling of fluid and structural solvers dedicated to yacht sails (Renzsh and [START_REF] Renzsh | Fluid Structure Interaction simulation of spinnakers -Getting closer to reality[END_REF][START_REF] Chapin | Performance optimization of interacting sails through fluid structure coupling[END_REF][START_REF] Trimarchi | Wrinkle development analysis in thin sail-like structures using MITC shell finite elements[END_REF][START_REF] Ranzenbach | Mainsail Planform Optimization for IRC 52 Using Fluid Structure Interaction[END_REF]. In past years, IRE-Nav and the K-Epsilon company have developed numerical tools dedicated to the simulation of the dynamic behavior of yacht sails. The FSI potential model ARAVANTI has been validated by full scale measurements [START_REF] Augier | Experimental validation of unsteady models for fluid structure interaction: Application to yacht sails and rigs[END_REF] and enables numerical studies of a yacht pitching in a head swell [START_REF] Augier | Dynamic Behaviour of a Flexible Yacht Sail Plan[END_REF][START_REF] Augier | Numerical study of a flexible sail plan submitted to pitching: Hysteresis phenomenon and effect of rig adjustments[END_REF], showing a clear break with the quasi-static approach. The recent RANS FSI coupling ARA-FINE TM /Marine [START_REF] Durand | FSI investigation on stability of downwind sails with an automatic dynamic trimming[END_REF] is required to simulate cases with strong separation for downwind simulations, but it is very time and CPU consuming.

Even though some advanced models are now available for sail aerodynamics, there is a real need for detailed validation of numerical simulations in order to provide reliable design tools for the sailing industry. Controlled experiments are also a great opportunity to understand the physics of FSI of yacht sails. Unfortunately, realistic and reliable experimental data is scarce and the validation of models in real conditions is difficult [START_REF] Augier | Experimental validation of unsteady models for fluid structure interaction: Application to yacht sails and rigs[END_REF][START_REF] Fossati | A novel full scale laboratory for yacht engineering research[END_REF]. In this context, wind tunnel testing and full-scale testing are required for comparison and validation [START_REF] Flay | A twisted flow wind tunnel for testing yacht sails[END_REF][START_REF] Renzsch | An experimental validation case for fluid-structure-interaction simulations of downwind sails[END_REF], Le Pelley et al., 2002). Wind tunnel testing has the advantage of being in a controlled environment where a balance can be used to measure the forces created by the sails on the boat frame [START_REF] Viola | On-water pressure measurements on a modern asymmetric spinnaker[END_REF][START_REF] Fossati | Aero-Hydrodynamics and the Performance of Sailing Yachts: The Science Behind Sailing Yachts and Their Design[END_REF][START_REF] Fossati | Sails Aerodynamic Behavior in dynamic condition[END_REF], 2010[START_REF] Wright | Off-wind sail performance prediction and optimisation[END_REF]. Pressure and flying shape measurements can also be performed in wind tunnels [START_REF] Lasher | Validation of Reynoldsaveraged NavierStokes simulations for international Americas Cup class spinnaker force coefficients in an atmospheric boundary layer[END_REF], Graf and Müller, 2009[START_REF] Viola | Sail pressures from fullscale, wind-tunnel and numerical investigations[END_REF][START_REF] Viola | Upwind sail aerodynamics: A RANS numerical investigation validated with wind tunnel pressure measurements[END_REF]. In a recent study, [START_REF] Gerhardt | Unsteady aerodynamics of two interacting yacht sails in twodimensional potential flow[END_REF] developed an analytical model to predict the unsteady aerodynamics of interacting yacht sails in 2D potential flow, and performed 2D wind tunnel oscillation tests with a motion range typical of a 82-foot (25m) racing yacht (1992 International America's Cup Class). Recently [START_REF] Fossati | An experimental investigation of unsteady sail aerodynamics including sail flex[END_REF], 2009, 2010[START_REF] Schafer | What Is a Savitzky-Golay Filter?[END_REF] studied the aerodynamics of model-scale rigid sails in a wind tunnel, and showed that pitching motion has a strong and non-trivial effect on aerodynamic forces.

A dedicated experiment has been developed in the Yacht Research Unit Twisted Flow Wind Tunnel, University of Auckland, to study the aerodynamics of dynamic trimming. The model was simplified to a simple model-sized IMOCA 60 mainsail and a mast with no shrouds. We measured the effect of dynamic trimming on the forces (F x , F y ) with the balance and the load in the sheet (F sheet ) for a given incoming wind (U ref = 3.5 m s -1 at 1.5 m height at model-scale location in an empty wind tunnel configuration) for 3 different sail design shapes.

In the first part of the paper, we describe the experimental set up and we define the optimum trimming. In the second part, the results are presented for different trimming oscillation amplitudes and frequencies and for different sail design shapes. Finally the influence of these different parameters on the global performance of the rig is discussed.

EXPERIMENTAL SETUP

Experiments were performed thanks to the Sailing Fluids collaboration program in the Twisted Flow Wind Tunnel of the Yacht Research Unit of the University of Auckland described in [START_REF] Flay | A twisted flow wind tunnel for testing yacht sails[END_REF].

An 1/13th scale IMOCA 60 foot design mainsail was designed and built by INCIDENCE SAILS, using SAIL-PACK software developed by BSG DEVELOPPEMENTS, for these experiments. A system of three stepper motors and a control card was used in order to modify the main sheet length L sheet and main car position L car (see Fig. 1). Therefore, the mainsail trimming was imposed remotely without any human contact in the wind tunnel. The uncertainty of imposed trimming was estimated to be ±2 mm through repeated measurements. Fig. 2 shows the 2.2 m long mast with the scaled mainsail in the 7.2 m wide by 3.5 m tall open jet test section of the YRU wind tunnel. Sail geometry is defined in Fig. 1. The rig is composed of a single 14 mm circular section carbon mast without spread- 2) through the VS-PARS acquisition system [START_REF] Le Pelley | VSPARS: A combined sail and rig recognition system using imaging techniques[END_REF]. The sampling frequency of the system measurement is 200 Hz and every run is recorded over 30 s.

The velocity profile follows the empty wind tunnel boundary layer profile and is not twisted (no vanes in the flow). The apparent wind speed (AW S) is U ref = 3.5 m s -1 ± 0.15 m s -1 -measured at 1.5 m high at the model-scale location in an empty configuration-and an apparent wind angle (AW A) set to 60 • ± 2 • .

A Pitot tube in the wind tunnel roof, was used to measure the dynamic pressure during each run. The mean value q(t) calculated for each test was used for the normalization of

Optimum trimming

Different sail design shapes were tested. Three sails, made from the same sail cloth were designed with different cambers:

• MSstd = camber of the full scale sail (9.19% at the reference stripe)

• MSflat = no camber

• MSmax = more camber than MSstd (11.67% at the reference stripe)

A first test was performed in order to determine the best trim for the studied AW A = 60 • . The model was placed on the balance and the sail was statically trimmed to the optimum CF obj = CF x -0.1|CF y |. This optimization target takes into account the contribution of the side force on the aerodynamic force and can be found in the design process of sailing yacht to consider the penalty due to the added hydrodynamic drag and leeway. For more details on the optimization function readers should refer to [START_REF] Sacher | Sail trimming FSI simulation -comparison of viscous and inviscid flow models to optimise upwind sails trim[END_REF]. Three stepper motors were used as winches to trim the sail: two motors used to trim the traveller position L car and one centered motor used to trim the main sheet length L sheet . Here we were looking for the best 2 trimming parameters (L sheet , L car ). Optimum trimming was extracted from the test using the algorithm described in [START_REF] Sacher | Experimental and numerical trimming optimizations for a mainsail in upwind conditions[END_REF] and used as the reference for the dynamic trimming described in the following sections.

Dynamic trimming

The dynamic trimming consists of an oscillation in the sheet length L sheet around the optimum trimming length obtained previously. The dynamic trimming was done with a fixed traveller position L car (obtained from the optimum trimming) and the instantaneous sheet length L sheet (t) could be calculated from the controlled and recorded angular position of the rotating plate (see Fig. 3 and1 ). L sheet (t), the instantaneous length of the sheet, is a function of A the amplitude of variation in mm, f the frequency of oscillation (rotation frequency of the stepper motor controlling the rotating plate) in Hz and the model-scale configuration geometry. The frequency f and amplitude A of oscillation were controlled by the rotating plate placed at the center-line of the boat as illustrated in Fig. 3. The sheet was connected to a pin fixed on the plate. The amplitude of oscillation depends on radial position of the pin. A = 10 mm stands for an eccentric of 10 mm and corresponds to a peak to peak amplitude of motion of 20 mm on L sheet (oscillation of L sheet of ±10 mm).

Rotating plate frequency f

Pin Amplitude A Lsheet Sail

RESULTS

We examine here the influence of the dynamic trimming on the aerodynamic forces of the sail. Three different sails were tested for 3 amplitudes of oscillation of 10 mm, 20 mm and 30 mm and 7 ordered frequencies from 0 Hz to 3 Hz. From these frequencies, non-dimensional reduced frequencies f r are defined in the post processing parameters.

Post processing parameters

In this study we define the reduced frequency f r = f.c/U ref , with f the frequency of oscillation in Hz, c the reference chord length c = S/h = 0.475 m and U ref = 3.5 m s -1 the reference flow speed. The reduced frequency is a non-dimensional indicator defined as the ratio of the oscillating motion to the reference convection time, from 0 to 0.38.

In the following part, aerodynamic forces and sheet loads are normalized as:

• the instantaneous aerodynamic driving force is defined using CF x (t) = Fx(t) q(t)S

• its mean value presented in our study

CF x = CF x (t) = Fx(t) q(t)S
• equivalent definition is used for the side force coeffi-

cient CF y = CF y (t) = Fy(t) q(t)S
• equivalent definition is used for the sheet load coeffi-

cient CF sheet = CF sheet (t) = F sheet (t)
q(t)S

• q(t) = 1 2 ρU (t) 2 is the dynamic pressure measured during the run by the pitot tube.

Forces were averaged over an integer number of period of oscillation regardless of the reduced frequency in order to compare relevant mean values. Time series were filtered with a low pass filter frequencies defined as a Savitzky-Golay filter of order 1 of span 21 samples [START_REF] Schafer | What Is a Savitzky-Golay Filter?[END_REF].

Effect of the reduced frequency f r

We focus here on the effect of the reduced frequency f r on the forces for the case of the standard mainsail (MSstd) for an oscillation amplitude A = 20 mm round the optimum L sheet . Coefficients were averaged over the maximum number of integer oscillation periods found in the 30 s recording. Results are presented in Fig. 4. Measurements were doubled and showed good repeatability. Up and down triangles represent the maximum amplitude i.e. the maximum and minimum value of the time series.

For the first oscillation frequency studied, f r < 0.02, the force coefficients decrease compared to the static situation f r = 0 values. The oscillation is very slow and could be considered as quasi-steady. This quasi-steady oscillation around the optimum L sheet degrades the performance because the sail is trimmed at a non-optimum point most of the time. For f r > 0.02, dynamic trimming increases the mean force coefficient, which reaches a maximum around f r = 0.255. The aerodynamic forces seem to benefit from an unsteady propulsion phenomenon due to the flapping of the sail. This unsteady propulsion is maximized for a defined range of frequencies and its effect decreases above f r = 0.255. Amplitudes of variation of the force coefficients, illustrated by the triangles in Fig. 4, increase significantly until f r = 0.255 and collapse dramatically at higher frequencies for CF x and CF sheet . In the case of CF y , the amplitude of variation keeps increasing with the frequency of oscillation. The results show the effect of dynamic trimming compared to the steady trimming maximizes CF x at a specific range of reduced frequency around 0.255.

Figs. 5 and 6 present the temporal evolution of the load in the sheet and the driving coefficient with the sheet length. This type of Lissajou representation was first proposed for a sailing yacht study by [START_REF] Fossati | Sails Aerodynamic Behavior in dynamic condition[END_REF], 2010[START_REF] Schafer | What Is a Savitzky-Golay Filter?[END_REF] in wind tunnel testing and was then used by [START_REF] Augier | Dynamic Behaviour of a Flexible Yacht Sail Plan[END_REF][START_REF] Augier | Numerical study of a flexible sail plan submitted to pitching: Hysteresis phenomenon and effect of rig adjustments[END_REF] in simulations and full scale measurements. For more clarity, signals are represented for only 12 s. We present 4 of the 8 studied frequencies, but the trends are identical. The top graph illustrates the static case. The number of cycles represented increases with the frequency f r . All the curves have been centered around their respective optimimum L sheet , which are slightly different for the different design shapes. L sheet = 0 mm is set at the static optimum trim L sheet static for the optimum of the optimization target CF obj .

CF sheet vs L sheet describes a loop which witnesses a hysteresis phenomenon (Fig. 5). In this case, the area inside the loop is the mechanical work exchanged with the rig system from the trimming stepper motor. The counterclockwise sense of rotation, indicated by the arrow on the figure, shows that the work is negative, i.e. given to the system. This confirms that the sail and rig system are forced by the motion of the sheet for the whole range of studied frequencies. The area in the loop increases slightly until f r = 0.255 where it reaches a maximum. The loop collapses at f r = 0.38. The work exchanged with the rig system is a maximum at f r = 0.255 which corresponds to the optimum CF x observed in Fig. 4a.

CF x vs L sheet describes a loop as well (Fig. 6). One should realise that the area inside the loops is not actual physical work however it follows the same trend as the work energy from F x along the x-direction. It is very interesting to observe that the sense of rotation switches for the different frequencies. For f r = 0.013 and 0.38, the system dissipates energy as it turns counter-clockwise. The system gains energy from the oscillation at f r = 0.255 (clockwise rotation). The f r = 0.127 case is a transition where the loop describes a figure 8 shape. Power is calculated at each reduced frequency and presented in Fig. 7. Power is proportional to the area in the loop illustrated in Fig. 5 and is defined as: The power value shown on Fig. 7 is the averaged value of each power value calculated using the previous equation on each entire period oscillation loop. The exchange of energy of the forcing is related to the maximum of force obtained at f r = 0.255 and the sudden collapse of the amplitude of variation of CF x and CF sheet at f r = 0.38. Nevertheless a part of the trend observed in the force coefficient needs to be explained by studying the other parameters of the other forcing parameters such as the sail camber i.e. the design shape.

P = q(t)S T

Effect of the design shape

The effects of the reduced frequency f r on the forces are presented for the 3 design shapes and the 3 amplitudes of oscillation. For each sail, the trimming oscillation is done around its specific optimum. L sheet Static are different for each case. Again, coefficients are averaged over the maximum number of full oscillation periods found in the 30 s recording. Results are presented in Figs. 8, 9 and 10, CF x , CF sheet and CF obj respectively. Oscillation amplitudes (maximum and minimum value) of force coefficients are not displayed for clarity but trends are identical to those described in the previous section. Due to the parameters of the optimum trimming (L sheet , L car ) for the flat mainsail design shape, high frequency oscillations could not be explored at A = 30 mm because the forcing was too strong.

The general trends described in the previous section are identical for the 3 studied sail design shape and the different amplitudes of oscillation. The tendencies observed at A = 20 mm are amplified at greater amplitude A = 30 mm and slightly minimized at A = 10 mm.

It is interesting to notice that the effect of the dynamic trimming is greater for the flat mainsail design MSflat. The CF x coefficient are nearly identical for the two cambered sails for A = 10 mm and A = 20 mm whereas the static performances are significantly worse. It seems that the dynamic behavior due to flapping catches/compensates for the defect of flat mainsail design MSflat in static conditions. The unsteady propulsion phenomenon is high enough to compensate for the poor aerodynamic performance of the flat sail in a steady trimming. The oscillation needs a minimum of amplitude of A > 10 mm to have a significant effect on the MSflat. However, the optimum of MSflat is reached for a specific frequency f r = 0.255 and decreases rapidly around this value, unlike the other sails MSmax and MSstd where the range of optimal frequencies is wider.

The load in the sheet in static situations i.e. f r = 0 is linked to the camber (Fig. 9). The static CF sheet is greater for the maximum camber mainsail MSmax and it is identical for the two other sails. Variations in the load in the sheet CF sheet for different frequencies are consistent with the effect of dynamic trimming observed on MSflat. The trends are identical with CF x . At low oscillation amplitudes, the sheet tension increases significantly for the flat sail until f r = 0.32, when the CF sheet reaches a maximum and decreases slightly after f r = 0.13 for the other sails. For A = 20 mm, the maximum load in the sheet is reached at lower frequencies but a greater load is still necessary to make the flat sail oscillate. It seems that at these amplitudes, the energy brought to the system by the forced oscillation is greater in the case of MSflat, which explains the important gain on the aerodynamic coefficients observed in Figs. 5 and6. The differences between the sails are smoothed at A = 30 mm in CF sheet , as illustrated by the energy brought to the system in CF x and CF y . Energy brought to the system by the oscillation of the sheet is illustrated in Fig. 7. The power exchanged is a maximum at f r = 0.255.

The CF obj evolution shown in Fig. 10 depends on both the camber of the sail and the amplitude of oscillation. For the low oscillation A = 10 mm in Fig. 10a, the optimization target shows a maximum for the flat mainsail design contrary to the standard and maximum camber designs which present a plateau from f r = 0.255. This plateau disappears for higher amplitude oscillations and all the curves present a maximum. The maximum camber design presents either the best optimization value or is fairly close to the maximum optimization target value regarding the different oscilation frequencies and amplitudes. For this AW A = 60 • this trend confirms the sailors' knowledge causing them to try to increase the camber by easing the outhaul of the sail to improve their performance in a situation such as a dogleg while sailing perpendicular to the true wind direction.

DISCUSSION

Dynamic contributions to aerodynamic forces can be decomposed into three components. The first one is due to the change in circulation around the profile. At this AW A = 60 • , the more camber, the more lift in static conditions. The second component is due to the unsteady propulsion caused by the forced oscillation. At a certain range of frequencies studied, flapping might produce vortices structures beneficial to the aerodynamic force produced by the sail. Vortices structures are linked to the sail area, the frequency and the amplitude of oscillation but do not depend on the sail profile. This flapping effect on the flat sail MSflat which suffers from a poor static aerodynamic contribution is then much more significant but on a narrow range of frequencies. The third component is the energy transferred to the system by the oscillation forcing. As mentioned in the section describing the effect of the reduced frequency f r , mechanical work is given to the system by the forcing in the sheet (Fig. 7). This work is dissipated at most of the frequencies but is beneficial to the thrust at f r = 0.255 (Fig. 8). In the case of a dynamic trimming, the aerodynamic force is composed of the three components with different effects depending on the frequencies and amplitudes of oscillation. It seems that the forcing at f r = 0.255 benefits from all three components, the reason for the local optimum at that frequency.

CONCLUSIONS

An innovative oscillating trimming experiment has been developed in the Twisted Flow Wind Tunnel at the Yacht Research Unit, University of Auckland. The oscillating trimming effect has been studied on different design shapes of IMOCA 60 type mainsails at AW A = 60 • with different input parameters: amplitude and reduced frequency. The dynamic oscillations clearly show that quasi-static measurements are not relevant for predicting aerodynamic forces even at quite low reduced frequencies. These results support previous findings that static or quasi-static approaches are not sufficient to capture the complexities of dynamic effects, even for the simplified oscillating trimming simulation.

The sheet load measurement enabled us to calculate the mechanical power transmitted from the trimming device to the entire rig and sail system and could be correlated with the aerodynamic force evolutions of the different sails.

The dynamic effect showed that there was an optimum reduced frequency f r = 0.255 that improved the performance function for the different sails related to a maximum power transmitted to the rig and sails by the sheet. The three different model sails presented the same trends, but the dynamic improvement was more significant for the flat sail: up to an increase of 40% of its CF obj at f r = 0.255 and A = 30 mm compared to the steady case.

Oscillations around optimum static trim have also been performed for AW A = 25 • and AW A = 40 • and will be compared to this paper's results in a future publication. Flying shape and rig part tracking analysis will also be performed. Further work will be done using those data for comparison and validation of unsteady numerical simulation tools.

Figure 1 :

 1 Figure 1: Experimental set up for dynamic trimming. Dimensions are in mm.

Figure 2 :

 2 Figure 2: Model mainsail in YRU Twisted Flow Wind Tunnel, University of Auckland

Figure 3 :

 3 Figure 3: Experimental set up for dynamic trimming: rotating disk (photograph taken without wind)

  CF sheet for A = 20 mm

Figure 4 :

 4 Figure 4: Effect of f r at amplitude A = 20 mm on CF x (fig 4a), CF y (fig 4b) and CF sheet (fig 4c) for the standard mainsail. Up and down triangles represent the maximum and minimu amplitude of the time series respectively

Figure 5 :

 5 Figure 5: Evolution of CF sheet with L sheet at amplitude A = 20 mm for the standard mainsail for different reduced frequencies. Signals are presented for 12 s. The steady part was done without load sensor, so no steady load sheet is available for this configuration.

Figure 6 :

 6 Figure 6: Evolution of CF x with L sheet at amplitude A = 20 mm for the standard mainsail for different reduced frequencies. Signals are presented for 12 s. Red crosses represent the steady state extrapolated from the 2D optimization part data.

Figure 7 :

 7 Figure 7: Power given to the system by the sheet oscillation forcing at A = 20 mm. Power is proportional to the area in the loop represented in Fig. 5

Figure 8 :

 8 Figure 8: Effect of f r on CF x for the 3 design shape at amplitude (a) A = 10 mm, (b) A = 20 mm and (c) A = 30 mm.

Figure 9 :

 9 Figure 9: Effect of f r on CF sheet for the 3 design shape at amplitude (a) A = 10 mm, (b) A = 20 mm and (c) A = 30 mm.

ACKNOWLEDGEMENTS

This project has received funding from the European Unions Seventh Program for research, technological development and demonstration under grant agreement N o PIRSES-GA-2012-318924 and from the Royal Society of New Zealand for the UK-France-NZ collaboration project, SAILING FLUIDS (see www.sailingfluids.org). This work was supported by the French Naval Academy, Brest Métropole Océane, Région Bretagne and the Marie Curie European Unions Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n o PCOFUND-GA-2013-609102 (PRESTIGE-Campus France). This work was supported by the "Laboratoire d'Excellence" LabexMER (ANR-10-LABX-19) and co-funded by a grant from the French government under the program "Investissements d'Avenir". The authors are grateful to K-Epsilon and VS-PARS company for their continuous collaboration and to Ronan Floch from Incidence Sails for designing and manufacturing the model-scale sails. The authors would like to thank the SEFER services for providing the electronical actuator and remote control parts, Mr David Le Pelley, wind tunnel manager, and Dr Nick Velychko for their help, guidance, their wise advice and comments.


Anne Estrade
email: anne.estrade@parisdescartes.fr

Julie Fournier
email: julie.fournier@upmc.fr

Anisotropic Gaussian wave models

Keywords: Gaussian field, random wave, nodal statistics, level set, crossing theory, anisotropy 2010 Mathematics Subject Classification: primary 60G60, secondary 60G15, 60K40, 62H11, 86A05

come

Introduction

For many centuries, physicists have been using wave models defined on a multidimensional space in various domains as different as acoustics, electronics, geophysics, oceanography or seismology. In order to take into account variability or uncertainty, it is useful to consider random wave models. It is the exact purpose of a pioneer exhaustive study by Longuet and Higgins [START_REF] Longuet-Higgins | The Statistical Analysis of a Random, Moving Surface[END_REF] that was concerned by sea waves modelized as a random moving surface. Another mathematical pioneer study was raised by Berry in several papers, [START_REF] Berry | Regular and irregular semiclassical wavefunctions[END_REF] or [START_REF] Berry | Phase singularities in isotropic random waves[END_REF] for instance.

These seminal works opened a wide area of research in the last decades, either for statistical purposes ( [START_REF] Azaïs | Geometrical Characteristics of Gaussian Sea Waves[END_REF], [START_REF] Lindgren | Slepian Models for the Stochastic Shape of Individual Lagrange Sea Waves[END_REF], [START_REF] Aberg | Palm Distributions of Wave Characteristics in Encountering Seas[END_REF], [START_REF] Azaïs | Rice formula and Gaussian waves[END_REF], [START_REF] Beliaev | Two point function for critical points of a random plane wave[END_REF], [START_REF] Nourdin | Nodal statistics of planar random waves[END_REF]), or more recently for topological purposes in link with number theory ( [START_REF] Rudnick | On the volume of nodal sets for eigenfunctions of the Laplacian on the torus[END_REF], [START_REF] Krishnapur | Nodal length fluctuations for arithmetic random waves[END_REF], [START_REF] Marinucci | Non-Universality of Nodal Length Distribution for Arithmetic Random Waves[END_REF]). Ten years ago, the interest for nodal sets or level sets also met the theory of crossings developed by Rice for one-dimensional stochastic processes fifty years before, yielding two inspiring books by Adler and Taylor [START_REF] Adler | Random Fields and Geometry[END_REF] and by Azaïs and Wschebor [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]. The present paper is clearly inspired by all the above references but to the best of our knowledge it is the first time that the different models are gathered in the same work and are studied under the same focus: anisotropy.

A big demand for anisotropic models is nowadays observed, in particular by practitioners in geostatistics, offshore engineering, heterogeneous material or medical imaging (see for instance [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF], [START_REF] Klatt | Morphometry of random spatial structures in physics[END_REF], [START_REF] Allard | Anisotropy models for spatial data[END_REF]), but also for more theoretical studies dedicated to image synthesis and analysis, optics, cosmology or arithmetic ( [START_REF] Biermé | A turning band method for the simulation of anisotropic fractional Brownian fields[END_REF], [START_REF] Olhede | Detectiong directionality in random fields using the monogenic signal[END_REF], [START_REF] De Angelis | Spatial Distributions of Phase Singularities in Optical Random Vector Waves[END_REF], [START_REF] Planck | Planck 2015 results XVI. Isotropy and statistics of the CMB[END_REF], [START_REF] Kurlberg | On probability measures arising from lattice points on circles[END_REF]).

In the present paper, we aim at exploring the anisotropy of stationary random fields that are defined on a d-dimensional space with d ≥ 2. The random fields that we consider are stationary centred Gaussian fields whose covariance function is given by t ∈ R d → E[cos(k • t)] for any d-dimensional random vector k. We call them random waves since each of them has the same covariance function as any single random wave prescribed by t ∈ R d → √ 2 cos(k • t + η), where η is uniformely distributed on [0, 2π] and independent from k. Let us stress that the class of Gaussian random waves actually coincides with the one of stationary Gaussian fields that are centred and have unit variance. Indeed, for any such a field, it is sufficient to consider any random vector k, called random wavevector, whose distribution is equal to its spectral measure. Our purpose is to link the geometric and anisotropic behaviour properties of any Gaussian random wave with the distribution of its random wavevector.

We like to mention that when the random wavevector k is equal to Au with A a deterministic matrix and u a random vector in R d whose distribution is invariant under rotations, the associated Gaussian random wave has the same distribution as an isotropic random wave deformed by the linear transformation A T . In that case, the study of anisotropy, either in the spectral domain, or in the parameter domain, is equivalent. In the general case where no linear deformation is involved, studing anisotropy in those two domains falls under two different approaches. The latter point of view is adopted in [START_REF] Allard | Anisotropy models for spatial data[END_REF] or in [START_REF] Fournier | Identification and isotropy characterization of deformed random fields through excursion sets[END_REF] for instance, whereas our paper definitively belongs to the former type as did [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF] or [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF].

A part of our study focuses on Gaussian random waves associated with wavevectors whose norm and direction are independent random variables. In order to illustrate our results, some models are specified. Precisely, we exhibit anisotropic versions of famous Berry's monochromatic random waves (see [START_REF] Berry | Regular and irregular semiclassical wavefunctions[END_REF]), which are solutions of Helmholtz equation and are prescribed by a random wavevector belonging to a sphere. Another famous model is studied in the present paper: a space-time model adjusted for sea waves (see [START_REF] Longuet-Higgins | The Statistical Analysis of a Random, Moving Surface[END_REF]). It is a Gaussian random wave indexed by R 2 × R that indicates the sea height at each point and each time instant. Its three-dimensional random wavevector is forced to live in a two-dimensional surface known as Airy surface. All over the paper, two specific distributions for the random wavevector in dimension two are examined. One is called "elementary model". It is described by a main direction and a bandwidth that quantifies the anisotropy. We call the other one "toy model". It is given by a positive probability density function only depending on a single parameter that carries out the whole quantified information on anisotropy.

Our major contribution to the study of anisotropic Gaussian random fields is twofold: qualitative with a new point of view based on the associated wavevector, and quantitative with closed formulas linking the features of a random wave with the features of its wavevector. Concerning the former, we give a necessar and sufficient condition on the wavevector for the random wave to solve a partial differential equation almost surely (see Theorem 2.6). Concerning the latter, we prove that the expected length of the nodal lines of planar random waves is a decreasing function of the anisotropy of its random wavevector (see Proposition 3.4). We extend our study by presenting a formula for the expected length of a crest, which is the locus of points whose gradient has a prescribed direction (see Proposition 4.3). It allows us to prove that the direction that maximises the expected length of the crest lines is not necessarily orthogonal to the mode of the random wavevector's direction. For this purpose, we properly quantify anisotropy through various concepts, such as coherency index or favorite direction which are borrowed from spatial directional statistics.

The paper is organised as follows. General facts are presented in Section 2, in particular the key point of spectral representation, the link with partial differential equations that are solved by the random waves and at last the presentation of various models of Gaussian random waves characterized by the distribution of their random wavevector. In Section 3, we focus on the nodal sets, their Hausdorff measure (with an expectation formula and an upper-bound on the variance) and their directional statistics. Section 4 is devoted to the study of crest lines and their expected length, in the planar case. The technical computations are detailed in the Appendix section.

Notations and definitions.

We write N 0 for the set {0, 1, 2, • • • } of the non-negative integers and N the set of positive integers.

Let an integer d ≥ 2. We fix an orthonormal basis of R d and we use the same notation for a vector z in R d and the vector of its coordinates in this basis. For any z and z in R d , we write z • z the canonical Euclidian scalar product of z and z , • the associated norm and I d the identity matrix of size d. For ϕ ∈ [0, 2π], u ϕ denotes the vector (cos ϕ, sin ϕ) in R 2 .

For j = (j

1 , • • • , j d ) ∈ N 0 d , we write |j| = d l=1 j l . Moreover, if λ ∈ R d and if F is a smooth map from R d to R, we write λ j = d l=1 λ j l l and ∂ j F = ∂ |j| F ∂ j1 λ 1 • • • ∂ j d λ d .
We also denote by F (t) and by F (t) the gradient vector and the Hessian matrix of F at point t, respectively. For any positive integer s, H s denotes the Hausdorff measure of dimension s. Let Z = (z i ) 1≤i≤d be a random vector in R d . We write E[Z] its d-dimensional expectation vector, Z Z T the d × d matrix (z i z j ) 1≤i,j≤d and E[Z Z T ] the matrix of the second moments of Z. The standard Gaussian probability density function on R d is denoted by Φ d .

Let X be a real Gaussian random field defined on R d . It is said to be stationary if for any translation τ of R d , X • τ and X have the same law. Then we can define its covariance function r on R d such that

∀t ∈ R d , r(t) = Cov(X(t), X(0)).
Furthermore, X is said to be isotropic if for any rotation ρ in R d (element of the special orthogonal group S0(d)), X • ρ and X have the same law, which is equivalent (if X is also stationary) to r being radial, that is to r(t) depending on t only through t .

2 Anisotropic Gaussian wave model

Definition

If X : R d → R is a centred and stationary Gaussian random field with unit variance, according to Bochner's theorem, there exists a symmetric probability measure F on R d and a symmetric random variable k in R d (with probability measure F ), such that the covariance function r of X is given by

∀t ∈ R d , r(t) = E[X(0)X(t)] = R d exp(iu • t) dF (u) (1) = E[cos(k • t)]. (2)
Note that the distribution of k satisfying (2) for a given Gaussian field X is no more unique if one drops the symmetry assumption. In the following, we will call random wavevector associated with X any random vector k in R d that satisfies [START_REF] Adler | Random Fields and Geometry[END_REF]. Besides, let us quote that for any random vector k, the map t → E[cos(k • t)] actually defines a covariance function. Indeed, the single random wave ( √ 2 cos(k • t + η)) t∈R d , where η is uniformely distributed in [0, 2π], defines a stationary centred random field with the prescribed covariance function. Therefore, we introduce the next definition. Definition 2.1 Let k be a d-dimensional random vector. We define G k , the Gaussian random wave associated with k, as the real, centred, stationary and Gaussian random field on R d with covariance function given by (2).

Due to Kolmogorov extension theorem (see [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] Sections 1.1 and 1.2 for instance), such a field exists and its distribution is unique. A first consequence of the definition is that G k has unit variance. As it is stated in the next proposition, deduced from general properties of stationary covariance functions, the random field G k is, generally speaking, anisotropic and its anisotropy depends on the law of k. Proposition 2.2 Let k be a d-dimensional random vector and let G k be the Gaussian random wave associated with k.

1. The spectral measure F of G k is equal to the probability measure of k s , the symmetrized random variable associated to k1 . Moreover, G k is isotropic if and only if the law of k s is invariant under rotations.

2. The covariance function r of G k admits derivatives up to order m (m ∈ N 0 ) if and only if k admits moments of order m. In this case, for any j ∈ N 0 d such that |j| ≤ m, we have

∂ j r(0) = 0 if |j| is odd ; ∂ j r(0) = (-1) |j|/2 E[k j ] if |j| is even. In particular, r (0) = -E[kk T ].
If the random wavevector k admits moments of any order, then according to the previous proposition, the covariance function r of G k is of class C ∞ and consequently there exists a version of G k with almost every realization of class C ∞ .

In order to build a Gaussian field with the same distribution as G k , one way is to deduce a spectral representation from the integral expression (1) of its covariance function. To do so, let W F be a complex Gaussian F -noise on R d , i.e. a C-valued process defined on the set B(R d ) of Borelians such that

• a.s. W F is a complex-valued measure on B(R d ), • ∀A ∈ B(R d ), W F (A) is a complex-valued Gaussian variable with E[W F (A)] = 0 and E[W F (A)W F (A)] = F (A)
, where • denotes the complex conjugation,

• for any sequence (A n ) n∈N of pairwise disjoint Borel sets, (W F (A n )) n∈N are independent random variables.

Moreover, we add the property that for any

A ∈ B(R d ), W F (A) = W F (-A).
Then, it is easy to check that the Gaussian stationary random field prescribed by

R d e it•u dW F (u) t∈R d (3)
is real-valued, centred and that its covariance function is given by (1).

Starting from the random wavevector k, we can also obtain an asymptotic representation for G k as a limit of a sum of single random waves with the same covariance function. For this let us introduce identically distributed versions (k j ) j∈N of k and uniformly distributed random variables (η j ) j∈N on [0, 2π] and assume that all these random variables are independent. According to the central limit theorem applied to finite-dimensional distributions, the distribution of the random field

  2 N N j=1 cos(k j • t + η j )   t∈R d
converges as N tends to ∞ towards a Gaussian random field with covariance function r given by (2).

Random wavevector

In this paper, we focus on two families of distributions for k. First, we are interested in cases where the random wavevector k is supported by the zero set of a multivariate polynomial P , {λ ∈ R d : P (λ) = 0}. In particular, Section 2.3 deals with this assumption. Besides, we will also concentrate on the case where k is separable, in the sense of the following definition.

Definition 2.3

Let k be a random vector in R d . We say that k is separable if a.s. k = 0 and if k and 1 k k are independent.

If k is separable, we write k = k k, where k is a random variable in S d-1 . For instance, if ||k|| is almost surely constant equal to κ > 0, then k is separable and it is as well supported by the zero set of P (x) = d i=1 x 2 i -κ 2 . In this case, we call κ = k the wavenumber of G k . In Section 2.4.1, we introduce properly this particular model corresponding to what we call anisotropic Berry's monochromatic random waves. We refer to Section 2.4 for specific examples of distributions of the random wavevectors.

If k is centred, which is for instance the case if k has a symmetric law, the coefficients of matrix E[kk T ] are the covariances between the coordinates of k in our basis. In the planar case, this matrix is involved in the definition of the coherency index of k. Definition 2.4 (Coherency index) Let k be a random vector in R 2 . We assume that the symmetric non-negative matrix E[kk T ] is non zero. We write λ -and λ + its eigenvalues, such that 0 ≤ λ -≤ λ + and λ + > 0. The coherency index of k is a real number in [0, 1] defined as

c(k) = λ + -λ - λ + + λ - .
Despite the fact that matrix E[kk T ] depends on the choice of the basis, note that the coherency index does not depend on it. The coherency index is a common tool in spatial statistics and physics to characterize the anisotropy of a model, see [START_REF] Mardia | Directional statistics[END_REF] and [START_REF] Klatt | Morphometry of random spatial structures in physics[END_REF] for instance. In [START_REF] Polisano | Waveletbased orientation of localizable Gaussian fields[END_REF], it is also computed for the so-named structure tensor in order to quantify the anisotropy of an anisotropic Gaussian self-similar planar field with stationary increments.

Let us consider two extreme cases. In the isotropic case, λ -= λ + and c(k) = 0. On the contrary, if k is totally anisotropic in the sense that k is a.s. directed along a single deterministic direction, then λ -= 0 and c(k

) = 1. Note also that if we write E[kk T ] = m 2,0 m 1,1 m 1,1 m 0,2
, then

λ ± = 1 2 (T ± √ ∆) and c(k) = √ ∆ T , (4)
where

T = m 2,0 + m 0,2 and ∆ = (m 2,0 -m 0,2 ) 2 + 4m 2 1,1 .
Remark 2.5 If k is separable then its coherency index only depends on the directional distribution of k. Indeed, writing

k = k k, we obtain E[kk T ] = E[ k 2 ] E[ kkT ], so c(k) = c( k)
and the coherency index of k is simply the difference between the eigenvalues of E[ kkT ] because the trace of this matrix is equal to one.

Link with partial differential equation

We are going to prove that G k satisfies a specific partial differential equation if and only if the random wavevector k is supported by a specific hypersurface of R d . Let P be an even d-multivariate polynomial. Then there exists a sequence of real numbers (α j ) j∈N0 d with only finitely many non-zero terms, such that

∀λ ∈ R d , P (λ) = j∈N0 d ; |j| even α j λ j . (5)
We associate with P the following differential operator:

L P (X) = j∈N0 d ; |j| even (-1) |j|/2 α j ∂ j X.
Theorem 2.6 Let P be an even multivariate polynomial defined by ( 5) and let X be a Gaussian random field defined on R d that is centred, stationary, with unit variance and almost surely of class C ∞ . The following properties are equivalent.

1. The Gaussian random field X almost surely solves the partial differential equation L P (X) = 0.

2. Any random wavevector k associated with X almost surely satisfies P (k) = 0.

3. The covariance function r of X solves the partial differential equation L P (r) = 0.

4. The spectral measure F of X is supported by {λ ∈ R d : P (λ) = 0}.

The above theorem allows us to exhibit random anisotropic solutions of some famous partial differential equations. Let us give an example, using the Laplacian operator ∆ on R d , defined by ∆ = 1≤j≤d

∂ 2 ∂t 2 j . If for some posi- tive κ, k ∈ κS d-1
, then G k is an almost sure solution of Helmholtz equation ∆X + κ 2 X = 0. In the same vein, the Gaussian random wave defined on R 3 , associated with a random wavevector with support in the Airy surface Items 2. and 4. are equivalent as for any random wavevector k associated with X, and for k s the symmetrized random variable associated to k,

{(x, y, z) ∈ R 3 ; x 2 + y 2 -z 4 = 0} is an almost sure solution of the partial differential equation ∂ 2 ∂x 2 X + ∂ 2 ∂y 2 X + ∂ 4 ∂z 4 X = 0. It is
E[P (k)] = E[P (k s )] = R d P (λ) 2 dF (λ).
At last, items 3. and 4. are equivalent as

L P (r)(t) = R d P (λ) e iλ•t dF (λ) = P dF (t),
where we have denoted by • the Fourier transform of any finite measure.

Presentation of the models

Anisotropic Berry's monochromatic waves

In this section, we focus on the case where the random wavevector k is such that, for some deterministic wavenumber κ > 0, k = κ a.s., that is

κ -1 k ∈ S d-1 a.s.. Since k is not necessarily isotropically distributed, the (unique in distribution) associated stationary centred Gaussian random field G k on R d is anisotropic. Since k is a.s. bounded, it is clear that G k is a.s. of class C ∞ . Hence, rephras- ing Theorem 2.6, we get that G k is the generic Gaussian solution of Helmholtz equation ∆X + κ 2 X = 0.
Equivalently, G k is an eigenfunction of the operator -∆, for the eigenvalue κ 2 . Therefore, extending the definition introduced by Berry in [START_REF] Berry | Regular and irregular semiclassical wavefunctions[END_REF] and intensively studied in the last years, we refer to G k as an anisotropic Berry's monochromatic wave with random wavenumber κ.

Applying the change of variables t → κt yields the scaling property that (G k (t)) t∈R d and (G κ -1 k (κt)) t∈R d have the same distribution, where we recall that the random vector κ -1 k takes its values in S d-1 .

Besides, we remark that k s is such that κ -1 k s is supported by S d-1 . Hence, we can deduce from Proposition 2.2 that the covariance function of G k is given by

r(t) = S d-1 e iκu•t dµ(u), t ∈ R d ,
where µ denotes the probability measure of κ -1 k s .

Planar and separable random waves

In this section, we set d = 2 and we assume that the wavevector k is separable, in the sense of Definition 2.3. The two following examples of parametric distribution for the unitary wavevector k = 1 k k will allow us to make computations to illustrate and comment the results of Sections 3 and 4 (see also Appendix Section A.1 for moments formulas). We write k = (cos Θ, sin Θ), where Θ is a random variable in [0, 2π] and we fix θ 0 in [0, 2π].

Example 1 (Toy model) Let α ≥ 0. The density of Θ with respect to Lebesgue measure on [0, 2π] is given by

θ → C α | cos(θ -θ 0 )| α , with C α = Γ(1 + α/2) 2 √ πΓ(1/2 + α/2) , ( 6
)
where Γ is the usual Gamma function. Parameter α is considered as an anisotropy parameter: α = 0 brings an isotropic model, whereas, at the opposite, α → +∞ corresponds with a totally anisotropic random field since k is a.s. oriented along the x-axis. Our toy model is inspired by [START_REF] Klatt | Morphometry of random spatial structures in physics[END_REF] Section 2.1.2, where it is introduced to represent anisotropic spatial structures in physics. It is also used in [START_REF] De Angelis | Spatial Distributions of Phase Singularities in Optical Random Vector Waves[END_REF] with α = 2 and θ 0 = 0 or θ 0 = π/2 to modelize the two coordinates of a two-dimensional electromagnetic wave.

Example

-h, θ 0 +h]∪[θ 0 +π -h, θ 0 +π +h] with 0 ≤ h ≤ π/2.
The elementary model is studied in [START_REF] Biermé | A turning band method for the simulation of anisotropic fractional Brownian fields[END_REF] and [START_REF] Polisano | Waveletbased orientation of localizable Gaussian fields[END_REF].

Up to a rotation of the basis, we can assume that in both examples θ 0 = 0. Thus we set θ 0 = 0 in the following.

Gaussian sea waves

In this section, we now concentrate on the case where the random wavevector k is 3-dimensional and a.s. belongs to Airy surface, i.e.

k ∈ A = {(λ 1 , λ 2 , λ 3 ) ∈ R 3 ; (λ 1 ) 2 + (λ 2 ) 2 = (λ 3 ) 4 } a.s..
The covariance function of the Gaussian random wave G k associated with k is

r(t) = E[cos(k • t)] = A cos(λ • t) dF (λ), t ∈ R 3 ,
where F is the probability distribution of k supported by A.

The random field G k coincides with the one used for the spatiotemporal random modelization of sea waves, assuming that the depth of the sea is infinite (see [START_REF] Longuet-Higgins | The Statistical Analysis of a Random, Moving Surface[END_REF] for the original idea, [START_REF] Azaïs | Geometrical Characteristics of Gaussian Sea Waves[END_REF] or [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] for more recent developments). More precisely, for (x, y, s) ∈ R 2 × R, G k (x, y, s) can be seen as the algebraic height of a wave at point (x, y) and time s. If the moments of k are finite up to order four, we recall that according to Theorem 2.6, G k solves the partial differential equation

∆G k + ∂ 4 t G k = 0,
with ∆ the two-dimensional spatial Laplacian operator and ∂ 4 t the fourth temporal derivative.

We use the following parametrization of A,

(z, θ) ∈ R × [0, 2π) → (z 2 cos θ, z 2 sin θ, z),
which provides a bijection φ from R\{0}×[0, 2π) onto A\{(0, 0, 0)}. Performing the appropriate change of variables yields r(x, y, s) = R×(0,2π) cos(xz 2 cos θ + yz 2 sin θ + sz) dF φ (z, θ), where F φ is the image of measure F by the map φ -1 . When k admits f as probability density function with respect to the surface measure on A, consequently to the coarea formula, we get r(x, y, s) = R×(0,2π) cos(xz 2 cos θ + yz 2 sin θ + sz)g(z, θ) dzdθ, where the map g is given by

g(z, θ) = f (z 2 cos θ, z 2 sin θ, z) z 2 (1 + 4z 2 ) 1/2 .
Following the literature, g is called directional power spectrum of G k (see [START_REF] Azaïs | Geometrical Characteristics of Gaussian Sea Waves[END_REF] and [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] Chapter 11). Experimental directional power spectra are exhibited in [START_REF] Azaïs | Geometrical Characteristics of Gaussian Sea Waves[END_REF], derived from sea data provided by Ifremer.

Let us fix time s = s 0 and look at the random field defined on R 2 ,

Z k (x, y) = G k (x, y, s 0 ) (x, y) ∈ R 2 ,
as a picture of the sea height at time s 0 . It is a two-dimensional stationary centred Gaussian random field, whose covariance function is given by

r 0 (x, y) = r(x, y, 0) = R×(0,2π) cos(xz 2 cos θ + yz 2 sin θ) dF φ (z, θ) = E[cos((x, y) • π(k))],
where the random vector π(k) is described in polar coordinates by (R 2 , Θ) with (R, Θ) distributed according to measure F φ . Thus π(k) is the random wavevector associated with Z k and it is nothing but the projection of the A-valued random wavevector k onto the plane of the first two coordinates. Consequently, the moments of π(k) are given, for any integers j and k in N 0 , by

m j,k = (0,2π)×R (z 2 cos θ) j (z 2 sin θ) k dF φ (z, θ),
assuming that the above integral is finite. Note that if F φ can be written as a tensorial product measure: dF φ (z, θ) = d Ξ(z) ⊗ dΛ(θ) then, according to Definition 2.3, π(k) is separable.

Level sets of Gaussian random waves

Let k be a random wavevector in R d admitting moments up to order four and let G k be the associated Gaussian random field defined on R d . From now on in this section, we assume that G k is almost surely of class C 2 . Let a ∈ R. We are interested in the level set

G -1 k (a) = {t ∈ R d : G k (t) = a},
which is a.s. a C 2 -submanifold of R d with dimension d -1, called nodal set in the case a = 0.

Expected measure

We are now interested in the expected measure of the level sets of G k . Let Q be a compact set in R d with non empty interior. We focus on the (d -1)-dimensional Hausdorff measure of the a-level set restricted to Q, namely

(a, k, Q) := H d-1 G -1 k (a) ∩ Q = H d-1 ({t ∈ Q / G k (t) = a}) .
For now on in Section 3, we assume that G k (0) is a non-degenerate Gaussian vector or, equivalently, that E[kk T ] is invertible. This allows us to apply Kac-Rice formula (see [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] Theorem 6.8 for instance). It yields

E[ (a, k, Q)] = Q E[ G k (t) | G k (t) = a] p G k (t) (a) dt,
where p G k (t) , the probability density function of G k (t), is actually given by the standard Gaussian distribution. Using the stationarity of G k and the fact that for a fixed point t, G k (t) and G k (t) are independent random variables, we have

E[ (a, k, Q)] = H d (Q) e -a 2 /2 √ 2π E[ G k (0) ].
Consequently, recalling that G k (0) is the Euclidean norm of a d-dimensional centred Gaussian vector with variance matrix -r (0) = E[kk T ] and that Φ d stands for the standard Gaussian probability density function on R d , we obtain

E[ (a, k, Q)] = H d (Q) e -a 2 /2 √ 2π R d (E[kk T ]x • x) 1/2 Φ d (x) dx.
In the separable case, we deduce from the above formula the following straightforward lemma.

Lemma 3.1 We assume that k is separable with k = k k and that E[kk T ] is invertible. Then E[ (a, k, Q)] = H d (Q) e -a 2 /2 √ 2π E[ k 2 ] 1/2 R d (E[ k k T ]x • x) 1/2 Φ d (x) dx. ( 7
)
The above formula applies to Berry's monochromatic isotropic random wave, i.e. to the case where k is a.s. constant equal to some positive constant κ and k is uniformly distributed in S d-1 . In this case,

E[ k 2 ] = κ 2 and E[ k k T ] = (1/d) I d . Hence, the involved integral becomes R d x Φ d (x)
dx, which is the mean of a χ-distributed random variable with d degrees of freedom and is known to be equal to √ 2 Γ((d+1)/2) Γ(d/2) , with Γ the usual Gamma function. Finally, we recover the well known formula expressing the expected measure of the a-level set of a Berry isotropic random wave (see [START_REF] Berry | Regular and irregular semiclassical wavefunctions[END_REF] in the planar case):

E[ (a, k, Q)] = H d (Q) e -a 2 /2 √ dπ κ Γ((d + 1)/2) Γ(d/2) .
We complete the computation of the expected measure of nodal sets by giving an estimate on the variance. Next proposition provides an upper bound when the compact domain Q shrinks to a small ball B(0, R).

Proposition 3.2 For R > 0 small enough, Var[ (a, k, B(0, R))] ≤ C e -a 2 /2 R 2d-1 E[ k 2 ] -λ min λ 1/2 min
, where λ min is the smallest eigenvalue of E[kk T ] and C is some universal positive constant. , depending on the covariance function of the random wave. For a fixed value of E[ k 2 , this factor is minimized in the isotropic case, where we can write E[kk T ] = λI d and it is equal to (d -1)λ 1/2 . In the separable case, writing λmin the smallest eigenvalue of E[ kkT ], this factor can been expressed as

E[ k 2 ] 1-λmin λ1/2 min .
Proof. Let us start with Kac-Rice formula (see [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] Theorem 6.9) to compute the second moment of (a, k, Q):

E[ (a, k, Q) 2 ] = Q Q E[ G k (s) G k (t) | G k (s) = G k (t) = a] p s,t (a, a) ds dt,
where p s,t denotes the probability density function of (G k (s), G k (t)). Using firstly the stationarity of G k and secondly Cauchy-Schwarz inequality, we get

E[ (a, k, Q) 2 ] = R d H d (Q ∩ Q -t) E[ G k (0) G k (t) | G k (0) = G k (t) = a] p 0,t (a, a) dt ≤ R d H d (Q ∩ Q -t) (F a,k (t) F a,k (-t)) 1/2 dt,
where

F a,k (t) = E[ G k (0) 2 | G k (0) = G k (t) = a] p 0,t (a, a).
Let us now evaluate F a,k (t). On the one hand, a straightforward regression shows that the conditional distribution of G (0) given {G(0) = G(t) = a} is Gaussian with mean a 1+r(t) r (t) and covariance matrix -r (0) -

1 1-r(t) 2 r (t)r (t) T . On the other hand, if Z is a Gaussian vector N (b, Σ), then E[ Z 2 ] = b 2 + T r(Σ) with T r(•) denoting the trace. Hence, since T r(r (t)r (t) T ) = r (t) 2 , we get F a,k (t) = a 1 + r(t) 2 r (t) 2 -T r(r (0)) - 1 1 -r(t) 2 r (t) 2 e -a 2 /(1+r(t)) 2π(1 -r(t) 2 ) 1/2 .
This formula shows that F a,k (t) = F a,k (-t) and it puts us in position to write a Taylor expansion of F a,k (t) as t goes to 0. In order to simplify computations, and without loss of generality, one can assume that an orthonormal basis of R d is given such that the Hessian matrix -r (0) is diagonal, with eigenvalues

λ 1 ≥ • • • ≥ λ d > 0.
Then, as t goes to 0,

r(t) = 1 - 1 2 1≤i≤d λ i t 2 i + o( t 3 ) and r (t) 2 = 1≤i≤d λ 2 i t 2 i + o( t 2 ).
Hence,

F a,k (t) =   1≤i≤d λ i - 1≤i≤d λ 2 i t 2 i 1≤i≤d λ i t 2 i   e -a 2 /2 2π( 1≤i≤d λ i t 2 i ) 1/2 + o( t ) = e -a 2 /2 1≤i≤d λ i ( j =i λ j )t 2 i 2π( 1≤i≤d λ i t 2 i ) 3/2 + o( t ) (8)
≤ e -a 2 /2 j =d λ j 2πλ

1/2 d t -1 + o( t ).
Now we integrate the latter upper bound of F a,k (t) on the set of points t such that B(0, R) ∩ B(0, R) -t = ∅, which is equal to B(0, 2R). It yields

E[ (a, k, B(0, R)) 2 ] ≤ C e -a 2 /2 j =d λ j λ 1/2 d R d+d-1 = C e -a 2 /2 E[ k 2 ] -λ d λ 1/2 d R 2d-1 , since E[ k 2 ] = -T r(r (0)) = 1≤i≤d λ i . As E[ (a, k, B(0, R))]
2 has order of magnitude R 2d , it is negligible with respect to the second moment, and we get the announced upper bound for the variance of (a, k, B(0, R)).

Planar case

In the planar case, i.e. d = 2, the level sets G -1 k (a) are one-dimensional. Furthermore assuming that k is separable, we will establish that the level curves mean length decreases with anisotropy. Our formula involves the coherency index, introduced in Definition 2.4.

To compute the integral in the right-hand side of ( 7), we use the following well known fact, that can be proved with simple algebra. If M is a symmetric positive definite matrix with eigenvalues γ -and γ + such that 0 < γ -≤ γ + , then

R 2 (M x • x) 1/2 Φ 2 (x) dx = 2γ + π 1/2 E (1 -γ -/γ + ) 1/2 , (9)
where E stands for the elliptic integral given by E(x) = π/2 0

(1 -x 2 sin 2 θ) 1/2 dθ, for x ∈ [0, 1]. In our case, we set M = E[ k k T ] and γ -+ γ + = 1 since k belongs to S 1 , a.s.. Hence, writing c = c( k), we have 2γ + = 1 + c and 1 -γ -/γ + = 2c 1+c . Consequently, the following proposition holds.

E[ (a, k, Q)] = H 2 (Q) e -a 2 /2 π √ 2 E[ k 2 ] 1/2 F c( k) ,
where the map

F : c ∈ [0, 1] → (1 + c) 1/2 E 2c 1+c
1/2 is strictly decreasing.

The proof of the decrease of mapping F is postponed to the Appendix section, see Lemma A.3. Another expression for the same expectation can be found in [START_REF] Longuet-Higgins | The Statistical Analysis of a Random, Moving Surface[END_REF] Formula (2.3.13), however our formulation highlights the effect of the wavevector's distribution on the mean length of level sets. Remark 3.5 Regarding the coherency index c( k) as a parameter that positively quantifies anisotropy, the above formula clearly indicates that the mean length of level curves is decreasing as the anisotropy of k increases. Remark 3.6 In the planar case, the proof of Proposition 3.2 allows us to obtain a better upper bound for the variance of (a, k, B(0, R)), if R is small enough, than the one stated in the proposition itself. To begin with, when d = 2, Equality (8) can be written

F a,k (t) = λ - λ 1/2 + t 2 t 2 1 + λ - λ + t 2 2 3/2 + o( t ). ( 10
)
We introduce another elliptic integral Ẽ(x) = π/2 0

(1 -x 2 sin 2 θ) -3/2 dθ, for x ∈ [0, 1] and

G : c ∈ [0, 1] → 1 -c (1 + c) 1/2 Ẽ 2c 1 + c 1/2 .
Then, after integrating [START_REF] Berry | Phase singularities in isotropic random waves[END_REF] in polar coordinates, we obtain

Var[ (a, k, B(0, R))] ≤ C e -a 2 /2 R 3 E[ k 2 ] 1/2 G c( k) ,
where C is a universal constant.

We now apply Proposition 3.4 to our separable examples, prescribing the directional distribution of the wavevector k. We refer to the Appendix Section A.1 for the detailed computation of their moments.

Example 1 (Toy model) Take k distributed on S 1 with probability density function given by ( 6) for some positive α. Then, E[ k k T ] = 1 α+2 α + 1 0 0 1 and consequently, c( k) = α α+2 , which is an increasing function of parameter α. As observed in Remark 3.5, the more anisotropic the model is, the smaller the expected length of level sets is.

Example 2 (Elementary model) We choose k to be uniformly distributed on

[-h, h] ∪ [π -h, π + h] for some 0 < h ≤ π/2. In that case, E[ k k T ] = 1 2 1 + sin(2h) 2h 0 0 1 -sin(2h) 2h
. Hence, c( k) = sin(2h) 2h , which is decreasing on (0, π/2]. Again, the mean length of level sets is decreasing with anisotropy, i.e. as h is decreasing.

Favorite orientation

In this section, we introduce the directional statistics features of the random waves.

Definition 3.7 We call favorite direction of a random vector

V in R d any u ∈ S d-1 that maximizes E[(V.u) 2 ]. Since E[(V.u) 2 ] = u.E[V V T ]
u, the favorite directions of V are the eigenvectors with norm one associated with the largest eigenvalue of the symmetric positive matrix E[V V T ]. If the largest eigenvalue is simple, i.e. if the dimension of the associated eigenspace is one, then there are exactly two favorite directions, which are opposite one another. Note that if a random wavevector k is separable then its favorite directions are exactly the ones of k.

We turn to the directional study of the level set G -1 k (a). For any t ∈ G -1 k (a), the tangent space at point t, T t G -1 k (a), is a (d -1)-dimensional linear subspace that is orthogonal to the vector G k (t). Using the previous definition, the favorite directions of G k (t) are given by the unitary eigenvectors associated with the largest eigenvalue of E[G k (t)G k (t) T ]. As the latter matrix is equal to E[kk T ] according to Proposition 2.2, the favorite directions of G k (t) coincide with those of k. Hence, we get the next statement that sounds physically intuitive: the favorite orientations of the level sets G -1 k (a) are orthogonal to the favorite directions of k. Actually, it can be written out as a precise proposition.

Directional derivative as a random wave

The crest line can be considered as the nodal line of the random field (G k • u ϕ ), which is the partial derivative of G k along vector u ϕ . This random field is Gaussian, stationary, centred and we denote by v 0,0 (ϕ) its variance:

v 0,0 (ϕ) = E[(G k (0) • u ϕ ) 2 ] = u ϕ • E[kk T ]u ϕ = E[(k • u ϕ ) 2 ].
Note that v 0,0 (ϕ) does not vanish since E[kk T ] is invertible.

By deriving the integral representation (1) of r, we obtain that the covariance function of (

G k • u ϕ ) is ∀t ∈ R 2 , E[(G k (t) • u ϕ )(G k (0) • u ϕ )] = R 2 cos(t • λ)(λ • u ϕ ) 2 dF (λ). (11)
Therefore, the following lemma holds Lemma 4.1 The covariance function of

G k • u ϕ v 0,0 (ϕ) 1/2 is given by t ∈ R 2 → E[cos(K ϕ • t)],
where K ϕ is a two-dimensional random vector whose probability distribution is equal to

(λ•uϕ) 2 v0,0(ϕ) dF (λ), with F the probability measure of k. In other words, K ϕ is a random wavevector associated to G k • u ϕ v 0,0 (ϕ) 1/2 .
We introduce the moments of measure F and the ones of measure (λ • u ϕ ) 2 dF (λ): for any (i, j) ∈ N 2 0 ,

m i,j := R 2 (λ 1 ) i (λ 2 ) j dF (λ) = E[(k • u 0 ) i (k • u π/2 ) j ], v i,j (ϕ) := R 2 (λ 1 ) i (λ 2 ) j (λ • u ϕ ) 2 dF (λ). (12)
By a mere development, the moments (v i,j (ϕ)) can be expressed as polynomial functions of (cos ϕ, sin ϕ), the coefficients depending linearly on the moments (m i,j ) (see [START_REF] Ley | Modern directional statistics[END_REF] in the Appendix section).

Until the end of this section and in the following one, we focus on the case of a separable random wavevector k, that we write k = k u Θ . We denote by Ξ the probability distribution of k on (0, +∞) and Λ the one of Θ on [0, 2π], assuming that k and Θ are independent. We introduce

∀j ∈ N 0 , M j = E[ k j ] = R + ρ j dΞ(ρ)
the moments of the random variable k and µ j,k the ones of Θ, that is for any (j, k) ∈ N 2 0 , µ i,j = [0,2π] cos i θ sin j θ dΛ(θ).

For any integrable function h,

E[h(k)] = [0,+∞)×[0,2π]
h(ρ cos θ, ρ sin θ) dΞ ⊗ dΛ(ρ, θ).

Therefore,

E[h(K ϕ )] = 1 v 0,0 (ϕ) (0,+∞)×[0,2π]
h(ρ cos θ, ρ sin θ)ρ 2 cos 2 (θ -ϕ) dΞ ⊗ dΛ(ρ, θ).

To end with, we write ν i,j (ϕ) = 2π 0 cos i (θ) sin j (θ) cos 2 (θ -ϕ) dΛ(θ).

We can directly state the following lemma.

Lemma 4.2 If k is separable, then K ϕ is also separable and ∀(i, j) ∈ N 0 2 , m i,j = M i+j µ i,j and v i,j (ϕ) = M 2+i+j ν i,j (ϕ), assuming that the above moments exist.

Mean length of a crest line in a fixed direction

The Hausdorff dimension of a crest line is clearly equal to one. We consider its length within a compact domain Q ⊂ R 2 such that H 2 (Q) > 0, l(k, Q, ϕ) := H 1 ({t ∈ Q ; G k (t) • u ϕ = 0}) .

Since the crest line of G k in direction ϕ is nothing but the nodal line of

E[l(k, Q, ϕ)] = H 2 (Q) 1 π √ 2 E[ K ϕ 2 ] 1/2 F(c( K ϕ )),
where function F has been introduced in Proposition 3.4,

E( K ϕ 2 ) = M 4 M 2 , (14)
c( K ϕ ) = (ν 2,0 (ϕ) -ν 0,2 (ϕ)) 2 + 4ν 1,1 (ϕ) 2 1/2 ν 0,0 (ϕ) . Proof. We will perform an asymptotic expansion of c( K π/2+ϕ ) 2 near ϕ = 0 in order to show that ϕ → c( K ϕ ) admits a local minimum at π/2.

In the following lines, we write g(ϕ) = O(h(ϕ)) as ϕ tends to 0, if there exists ϕ 0 ∈ (0, 2π) and M > 0 such that After some algebra, it gives c( K π/2+ϕ ) 2 = (α -2) 2 + 24α(α + 1)ϕ 2 + O(ϕ 4 ),

  used as a spatiotemporal random sea wave model, which is thoroughly presented in Section 2.4.3. Proof of Theorem 2.6. Items 1. and 4. are equivalent as Var(L P (X)(t)) = Var(L P (X)(0)) = j,k∈N0 d ; |j|,|k| even (-1) (|j|+|k|)/2 α j α k ∂ (j+k) r(0) = R d P (λ) 2 dF (λ).

Remark 3 . 3

 33 In the above upper bound, let us comment the factor E[ k 2 ]-λmin λ

Proposition 3 . 4

 34 Let k be a separable random wavevector in R 2 such that k = k k and E[kk T ] is invertible. Let us denote by c( k) = c(k) the coherency index of k. Then,

G

  k •uϕ v0,0(ϕ) 1/2 , Lemma 4.1 allows us to apply Proposition 3.4 with the random wavevector K ϕ . It yields the following proposition. Proposition 4.3 We assume that k is separable and thatE[kk T ] is invertible. Then for any ϕ ∈ [0, π) such that E[K ϕ K T ϕ ] is invertible,

Figure 1 :

 1 Figure 1: Graph of ϕ → c( Kϕ) 2 for α = 1 (black), α = 2 (blue) and α = 3 (green).

From ( 18 ) 1 + 2 -

 1812 ∀ϕ ∈ [0, 2π], |ϕ| < |ϕ 0 | ⇒ |g(ϕ)| ≤ M |h(ϕ)|. and Lemma A.1, we getν 0,0 (π/2 + ϕ) = µ 0,2 + (µ 2,0 -µ 0,2 )ϕ 2 + O(ϕ 4 ) αϕ 2 ) + O(ϕ 4 ) ν 2,0 (π/2 + ϕ) = µ 2,2 + (µ 4,0 -µ 2,2 )ϕ 2 + O(ϕ 4 ) = α + 1 (α + 2)(α + 4) (1 + (α + 2)ϕ 2 + O(ϕ 4 ) ν 0,2 (π/2 + ϕ) = µ 0,4 + (µ 2,2 -µ 0,4 )ϕ 2 + O(ϕ 4 ) α -2)ϕ 2 ) + O(ϕ 4 ) ν 1,1 (π/2 + ϕ) = 2µ 2,2 ϕ(1ϕ 2 ) + O(ϕ 4 ).

  2 (Elementary model)The random variable Θ is uniformly distributed on [θ 0 -h, θ 0 + h] with 0 ≤ h ≤ π. Parameter θ 0 indicates the main direction whereas parameter h negatively quantifies anisotropy, in the sense that the more anisotropic the model is, the smaller parameter h is. Actually, h = 0 corresponds with a totally anisotropic model, h ≈ 0 corresponds with what is named narrow spectrum model in[START_REF] Longuet-Higgins | The Statistical Analysis of a Random, Moving Surface[END_REF] Section I.6, and h = π corresponds with the isotropic model. If one wishes a symmetric model, one can also consider Θ uniformly distributed on [θ 0

If F k and F -k are respectively the probability measures of k and -k, then the symmetrized random variable associated with k is defined as the random variable with probability measure F = 1

2 (F k + F -k ).

 Proposition 3.8Let τ be a d-dimensional vector field defined on the level set G -1 k (a) such that, at any point t ∈ G -1 k (a), τ (t) is orthogonal to T t G -1 k (a). Then, at any point t, the favorite directions of τ (t) are given by the favorite directions of k.

This formalizes an assertion in [START_REF] Longuet-Higgins | The Statistical Analysis of a Random, Moving Surface[END_REF] Section 2.3, according to which the direction of the contour is near the principal direction. In this statement, the principal direction corresponds to our favorite direction.

Let us illustrate the notion of favorite direction on two-dimensional separable random wavevectors. We write k = k u Θ , k and Θ being independent random variables. Note that for any ϕ ∈ [0, 2π], k • u ϕ = k cos(θ -ϕ). On the one hand, if Θ is uniformly distributed on [0, 2π], then E[kk T ] = Example 1 (Toy model) If Θ admits a probability density function given by ( 6), for a given α > 0, then

and the favorite directions of k are again ±u 0 .

Crest lines of planar Gaussian waves

Let k be a two-dimensional random wavevector. We write F its probability law and G k its associated planar Gaussian random wave that we assume to be almost surely of class C 3 . We assume that E[kk T ] is invertible, which equivalently excludes the case of k almost surely oriented along a fixed direction. We fix a direction ϕ ∈ [0, π) and we consider the crest line in direction ϕ. More precisely, we introduce the random set

which contains all the points t in R 2 such that the gradient of G k at point t is orthogonal to direction ϕ. This set can be compared with the one of specular points in a fixed direction, as defined in [START_REF] Longuet-Higgins | The Statistical Analysis of a Random, Moving Surface[END_REF]. Crest lines intersected with nodal lines are also studied in [START_REF] Rudnick | Points on nodal lines with given direction[END_REF], in the case of arithmetic random waves, which are random eigenfunctions of the Laplacian on the two-dimensional torus.

We refer to the Appendix Section A.3 for the proof of ( 14) and [START_REF] Krishnapur | Nodal length fluctuations for arithmetic random waves[END_REF]. See also Formula [START_REF] Ley | Modern directional statistics[END_REF] in that section to get an expression of c( K ϕ ) in terms of the fourth-order moments (m i,j ) (i,j)∈N 2 0 , i+j=4 . According to Proposition 4.3, E[l(k, Q, ϕ)] depends on the directional distribution of k through the factor F(c( K ϕ )) (involving the fourth-order moments of k), and on its radial distribution through the factor E[ K ϕ 2 ] (involving the second-order and the fourth-order moment of Ξ). Consequently, the mean length of crest lines is a decreasing function of the anisotropy of K ϕ . The variations of the expected length of the crests with respect to the direction can be studied through the variations of the map ϕ → c( K ϕ ). For more detailed results, see the Appendix Section A.4.

Let us remark that a similar formula is also derived in [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] (Proposition 11.4) and in [START_REF] Azaïs | Geometrical Characteristics of Gaussian Sea Waves[END_REF] (Assertion 3). It is apparently different from ours, since the gradient of the random field G k • u ϕ is not computed according to the directions of the canonical basis but according to the ones of (u ϕ , u ϕ+π/2 ). Nevertheless they actually coincide. Now we apply Proposition 4.3 to our separable examples. All is about computing the coherency index of K ϕ . For each case, we examine the crest direction(s) that maximize(s) the expectation of the crest length. In [START_REF] Longuet-Higgins | The Statistical Analysis of a Random, Moving Surface[END_REF] or in [START_REF] Azaïs | Geometrical Characteristics of Gaussian Sea Waves[END_REF], a rule of thumb is suggested claiming that the direction [that maximises the expected length of crests] is orthogonal to the direction for the maximum integral of the spectrum, i.e. is the most probable direction for the waves. In this statement, the "most probable direction for the waves" refers to the mode of the random wavevector k. However, according to our examples, such a rule is not necessarily satisfied. This can be explained by the expectation formula of Proposition 4.3 itself, which shows a dependency on both the second-order and the fourth-order moments of k, and not on the mode of k.

Examples We consider a random wavevector k that we write k = k u Θ as before and we use the formulas of Appendix Sections A.3 and A.4.

• Toy Model. The probability density function of Θ is given by ( 6), for some fixed α > 0 and θ 0 equal to zero. The mode of k is 0 in that case. An asymptotic expansion of ϕ → c( K ϕ ) 2 near ϕ = π/2 is performed in Lemma A.5 in the Appendix section. It shows that the expected length of crests admits a local maximum at ϕ = π/2, which is precisely orthogonal to the most probable direction of k and to its favorite direction as well.

• Elementary wave. We assume that Θ is uniformely distributed on

The moments coherency index of K ϕ can be computed thanks to Formula (18) and Lemma A.2. Figure 2 shows the graph of ϕ → c( K ϕ ) 2 . The expected length of crests appears to admit a maximum at ϕ = π/2, thus it is orthogonal to the favorite direction of k.

• A very special example. Let 1 4 3 j=0 δ jπ/2 be the distribution of Θ on [0, 2π], δ a being the Dirac measure at point a. It is called Cilleruelo 20 measure in [START_REF] Rudnick | Points on nodal lines with given direction[END_REF]. The modes of k are {u jπ/2 : j = 0, 1, 2, 3}. We can mention that it does not admit any favorite direction. Computing the moments of F , we get

Since F is strictly decreasing on [0, 1], the mean length of crests is maximal when cos(2ϕ) = 0 , i.e. for ϕ = π/4 or 3π/4 modulo π. These directions are not orthogonal to the modes of k.

A Appendix

A.1 Moments of two specific planar random wavevectors

Let k be a planar separable wavevector that we write k = k k with k and k independent and k = (cos Θ, sin Θ). In the two following lemmas, we compute the moments of k, i.e.

for any non negative integers j and k, in two specific cases, namely the toy model and the elementary model as introduced in Section 2.4.2.

Lemma A.1 If Θ has a probability density function given by

for some nonnegative constant α, then the following formulas hold:

• for any even integers j and k, µ

In particular, it yields the non-zero second and fourth-order moments of k:

.

Proof. It is clear that µ 0,0 = 1, µ j,k = 0 whenever j or k is odd and that µ j,0 = C α /C α+j for any even integer j. Using the explicit value of C α yields the value of µ j,0 . Finally, for any even integers j and k, writing sin 2 θ = 1 -cos 2 θ yields the formula for µ j,k .

for some constant h ∈ (0, π/2), then the following formulas hold:

• µ j,k = 0 whenever j or k is odd,

where sinc(θ) = sin(θ) θ for any θ = 0 . In particular, this implies

Proof. Symmetry arguments explain the vanishing moments, while the non-zero ones can be evaluated by linearizing trigonometric functions.

A.2 Variations of map F

Lemma A.3 Let E be the elliptic integral given by E(x) = π/2 0

(1 -

(1-k 2 sin 2 θ) 1/2 dθ. Therefore, for any c ∈ [0, 1),

It remains to show that the above integral, which we call J(k) with k = ( 2c 1+c ) 1/2 , is negative. Splitting the integral J(k) := π/2 0 cos(2θ) (1-k 2 sin 2 θ) 1/2 dθ into two parts, on [0, π/4] and on [π/4, π/2], and performing the change of variables θ = π/2-θ within the second part, we get

which is negative since cos θ > sin θ for θ ∈ (0, π/4).

A.3 Coherency index of K ϕ

In this section, we prove Formulas ( 14) and ( 15) that are stated in Proposition 4.3. We start with a general result that allows us to compute the coherency index of K ϕ .

where

with

2.

.

Proof. As a consequence of Lemma 4.1 , E[K ϕ K T ϕ ] is the covariance matrix of the gradient vector of

and Point 3 derives from Formula (4). Now, let us assume that k is separable. Lemma 4.1 ensures that K ϕ is also separable. Then

, where the moments (ν i,j (ϕ)) have been defined by [START_REF] Fournier | Identification and isotropy characterization of deformed random fields through excursion sets[END_REF]. It results from Lemma 4.2 and from ν 2,0 (ϕ) + ν 0,2 (ϕ) = ν 0,0 (ϕ) that E[ K ϕ 2 ] = M4 M2 (which is actually equation ( 14)) and that the trace of

23 also be seen as a consequence of K ϕ ∈ S 1 ). Consequently, its coherency index is

which is actually equation [START_REF] Krishnapur | Nodal length fluctuations for arithmetic random waves[END_REF].

A.4 Variations of the mean length of crests

Thanks to Proposition 4.3, we know that the mean length of a crest of direction ϕ is given as a decreasing function of c( K ϕ ). Recall that Formula (15) allows us to write out c( K ϕ ) 2 as

where the ν i,j (ϕ)'s are defined by [START_REF] Fournier | Identification and isotropy characterization of deformed random fields through excursion sets[END_REF]. By Lemma 4.2, the moments ν i,j (ϕ) are such that ν i,j (ϕ) = 1 M2+i+j v i,j (ϕ). We use [START_REF] Ley | Modern directional statistics[END_REF] to express v i,j (ϕ) in terms of the moments of k denoted by m i,j . Since m i,j = M i+j µ i,j , Lemma 4.2 also yields

In the case where the moments µ 1,1 , µ 1,3 and µ 3,1 simultaneously vanish, which is the case in our two favorite examples (see Lemmas A.1 and A.2), we like to remark that there exist two polynomials of degree two, say P and Q, such that ∀ϕ ∈ [0, 2π], c( K ϕ ) 2 = P (cos(2ϕ)) Q(cos(2ϕ)) .

Consequently, ϕ = 0 and ϕ = π/2 appear as stationary points of the π-periodic map ϕ → c( K ϕ ) 2 , whose graph is symmetric with respect to π/2.

A.4.1 The toy model case

We assume that the direction of k is given by the toy model (see Example 1 in Section 2.4.2). Using [START_REF] Lindgren | Slepian Models for the Stochastic Shape of Individual Lagrange Sea Waves[END_REF] and the expression for the µ i,j 's given by Lemma A.1, we can compute c( K ϕ ) 2 . Figure 1 shows the graph of ϕ → c( K ϕ ) 2 for various values of α. We observe that ϕ → c( K ϕ ) 2 is minimal at ϕ = π/2, which is substantiated by the next lemma.

Lemma A. [START_REF] Azaïs | Geometrical Characteristics of Gaussian Sea Waves[END_REF] The map ϕ → c( K ϕ ) admits a local minimum at ϕ = π/2.

which clearly admits a minimum at ϕ = 0.

A.4.2 The elementary model case

We assume that the direction of k is given by the elementary model (see Example 2 in Section 2.4.2). Using [START_REF] Lindgren | Slepian Models for the Stochastic Shape of Individual Lagrange Sea Waves[END_REF] and the expression for the µ i,j 's given by Lemma A.2, one can compute c( K ϕ ) 2 . Figure 2 shows the graph of ϕ → c( K ϕ ) 2 for various values of h. We again observe that it is minimal at ϕ = π/2.