A Orozco Estrada

Raúl C Cruz Gómez

A Cros

P Le Gal
email: legal@irphe.univ-mrs.fr

A Orozco Estrada

Coalescence of lenticular anticyclones in a linearly stratified rotating fluid

This study is devoted to laboratory experiments on the coalescence of two lenticular anticyclones in a linearly stratified rotating fluid. These anticyclones are generated by injecting small volumes of fluid at the center of a rotating tank where a linearly stratified layer was previously prepared with salt. The characteristics of the interaction between the vortices are studied by visualization and Particle Image Velocimetry (PIV) as a function of the initial separation distance between the vortices, the Coriolis parameter of the rotating table and the Brünt-Väisälä frequency of the density stratification. Our results show that the merging critical distance depends drastically on the Rossby radius of deformation of the vortices and are in complete agreement with previous numerical modeling of vortex coalescence. We have also observed that mergers involve three-dimensional processes as the vortices intertwine together possibly because of the presence of an elliptic instability that tilts the vortex cores. They are also accompanied by the emission of vorticity filaments and internal gravity waves radiation although we cannot prove that in our experiments these waves are solely due to the merging process.

Introduction

Oceanic meso-scale lenticular vortices play an important role in the redistribution of momentum, heat, salt, nutrients and plankton in oceans and thus contribute to the climate equilibrium and ecological diversity on Earth. These vortices are governed by geostrophic and hydrostatic balances between pressure gradients, Coriolis and buoyancy forces from where they get their shape and aspect ratio as shown by [START_REF] Hassanzadeh | The universal aspect ratio of vortices in rotating stratified flows: theory and simulation[END_REF], [START_REF] Aubert | The universal aspect ratio of vortices in rotating stratified flows: experiments and observations[END_REF] and by [START_REF] De La | A laboratory study of floating lenticular anticyclones[END_REF] for surface vortices. Their chaotic motions in oceans give rise to a quasi-geostrophic turbulence characterized by a reverse energy cascade and a forward enstrophy cascade [START_REF] Kraichnan | Inertial ranges in two dimensional turbulence[END_REF][START_REF] Charney | Geostrophic Turbulence[END_REF]. Understanding the way energy escapes from this mesoscopic turbulence to feed the smallest oceanic scales where dissipation occurs, is the subject of an intense research. In this context, vortex coalescence as observed in oceans [START_REF] Schultz Tokos | Merging and migration of two meddies[END_REF]) can be seen as a building block of these reverse and forward cascades: vortex mergers participate to the reverse cascade towards the large scales of the oceanic turbulence while they generate at the same time fine filaments of vorticity [START_REF] Cushman-Roisin | On the role of filamentation in the merging of anticyclonic lenses[END_REF] as well as internal waves (Viúdez 2006, Reinaud and[START_REF] Reinaud | The merger of geophysical vortices at finite Rossby and Froude number[END_REF]) that could feed small dissipative scales. The aim of the present study is to describe and parametrize the merging of two lenticular anticyclones by means of stratified flow experiments performed on a rotating table. For this purpose, we generate pairs of anticyclonic vortices by the gentle injection of two small volumes of water inside a continuously stably stratified rotating layer in a similar way that [START_REF] Griffiths | Coalescing of geostrophic vortices[END_REF] did thirty years ago but for a two layer system. To our knowledge, this problem has never been revisited experimentally for the case of lenticular vortices imbedded in a continuously stratified layer.

Symmetric vortex pairing has been widely studied in two-dimensional, inviscid, incompressible flows [START_REF] Zabusky | Contour dynamics for the Euler equations in two dimensions[END_REF][START_REF] Melander | Symmetric vortex merger in two dimensions: causes and conditions[END_REF][START_REF] Waugh | The efficiency of symmetric vortex merger[END_REF][START_REF] Nielsen | Vortex merging and spectral cascade in two-dimensional flows[END_REF]. It was shown that, in this case, the vortices merge when the ratio of the initial distance d 0 between their centers and their radius R is small enough (typically d 0 /R ≤ 3.3, [START_REF] Waugh | The efficiency of symmetric vortex merger[END_REF][START_REF] Meunier | Physics of vortex merging[END_REF][START_REF] Leweke | Dynamics and instabilities of vortex pairs[END_REF]. Nevertheless, when the vortices are baroclinic, the Rossby radius of deformation may drastically modify the critical distance d c for which merging occurs. The Rossby radius of deformation λ gives an order of magnitude of the scale for which the Coriolis effect becomes as important as the buoyancy. It gives the effective length of the horizontal distance between the region of generation of a disturbance and its outermost extent. This Rossby radius of deformation is equal to

λ = N f H , ( 1
)
where H is the vortex height, f the Coriolis parameter and N the Brünt-Väisälä frequency. For a continuously stratified fluid, N = -(g/ρ 0 ) ∂ρ/∂z represents the oscillation frequency of a parcel of fluid with density ρ 0 , vertically displaced in a fluid layer of density ρ(z) and where g is the gravity.

As far as we know, the first and unique experimental study on baroclinic vortex pairing in a two-layer stratification was carried out by [START_REF] Griffiths | Coalescing of geostrophic vortices[END_REF] for various values of λ/R and d 0 /R. The vortex height is fixed by the top layer depth and it was found that a strong enough stratification λ/R > 2.0 enhances the merging, and in consequence increases the critical distance d c that separates pairing to non pairing cases. Following this experimental work, two numerical studies evidenced the importance of the initial conditions on the influence of stratification in a two-layer model. First, [START_REF] Polvani | Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger[END_REF] found that stratification does not modify the critical initial distance d c ; but this contradictory result was explained a year later by [START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF] who proved that density stratification has indeed a drastic effect on d c /R if the relative vorticity is zero in the lower layer. This condition was not fulfilled in the work of [START_REF] Polvani | Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger[END_REF] who enforced a zero potential vorticity in the lower layer (a condition which is called "PVI" for Potential vorticity initialization). [START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF] found that the barotropic limit is reached when λ/R → 0 as well as for large values of λ/R (λ/R > 4). In the case of zero relative vorticity initialization (called "RVI" conditions), the merging is more efficient when λ/R 2.1 as the critical distance reaches a maximum value d c 6.8 R. Nevertheless, it was noted later by [START_REF] Verron | Scale-dependent merging of baroclinic vortices[END_REF] that this maximum can be modified if mixed conditions between zero potential and zero relative vorticity initialization are taken into account. Indeed the peak in the initial distance critical values can shift towards lower values of λ/R and d c /R.

Merger observations in the Atlantic Ocean of strongly interacting meddies were reported by [START_REF] Carton | Meddy dynamics and interaction with neighboring eddies southwest of Portugal: observations and modeling[END_REF][START_REF] L'hégaret | Evidence of Mediterranean water dipole collision in the Gulf of Cadiz[END_REF]. Some mergers were also described by Schultz [START_REF] Schultz Tokos | Merging and migration of two meddies[END_REF], [START_REF] Carton | Meddy coupling with a deep cyclone in the Gulf of Cadiz[END_REF], [START_REF] Rodríguez-Marroyo | Vortex merger in oceanic tripoles[END_REF] and [START_REF] Barbosa | A census of meddies in a long-term high-resolution simulation[END_REF]. Besides these oceanographic observations, merging of two giant anticyclones in the atmosphere of Jupiter is also described by [START_REF] Sanchez-Lavega | The merger of two giant anticyclones in the atmosphere of Jupiter[END_REF]. We can moreover easily imagine other astrophysical applications such as the coalescence of baroclinic vortices in accretion disks for instance. Whereas the two-layer model is a good first order approximation for these natural cases, oceans or atmospheres are continuously stratified. To our knowledge, only one laboratory study has been reported up to now in a linearly stratified rotating fluid but only for pairs of barotropic vortices [START_REF] Cariteau | Etude de la stabilité et de l'interaction de cyclones intenses en fluide stratifié[END_REF]. This laboratory experiment showed that the critical distance for the merging of long columnar vortices increases when stratification increases in agreement with the numerical prediction of [START_REF] Dritschel | Vortex merger in rotating stratified flows[END_REF], who extended the results of Von [START_REF] Hardenberg | Vortex merging in quasi-geostrophic flows[END_REF] to vortices with large aspect ratio H/R = 20f /N . In these last studies, three-dimensional quasi-geostrophic vortices are modeled and the barotropic critical distance is recovered for flat vortices (H/R → 0), whereas unexpectedly d c /R continues to increase with H/R. This enhancement of coalescence for large separating distances is due to the existence of a bending instability (most probably the zig-zag instability of [START_REF] Billant | Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations[END_REF][START_REF] Otheguy | Pairing of two vertical columnar vortices in a stratified fluid[END_REF] that brings the vortex cores closer together. Another ingredient that was pointed by [START_REF] Valcke | Interactions of baroclinic isolated vortices: the dominant effect of shielding[END_REF] comes from the possible presence of shielding rings around the vortices. Indeed, as observed numerically in the pairing of realistic oceanic meddies by [START_REF] Ciani | On the merger of subsurface isolated vortices[END_REF], the cyclonic ring around each anticyclone can prevent the merger by a redistribution of the shield cyclonic potential vorticity that generates a repulsive effect on the cores. As we will see later, our laboratory vortices are indeed shielded and our experiments reveal some interesting features already detected in numerical simulations although the shields seem not to prevent coalescence in our present observations.

Our study explores the (λ/R and d 0 /R) space in order to determine the critical merging distance d c as a function of the Rossby radius of deformation λ. Both the global rotation rate and the density stratification will be changed. Around ninety experimental runs (including Particle Image Velocimetry (PIV) measurements and dye visualizations) have been realized. Each experiment is labeled with a number which will be used to identify them. Pairing and non pairing situations are discriminated by measuring the evolution in time of the distance separating the vortex cores. When this separation distance becomes less than a vortex diameter, the merging process will be declared effective. In most of the runs, there are no ambiguities to decide if the pairs have merged or not after a duration of about 20 minutes per run. Besides determining the critical initial distance d c , we have also observed that the injection flux that controls the Rossby number of the vortices, plays a role on the merging time and on the rotation rate of the pair. These results are consistent with the recent numerical simulations of [START_REF] Reinaud | The merger of geophysical vortices at finite Rossby and Froude number[END_REF]. Finally, we will show that a Fourier analysis permits the detection of Internal Gravity Waves (IGWs) that radiate energy away during the vortex generation by injection and during the first instants of coalescence.

Experimental set-up and measurement techniques

2.1.

Rotating table installation and methodology

All the experiments were performed in the Geophysical Fluid Dynamics Laboratory of Guadalajara University. The experimental set-up is shown in figure 1. The rotating transparent Plexiglass tank dimensions are 40 × 40 × 40 cm 3 which are at least ten times larger than the vortex diameters and twenty times deeper that the vortex heights. Two constant density 5 cm deep layers are generated at the bottom and at the top of the total fluid layer, while the stratified region has a vertical extension of 20 cm. The upper layer water is fresh while the bottom layer water is salty and has a density whose value depends upon the desired stratification. The linear stratification is made by the two-bucket method [START_REF] Oster | Density gradients[END_REF] which is a classical method that consists in mixing progressively the water from a bucket of pure water with the brine of a second bucket. Then the Brünt-Väisälä frequency N is determined from the measurement of the density gradient after the fluid was left at rest for more than 12 hours before performing the experiments. The diffusion of salt in water being slow (Schmidt number equal to 700), the stratified layer can be kept for days if not perturbed. The whole device shown in figure 1 is then slowly set into rotation for more than 2 hours until the fluid layer reaches a solid body rotation at a rotation rate Ω. The Coriolis parameter is given by f = 2Ω. A side-view camera allows to determine the depth H of the vortices as shown in figure 1, while the top-view camera records the two vortices evolution after they have been created. The two anticyclones are generated by injecting a small volume of fluid, between 5 ml and 20 ml, at the center of the vertical stratified layer (see for instance [START_REF] Hopfinger | Vortices in rotating fluids[END_REF] as it is sketched in figure 1. In order to generate controlled vortices, two micro-pumps suck dyed salty water from two distinct tanks whose density is equal to the density in the middle-plane of the stratified layer. Two flow meters allow to measure the flow rate. The dyed water is injected through two rigid 4 mm inner diameter pipes which are fixed on the rotating tank. The separation distance d 0 between the pipes sets the initial distance between the two vortices. Note that some of our experiments were realized by hand injection, using syringes (instead of the pumps), of a 5 ml volume of water in approximately 12 s. As already explained in the introduction, each run is labelled by a number and is characterized by its rotation Coriolis parameter f , its Brünt-Väisälä frequency N , the initial separating distance d 0 and also the flow rate Q of the injection that has, as we will show later, a great influence on the Rossby number of the vortices. Most of the time, both vortices in the pairs were of equal strength and size. However, in about one third of the total number of experiments, the symmetry between both vortices is not perfectly respected (because of uncontrolled experimental uncertainties like tiny bubbles in the injection pipes for instance) and asymmetric fusion takes place [START_REF] Dritschel | Vortex merger in rotating stratified flows[END_REF]. In this case, one of the vortices wraps around the other one as seen in figure 2. Therefore in the following we have removed all the data corresponding to these asymmetric experimental configurations and we are able to produce a state diagram using more than forty data points.

Visualization and PIV measurements

Most of our experiments have been realized with vortices colored with dye in order to measure their core diameters and heights. One vortex is colored in red and the other one in green (or blue) in order to facilitate the use of simple algorithms to differentiate and track the vortices by video image analysis. Their radius and separating distance are recorded by the top-view camera. Their height is measured by the side-view camera. Radii and heights will be then used to compute the Rossby radius of deformation λ of the anticyclones. In order to calculate the Rossby number of the vortices, but also to describe precisely their shape, Particle Image Velocimetry (PIV) measurements were performed in both horizontal and vertical planes for some experiments. To do so, small polyamide particles (diameter of 50 µm) were added in the water prior to the stratification process. These particles are also added in the salt water that is injected to generate the vortices. A green light laser beam is then used to illuminate the particles in planes crossing the vortices. The top view camera is used to record the movement of the particles with a frame rate of 1 Hz. Two frames of the video sequences are used to estimate the particle displacement using PIVlab [START_REF] Thielicke | PIVlab-towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB[END_REF] or Flow Manager (Dantec Dynamics c ) softwares. Each image is subdivided in 32 pixels interrogation windows to calculate the instantaneous velocity fields as a function of time at the rate of 1 field per second.

Description of a single anticyclone

The geostrophic balance of the anticyclonic lenses

In order to describe a single vortex before proceeding to the merging experiments, we have studied its velocity field by performing PIV measurements in its equatorial plane. Figure 3(a) presents an example of the vorticity field recorded in and around an anticyclone. As can be clearly seen, a cyclonic ring is present around the vortex core. The Rossby number of the vortex, i.e. its magnitude, can be measured by the minimum of the vorticity field ω 0 in its center: Ro = ω 0 /f . This kind of shielded vorticity field is quite classic for oceanic vortices and can be modeled by a modified Gaussian shape

ω(r) = ω 0 1 - α 2 r L α exp - r L α , ( 2
)
where α and L are free parameters. When α = 2, the vortex is said to be Gaussian. Figure 3(b) shows the data from the previous field, plotted along the radius for different azimuthal angle. A quite nice collapse of the vorticity radial profiles proves that they can be fitted with a modified Gaussian function where α and L are determined by best fit. In this experimental run, α lies between 2 and 1.5 as time varies and this will always be the case whatever the parameter values. L is the vortex typical horizontal scale corresponding to the radial location where the vorticity goes through zero. This value also varies in time as the vortex expands when slowing down. Figure 4 presents the temporal variation of the Rossby number Ro, of L and of α respectively. This slow damping of the vortex has been analysed in great details by [START_REF] Facchini | On the lifetime of a pancake anticyclone in a rotating stratified flow[END_REF] and is due to a subtle interplay between viscous dissipation and secondary circulations which advect the density anomaly.

As it was shown in previous studies, vortices are governed by geostrophic and hydrostatic balances from where they get their shape that obeys scaling laws (see De la Rosa Zambrano et al. (2017) for instance). Moreover, we have also observed that the Rossby number depends upon the injection rate as the vorticity is a direct consequence of the horizontal radial flow, which by action of the Coriolis effect generates an azimuthal velocity field. Figure 5 shows this dependence for four values of Q, where a linear behaviour is observed.

Finally, to complete our description of a single vortex, we have also realized PIV measurements in vertical planes going through the vortex at mid-radius. As this plane is off centered, because of its thickness (1 mm), the projection onto the plane of the particle trajectories can be recorded and analyzed. As presented in figure 6, the evolution of the azimuthal velocity along the vertical direction can be quite nicely fitted by a modified Gaussian function with a typical vertical scale Z and an exponent β. Therefore our laboratory vortices created by injection of fluid in a rotating and stratified layer have a similar three-dimensional shielded structure as the real oceanic vortices [START_REF] Ciani | On the merger of subsurface isolated vortices[END_REF])

ω(r) = ω 0 1 - α 2 r L α exp - r L α exp - z Z β .
(3)

Measurements of the radius and height of the vortices

Most of our experiments were realized using visualizations of the pair of vortices. After injection, we waited several table rotation periods to get a quasi-geostrophic adjustment of the injected patches during which these get their circular shape. Their radii R were then obtained at this initial time by a measurement of the horizontal size of the circular patches visualized by dye (see section 4.3 for details of image prcessing). To compare R with L, we performed a complementary experiment where both techniques are applied together: visualization of the injected water and PIV measurement. Figure 7 shows that the two values agree by 10%, R being larger than L. Thus a dividing factor of 1.1 has been applied in the following to all our radius R measurements of the dye patches in order to transform these radii R into the proper typical horizontal scales L.

In particular, to build the state diagram of figure 16, the initial values of R will be transformed in initial values of L in order to compare our experimental data to former published results.

The height H of the vortices are directly measured by image analysis of the videos recorded by the side-view camera to calculate the Rossby deformation radius λ. We will keep the initial values of these heights i.e. after the transients due to the time needed for the anticyclones to reach their geostrophic balance (neglecting the presence of the other vortex in the pair) to perform this calculation.

Coalescence of identical anticyclones

Visualization of the merging process

As explained previously, the vortices are created by injection of 5 to 20 ml of salt water in a layer that was stratified in density with salt and set into rotation for two hours before performing each experiment. The density of the injected water was chosen to be the value of the density at the injection location so that the anticyclones stay at the desired altitude. The top-view camera records the complete process from the injection to the final time when all the fluid motions are damped out, while the side-view camera records only the first minutes of the runs. A complete sequence lasts from 3 to 30 minutes depending upon the initial conditions. Figure 8 presents for f = 1.57 rad/s, N = 2.8 rad/s, d 0 = 8 cm and Q = 45 ml/min, four images from a typical sequence where two anticyclones merge. The first effect of the mutual influence of one vortex onto the other is a global rotation of the pair. Then, when getting closer to each other, the vortices are deformed and a cusp is formed on the inner sides of both of them. The streaklines around these cusps are then elongated, captured and then wrapped around the opposite core to form a Yin and Yang Chinese symbol. As observed on the pictures, the process is threedimensional as the vortices intertwine with overlapping of both cores seen as a superimposition of red and green colors. We also note the presence of dye filaments repulsed away from the final merged vortex core. Figure 9 presents another illustration of the merging process in the case where the cyclonic shields are repulsed in the form of two side cyclonic vortices. For this sequence, f = 1 rad/s, N = 3.12 rad/s, d 0 = 7.5 cm and Q = 25 ml/min. These visualizations are perfect illustrations of the numerical simulations of [START_REF] Valcke | Interactions of baroclinic isolated vortices: the dominant effect of shielding[END_REF] for two layers quasi-geostrophic models and of [START_REF] Ciani | On the merger of subsurface isolated vortices[END_REF] for computations of realistic oceanographic eddy coalescence.

The three-dimensional features of the merging of baroclinic vortices can also be observed by the side-view camera. Figure 10 presents two images where the pair of anticyclones is viewed from the side for two different instants. It is very clear on these images that the vortices are tilted (versus the vertical axis) during coalescence. Although it is difficult to prove, we speculate that an elliptic instability may take place here in the same way it appears in the case of barotropic columnar vortices merging [START_REF] Meunier | Physics of vortex merging[END_REF]. Unlike the elliptic instability of these columnar vortices, the most unstable mode of the elliptic instability of fluids inside rotating ellipsoids [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF]) is a rotation around a tilted axis (the spin-over mode). From the top-view images, it is clear that the mutual interaction between the anticyclonic lenses can deform sufficiently each vortex such that their ellipticity is large and of the order of their diameter, enhancing the possible appearance of the instability in this free of boundary layers flow.

PIV measurements of coalescence

Besides visualization, we have also performed PIV measurements of velocity fields during coalescence. Figure 11 presents an example of the temporal evolution of a vorticity field. At initial times, soon after the vortex generation, we can see the separated cores and their shielding cyclonic rings which, in the present case, have merged in a figure eight shape. Rapidly this shield disappears and as the vortices rotate one around the other, two vorticity arms are formed at the external tips of the pair. These arms wrap around the vortices and become thinner as they are elongated by the core rotation. Finally, these filaments are dissipated while the cores are completely merged.

Image analysis and measurements of the global induced rotation rate and the merging time of the pair

As mentioned before we have tracked the vortices visualized with dye by image analysis. We first determine the center position of each vortex and measure their radius by extracting one frame per second from the video. A red and then a green (or blue, according to the dye color we used) filter is applied to each frame. An intensity filter for each color is then used in order to highlight each vortex. These highlighted images are then binarized such that only the vortices are visible. From these images we can now extract different parameters like the radius and the center position of each vortex. These radii are calculated such that the corresponding disks have the same areas as the binarized patches. The vortex common rotation center is then determined and we can compute the global orbital rotation rate ω orb of the pair and the separation distance between both vortices. Figure 12 shows an example of the merging process for f = 0.5 rad/s, N = 1.75 rad/s, Q = 60 ml/min and d 0 = 8 cm on which we have applied our image analysis. As can be observed, in this case, both vortices have ejected their cyclonic rings in the form of two thin cyclonic filaments. We have also represented on the figure the white circles whose centers and radii have been determined by our image processing.

Figure 13(a) shows the spiraling trajectories of each vortex in the rotating frame of the table; the numbers beside the plotted points indicate the elapsed time in seconds. From these, we can easily calculate the azimuthal positions θ 1 and θ 2 of each vortex core versus time, and deduce by a simple linear fit the orbital rotation rates ω 1 and ω 2 of each vortex as shown in figure 13(b). As ω 1 and ω 2 have nearly constant and very close values in all the runs, we will, in the following, average them to obtain a measurement of the global orbital rotation rate ω orb . We note a systematic slight increase of ω orb at the end of the merging processes. In the same way, we can measure the distance between the cores as a function of time. Figure 13(c) presents these results from the former example.

As can be observed, we can systematically identify three different stages in the merging process in accordance to what has been previously described in the literature for barotropic vortex coalescence [START_REF] Melander | Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation[END_REF][START_REF] Meunier | Physics of vortex merging[END_REF][START_REF] Leweke | Dynamics and instabilities of vortex pairs[END_REF]. At the beginning of the interaction we see that the distance between centers stays constant or varies slightly; this first period of time could be identified as a first diffusive stage. In the second or convective stage a rapid decrease of the separation distance is observed until it reaches a minimum. In the third stage, the vortices have merged and we no longer detect any variation of the distance between their centers. To characterize the merging time, we choose to fit the intermediate (or convective stage, where each vortex feels the velocity field from the other vortex that induces the merging) by a decreasing exponential function of time exp (-t/τ ) where τ represents the characteristic time of coalescence. This method permits to deduce easily and automatically τ in order to describe the duration of the process as function of the experimental parameters.

We have found that both the global rotation rate ω orb of the vortex pair and the merging time τ are influenced by the injection flux Q that creates the vortices. A crude two-dimensional model can be built by applying Stoke's theorem on a circle passing through the centers of both vortices: it is straightforward to show that the global orbital rotation rate ω orb is proportional to the vorticity contained essentially in the vortex cores of radius R and inversely proportional to the square of the initial separating distance d 0 . As the Rossby number of each vortex is itself a linear function of the injection rate Q (see figure 5), we can plot ω orb as a function of QR 2 /d 0 2 to test this prediction. Figure 14 shows that this trend is verified by our experimental data: the larger the radii and the smaller the initial distances are, the higher the orbital angular velocity of the vortices around their barycenter is. We can also observe on figure 14 that high injection fluxes Q correspond to fast angular velocities. This can be understood using the already mentioned link between the Rossby number of each vortex and the injection rate through the Coriolis force action. We can thus conclude that the individual angular velocity measured by the Rossby number of each vortex has a direct effect on the global rotation rate of the pair of vortices.

In the same spirit, we have also found that the injection flux Q directly influences the merging time τ . This is shown in figure 15. As expected, the larger the initial distance and the smaller the radii are, the longer it takes for the vortices to merge. Interestingly, we can also observe that mergers are faster (τ small) for high injection flux Q, i.e. is for large individual Rossby numbers. The role of the Rossby number in the coalescence process has also been discovered by the recent numerical observations of [START_REF] Reinaud | The merger of geophysical vortices at finite Rossby and Froude number[END_REF].

The state diagram

Having varied the initial distance d 0 separating the vortices at their creation, the rotating rate Ω of the rotating table, the density gradient that implies different Brünt-Väisälä frequencies and also the injection rate Q that permits to change the Rossby number and thus the initial dimensions H and L of each vortical pairs, we can build the state diagram in the (λ/L and d 0 /L) plane. When pairing is observed, we represent the result at its corresponding location with a blue star next to its label. When the vortices do not merge, we plot the result as a red circle. Each experiment is labeled with a number which is smaller than 100 for hand injection runs and above 100 for pump injection runs. Figure 16 presents this synthesis. As expected, the blue stars are concentrated at the bottom of the figure, i.e. when the initial separating distance d 0 is small. We have shaded this area in blue. On the contrary, when the initial distance is large, the vortices have more difficulty to merge and the red stars are mostly located in the upper part of the figure. However, it is also clear from our results that the Rossby radius of deformation has a tremendous influence on coalescence. Although we cannot perform experiments at too small values of λ/L (our lowest λ /L is around 0.6) because of the presence of centrifugal or baroclinic instabilities that destroy individual vortices, we observe that our experimental results are compatible with the two-dimensional limit. Together with our experimental data points, we represent also the different results from diverse authors. First, the Griffiths and Hopfinger's curve (1987) is plotted (dashed blue curve). It presents a clear increase of the critical distance when λ/L is between 2 and 4. As can be observed on figure 16, our experimental data points show a similar trend when λ/L is between 1 and 3. Around a value close to 2.5, the vortices are prone to merge even if they are separated by a distance more than twice the two-dimensional critical distance (7.5 × L compared to 3.3 × L). However, contrary to Griffiths and Hopfinger's results but in excellent agreement with the numerical simulations of [START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF] (dashdotted red curve), our experimental critical curve obviously decreases back when λ/L is larger than 3. Then for λ/L larger than 4, i.e. with strong stratification effects compared to rotation, the vortices and their motions are confined in very shallow layers and the frontier d c between pairing and non pairing dynamics decreases back to its expected two-dimensional value close to 3.3. Note that our results do not support (as expected) the results from the three-dimensional quasi-geostrophic model of [START_REF] Dritschel | Vortex merger in rotating stratified flows[END_REF] (solid black curve) where a secondary instability on tall vortices, not seen in our experiments, was suspected to help coalescence. Amazingly, 2002) (-). Note that in accordance with the definition of the Rossby radius of deformation used by [START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF] and explained in [START_REF] Hopfinger | Vortices in rotating fluids[END_REF], the red dash-dotted curve has been shifted by a multiplicative factor √ 2. The yellow area represents the "partial pairing" region detected by [START_REF] Dritschel | Vortex merger in rotating stratified flows[END_REF]. The experimental two layer experiment results of [START_REF] Griffiths | Coalescing of geostrophic vortices[END_REF] are represented by the blue dash curve (---) and our estimation of the extension of the barotropic criterion by the grey dotted curve (• • • ). (Colour online) our results appear to match better with the RVI (see the definition of RVI and PVI in our introduction) two-layer model of [START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF] than with the two-layer Griffiths and Hopfinger's experiments. Verron et al. discuss that the initial conditions of the Griffiths and Hopfinger's experiments are more consistent with PVI conditions because the injection of fluid in the upper layer would deflect the interface and according to potential vorticity conservation, some relative vorticity should be created in the lower layer. With this argument, they reach some kind of contradiction in the sense that the results from the Griffiths and Hopfinger's experiments are more consistent with RVI type conditions contrarily to their initial conditions which are more consistent with PVI type conditions. They try to explain this contradiction by stating that the vorticity induced in the lower layer is weaker than in the upper layer thus the vortices do not present a barotropic vertical structure. They think that the vorticity field in the lower layer is weak enough to prevent the initial conditions to be entirely of PVI type and that would be the reason why the Griffiths and Hopfinger's experimental results differ from the PVI results. In our case, the linear stratification might act as a shield, isolating the vortices from the fluid above and under the vortices. Therefore the relative vorticity stays zero outside the vortices as it can be observed on figure 6. As a consequence, our initial conditions are very close to RVI conditions, and this explains why our experimental results are so similar to the RVI numerical simulations of [START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF].

To our point of view, the final picture of baroclinic vortex coalescence could be interpreted as follows. For λ/L lower than unity (in this case the Rossby radius of deformation stands inside the vortex core), the influence of density stratification is weak and the critical distance d c stays close to the two-dimensional limit 3.3 × L. Note that in the experiment by [START_REF] Griffiths | Coalescing of geostrophic vortices[END_REF] as well as in the numerical simulations of [START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF] and also of [START_REF] Dritschel | Vortex merger in rotating stratified flows[END_REF], there is a small decrease of this limit down to 2.5 × L. Then, for λ > L, as we saw previously, stratification helps coalescence. If we consider that the radius of influence gives a rather good estimation of the frontier separating pairing to non pairing vortices. Finally, as already noted, coalescence being a three dimensional process, a further increase of the density gradients (and thus of N ) makes vortex pairing more and more difficult. The limit is reached in our experiment for λ/L 3 where the critical distance starts to decrease back towards its two-dimensional value, reached as expected for very flat pairs of anticyclones in highly stratified fluid layers. This return to the barotropic value at large λ/L was not observed in the experiment of [START_REF] Griffiths | Coalescing of geostrophic vortices[END_REF] possibly because of the two layer configuration that confines the vortices in the top layer.

Fourier modes of the merger and Internal Gravity Waves radiation

Internal Gravity Waves (IGWs) play a major role in the atmosphere and in the ocean where they can be seen as a conveyer of energy from large scale flows to small scales so that they participate to mixing and energy dissipation. Therefore, many studies focus on the spontaneous generation of IGWs from nearly balanced flows (see the review by Vanneste ( 2013)). Among these, several numerical simulations based on primitive equation simulations [START_REF] Sugimoto | Generation and backreaction of spontaneously emitted inertia-gravity waves[END_REF] or based on shallow water models [START_REF] Sugimoto | Cyclone-anticyclone asymmetry in gravity wave radiation from a co-rotating vortex pair in rotating shallow water[END_REF][START_REF] Sugimoto | Inertia-gravity wave radiation from the merging of two co-rotating vortices in the f-plane shallow water system[END_REF] show that vortex dipoles or vortex pairs are prompt to generate IGWs. The ultimate goal of these researches is to quantify the amount of ageostrophic energy loss when the vortices are propagating in the case of a dipole, or merging in the case where they are of the same sign. Therefore, in addition to describing the different regimes that lead or not to coalescence, we will address now the problem of the generation of IGWs during the transient time of the fusion of our lenticular anticyclonic pairs. These were already observed in the numerical simulations of [START_REF] Viúdez | Spiral patterns of inertia-gravity waves in geophysical flows[END_REF], [START_REF] Sugimoto | Inertia-gravity wave radiation from the merging of two co-rotating vortices in the f-plane shallow water system[END_REF] and of [START_REF] Reinaud | The merger of geophysical vortices at finite Rossby and Froude number[END_REF]. A classical procedure to detect Internal Gravity Waves (IGWs) is to calculate the horizontal divergence of the PIV velocity fields [START_REF] Sugimoto | Inertia-gravity wave radiation from the merging of two co-rotating vortices in the f-plane shallow water system[END_REF][START_REF] Rodda | Kelvin and inertia-gravity waves in the barostrat instability experiment[END_REF]). Then a space-time diagram can be computed by recording the fluctuations of the horizontal divergence along a line going through the pair of vortices (see figure 17(a) for an example with f = 0.5 rad/s and N = 0.95 rad/s). As can be observed on figure 17(b), waves clearly propagate away from the merging pair, as expected for f < N . The calculation of the average period of these waves (about 10 seconds) show that their frequency is equal to 0.62 rad/s, a value between f and N as requested by the dispersion relation of IGWs. Note that their wavelength is comparable with the diameter of the initial vortices. As the sign of the phase velocity is a function of f -N , we also present in figure 17(c) an example of inwards propagation for f = 1.2 rad/s, N = 0.95 rad/s, Q = 60 ml/min and d 0 = 8 cm, i.e. in the case where f > N . Figure 18 shows the temporal Fourier spectrum of a time series recorded at a point in one of our PIV horizontal divergence fields for f = 1 rad/s, N = 0.7 rad/s, Q = 25 ml/min and d 0 = 5 cm. The low frequency peaks correspond to the different Fourier modes with different azimuthal wave numbers m associated with the double structure of the field during the vortex pairing. In addition a well defined wave packet is clearly visible in the IGWs authorized band between the buoyancy frequency N and the Coriolis frequency f . Snapshots of these modes and of the propagating IGWs fields are calculated by band-pass filtering the global vorticity field. The low frequency modes are associated with the deformation of the vorticity field due to the mutual induction of the vortices in the pair. All these modes rotate with an identical rotation rate which is that of the vortices orbiting one around the other at the rate ω orb while the IGWs propagate radially with their own phase velocity. Let us note however that if our observations of IGWs wave packets are in agreement with recent numerical computations of [START_REF] Viúdez | Spiral patterns of inertia-gravity waves in geophysical flows[END_REF] and of [START_REF] Reinaud | The merger of geophysical vortices at finite Rossby and Froude number[END_REF], we are not able at the stage of our study to separate the IGWs wave trains created by the injection process when generating the vortices to those which can be naturally emitted during merging by the rotating non axi-symmetric vortical patterns.

Conclusion

In this experimental work, we have for the first time studied the coalescence of baroclinic anticyclones that were generated in a rotating and continuously stratified layer. Our results clearly show that the merging process is highly dependent on the Rossby radius of deformation λ. First, when λ is smaller than the vortex core, stratification seems not to play an important role. Then, in a range of λ between 1 and 3, coalescence is greatly enhanced. Our interpretation of this enhancement suggests a modification of the two-dimensional criterion by the replacement of the anticyclones radius by their Rossby radius of deformation as this horizontal length is the effective scale associated to baroclinic vortices. But this positive effect due to stratification is only efficient in a restricted region of λ/L. As coalescence of baroclinic vortices involves three dimensional flows during which the vortices intertwine possibly because of the elliptic instability, it is expected that very flat vortices in highly stratified fluid resist against merging and the twodimensional limit is recovered. As a consequence, the barotropic criterion is recovered at large λ. We have also shown that the merging time and the global rotation rate of the pair depend on the Rossby number of each vortex. Besides the description of the state diagram that confirms the numerical simulation of [START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF], we have also discovered that internal gravity waves are emitted during mergers. Unfortunately, we cannot discriminate at this point the waves generated by the injections used to create the vortices, to those that are emitted by the non balanced vortex pair during coalescence. As a final point, let us mention the visual analogy of our hydrodynamical system (see our figure 8) with the merger of two black holes associated with the radiation of gravitational waves (see figure 2 of [START_REF] Abbott | Observation of gravitational waves from a binary black hole merger[END_REF].

Acknowledgments

This work has been carried out in the framework of the Labex MEC Project 410 (No. ANR-10-LABX-0092) and of the A*MIDEX Project (No. ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR). The authors also acknowledge support from the American Physical Society's International Research Travel Award Program (IRTAP). We also thank for their financial support, the University of Guadalajara and Aix Marseille Université for the fundings "Appel à Projets Actions Internationales" 2016. A.O.E. acknowledges CONACyT for his scholarship.

Figure 1 .Figure 2 .

 12 Figure 1. Experimental system scheme with the pipes, tubes and injection pumps that permit the creation of the pair of anticyclones. (Colour online)

Figure 3

 3 Figure 3. (a) The vorticity field of a single shielded anticyclone in the laboratory for f = 1 rad/s, N = 1.02 rad/s and Q = 80 ml/min. (b) Vorticity profile showing 1888 data points together with their modified Gaussian fit with α = 1.9 and L = 3.5 cm as described in equation (2). (Colour online)

Figure 4 .

 4 Figure 4. For the experimental run of figure 3, the temporal evolution of (a) the Rossby number Ro, (b) the horizontal typical scale L and (c) α.

Figure 5 .

 5 Figure 5. Variation of the Rossby number Ro for a single vortex as a function of the injection flow rate Q. Data points for five experiments are plotted. Circles show the mean Rossby number for each experiment.

Figure 6

 6 Figure 6. (a) The velocity field in a vertical section going through the anticyclone at mid-radius. (b) The velocity profile showing the data points together with their modified Gaussian fit with β = 1.87 as described in equation (3). (Colour online)

Figure 8 .Figure 9 .

 89 Figure 8. Temporal sequence showing the coalescence of two anticyclones for f = 1.57 rad/s, N = 2.8 rad/s, Q = 45 ml/min and d 0 = 8 cm. (Colour online)

Figure 10 .Figure 11 .

 1011 Figure10. Two observations of the merging process from the side view camera for f = 1 rad/s, N = 3.16 rad/s, Q = 30 ml/min and d 0 = 14 cm. The anticyclonic lenses are clearly tilted when merging, enhancing the three-dimensional process as they intertwine each other. We suspect the elliptic instability to be at the origin of this tilting. (Colour online)

Figure 12 .

 12 Figure 12. Temporal sequence showing the coalescence of two anticyclones for f = 0.5 rad/s, N = 1.75 rad/s, Q = 60 ml/min and d 0 = 8 cm. The white circles and stars indicate the vortex centers and their corresponding radii found via image processing. (Colour online)

Figure 13

 13 Figure 13. (a) The spiral trajectories of the vortices as function of time for the merger of figure 12. (b) Temporal evolution of the azimuthal angle of both vortices for the merger of figure 12; time is given in rotating table period and -ω 1 and -ω 2 are the global rotation rates of each vortex measured by the linear fit at the initial time (dashed lines). (c) Temporal evolution of the distance d between both vortices for the merger of figure 12; We have also represented the exponential fit of this evolution that gives an estimation of the merging time τ (in this case τ = 10.5T ). (Colour online)

Figure 15 .

 15 Figure 15. The evolution of the merging time as function of Qd 0 /R where R is the radius of the vortex cores, d 0 the initial separating distance, and Q the injection flux which is proportional to the Rossby number. This shows in particular that the merging time depends on the vortex individual Rossby number.

Figure 16 .

 16 Figure 16. State diagram showing a synthesis of our experiments with the pairing ( ) and the non pairing (•) events together with the different numerical predictions of Verron et al. (1990) (-• -) and of Dritschel (2002) (-). Note that in accordance with the definition of the Rossby radius of deformation used by[START_REF] Verron | Sensitivity to initial conditions in the merging of two-layer baroclinic vortices[END_REF] and explained in[START_REF] Hopfinger | Vortices in rotating fluids[END_REF], the red dash-dotted curve has been shifted by a multiplicative factor √ 2. The yellow area represents the "partial pairing" region detected by[START_REF] Dritschel | Vortex merger in rotating stratified flows[END_REF]. The experimental two layer experiment results of[START_REF] Griffiths | Coalescing of geostrophic vortices[END_REF] are represented by the blue dash curve (---) and our estimation of the extension of the barotropic criterion by the grey dotted curve (• • • ). (Colour online)

Figure 17

 17 Figure 17. (a) Horizontal divergence of a PIV velocity field for f = 0.5 rad/s, N = 0.95 rad/s, Q = 60 ml/min and d 0 = 8 cm, with the intersecting line along which the space-time diagram is computed. (b) Space-time diagram computed along the preceding line passing through the vortex pair. Wave trains are clearly observed with a period of 10 seconds, compatible with the outwards propagation of IGWs. (c) Space-time diagram showing inwards propagation of waves for f = 1.2 rad/s, N = 0.95 rad/s, Q = 60 ml/min and d 0 = 8 cm. (Colour online)

Figure 18 .

 18 Figure 18. Fourier spectrum of a temporal series recorded at a point in the PIV horizontal divergence field together with the spatial structures of the corresponding modes labeled by their azimuthal wave numbers m involved in the vortex merging. The internal gravity waves (IGWs) packet emitted during the merging is visible in the authorized spectral band confined between N and f . (Colour online)

GGAF-2019-0050-OrozcoCruz GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS

GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS


R Kumar

Y Jin

S Marre

O Poncelet

T Brunet

J Leng

O Mondain-Monval

Drying kinetics and acoustic properties of soft porous polymer materials

We describe a method for the fabrication of acoustic metasurfaces, which is based on soft porous polymer materials.

The materials are obtained using an emulsion templating technique, which allows for the fabrication of soft porous polymers with fully controlled porosity values between 0 and 30%. Our approach involves the polymerization of water-in-silicone emulsions with controlled water volume fractions. The obtained wet solid monolith samples are dried using three different methods. Due to the softness of the polymer matrix such as, polyHIPE hydrogels or like in silica aerogels, the first methodregular air drying -leads to a collapse of the material and we present a complete experimental study of the observed kinetics as well as a model to account for the observed results. We show that this model can catch the kinetics characteristics. Then, using two alternative drying techniques, H 2 O 2 -assisted and supercritical drying, we show that it is possible to obtain materials with fully controlled porosities. The speed of sound -or equivalently the material acoustic index -inside the material being dependent on its porosity, we obtain a gradient-index acoustic material by spatially controlling the porosity distribution along the two dimensions of these metasurfaces. Their ability in terms of wavefront shaping is then demonstrated through a deflecting experiment performed in water with a sample having a thickness 5 times smaller than the incident acoustic wavelength at ultrasonic frequencies.

Introduction

Metasurfaces are 2D materials of sub wavelength thickness that can provide non-trivial local phase shifts or amplitude modulation (or gradient) of the waves. Such devices enable a full spatial control of the wavefront. For example, a metasurface with appropriate properties can deflect or focus a beam. In other cases, it may also be used to produce twisted beams [START_REF] Xu | Planar Gradient Metamaterials[END_REF]. Initially introduced for electromagnetic waves, the concept has been extended to acoustics [START_REF] Assouar | Acoustic metasurfaces[END_REF] and other areas of wave physics. Following those tracks, acoustic gradient index (GRIN) metasurfaces, i.e. surfaces in which the acoustic index varies as a function of position in the material, have been the object of intense research in the last years, leading to many realizations, mainly for air-borne acoustic waves [START_REF] Assouar | Acoustic metasurfaces[END_REF][START_REF] Xie | Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface[END_REF][START_REF] Li | Three-dimensional Ultrathin Planar Lenses by Acoustic Metamaterials[END_REF][START_REF] Xie | Coding Acoustic Metasurfaces[END_REF][START_REF] Tang | Hollow-Out Patterning Ultrathin Acoustic Metasurfaces for Multifunctionalities Using Soft fiber/Rigid Bead Networks[END_REF]. The surfaces are obtained by micromechanical approaches, e.g. 3D printing techniques, and are based on the use of labyrinthine structures (with sizes on the order of 1 cm) allowing for a fine control of the wavefront in air at audible frequencies. Unfortunately, this smart labyrinthine approach cannot be used in water owing to the much larger acoustic impedance of water with respect to air [START_REF] Su | Broadband focusing of underwater sound using a transparent pentamode lens[END_REF]. Therefore, only few devices working in a very narrow frequency range [START_REF] Calvo | Thin Fresnel zone plate lenses for focusing underwater sound[END_REF] have been produced for water-borne acoustics, a domain of great interest for underwater applications and medical ultrasound devices. In a recent paper [START_REF] Jin | Flat acoustics with soft gradient-index Metasurfaces[END_REF], we demonstrated that very thin sheets composed of soft porous polymers can be used as acoustic metasurfaces for water-borne ultrasonic waves, which paves the way to the realization of many acoustic devices working in fluid phases. However, as previously observed, the fabrication of homogeneous soft porous polymer materials is difficult due to the collapse of the structure upon drying [START_REF] Kovalenko | Tailoring of the porous structure of soft emulsion-templated polymer materials[END_REF], a problem also encountered for the synthesis of silica aerogels [START_REF] Bisson | Drying of Silica Gels to Obtain Aerogels: Phenomenology and Basic Techniques[END_REF], polyHIPEs [START_REF] Lumelsky | Biodegradable Porous Polymers through Emulsion Templating[END_REF] or soft hydrogels [START_REF] Wheeler | The transpiration of water at negative pressures in a synthetic tree[END_REF]. In this paper, we describe the experimental procedure allowing for the production of broadband acoustic GRIN metasurfaces made of soft porous materials achieved using an emulsion templating technique (the so-called polyHIPEs approach [START_REF] Silverstein | Recent advances in emulsion-templated porous polymers[END_REF][START_REF] Pulko | High Internal Phase Emulsion Templating -A Path To Hierarchically Porous Functional Polymers[END_REF]) and specifically adapted drying techniques. We focus on the understanding of the drying mechanism and on the structural properties of the obtained samples. We adapt two drying techniques, which allows to produce soft samples with fully controlled porosity between 0 and 30 %. Then, we show one examples of a device capable of manipulating sound waves with a thickness much smaller than the wavelength   (typically   /5).

Synthesis of samples

The wet macroporous monolith samples were obtained by the UV polymerization of a water-in-Poly(dimethyl siloxane) (PDMS) emulsion as previously reported by our group [START_REF] Zimny | Soft Porous Silicone Rubbers as Key Elements for the Realization of Acoustic Metamaterials[END_REF][START_REF] Kovalenko | Tuning the sound speed in macroporous polymers with a hard or soft matrix[END_REF]. The continuous phase of the emulsions was obtained by mixing 12.45 g of silicone-based UV cross-linkable PDMS fluid (Silcolease UV Poly200™ from Bluestar Silicones) together with 90 mg of catalyzer (Silcolease UV Cata211™ from Bluestar Silicones), 10 mg of UV sensitizer (2-isopropylthioxanthone from Rahn AG) and 50 mg of surfactant (2-octyl-1-dodecanol from Aldrich). In case of regular and supercritical drying, water phase containing 1.5 wt. % NaCl was introduced under mechanical stirring with a home-made helical device. The size distributions of emulsions were characterized by static light scattering using a Mastersizer S apparatus (Malvern) and optical microscopy (Leica DM 2500P microscope). The emulsions were introduced into Teflon (PTFE) molds of various thicknesses and maintained between two quartz plates, to avoid water evaporation during the polymerization process and achieve samples with smooth surfaces. UV was applied for typically 5 min to each faces of the samples. For H 2 O 2 -assisted drying, the dispersed phase included various proportion of H 2 O 2 .

Drying of samples

Regular drying

In a recent study [START_REF] Kovalenko | Tailoring of the porous structure of soft emulsion-templated polymer materials[END_REF], we showed that the drying of porous samples could lead to collapse phenomena due to the negative pressure that takes place inside water droplets as the liquid leaks out of the materials during the drying step. Such collapse was only observed in systems where the inside water droplets do not stick to each other before the polymerization. In such case, the obtained polymerized materials is of the closed cell type, with no direct connections between the pores containing the water. In this case, the permeation of the water from the inside of the pores to the exterior occurs through the polymer walls and the materials collapse during drying. As we showed before, such a collapse is not observed when the porosity of the sample is connected through micrometric holes appearing between the pores during the polymerization process. One possible way to interpret this phenomenon is to consider that during the drying, the sample is submitted to a drying pressure:

P drying = 2/R ( 1
)
where  is the surface tension of water and R the typical size of the channels by which the liquid is permeating through the matrix. For connected droplets such as the ones obtained with aggregated emulsions, one gets R ~ 1 µm and P drying ~ 0.1 MPa. If the droplets are isolated in the material, water is permeating through the sample and the characteristic size is almost molecular and typically smaller than 1 nm, so P drying ~ 100 MPa. These values should be compared to the Young's modulus E 0 of the PDMS matrix (~ 7 MPa in our case) and explains why the pores collapse in the system with a closed porosity (P drying >> E 0 ) but not with the opened one (P drying << E 0 ). To avoid this phenomenon, we identified two different methods that will be presented below.

First, to further understand the underlying mechanisms of drying, we performed a complete experimental study of the drying kinetics and we propose a model to account for the obtained results.

Drying kinetics measurements

First, we examine the spontaneous drying kinetics of monoliths left in an oven at a given temperature , constant humidity). The monoliths have the shape of disks with diameter 32 mm and thicknesses varying between 1 and 5 mm. In order to simplify the geometry of drying, the edges of the disks were covered by a Teflon tape while their faces were freely exposed to the ambient hot air. The drying kinetics was acquired by regularly weighing the samples over time for a set of samples with varying water volume fraction (2 to 30%). To quantify the drying rate, we express the relative mass loss of water W:

(

where , stand for the mass of the samples at times , and infinity respectively. We also checked that the hygrometry of the oven is not altered by the release of water during drying by evaporating a small volume of pure water ( ) also regularly weighted over time; it leads to the linear decrease of the mass (data not shown) which is consistent, we believe, with a constant humidity. In all cases, the relative mass loss fits reasonably well a simple law , or equivalently:

(

where the final drying time is defined as . Experiments suggest that , see Fig. 1, which we will interpret below with a simple model.

To follow further the drying of the materials, we performed Scanning Electron Microscopy (SEM) observations (Hitachi TM-1000) of samples sacrificed during the course of drying: as time passes, we cut several slabs and observe their side-views. Interestingly, we observe the progression of a front that separates a zone near the outer surface

where the porosity has collapsed and an inner zone with preserved porosity, see Fig. 2. Also interesting is the shape of the collapsed pores after drying, which are more visible on Fig. 2 (Bottom) and can be compared to the one observed by Milner et al. [START_REF] Milner | Creasing in evaporation-driven cavity collapse[END_REF]. These authors studied the behavior of an individual water droplet initially stuck into PDMS matrices of various softness. As the water leaks out of the internal pore through the matrix, they observe various types of deformation of the PDMS material depending on the toughness of the matrix. In "tough" matrices (i.e. in their case, as the polymer shear modulus exceeds around ), they observe a deformation of the pore that is very similar to the one observed in our sample (see the Fig. 3 of their paper). In our case, the shear modulus was measured to be ) and we do observe the same kind of deformation.

Drying kinetics model

We now derive a very basic view of what is likely to happen during drying. In brief, the slab containing some volume fraction of dispersed water dries because the water permeates from the embedded drops to the surface of the slab in contact with an unsaturated atmosphere: the difference of chemical potential of water inside and outside the slab is the driving force of the permeation process. Yet, the kinetics is extremely complex because it involves describing the concentration field of water solubilized in the silicone everywhere in between the drop, in space and time, with droplets behaving as water source up to a point they cavitate and collapse. It is clearly beyond our goal and we circumvent such a hard problem with simplifying assumptions. First, we define in Fig. 3 the geometry of the slab by its thickness and assume the diffusion problem reduces to a one-dimensional problem along the dimension perpendicular to the faces of the slab. Then, we assume that the boundary condition (BC) at the surface of the slab is of fixed concentration of water in the polymer , in local equilibrium with the outer concentration of water in air. In principle, we should set a flux BC but basic estimates evidence that the mass transport of water is strongly limited by diffusion inside the slab, not outside, therefore justifying a concentration BC. Then, we assume that everywhere deep in the sample-in the wet zone, see Fig. 3the silicone is fully saturated with solubilized water at concentration provided by water droplets acting as sources; near the interface however, the permeation-induced drying process empties the drops. We therefore split the slab into a dry and a wet zone with a well-defined front, as indeed observed experimentally, see Fig. 2, and also confirmed by finite element calculations (not shown here). Consequently, the concentration field in the dry zone falls (about) linearly from saturation to surface concentration, and induces a drying flux:

(4)
with the diffusion coefficient of water in the silicone ( at room temperature). Now, we assume that the drying front evolves slowly (justified below), which sets a quasi-static model for the mass conservation and provides a kinetic equation for the propagation of the front. The latter moves because the drying flux pumps the water, the source being the drop which deliver a molar mass where is an equivalent concentration ( is the molar volume of liquid water, i.e., the ratio of molar mass and density of water:

).

Equating these two fluxes yields:

(5) assuming that at . We thus recover a diffusion equation for the drying front with an effective diffusion coefficient . We define the final drying time such as , leading to , which leads to a more compact writing: . It is worth mentioning that this quasi-static approach is not universal and requires the wet-to-dry interface to move slowly as compared to the time it takes for the solubilized water concentration field to equilibrate to a linear profile by diffusion in the dry zone [START_REF] Daubersies | Evaporation of solutions and colloidal dispersions in confined droplets[END_REF]. [START_REF] Dollet | Drying of channels by evaporation through a permeable medium[END_REF], we get . We believe the agreement is satisfying with our value because a higher temperature should imply a higher solubility along with a larger mobility of water molecules.

Supercritical drying

An alternative method to avoid the collapse of the materials upon drying one consists in eliminating the existence of the water/gas interface during drying by using supercritical CO 2 as performed by Martina et al. [START_REF] Della Martina | Siloxane Elastomer Foams[END_REF]. This technique is commonly used for the drying of silica to obtain aerogels [START_REF] Bisson | Drying of Silica Gels to Obtain Aerogels: Phenomenology and Basic Techniques[END_REF]. However, since liquid CO 2 is not miscible with water, water is progressively exchanged with ethanol by plunging the wet material in a water bath and slowly enriching the medium with liquid ethanol using a peristaltic pump. At the end of this exchange process, one obtains ethanol droplets embedded in the PDMS matrix. The material is then introduced in a tank reactor in which the pressure and temperature can be adjusted. The reactor is then slowly pressurized with CO 2 at a constant flow rate of 11 g/min and the pressure and temperature are adjusted to P = 11 MPa and T = 50 °C and kept constant during the drying process.

CO 2 mixes with ethanol resulting in a single supercritical phase in these conditions. Pure CO 2 being constantly introduced into the reactor, the ethanol is slowly extracted, and the system finally only contains supercritical CO 2 .

The pressure is then slowly decreased to 0.1 MPa (over 1 hour) and the temperature is decreased to ambient conditions. Thus, the system never crosses any phase transition lines and no liquid-gas interface is present in the system.

H 2 O 2 -assisted drying

The third method consists of introducing a proportion of hydrogen peroxide (H 2 O 2 ) in the dispersed water phase [START_REF] Bae | Fabrication of covered porous PLGA microspheres using hydrogen peroxide for controlled drug delivery and regenerative medicine[END_REF]. H 2 O 2 is spontaneously decomposed in H 2 O and O 2 (gas) and the nucleation of the gas bubbles inside the pores allows for a release of the negative pressure endured by the material during the drying process. As a consequence, as drying occurs, one expects the nucleated gas bubbles to expand and replace the liquid that permeates through the polymer matrix. However, the decomposition reaction of hydrogen peroxide in oxygen and water has a very low rate constant at room temperature and in the absence of any catalyst. As a consequence, the decomposition of H 2 O 2 does not occur during the emulsification process and only starts during the enlightening of the material with the UV lamp and the following drying step that is completed at 80°C. Indeed, kinetic measurements of oxygen production (not shown here) show that the rate constant of the reaction increases by more than one order of magnitude when increasing the temperature from 20°C to 80°C.

To observe the drying process, , the samples were UV polymerized and then rapidly put in a thermally controlled

Peltier microscope cell at a drying temperature of 80°C and observed at different times. Due to the difference in refractive indices of PDMS, water, and air, the drying phase fronts are visible, and the change in the drying front can be clearly observed. We performed these observations for both the OH457-H 2 O and OH457-H 2 O 2 systems. The optical microscopy pictures obtained at different times after the UV polymerization in these two cases are presented on Fig. 4.

At t = 0, the images were taken just after the sample's introduction in the Peltier cell at a temperature of 80 °C.

Initially, both samples appear very dark, which is due to the strong scattering of light induced by the difference of optical index between the water that is contained inside the droplets and the continuous PDMS phase. As time passes, a drying front is observed between a semi-transparent region that appears at the edge of the sample and moves towards the center. The red arrow shows the direction of propagation of this drying front. The undried region appears dark due to the light scattering by water-filled pores.

In both samples (and up to t = 10 min), the presence of the semi-transparent region indicates that the system becomes less scattering, which is due to the collapse of the pores as the water phase escapes the material. At t = 10 min, a difference of behavior is observed between the two systems: in the OH457-H 2 O 2 system, a dark zone starts to appear again at the edge of the sample, which indicates the reappearance of a strong scattering of light by the sample. This means that upon drying pores opens again in the first system but not in the case of the OH457-H 2 O, which still remains almost transparent up to 35 minutes after the beginning of the experiment. By contrast the OH457-H 2 O 2 system remains completely dark, which is due to the presence of air inside the pores. This means that, after a given amount of time (about 10 minutes in our case), the nucleation of O 2 gas bubbles occurs in the system, which leads to a release of this internal stress that applies to the pores during drying and the pores reopen to reach back their initial size and shape. Such method allows the system to be simply dried at room pressure while avoiding the step of exchange of water with ethanol. To achieve the targeted porosity, we optimized the concentration of H 2 O 2 and found out the concentration of 15% to be the optimal one to achieve the targeted porosity.

Characterization of the porous samples

The microstructures of obtained dried porous samples with the two above methods were again characterized by SEM. Fig. 5 (A) and (B) show the images of dried porous samples obtained with the H 2 O 2 -assisted (A) and the supercritical drying (B) techniques. The images are very similar and exhibit, in both cases, perfectly circular pores with a size distribution (Fig. 5 (C)) very close to the one of the initial emulsions. The sizes were obtained using ImageJ software for the analysis of the SEM pictures, while static light scattering experiment was performed to characterize the initial emulsion droplets size distribution. The porosity of the samples performed with the three drying techniques were then measured using a home-made setup measuring the Archimedes force acting on samples immerged in a water-glycerol mixture. The obtained values were more precise than the ones obtained by direct calculation from the mass and the dimensions of the samples. Fig. 5 (D) shows the final porosity (%) as a function of the dispersed water phase volume fraction (%) in emulsion for the three drying techniques. In the case of regular drying, a large difference between the final porosity of dried sample (that never exceeds 5% whatever the initial water volume fraction in the emulsion) and the dispersed water phase volume fraction in the initial emulsion is measured, which is a direct consequence of the pores collapse during drying, as observed in a previous study [START_REF] Kovalenko | Tailoring of the porous structure of soft emulsion-templated polymer materials[END_REF]. In contrary, supercritical and H 2 O 2 -assisted drying allow for a remarkable control over the samples porosity as evidenced by the data of Fig. 5 (D) where the porosity is roughly equal to the initial water volume fraction.

To determine the surface area of the samples, they were also all submitted to nitrogen adsorption measurements.

The BET analysis of our samples (not shown here) led to very low values of the specific surface area, i.e. less than 0.5 m 2 /g for all our samples, which is the detection limit of the apparatus. This means that these samples do not exhibit any microporosity (pores in the nanometer range). Thus, a rough calculation of the expected surface area for such samples can be made by considering the contribution from the macropores that were observed using SEM. In this case, the surface specific area S S of a sample of volume V tot , porosity and mass density  should be equal to:

Where V pore = 4/3  R 3 and S pore = 4 R 2 are the volume and the surface of one pore of radius R, respectively. For the samples with the highest porosity ( = 30 %) and having a radius R ~2.

Acoustic properties

Acoustic properties of the bulk materials

The experimental device used to measure the acoustic velocity in the samples consists of two identical piezoelectric transducers with diameters 30 mm (Olympus 301). One transducer serves as a source and the other one as a receiver. These are placed face to face on a manual device, graduated, linearly sliding for varying the distance between the transducers, thereby enabling an accurate measurement of the sample thickness. The transducers are broadband with a central frequency of 500 kHz. For each given porosity, the measurements are performed on two samples with thicknesses of 2 and 3 mm (with a constant diameter of 32 mm). Using a pulse generator (Olympus 5077PR-40-E), the transmitting transducer is excited by an electrical pulse of amplitude 400V with a repetition rate of 100 Hz allowing thereby the generation of a wideband acoustic signal. The acoustic signal transmitted and detected by the receiver is displayed on an oscilloscope (type Lecroy 9450A). The thickness of the sample is equivalent to the wave propagation distance. Because of the high dissipation of waves in the sample that is due to the macroporous nature of the materials, the excitation signals with frequencies centered around 500 kHz are totally absorbed before reaching the receiver. So, the receiver thus only detects the waves of lowest frequencies (normally 20 to 200 kHz), which are less dissipated than the one of highest frequency.

We then performed the acoustic characterizations of the obtained samples and measured the evolution of the longitudinal phase velocity (C L ) as a function of the materials porosity (Fig. 6). From this curve, the evolution of the acoustic refraction index (n) as a function of porosity ( ), was deduced as ratio of phase velocities in porous material and water (n = C L, material /C L , water ). As in previous papers [START_REF] Kovalenko | Tuning the sound speed in macroporous polymers with a hard or soft matrix[END_REF][START_REF] Ba | Soft porous silicone rubbers with ultra-low sound speeds in acoustic metamaterials[END_REF], we compared our data to theoretical predictions [START_REF] Kuster | Velocity and attenuation of seismic waves in two-phase: Part I. Theoretical formulations[END_REF], expecting the following equation to describe the behavior of C L as a function of porosity :

(7) (a) (b) (c)
Where C L (0), K 0 and G 0 are, respectively, the sound speed (≈ 1100 m/s) and the bulk isotropic and shear moduli of the pure non porous matrix. We measured G 0 ≈ 3 +/-1 MPa and estimate K 0 =  0 C L (0) 2 ≈ 1 000 MPa, where  0 is the mass density of the pure matrix (≈ 10 3 kg/m 3 ). Using such values and equation ( 7), we get the data plotted on Fig. 6, in reasonable agreement with our experimental results.

It clearly comes out that the velocity only depends on the material's final porosity whatever the used technique of drying. The curve also shows that porosity larger than 5% cannot be obtained using classical drying due to the partial collapse of the material. From these studies, we deduce that porous materials with very well controlled porosity and consequently very well-defined acoustic index can be obtained. In the next section, we use these materials to fabricate an acoustic device, as an example of what can be achieved with these materials.

Fabrication of the final GRIN samples

In a recent paper, we showed that such well-defined porous materials can be assembled to obtain gradient index (GRIN) acoustic metasurfaces [START_REF] Jin | Flat acoustics with soft gradient-index Metasurfaces[END_REF]. In this section, we briefly show how the samples were technically assembled and present one example of acoustic device as wave deflector.

On Fig. 7 (B), we present the sample that was used for beam deflection, which requires a linear gradient index along the X direction:

where X indicates the axis of the porosity gradient, = 5 degrees is the chosen deflecting angle and d = 2 mm is the sample thickness, We define the width of each stripe (interval in X) as 10 mm. Based on the results plotted on fig. 6, the porosity of each stripe is then determined.

The GRIN sample was obtained using the procedure described in fig. 7 (A), by layer by layer deposition of emulsions having required initial water volume fraction. Thanks to the immediate UV polymerization of each layer, no mixing of the emulsions was observed, and the porosity domains are very well separated. When the US transducer is covered by the linear GRIN metasurface, the wavefronts passing through this subwavelength coating are still planar but slightly tilted from the initially horizontal X-axis as shown in Fig. 8 (right). The measured angle of deflection is 5° in good agreement with the theoretical design of the metasurface index-profile.

These results inspire us to realize materials exhibiting spatial modulation of refractive index by modulating the porosity inside the materials. We explore this opportunity to the realization of the devices that are referred to as "soft acoustic gradient index metasurfaces", which can be used to focus, steer, or to create acoustic vortex beam.

These results are discussed in detail elsewhere [START_REF] Jin | Flat acoustics with soft gradient-index Metasurfaces[END_REF].

Conclusion

Using an emulsion templating method coupled to different drying techniques, we show that obtaining soft porous materials with closed cells geometry and controllable values of their porosity can only be obtained using specific drying techniques. The regular air drying of these materials with no specific precautions leads to collapsed materials, in agreement with previous results [START_REF] Kovalenko | Tailoring of the porous structure of soft emulsion-templated polymer materials[END_REF][START_REF] Milner | Creasing in evaporation-driven cavity collapse[END_REF] with uncontrolled values of their porosity. By contrast, and in agreement with results from the literature [START_REF] Della Martina | Siloxane Elastomer Foams[END_REF][START_REF] Bae | Fabrication of covered porous PLGA microspheres using hydrogen peroxide for controlled drug delivery and regenerative medicine[END_REF], drying using supercritical CO 2 or with systems containing H 2 O 2 in the cells lead to materials with fully controlled porosity. In these two cases, we show that the material final porosity is equal to the initial water content in the emulsion.

When the samples are dried in atmospheric conditions (regular drying), our data show that the drying kinetics obeys a simple law of the type:

in good agreement with the model that is developed in this paper. The required drying time is also shown to scale as the square of the sample thickness and linearly with the initial water volume fraction, which is also depicted by our model.

Using an adapted ultrasound acoustic setup, we show that the longitudinal wave acoustic velocity in the porous samples, and consequently the materials acoustic index, depend critically on their porosity, following a wellestablished model [START_REF] Kuster | Velocity and attenuation of seismic waves in two-phase: Part I. Theoretical formulations[END_REF]. By a spatial control of the porosity distribution, along the lateral dimensions of the samples, we show that it is possible to obtain gradient index acoustic metasurfaces allowing acoustic wavefront shaping while being much thinner than the incident wavelength. The procedure is simple and easily up-scalable, adaptable to any types of acoustic transducers and exhibits a unique versatility in terms of wavefront shaping. The obtained devices are broadband and active for acoustic waves travelling in water.

Fig. 1

 1 Fig. 1 collects the two main sets of data concerning the drying kinetics: First (Fig. 1, top left) a drying kinetics at

Figure 1 :

 1 Figure 1: (Top Left) Example of relative mass loss against time for four different initial volume fractions of water dispersed in the silicone (slab of thickness ); the solid line is a fit according to the model , see text (eq. (3)), from which we extract the final drying time at which . The shaded zones indicate the error on the fit, i.e., on . (Insert). All eleven drying kinetics are

  Right) Final drying time against the volume fraction of dispersed water in silicone and a linear fit with . (Bottom Left) Relative water mass loss against time for four different thicknesses of the slab (volume fraction of dispersed water ); (Bottom Right) Final drying time versus thickness of the slab where the line is a fit with .

Figure 2 :

 2 Figure 2: (Top) SEM side-views pictures of slabs with thickness 2mm and

Figure 3 :

 3 Figure 3: Sketch of the model: side view of the slab, symmetrical with respect to the plane and initially filled with a water emulsion at volume fraction . Due to the permeation of water across the silicone to the interface in contact with unsaturated air, the drops empty up with time. We assume that there is a drying front at a position before which all drops are empty (dry zone) and after which they remain full (wet zone).

Figure 4 :

 4 Figure 4: In-situ optical microscopy observations of the drying kinetics of cross-linked emulsions stabilized with OH457 containing (left column) or not (right column) H 2 O 2 in the dispersed phase. The samples are maintained at T = 80°C, squeezed between two glass slides and drying occurs from the border of the sample towards its center following the direction of the red arrow. Scale bar: 200 µm

  5 µm (see Fig 5C) and a density of around 700 kg/m 3 , we thus obtain a very low specific surface area S S ~ 0.5 m /g, in good agreement with the BET results.

Figure 5 :

 5 Figure 5: Characteristics of the final porous materials: (A-B) SEM pictures of the final materials obtained with (A) H 2 O 2 -assisted drying (B) supercritical drying ; (C): Pore size distributions(PSD) obtained from the SEM pictures for supercritical drying (the same is obtained for H 2 O 2 -assisted drying, not shown here) and water droplet size distribution (DSD) from static light scattering experiment on the initial emulsions; (D): evolution of the materials final porosity as a function of initial water volume fraction of the emulsions for the three drying methods. The solid lines on (D) are simple guides to the eyes.

Figure 6 :

 6 Figure 6: Evolution of the phase velocity as a function of the porosity at a fixed frequency of 90 kHz for the materials prepared with the three different drying methods. Solid line: Calculation of the acoustic velocity using equation (7) in the text with K 0 = 1000 MPa and G 0 = 3 MPa.

Figure 7 : 5 4. 3 .

 753 Figure 7: (A) Schematic of the procedure used for the fabrication of GRIN metasurfaces. From top to bottom: a thin polymer mold (nylon of polylactic acid (PLA) for example) is fabricated using a 3D printer. Emulsions with chosen volume fractions are introduced in each compartment and the system is then polymerized and dried (using supercritical CO2). (B) The final material is obtained and exhibit acoustic indices, which are respectively for each stripe (from right to left in the picture): 2.18; 2.69; 3.06; 3.61; 4.16; 4.41; 4.69; 5.11; 5.68; 6.25; 6.53; 6.81; 7.04; 7.27; 7.5

Figure 8 :

 8 Figure 8: Measured pressure fields at 150 kHz coming from a large flat ultrasonic transducer with no sample (left), with the deflecting metasurface (right). The black bar indicates 1 cm.

Acknowledgements

We thank Frédéric Marchal and Tania Ireland from Elkem Company for fruitful discussions and for providing us with silicone rubber. Special thanks to Marco Faustini and Philippe Barois for fruitful suggestions and discussions. Special thanks to Marie-Anne Dourges, from Institut des Sciences Moléculaires -CNRS -Univ. of Bordeaux, for her help on BET analysis. This work was partially funded and performed within the framework of the Labex AMADEUS ANR-10-LABEX-0042-AMADEUS with the help of the French state Initiative d'Excellence IdEx ANR-10-IDEX-003-02 and project BRENNUS ANR-15-CE08-0024.