L F Zagonel

M Bäurer

A Bailly

O Renault

M Hoffmann

S-J Shih

D Cockayne

N Barrett
email: nick.barrett@cea.fr

Orientation dependent work function of in situ annealed strontium titanate

Keywords: photoelectron spectroscopy, XPEEM, STO, work function, grain orientation

We have used energy filtered x-ray photoelectron emission microscopy (XPEEM) and synchrotron radiation to measure the grain orientation dependence of the work function of a sintered niobium doped strontium titanate ceramic. A significant spread in work function values is found. Grain orientation and surface reducing/oxidizing conditions are the main factors in determining the work function. Energy filtered XPEEM looks ideally suited for analysis of other technologically interesting polycrystalline samples.

Introduction

Oxide ceramics such as SrTiO 3 (STO) are widely used for passive devices in high frequency systems such as filters and antennas, and also in active devices such as tuneable rf filters and phase switchers. STO based compounds are also used in grain boundary layer capacitors, in which conductivity differences increase the device performance. The electronic structure of strontium titanate can be changed by the introduction of dopants, e.g. changing it from insulator to metallic conductor. [1] Recently, polycrystalline STO has been under discussion in the semiconductor industry as a candidate material for DRAM memories. [2] STO is also important as a model system for more complex oxides. The solid solution STO-BaTiO 3 , for example, is used as a capacitor material. For applications in electronic devices, the contact between the ceramic and the electrode material plays an important role in device performance. [START_REF] Rhoderick | Metal-Semiconductor Contacts[END_REF] In particular, band alignment at the metal-oxide interface can determine the electrical properties of the device. [START_REF] Nagaraj | [END_REF] The work function difference at an electrode-insulator contact [5] can give rise to the formation of electrical double layers, especially important in miniaturized devices where the ratio of surface to volume is very large. Due to work function differences at the interface, a positive or negative interface charge may be expected, thus directly affecting the performance of field effect devices and the oxidation of the metallic contact [6]. For example, in YBaCuO/STO, excess electrons are trapped on the YBCO side of the interface. [7] Shifts in the band alignment can lead to variations in the spacecharge and thus in the interfacial electrostatic fields, which in turn influence the rate of diffusion of O 2- ions and the electrode oxidation rate at the interface.

The work function is sensitive to the surface chemistry and STO surface stoichiometry is orientation dependent. Thus, the work function is expected to depend on the crystal orientation of the grains. Vlachos et al. showed that heating STO(100) above 1300 K reduces the surface via the creation of oxygen vacancies, provoking significant changes in the work function up to 0.82 eV. [8] Maus-Friedrichs et al observe surface defects after annealing in UHV at 1000K, giving rise to a density of states at 1-2 eV below the Fermi level with an intensity of 1% of the oxygen 2p valence band emission. [9] In the literature, the work function of undoped STO (100) is given as 4.2 eV. [10] Similar values were reported for undoped and for 1at. % Nb doped material. [START_REF] Nagaraj | [END_REF] However, little data exists on the work function of sintered ceramics. [11] The work function may be measured using ultra-violet photoelectron spectroscopy (UPS) and a photon source such as He I. Applying a negative polarisation to the sample allows one to measure the spectral width between threshold and the Fermi level and thus deduce the work function, provided that the offset between the sample and spectrometer work functions is known. However, such UPS analysis will give, at best, an area averaged value for polycrystalline samples such as ceramics. In fact, different crystallographic grain orientations should give rise to a range of work function values. Furthermore, area averaged measurements cannot, by definition, distinguish contributions from the grain boundaries to the work function. In order to analyse single grain work functions it is necessary to combine spatial resolution of individual grains with the complete spectral information from UPS. Photoelectron emission microscopy with full energy filtering provides such a tool. If samples are sufficiently conducting, the work function is directly measured from the position of the photoemission threshold since the photoelectron energy E in PEEM is referenced with respect to the sample holder Fermi level E F . Thus, E-E F =E k + WF , where E k is the kinetic energy and  WF the sample work function. [12] The technique therefore seems well adapted for a quantitative analysis of the grain orientation dependence of the work function in sintered niobium doped STO ceramics.

First, we describe the sample preparation and measurement methods. Then, the scanning electron microscopy and electron back-scattering diffraction results are presented on the morphology and crystal structure of the ceramic. Area averaged UPS valence band spectroscopy gives a first insight into the work function behaviour. Finally, a quantitative energy-filtered XPEEM analysis provides the orientation and chemical state dependent grain work function.

Experiment

High purity strontium carbonate (99.9%, Sigma Aldrich), titania (99.9%, Sigma Aldrich) and niobium oxide (99.9%, ChemPur) powders were used to prepare polycrystalline Nb doped STO powder. After milling and calcination, the powder was uniaxially pressed into discs of 15 mm diameter and 5 mm thickness in a steel die and subsequently cold isostatically pressed at 400 MPa. The discs were heated to 1700 K at 20 Kmin -1 in a 95% N 2 +5% H 2 atmosphere and then kept at temperature for 20 h. After sintering the sample was quenched at more than 200Kmin -1 . Further details on the synthesis procedure have been published elsewhere. [1] The sample diameter was reduced to 10.5 mm and then cut into discs of 1 mm thickness allowing the analysis of the central part. Finally, the sample was polished down to 0.25 µm diamond paste (no chemical polishing was used). Once in the vacuum chamber (base pressure of 2×10 -8 Pa), the sample was annealed for 3 hours at 1073 K. This annealing procedure has already been shown to yield a clean, ordered surface in single crystal test samples oriented in the major directions.

The X-ray photoelectron emission microscopy (XPEEM) was carried out using an energyfiltered microscope (NanoESCA, Omicron Nanotechnology GmbH). The instrument was installed at the ID08 line of the European Synchrotron Radiation Facility (ESRF) which delivers soft x-rays over the 400-1500 eV photon energy range. The threshold measurements were carried out using h = 400 eV and an analyser pass energy of 100 eV, giving an overall energy resolution of 0.5 eV. All results were taken using a 35 m field of view (FoV). The lateral resolution estimated from a fit of the 16-84% intensity rise across a grain boundary is 0.25 µm.

After XPEEM experiments the same sample area was observed using Scanning Electron Microscopy (SEM) and the grain orientation was determined by electron back-scattering diffraction (EBSD). The EBSD measurements were carried out at the Materials Department of the University of Oxford. The microstructure was studied using a scanning electron microscope (FEG LEO 1525) at the CEA Saclay. High quality Kikuchi patterns were easily obtained without any further sample preparation. Reference marks placed on the surface allowed individual grains to be identified in the XPEEM experiments and in the EBSD observations, in order to determine the grain orientation for each work function measurement.

The area averaged electronic structure was studied with standard laboratory ultra-violet photoelectron spectroscopy (UPS) using He I light (h = 21.2 eV). The probed area was of about 1 mm 2 , thus averaging the grain orientation distribution. The electron energy analyser was set to a resolution of 50 meV and operated at normal detection angle. The photoemission threshold region was measured using -6 V sample polarization with respect to the analyzer.

Results and discussion

Morphology

Figure 1(a) shows the microstructure of the sample as determined by SEM. This is a backscattered electron image (secondary electron images showed no contrast), taken at an electron acceleration voltage of 15kV. The sintering in hydrogen atmosphere considerably inhibits the grain growth. As a consequence, the grain size is rather small compared to the average size observed on samples prepared in similar conditions but in air atmosphere. [1] A higher porosity is also observed again with respect to samples prepared in air.

Average valence band structure

The valence band of the STO ceramic is dominated by the O 2p band at a binding energy of ~ 7 eV, as shown in figure 2. Auger electron spectroscopy shows that annealing eliminates the surface carbon contamination, however, annealing can also generate oxygen vacancies. Comparison of annealing in oxygen and in ultra high vacuum (UHV) shows clearly the reducing influence of the latter. [13,14] Features at low binding energies are present in the bulk gap due to oxygen vacancies induced by UHV annealing.

[15] Thus, we expect that those grain orientations with oxygen rich surface terminations will be likely to show more important variations with respect to the nominal work function for stoichiometric surfaces. In a careful UPS study of single crystal STO (100) surfaces prepared under reducing and oxidising conditions, Aiura et al [16] observe a sharp metallic peak at the Fermi cut-off in the --X direction with a resonant behaviour. This metallic peak is well off resonance for He I radiation, furthermore it is unlikely that one of the STO grains in our ceramic sample is [001] oriented with the same angular precision as that employed in UPS. However, we do observe a non-zero density of states in the bulk band gap, with similar intensity to that reported by Kido et al. [15] and Aiura et al. [16] Figure 2. Area averaged UPS spectra of the STO ceramic as a function of surface treatment. In black, the spectrum after annealing at 1073 K for 30 minutes in 210 -8 mbar O 2 (36 Langmuir). In red, the spectrum after annealing for 90 minutes in UHV. The inset shows a zoom on the region around the Fermi level. The density of states in the bulk band gap between the oxygen 2p and the Fermi level of the oxidized surface points to residual carbon contamination on the STO grains or due to the porosity, whereas the much higher density of states and the clear Fermi level cut-off of the reduced surface suggests the presence of oxygen vacancies at the surface of the grains.

Work-Function

The work function was determined for 62 grains using XPEEM just after an in situ UHV annealing in UHV at 1073 K, i.e. in the same conditions as for the "reduced surface" of Fig. 2. For this measurement a series of images was taken by sweeping E-E F through the photoemission threshold region from 3 to 10 eV in 0.05 eV steps. One example of such an image is show in Figure 3 2 eV is due to the band pass window of the energy analysis (0.5 eV). The grain spectra are extracted from the image series by defining an area of interest within each grain. After correction for the Schottky effect due to the high extractor field, E=98 meV for 12 kV, [17] the position of the photoemission threshold corresponds to the work function. This is a calculated correction which assumes that the Nb doping makes the STO sufficiently conducting to provide a metal like image potential; the actual values may be a few tens of milli-electronvolts higher. Secondly, one would expect rumpling for the grains with a mixed oxygen-cation termination, [18] leading to an outwards relaxation of the oxygen atoms, hence further compensating the Schottky effect. On the other hand, purely cation terminated surfaces should enhance the work function reduction. In both cases, it seems reasonable to treat the reported work function values as lower limits.

The opening angle  of the energy analyser gives rise to a parabolic energy dispersion or isochromaticity, in the vertical plane as a function of position in a given FoV as shown in figure 3(b).

The isochromaticity correction to the E-E F scale in the image series is obtained by integrating the image intensities over the FoV for constant y values and includes a vertical offset due to the instrumental alignment, apparent in figure 3(a) where the upper part of the FoV is brighter than the lower part. The linear increase of the STO grains work function as function of the orientation angle should be treated as an overall consequence of surface layer structural and chemical changes which determine the surface electronic properties. Each grain surface will have its own particular termination (possibly with a reconstruction) and stoichiometry as observed for different single crystals. [6,[19][20][21] In an XPEEM study of the surface termination chemistry of the same ceramic, we concluded that the [100] The transition in work function values between neighbouring grains is smooth, making the existence of a distinct phase within the grain boundary unlikely. For the area averaged UPS work function measurement such a contribution would in any case be negligible since the grain boundaries account for less than 1% of the analyzed area. With the spatial resolution of XPEEM the GB contribution could attain 5-10% of the extracted signal. In the extreme case of two distinguishable work functions one would expect structure in the extracted threshold spectra, as we have already observed on single gold-covered Si nanowires [12] or a significant broadening of the standard deviation in the error function fit to the rise in the photoemission signal. This is clearly not the case; thus we can exclude the grain boundaries as the source of the difference between the work function measured by XPEEM and that measured a posteriori by area averaged UPS. As in the case of single crystal STO studies, [START_REF] Nagaraj | [END_REF]10] it is the oxidizing/reducing conditions during the surface preparation and the chemistry of the terminating layer of each grain which play the dominant roles. The transition region between the two grains is 0.30-0.35 m wide, slightly larger than the spatial resolution (0.25 m). Thus, we cannot exclude the existence of a concentration gradient in charged defects due to a space charge region around the grain boundary. However, visualizing the space charge is outwith the spatial resolution used in these experiments. Nevertheless, there is a wealth of microscopic information well beyond a classical area averaged UPS analysis, underlining the importance of being able to carry out high resolution spatially resolved electron spectroscopy on a significant field of view of the sintered ceramic.

Conclusions

We have used fully energy filtered X-ray photoelectron emission microscopy and synchrotron radiation to measure the gain orientation dependent work function  WF of a niobium doped STO sintered ceramic. Photoemission threshold spectra are extracted from 62 distinct grains. The mean work function value is 4.26 eV with a standard deviation of 0.06 eV. The grain orientation is determined by electron back-scattered diffraction and correlated with the XPEEM work functions. A good linear correlation is obtained between  WF and the polar EBSD angle . In particular, the work functions determined for the three principal crystal faces [100], [110] and [111] are 4.13, 4.32 and 4.34 eV, respectively. The oxygen rich SrO terminated [100] face is particular sensitive to the surface preparation conditions used in-situ. In our work, the UHV annealing has produced surface defects, probably oxygen vacancies, evidenced by the existence of a non-zero density of states in the bulk band gap, as measured by a posteriori UPS. Further confirmation of the reduced nature of the XPEEM analyzed surface is obtained from the area averaged work function of 4.48 eV as determined by UPS on the ceramic after oxidation. Energy filtered XPEEM thus appears to be a powerful tool for quantitative analysis of spatially varying work functions in technologically interesting polycrystalline materials.

It would be particularly interesting to combine in a single experiment spatially resolved work function and chemical state analysis with either grain sensitive photoelectron diffraction an/or low energy electron microscopy (LEEM), in order to correlate the finer aspects of the links between surface electronic and atomic structure.

Figure 1 .

 1 Figure1(a) shows the microstructure of the sample as determined by SEM. This is a backscattered electron image (secondary electron images showed no contrast), taken at an electron acceleration voltage of 15kV. The sintering in hydrogen atmosphere considerably inhibits the grain growth. As a consequence, the grain size is rather small compared to the average size observed on samples prepared in similar conditions but in air atmosphere.[1] A higher porosity is also observed again with respect to samples prepared in air. Figure1(b)shows the EBSD results from the same region of the sintered STO ceramic, the measured grain orientations follow the colour code in the inset.

  (a), recorded at the very beginning of the photoemission threshold with E-E F = 3.85 eV. Darker grains indicate higher values of work-function while brighter grains have lower values of work-function. The clear presence of an energy distribution of photoelectron emission around the generally accepted work function values of 4.

Figure 3 (

 3 c) shows an example of the fit to the corrected experimental data using an error function to simulate the rising edge of the photoemission threshold. The uncertainty in the error function fits was ~ 0.01 eV. The average work function extracted from all 62 grains was (4.26 ± 0.06 eV). Thus the spread in work function values as a function of grain orientation is clearly significant with respect to the uncertainty in the fits to the individual threshold spectra.The EBSD data of figure 1(b) can be represented in terms of polar () and azimuthal () angles. No correlation between the extracted work function values from XPEEM and  was found. However, there is a high linear correlation (0.85) between  and the work function, as can be seen from figure4(a). In particular, the work function increases as the grain orientation passes successively through the three principal crystallographic directions <100>, <110> and <111>. Figure 4(b) is a histogram showing the number of grains observed in each 0.05 eV work function interval. As indicated, the average value is 4.26 eV with a standard deviation of 0.06 eV while grains were observed with work functions ranging from 4.10 eV up to 4.35 eV. The median work function is observed in the 4.25-4.30 eV interval and the entire histogram is skewed, with a longer tail at low work function values. The work function obtained from the area averaged UPS after annealing in UHV was 4.34 eV. The small difference with the mean XPEEM measured value of 4.26 eV may be due to slightly different reducing conditions, or possibly to the approximation used to correct for the Schottky effect on the measured work function

Figure 4 .

 4 Figure 4. (a) Experimentally determined work function as a function of the  angle as defined by electron back-scattering diffraction. The error bars correspond to a standard deviation of 0.06 eV for the error function fit to each photoemission threshold spectrum. The red line is the best linear correlation between work function and , with an R-value of 0.85. (b) Histogram of the distribution of work function values for the 62 analysed grains. The mean value is 4.26 eV.

Figure 3 .

 3 Figure 3. (a) Raw data image acquired at the photoemission threshold (E-E F = 3.85 eV) with a field of view 35 m. Due to the energy resolution of 0.5 eV some grains already emit photoelectrons. The isochromaticity in the dispersive plane of the analyser is apparent in the upper, lighter, part of the figure. Grains G1 and G2 are used for the grain boundary analysis below. (b) Schematic showing the dispersive plane in the NanoESCA giving rise to a parabolic correction to the FoV energy dispersion applied to all images before spectral extraction. (c) Typical threshold spectrum extracted from a 150 nm × 150 nm area of interest and the best least squares fit using an error function.

Figure 5 .

 5 Figure 5. Work function obtained from least square error function fit to the extracted threshold spectra from 150 nm wide regions at 170 nm intervals across the grain boundary G1/G2 shown in figure 3(a). The maximum error bar in the work function value is ±0.01 eV

Acknowledgments

This work was supported by the European commission under contract Nr. NMP3-CT-2005-013862 (INCEMS) and by the French national research agency project 05-NANO-065 XPEEM. The authors thank Patrick Bonnaillie for his skilled operation of the SEM and the ESRF-ID08 beam staff for their valuable technical assistance. We also thank B. Krömker for useful discussions on the intrinsic energy dispersion in the PEEM field of view.


Yvan Dossmann

Alexandre Paci

Francis Auclair

Mathieu Lepilliez

Emmanuel Cid

Topographically induced internal solitary waves in a pycnocline: Ultrasonic probes and stereo-correlation measurements

Internal solitary waves (ISWs) are large amplitude stable waves propagating in regions of high density gradients such as the ocean pycnocline. Their dynamics has often been investigated in two-dimensional approaches, however, their three-dimensional evolution is still poorly known. Experiments have been conducted in the large stratified water tank of CNRM-GAME to study the generation of ISWs in two academic configurations inspired by oceanic regimes. First, ultrasonic probes are used to measure the interfacial displacement in the two configurations. In the primary generation case for which the two layers are of constant density, the generation of ISWs is investigated in two series of experiments with varying amplitude and forcing frequency. In the secondary generation case for which the lower layer is stratified, the generation of ISWs from the impact of an internal wave beam on the pycnocline and their subsequent dynamics is studied. The dynamics of ISWs in these two regimes accords well with analytical approaches and numerical simulations performed in analogous configurations. Then, recent developments of a stereo correlation technique are used to describe the three-dimensional structure of propagating ISWs. In the primary generation configuration, small transverse effects are observed in the course of the ISW propagation. In the secondary generation configuration, larger transverse structures are observed in the interfacial waves dynamics. The interaction between interfacial troughs and internal waves propagating in the lower stratified layer are a possible cause for the generation of these structures. The magnitude of these transverse structures is quantified with a nondimensional parameter in the two configurations. They are twice as large in the secondary generation case as in the primary generation case.

I. INTRODUCTION

Internal solitary waves (ISWs) are nonlinear internal gravity waves commonly observed in geophysical flows. They propagate in regions of strong density vertical gradients that are present in the atmosphere (Tsai, Kanamori, and Artru, [START_REF] Tsai | The morning glory wave of southern California[END_REF] Rottman and Grimshaw 2 ) and in the ocean (Halpern, [START_REF] Halpern | Semidiurnal internal tides in Massachusetts bay[END_REF] New and Pingree [START_REF] New | Large-amplitude internal soliton packets in the central Bay of Biscay[END_REF] ). Their remarkable stability is due to the balance between nonlinear effects and dispersive effects induced by nonhydrostaticity and rotation (Helfrich and Melville 5 ). a) yvan.dossmann@anu.edu.au

In the ocean, these waves are trapped at the bottom of the oceanic mixed layer, namely the pycnocline, where they can propagate horizontally over hundreds of kilometers before decaying (New and da Silva [START_REF] New | Remote-sensing evidence for the local generation of internal soliton packets in the central Bay of Biscay[END_REF] ). Hence, they play a role in the dynamical and biological properties of the mixed layer (Lai et al., [START_REF] Lai | Impact of high-frequency nonlinear internal waves on plankton dynamics in Massachusetts bay[END_REF] Kantha and Clayson 8 ). To describe the role played by ISWs on the mixed layer properties, it is necessary to better understand their generation process and their dynamics.

At least two types of ISWs generation have been observed in the ocean. First, they can be induced by the direct interaction between an oscillatory tidal flow and a steep topography, such as the continental shelf in the Celtic Sea (Halpern, [START_REF] Halpern | Semidiurnal internal tides in Massachusetts bay[END_REF] Pingree and Mardell [START_REF] Pingree | Solitary internal waves in the Celtic sea[END_REF] ) or Pearl Bank in the Sea of Sulu (Apel et al. [START_REF] Apel | The Sulu sea internal soliton experiment[END_REF] ). This interaction provokes large amplitude vertical motions in the pycnocline above the topography. Due to the nonlinearity/dispersion balance, these vertical motions organize in trains of propagating ISWs. The term "Primary generation" and ISW1 is used in the remainder of the paper to refer to this generation mechanism and this ISW type, respectively.

Using observational measurements in the Bay of Biscay, New and Pingree [START_REF] New | Large-amplitude internal soliton packets in the central Bay of Biscay[END_REF][START_REF] New | Local generation of internal soliton packets in the central Bay of Biscay[END_REF] showed the existence of a different ISWs generation mechanism occurring 150 km away from the continental shelf. By tracking the path followed by internal waves in the lower layer, they concluded that ISWs are generated when internal wave beams emitted at the shelf impinge on the pycnocline. The same mechanism has been observed near the Sofala shelf in the Mozambique Channel (da Silva, New, and Magalhaes [START_REF] Da Silva | Internal solitary waves in the mozambique channel: Observations and interpretation[END_REF] ) and near the Ortegal Promontory in the Southern Bay of Biscay (Azevedo, da Silva, and New [START_REF] Azevedo | On the generation and propagation of internal solitary waves in the southern Bay of Biscay[END_REF] ). We adopt the terms "Secondary generation" and ISW2 to refer to this mechanism and this ISW type, respectively. This mechanism is also referred to as "local generation" in the literature.

These two generation mechanisms, and the dynamics of the associated ISWs, have been extensively described in two-dimensional configurations. In particular, the analytical studies of Gerkema, [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF][START_REF] Gerkema | Development of internal solitary waves in various thermocline regimes: A multi-modal approach[END_REF] and Maugé and Gerkema [START_REF] Maugé | Generation of weakly nonlinear nonhydrostatic internal tides over large topography: A multi-modal approach[END_REF] extracted important differences in the dynamics of ISW1s and ISW2s. Using a normal mode approach, they observe that ISW1s propagate in a stable unimodal manner. This stable dynamics was observed in 2D direct numerical simulations of the primary generation process (Dossmann, Auclair, and Paci [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Primary generation and topographic control[END_REF] ).

On the contrary, ISW2s are subject to more important energy exchanges with other normal modes while they propagate. These energy transfers lead to a downward leaking of energy from the propagating ISW2 to the stratified lower layer (Gerkema, [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF] Akylas et al. [START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF] ). The internal wave beam/pycnocline interaction and the subsequent ISW2 propagation and decay were observed in 2D direct numerical simulations using a forcing by internal wave beams with an analytical shape (Grisouard et al. [START_REF] Grisouard | Generation of internal solitary waves in a pycnocline by an internal wave beam: A numerical study[END_REF] ) or generated over a topography (Dossmann,Auclair,and Paci 20 ).

Selection criteria relying on the topography shape for the primary generation process and on the internal wave structure in the secondary generation process have been proposed. The reader can refer to Dossmann, Auclair, and Paci [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Primary generation and topographic control[END_REF][START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Secondary generation and spatial selection criteria[END_REF] for an extensive description of these criteria and complementary references about these two processes.

Laboratory experiments have permitted to study the dynamics of ISWs in real flows. Delisi and Orlanski [START_REF] Delisi | On the role of density jumps in the reflexion and breaking of internal gravity waves[END_REF] performed the first experimental study of the interaction of an internal wave beam with the pycnocline. They observed that large interfacial displacements are induced by the internal wave beam impact in a moderate pycnocline regime. 2D Synthetic Schlieren and PIV measurements have been used to characterize the linear scattering of the impinging internal wave ray on the pycnocline (Mathur and Peacock [START_REF] Mathur | Internal wave beam propagation in non-uniform stratifications[END_REF] ) and the effects of a shear in the pycnocline (Wunsch and Brandt [START_REF] Wunsch | Laboratory experiments in internal wave interactions with a pycnocline[END_REF] ). Mercier et al. [START_REF] Mercier | Soliton generation by internal tidal beams impinging on a pycnocline: Laboratory experiments[END_REF] combined the use of 2D PIV measurements and ultrasonic probes aligned in the tank axis to describe the second generation process.

Recent experimental developments have been used to study the three-dimensional (3D) structure of internal waves in quasi-linear regimes using stereoscopic PIV technique (Ghaemsaidi and Peacock [START_REF] Ghaemsaidi | 3D stereoscopic PIV visualization of the axisymmetric conical internal wave field generated by an oscillating sphere[END_REF] ) and tomographic synthetic Schlieren (Hazewinkel, Maas, and Dalziel 26 ). Lacaze et al. [START_REF] Lacaze | Wave patterns generated by an axisymmetric obstacle in a two-layer flow[END_REF] used a stereo-correlation technique to study the three-dimensional structure of interfacial waves generated by an axisymmetric obstacle. In a nonlinear regime, Maxworthy [START_REF] Maxworthy | A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge[END_REF] qualitatively described the structure of ISWs generated over a three-dimensional ridge in a horizontal plane using photographies. Structural variations seemingly caused by the ridge shape were observed in the direction transverse to the flow.

It is necessary to provide tools to quantitatively describe the three-dimensional structure of propagating ISWs. It is a step towards a better localization and quantification of the mixing induced in the pycnocline by the ISW propagation and breaking. Moreover, these results can provide insights to assess the need of three-dimensional simulations to model the dynamics of ISWs.

The present article aims at describing the three-dimensional dynamics of ISW1s and ISW2s in laboratory experiments. It relies on measurements of local interfacial displacements and of the three-dimensional structure of the interfacial displacement. This work is in continuity with numerical studies of the primary and secondary generation processes (Dossmann, Auclair, and Paci [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Primary generation and topographic control[END_REF][START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Secondary generation and spatial selection criteria[END_REF] ). While these studies inspired the choice of physical parameters in the present configurations, regimes of higher nonlinearity are expected in the laboratory experiments. In fact, the experimental ridge shape is 8 times steeper than the numerical one for similar amplitude forcings.

Ultrasonic probes are used to study the ISWs local dynamics in two series of experiments in the primary generation configuration, one with varying amplitude and the other with varying forcing frequency. In the secondary generation configuration, probes are disposed longitudinally to describe the different steps of the ISW dynamics and confront them with previous numerical and experimental studies.

Stereo correlation measurements are used for the first time to capture the dynamics of ISWs in a horizontal field, for the two processes. This experimental configuration allows to quantitatively estimate the magnitude of transverse effects in the dynamics of ISW1s and ISW2s, and to provide hints to understand their origin better.

The paper is organized as follows. Section II introduces the experimental configuration and a validation of the stereo correlation technique. Results issued from local measurements of the interfacial displacements for the primary and secondary generation configurations are respectively presented in Secs. III and IV. The three-dimensional structure of interfacial waves in the two configurations is discussed in Sec. V. Conclusions are drawn in Sec. VI.

II. EXPERIMENTAL SETUP

A. Experimental arrangement

The experiments have been carried out in the large stratified water flume at the geophysical fluid dynamics laboratory of CNRM-GAME (Météo-France and CNRS) in Toulouse (France). This facility has been used as a 22 m long, 1 m high, 3 m wide closed glass tank for the present experiments. The stratification is controlled by the salinity, while the laboratory air temperature is regulated at 20 • . Water is supplied in the water tank by two pumps connected to two reservoirs, respectively filled with freshwater and brines. The flow of each pump is controlled by a computer to obtain a given density at a given depth.

Two types of stratification have been used in the experiments. For the primary generation experiments (designed by PG0XX, with XX the experiment number), a two-layer stratification, with constant densities (ρ 1 , ρ 1 + ρ) and depths (h 1 , h 2 ) in the upper and lower layers, respectively, is used in a fluid of total depth H = 40 cm, as shown in Figure 1(a). The density profile is measured with a densitymeter AntonPaar that measures the density at a given depth with a precision of 0.5 kg/m 3 . The term pycnocline refers to the thin layer of high vertical density gradient observed in Figure 1(a) that supports the propagation of quasi-interfacial internal waves. The pycnocline center is located at z = -h 1 = -10 cm, where z is the vertical coordinate directed upwards, with the origin at the free surface. Its typical thickness varies from δ p = 2.5 ± 0.5 cm for the largest density jumps ( ρ = 90 kg/m 3 , PG038 to PG045) to δ p = 3.5 ± 0.5 cm for the smallest density jumps ( ρ = 26 kg/m 3 , PG046 to PG050).

For the secondary generation experiments (designed by SG0XX, with XX the experiment number), a two-layer stratification is also used in a fluid of total depth H = 40 cm or H = 80 cm, but this time the density varies linearly in the lower layer, while the upper layer remains homogeneous, as displayed in Figure 1(b). The density jump in the pycnocline, located at z = -h 1 = -4 cm varies between ρ = 28 kg/m 3 (SG014 to SG017) and ρ = 50 kg/m 3 (SG010 to SG013), while the Brunt-Väisälä frequency N 0 = √ -g/ρ 0 × d ρ/dz varies between N 0 = 0.93 rad/s (SG042 to SG053) and N 0 = 1.59 rad/s (SG005 to SG009). In both types of stratification, the water tank is filled from above using eight floating devices. The total flow of 10 m 3 /h used to fill the lower layer is reduced to 3-4 m 3 /h a few centimeters before reaching the pycnocline rest position, in order to limit the vertical extent of the pycnocline. Diffusive effects as well as mixing induced by propagating waves cause a progressive widening of the pycnocline in the course of the experiments. To control the sharpness of the pycnocline, density measurements are performed between series of two to three experiments. The water tank is emptied and refilled when the pycnocline thickness reaches a typical value of 4 cm. For a given stratification five or six experiments can be performed in one to two days.

The forcing is performed by the oscillation of a ridge of sinusoidal shape whose profile is given by

h(x) = 1/2h 0 1 + cos 2π x λ r , ( 1
)
where x is the horizontal coordinate with the origin at the center of the ridge base, h 0 = 10 cm is the ridge height, and λ r = 1.25 m is the ridge bottom width. The ridge spans 2.94 m in the transverse direction, while the tank width is 2.98 m. The ridge is suspended by four rods, attached on a mobile bench on rails. The mobile bench is linked to a motor forcing a harmonic back and forced motion at precisely controlled frequency and amplitude. Braces are used to prevent supplementary ridge transverse and longitudinal displacements. The ridge displacement is evaluated with a LASER telemeter positioned on one side of the water tank and aiming at a reflector placed on the mobile bench. The Fourier transform of the ridge displacement shows that the forcing is sinusoidal in all experiments, with no visible harmonic peaks.

To study the dynamics of interfacial waves propagating in the pycnocline, local acoustic measurements and three-dimensional optical measurements have been used.

B. Ultrasonic probes

Ten ultrasonic probes have been used to measure the mean pycnocline displacement z i (t) at a frequency of 200 Hz. These probes consist in an emitter-receiver of ultrasonic pulses, and a reflector at a fixed distance l = 15 cm from the emitter, as shown in Figure 2(a). The device measures the timelag θ for the pulse to travel on the path emitter-reflector-receiver, which expresses as

θ = 2( l -z i c 1 + z i c 2 ), (2)
for a pycnocline of zero thickness, where z i is the pycnocline displacement, and c 1 and c 2 > c 1 are the speeds of sound in the upper and lower layers, respectively. The probes deliver a voltage V s that is proportional to θ , and consequently to the relative distance between the emitter and the pycnocline. In the experimental case, the pycnocline is of finite thickness and the probe measures the mean vertical displacement between the emitter and receiver, which is slightly lower than the maximal vertical displacement in the pycnocline.

Probes are fixed on a bench during the measurements. To calibrate them, the bench is moved 1 cm up from the measurement position for 1 min, and then 1 cm down for 1 min using wedges, at the beginning of a series of experiments. The bench is then placed back to the measurement position before starting the experiment. The Voltage/displacement conversion is then carried out with the calibration measurements, as shown in Figure 2(b). Voltage outputs exhibit weak noise, and the relative uncertainty of the pycnocline displacement estimated at 2% is mainly due to the calibration step.

During the primary generation experiments, a bench with a "L" shape is used to support the probes as shown in Figure 3. Seven probes are aligned transversally in order to evaluate potential transverse effects in the pycnocline displacement. They are positioned at x = -3.58 m and y = 0.12; 0.34; 0.48; 0.56; 0.75; 0.82; 0.98 m where x and y are the longitudinal and transverse coordinates, respectively. x = 0 (resp. y = 0) is at the center of the ridge average position (resp. at the left edge of the tank in the direction of the wave propagation).

In the secondary generation configuration, eight probes are aligned in the tank axis at x = -0.23; -0.88; -1.68; -2.08; -2.53; -2.98; -3.43; -5.38 m and y = 1.49 m as indicated in Figure 3.

C. Stereo-correlation technique

This method has been developed to study the evolution of a sediment bed at the bottom of a hydraulic flow by recording and analyzing the structure of a regular pattern projected at the bottom (Astruc et al. [START_REF] Astruc | A stereoscopic method for rapid monitoring of the spatio-temporal evolution of the sand-bed elevation in the swash zone[END_REF] ). It has been adapted to determine the displacement of the pycnocline by observing a random pattern formed by particles seeded in the pycnocline, in experiments performed in the present water tank (Lacaze et al. [START_REF] Lacaze | Wave patterns generated by an axisymmetric obstacle in a two-layer flow[END_REF] ). The reader may refer to these two articles for a detailed description of this method, for which we recall the main steps here.

The principle of this method is to triangulate the positions of the particles seeded in the pycnocline, observed by two cameras PCO 2000 (4 MPixels) located on both sides of the water tank, and at a height of 4.10 m with respect to the pycnocline rest position. Cameras are equipped with Nikon Nikkor lenses with a focal length f = 35 mm and an aperture of 4. They are inclined with an angle of 20 • with respect to the vertical and the size of the observed field is approximately 3 m× 3 m. Cameras are first calibrated by recording several pictures of a test pattern at the bottom of the empty water tank, in different positions and inclinations. These pictures are used to determine internal parameters of the cameras, such as the focal length and the scale factor, and external parameters such as the position and inclination of one camera with respect to the other.

Before starting an experiment, the pycnocline is seeded with polystyrene particles. A surfactant is added in order to avoid the formation of particles aggregates at the free surface. Pictures are synchronously recorded during the experiments, using a low frequency generator sending a square wave at the sampling frequency to the two cameras. The sampling period varies between 200 ms and 500 ms so as to obtain 10-20 images pairs per forcing period.

Two steps are then necessary to determine the pycnocline displacements from experimental data. First, images are projected on a common plan to perform an unidimensional particle pairing, which is then used as a forecaster to pair the particles in the raw images. After this pairing step, particles positions are triangulated. Two experiments have been treated with the stereo-correlation technique, one in the primary generation configuration, and the other in the secondary generation configuration.

In the latter case, simultaneous ultrasonic and stereo measurements have been carried out to ensure the consistency of these techniques. For this purpose, the bench supporting the probes has been placed downstream, and just after the stereo field, in experiment SG019. The comparison between time series of the pycnocline displacement measured with one ultrasonic probe on the one hand, and at the closest position of the stereo field to the probe on the other hand, is shown in Figure 4. The two curves are similar, exhibiting periodic upwards and downwards motion with an amplitude of a half-centimeter due to the internal wave propagation in the pycnocline. A short lag of 1.5 ± 0.1 s is observed between the two data sets, with the stereo-correlation measurement slightly forward the probe measurement. It corresponds to the theoretical time taken by the wave to propagate over the distance of 20 cm between the probe and the position of the stereo-correlation measurement, calculated using the wave group speed of 0.15 m/s.

The correlation coefficient after lag correction and the mean difference between the two measurements are 0.89 and 1 mm, respectively. The stereo correlation technique is therefore sufficiently accurate to capture the dynamics of interfacial displacements on the order of centimetres.

Note that both techniques give access to mode-1 like displacements of the pycnocline. More complex pycnocline displacements are not accessible, however such dynamics has not been observed in video recordings in the present experiments.

In the remainder of the paper, experimental results relying on these two techniques are presented. Experimental parameters of the described experiments and the measurement techniques are recalled in Table I.

III. PRIMARY GENERATION

A. Nondimensional parameters

Nondimensional parameters introduced in previous works (Dossmann, Auclair, and Paci, [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Primary generation and topographic control[END_REF] Gerkema [START_REF] Gerkema | Nonlinear dispersive internal tides: Generation models for a rotating ocean[END_REF] ) are used to quantify the strength of nonlinear effects ( ), dispersive effects (δ), and the

= 4.5 A λ r h 0 H , ( 3
) δ = H ω c * 2 , (4)
and

λ nd = c * × T λ r = λ λ r , ( 5
)
with c * = g ρ ρ 0 h 1 h 2
H the linear longwave speed for interfacial waves in the case of a zero thickness pycnocline, ω = 2π/T the forcing pulsation, and λ = c * × T the theoretical linear longwave speed. In a given series of experiments, the typical nondimensional distance between two successive interfacial waves λ nd is varied via the forcing period T. Larger (smaller) wavelengths correspond to weaker (stronger) dispersive effects.

The length scale λ r /4.5 is used in the nonlinearity parameter . It is the e-folding width of a Gaussian ridge with the same maximum slope as the experimental ridge described by Eq. (1). Using this length scale, the parameter can be directly compared to the one used in simulations with a Gaussian ridge (Dossmann, Auclair, and Paci 17 ).

B. Strong dispersion

In series 9, the effect of an increasing nonlinearity parameter for a constant dispersion parameter is examined. Only weak transverse variations in the interfacial displacements are observed with the different probes, therefore the focus is put on the interfacial displacement measured by the probe located at x = -3.58 m and y = 0.48 m. In Figure 5 is shown the interfacial displacement for the four experiments of this series along with their corresponding Fourier Transform.

In all experiments of the series, the interfacial wave increases in amplitude in the two first forcing periods. The wave structure is then conserved over 15 periods. After that time, interactions between the emitted and reflected waves affect significantly the wave structure. The time window for the Fourier Transforms starts four periods after the interfacial wave reaches the probe and includes 10 periods in the steady regime.

The interfacial displacement has a similar evolution in the four experiments. About 2 s after starting the ridge oscillation, a periodic interfacial displacement on the order of millimetres reaches the probe. It corresponds with the propagation of surface waves at a speed √ g H ≈ 2 m/s. Three periods later, periodic pycnocline displacements on the order of centimetres are observed. This delay of three periods corresponds with the theoretical time of 18 s needed for the pycnocline wave to reach the probe, calculated from c * and the ridge-probe distance. The wave amplitude increases with the forcing amplitude from 2.0 ± 0.1 cm (PG031, = 0.28) to 4.7 ± 0.1 cm (PG034, = 0.71). The pycnocline waves are weakly asymmetrical, as the waves fronts are slightly steeper than the back of the waves. The Fourier spectra indicate that the pycnocline wave is overall sinusoidal, as the first harmonics contributes to less than 5% of the total signal, and higher order harmonics are hardly visible. In the two last experiments, interfacial waves are less regular in shape than for experiments with a smaller amplitude. This can be caused by wave breaking observed during the experiments.

As the waves amplitude are at the same order of magnitude as the pycnocline depth, nonlinear effects are at play in these experiments. The nonlinearity and dispersion parameters are at least one order of magnitude greater than the ones obtained in the study of Dossmann, Auclair, and Paci 17 for which trains of at least two ISW1s were observed (e.g., their Figure 3). In series 9, one stable interfacial wave is formed per forcing period. We infer that the strong nonlinearity of the waves is balanced by strong dispersive effects, preventing them to break into ISW1s trains. Regimes with weaker dispersion, for which ISW1s trains are likelier to be observed, are studied in Subsection III C.

C. Weak dispersion

In order to investigate the potential breaking of interfacial waves into trains of ISW1s, the study of the series 12 is carried out. In Figure 6 is shown the interfacial displacements for the six experiments in the series 12. The dispersion parameter decreases from Fig. 6(a) (PG045, δ = 1.49) to Fig. 6(f) (PG040, δ = 0.15), while the nonlinearity parameter is constant ( = 0.57).

In experiment PG045, a periodic asymmetric wave showing flat tops and narrow troughs is observed after a transient regime lasting 1 T. The troughs have a depth of 6.4 ± 0.1 cm, which is of the same order of magnitude of the upper layer depth (h 1 = 10 cm). When decreasing the dispersion parameter, the interfacial wave splits into periodic trains of two or three troughs ranked by decreasing depth as seen in PG044 (δ = 0.66) and PG042 (δ = 0.56). For these two experiments, the depth of the first trough is twice as small as in experiment PG045, while the second trough is typically 5 mm deep. Note that the number of troughs can vary after three forcing periods due to transient effects and interactions with reflected waves. The number of troughs in a train increases with decreasing dispersion parameter: three to four troughs are observed in PG043 (δ = 0.42) and PG041 (δ = 0.28) and four troughs are observed in PG040 (δ = 0.15). For these three experiments, the first troughs are typically 1 cm deep, and the following troughs in the train have depths on the order of millimetres.

Figure 7 shows the evolution of the depth of the first trough in a train with respect to δ. The wave symmetry is evaluated by measuring the trough depth with respect to the front and to the back of the trough. The trough depth increases monotonically with the dispersion parameter and it becomes more symmetrical.

D. Nature of the interfacial waves

In series 9 and 12, the amplitude of the wave or the depth of the first trough in a train are comparable to the pycnocline depth. Therefore, substantial nonlinear effects are at play in the propagation of interfacial waves. These effects are associated with a steepening of the interfacial wave that splits up into a trains of depth-ordered troughs in the experiments with relatively small dispersion parameter (PG040, PG041, PG042, PG043, PG044). Probe measurements carried out 1 m away in the direction of propagation and the stereo-correlation measurements presented in Sec. V showed that the train structure is overall conserved during the propagation. The balance between nonlinear and dispersive effects can explain this stability. In a qualitative way, these interfacial troughs propagate as ISW1s.

In experiments with larger dispersion parameter (PG045, PG031, PG032, PG033, PG034), we infer that dispersive effects are sufficiently strong to sustain the propagation of a periodic nonlinear interfacial wave. Its structure, consisting in a flatter top and a narrower trough, is similar to the one of the cnoidal waves described in Gerkema, [START_REF] Gerkema | Nonlinear dispersive internal tides: Generation models for a rotating ocean[END_REF] Chap. 1.2.3.

For series 12, we compare the number of observed troughs per train with the expected number of ISW1s N th , expressed in terms of and δ in Eq. ( 6), obtained by Gerkema [START_REF] Gerkema | Nonlinear dispersive internal tides: Generation models for a rotating ocean[END_REF] show the depth measurements with respect to the front (resp. to the back) of the wave. + show the depth average values. For each experiment, the first number is the number of ISW1s observed in a train. The second number is the expected number of ISW1s per train using Eq. ( 6).

an infinitely thin pycnocline

N th = 1 2 1 + 6a (1 -2α) α(1 -α)δ + 1 , (6)
with α = h 1 /H and a = 0.5, the typical ISW1 depth of 5 cm nondimensionalized by h 1 . Green numbers in Figure 7 stand for N th , whose round value corresponds with the theoretical ISW1s number in a train. The actual number of troughs in a train is in good accordance with N th , although it is slightly overestimated in the theoretical approach. Viscous effects and the pycnocline finite thickness, that are not accounted for in Gerkema's [START_REF] Gerkema | Nonlinear dispersive internal tides: Generation models for a rotating ocean[END_REF] theory, could explain this discrepancy.

In the numerical study of Dossmann, Auclair, and Paci, [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Primary generation and topographic control[END_REF] with an identical configuration except for the ridge steepness, similar trains are observed with a nonlinearity parameter one order of magnitude lower (e.g., Figure 6(b) and their Figure 3). These interfacial waves were identified as ISW1s by comparing their shape with outputs of a numerical solution of the Korteweg-de Vries equation.

The overall observed dynamics and the strong similarities with one analytical approach and the associated numerical study lead us to associate the propagating troughs with ISW1s in the present experiments. We choose to use this term to describe cnoidal waves as well, since their dynamics is also led by the nonlinearity/dispersion balance.

The typical ISW1s depths and nonlinearity parameters are 5 cm and = 0.5 in the present experiments and 2 cm and = 0.05 in the numerical simulations of Dossmann, Auclair, and Paci. [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Primary generation and topographic control[END_REF] Therefore, the primary generation process appears to be more efficient in the latter case. The mixing observed in the pycnocline above the ridge during the experiments could explain this efficiency decrease in highly nonlinear regimes. In fact, one substantial fraction of the forcing energy is directly transferred to the turbulent scales near the ridge, and is consequently not transferred to the propagating ISW1s.

IV. SECONDARY GENERATION A. Pycnocline strength

In the secondary generation configuration, the stratified lower layer can support the propagation of topographically generated internal wave beams. When the beam impinges on the pycnocline, it can induce large vertical displacements that can evolve into ISW2s. The efficiency of the secondary generation process is controlled by the "strength" of the density jump in the pycnocline with respect to the lower layer stratification. This quantity has been quantified using different nondimensional parameters in previous analytical approaches (Gerkema, [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF] Akylas et al. [START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF] , Delisi and Orlanski [START_REF] Delisi | On the role of density jumps in the reflexion and breaking of internal gravity waves[END_REF] ). The reader can refer to Dossmann, Auclair, and Paci 20 for a description of these parameters and their relation to the efficiency of the secondary generation process.

In the present article, we adopt the nondimensional parameter γ = c * /(N 0 H) introduced by Gerkema [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF] to describe the pycnocline strength, where c * is the linear interfacial longwave speed in the pycnocline and N 0 is the Brunt-Väisälä frequency in the lower layer. The generation of large amplitudes ISW2s is observed in regimes of moderate pycnoclines correspond to γ ≈ 0.1 (Grisouard and Staquet, [START_REF] Grisouard | Numerical simulations of the local generation of internal solitary waves in the Bay of Biscay[END_REF] Mercier et al., [START_REF] Mercier | Soliton generation by internal tidal beams impinging on a pycnocline: Laboratory experiments[END_REF] Dossmann, Auclair, and Paci 20 ) as predicted by analytical studies (Gerkema, [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF] Akylas et al. [START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF] ). In the seven series of experiments carried out in the secondary generation configuration, the stratification profiles correspond to moderate (γ = 0.15) to relatively strong (γ = 0.31) pycnocline strengths. The secondary generation process has been observed in this range of pycnocline strengths (Dossmann, Auclair, and Paci 20 ). For weaker density jumps, the pycnocline thickness is similar to the upper layer thickness, and cannot be considered as an quasi-interface separated the upper and lower layers. One selected experiment is presented in this section.

B. Longitudinal structure

Several steps are involved in the secondary generation process: the generation of large vertical displacements at the pycnocline by the beam impact, the subsequent propagation of ISW2s in the pycnocline and a progressive downward leakage of energy. To potentially observe these different steps, eight probes aligned in the direction of the forcing are used in the experiment SG033.

The total depth is twice as large as the depth in the primary generation configurations. It permits to reach a moderate pycnocline strength γ = 0.15 with a sufficient density jump. Moreover, as the ridge is located further away from the pycnocline, vertical displacements directly induced by the ridge oscillation are expected to be weaker than in the primary generation configuration. One can focus better on the secondary generation process.

Figure 8 displays the interfacial displacement and the corresponding Fourier Transform at each position. At x 1 , the ridge oscillation forces a quasi-harmonic interfacial displacement of millimetric amplitude. The pycnocline dynamics is strikingly different at x 2 , where periodic trains of steep troughs are measured. The number of troughs varies between 1 and 4 during the experiment. The depth of the first trough is 6 cm, and the following troughs, when present, are about 5 mm deep. In the Fourier spectrum, a strong signal at the forcing frequency is measured, and the four following harmonics have a significant amplitude. When propagating away from the ridge, the first trough progressively weakens from a depth of 4.7 cm at x 2 to 2 cm at x 7 , while the wave structure is overall conserved. The trough structure is hardly distinguishable at x 8 .

In order to assess whether the interfacial troughs can be associated with ISW2s, characteristics of the internal wave field in the lower layer are qualitatively estimated in a linear regime. As the Brunt-Väisälä pulsation is constant in the lower layer, linear internal waves propagate as upgoing and downgoing beams. The theoretical angle of the internal wave beams with respect to the horizontal θ r can be estimated using the dispersion relation ω = N 0 sin θ r , where ω is the forcing pulsation. One finds θ r = 31 for experiment SG033, an angle close to the maximal ridge angle θ m = 32. This angle matching corresponds to the near-critical regime of internal wave beams generation, for which large amplitude upgoing, and potentially downgoing, internal wave beams are emitted on either sides of the ridge (Dossmann et al. [START_REF] Dossmann | Simultaneous velocity and density measurements for an energy-based approach to internal waves generated over a ridge[END_REF] ).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: The initially upgoing internal wave beam propagating to the left of the ridge impinges on the pycnocline at x i, 1 = -0.75 ± 0.05 m. This position is found between x 1 , where the pycnocline is almost non-moving, and x 2 , where nonlinear propagating troughs are observed. The generation of these troughs is likely due to the internal wave beam impact on the pycnocline. They are ranked by decreasing depth and directed towards the thicker layer, which is characteristic of propagating ISWs (e.g., Dauxois and Peyrard [START_REF] Dauxois | Physics of Solitons[END_REF] ).

The secondary generation process is apparently at play in the experiment SG033. In this moderate pycnocline regime (γ = 0.15), a strong linear scattering of the internal wave beam is expected when it impinges on the pycnocline (Gerkema, [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF] Dossmann, Auclair, and Paci, [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Secondary generation and spatial selection criteria[END_REF] Mercier et al., [START_REF] Mercier | Soliton generation by internal tidal beams impinging on a pycnocline: Laboratory experiments[END_REF] Grisouard and Staquet [START_REF] Grisouard | Numerical simulations of the local generation of internal solitary waves in the Bay of Biscay[END_REF] ). This process can provoke the generation of large vertical displacements in the pycnocline that organize into ISW2s trains. In a similar (numerical) configuration, ISW2s were generated by the impact of a topographic internal wave beam on the pycnocline (Dossmann, Auclair, and Paci, [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Secondary generation and spatial selection criteria[END_REF] their Figure 3). The ISW2s amplitude was progressively decaying owing to a downward leaking of energy (Gerkema [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF] ), over a typical length scale of 1 m. An amplitude decay is also observed in the present experiment, over a similar scale of 2 m.

Overall, the dynamical properties of the propagating troughs permit to qualitatively identify them as ISW2s.

The evolution of the fundamental and the five first harmonics amplitudes with respect to the measurement position is plotted in Figure 9(a). The amplitudes are relatively small at x 1 , where the internal wave beam has not impacted the pycnocline. The largest value for the fundamental amplitude is reached just after the first impact of the internal wave beam on the pycnocline. After that, it progressively decreases when moving away from the first impact due to the amplitude decay.

If an initially downgoing internal wave beam is emitted, it is supposed to impact on the pycnocline at x i, 2 = -1.85 ± 0.05 m after reflecting at the bottom. However, no substantial evolution of the interfacial displacement is observed between x 4 = -1.68 m and x 5 = -2.08 m, apart from the amplitude decay. This dynamics suggests that the initially downgoing beam is either absent, or too weak to generate interfacial displacements on the order of centimetres. Similarly, there is no noticeable influence of the subsequent impacts of the initially upgoing internal wave beam on the interfacial displacement. It can be explained by the strong scattering of the beam at the first impact. The beam energy is spread over a larger distance and is therefore not as efficient to generate large interfacial displacements.

ISW2s observed at x 4 = -1.68 m are similar in shape to the one observed by Mercier et al. [START_REF] Mercier | Soliton generation by internal tidal beams impinging on a pycnocline: Laboratory experiments[END_REF] in moderate pycnocline regimes (their Figures 11(a) and 11(b)), apart from the ISW2 depth which is one order of magnitude larger in the present experiment. In their experiment with γ = 0.11, they observe a bulge in the Fourier spectrum of the interfacial displacement (their Figure 12(a)). It spreads approximately from 5ω 0 to 10ω 0 . This bulge is associated with time scales present in the ISW2s that are not necessarily harmonics. In their experiment with γ = 0.17, a weaker bulge is observed in the Fourier spectrum. The amplitude decrease with frequency is smaller after, than before the beam impact.

In the experiments performed in the secondary generation configuration, the ISW2s spectra are closer to the latter case. In fact, the amplitude decrease for short time scales (ω > 5ω 0 ) is smaller after the beam impact. The present experiments are associated with γ ≥ 0.15, while the bulge is observed by Mercier et al. [START_REF] Mercier | Soliton generation by internal tidal beams impinging on a pycnocline: Laboratory experiments[END_REF] for γ = 0.11. Hence, it could be present only in a short range of pycnocline regimes around the latter value.

V. STEREO-CORRELATION

The interfacial wave dynamics has been locally studied in the primary and secondary generation configuration using ultrasonic probes measurements. Stereo correlation measurements performed in the experiments PG052 and SG019 allow to assess and compare the importance of three-dimensional structures associated with these two processes. These two experiments have similar physical parameters to the ones of PG045 and SG033, respectively, for which ISWs trains have been measured with ultrasonic probes.

A. Mean longitudinal structures

First, the mean transverse isopycnal displacement η m (x, t) = η(x, y, t) y is examined in the two regimes and in the steady regime, with η(x, y, t) and y , respectively, standing for the interfacial displacement and transverse mean.

In the primary generation case (exp. PG052, Figure 10(a)), the propagation of a 6 cm deep ISW1 followed by a 2 cm deep ISW1 is observed. The first (deeper) trough propagates faster (0.25 ± 0.03 m/s) than the second (shallower) trough (0.14 ± 0.05 m/s), what causes a progressive separation of the troughs. The propagation speed of the first ISW1 is close to the theoretical value of 0.26 m/s for the linear propagation of an internal wave in an infinitely thin pycnocline. Apart from the progressive separation between the two ISW1s, the mean structure of the train is stable in the course of the propagation. In fact, the number of ISW1s per train is constant and the geometry of a given trough (depth and width) is conserved.

In the secondary generation case (exp. SG019, Figure 10 the troughs as they propagate. At the right of the field, a progressively decaying propagating trough is observed from t = 8.5 T to t = 9 T. Overall, the mean interfacial wave structure does not consist in shape-conserving propagating troughs trains that can directly be identified with ISW2s. In fact, the generation of ISW2s may not be as efficient as in the experiment SG033. The value of γ that partly controls the efficiency of the secondary generation process is slightly higher in the present experiment (γ = 0.21 versus γ = 0.15), what can explain a different interfacial dynamics in the two experiments. The interaction of internal wave beams with the pycnocline may generate large amplitude interfacial waves that do not organize into ISW2s trains. Energy transfers from interfacial waves to internal waves propagating in the lower, stratified layer are another explanation for these variations in the longitudinal structure of the troughs. In the case of a stronger pycnocline corresponding to a primary generation configuration, the ISW1s trains are less subject to such energy transfers and therefore conserve their shape.

The term trough, which is more general than ISW2, is used in the remainder of the paper to refer to the interfacial waves observed in exp. SG019. Stereo-correlation measurements with a larger field of view in the longitudinal direction could permit to identify their nature more thoroughly.

The study of the 3D structure of the propagating interfacial waves can provide better insights about their dynamics.

B. 3D structure

The time evolution of the interfacial displacement field η(x, y, t) is studied for the two generation processes.

In the primary generation case, the propagation of one large trough at a speed 0.25 ± 0.03 m/s is observed in Figure 11. Apart from millimetric variations, it has a constant structure in the transverse direction. A second shallower trough with a varying transverse structure is observed. In fact, it is only present in the middle part of the field (-225 cm <y < -75 cm) and vanishes close to the edges of the tank. These two depth-ranked troughs correspond to the ISW1s train characterized using the mean interfacial displacement. In the 1.5 m plateau separating two ISW1s trains, the interfacial displacement is overall constant in the longitudinal and transverse directions, except from millimetric variations.

The pycnocline displacement in the course of SG019 is shown in Figure 12. Troughs on the order of centimetres and propagating leftwards are observed (blue circles at t = 8.75 T). They are in the middle of the field at t = 8 T and t = 9 T, and correspond to the troughs observed in Figure 10(b). However, transverse variations of comparable magnitude are also present, what makes them harder to identify at all times. These transverse variations are observable in both the troughs and in the plateau separating them. They extend over typically 50 cm in the transverse direction. A space-time diagram at a given longitudinal position x = -150 cm permits to observe the progressive formation of transverse structures (Figure 13). At t = 3 T, a single trough showing millimetric variations in the transverse direction is observed. The magnitude of transverse variations in the interfacial displacement increases with time. It results in a progressive separation of the trough into several troughs of limited transverse extensions. Five troughs reaching the measurement position x = -150 cm at different times are observed between t = 8.5 T and t = 9 T. Such dynamics is not observed in the primary generation configuration: ISW1s conserve their shape and do not break up in smaller troughs.

C. Magnitude and sources of the transverse effects

In order to provide a quantitative evaluation of the transverse effects in the two regimes, the normalized mean transverse deviation E d = |η(x, y, t) -η m (x, t)| y /d is calculated, where d is the depth of the first trough of the train. One finds E d = 0.08 for exp. PG052, and E d = 0.14 for exp. SG019. The magnitude of transverse effects are twice larger in the secondary generation configuration than in the primary generation configuration.

Considering the role of the lower layer can help at understanding better why transverse structures are more substantial in the secondary generation case. As the lower layer is stratified, it can support the propagation of internal waves in various forms after the beam impact. First, an energy fraction of the incoming internal wave beam is reflected down in the form of a scattered beam after the impact (Dossmann, Auclair, and Paci, [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Secondary generation and spatial selection criteria[END_REF] Mathur and Peacock, [START_REF] Mathur | Internal wave beam propagation in non-uniform stratifications[END_REF] Grisouard and Staquet 31 ). Second, as interfacial troughs progress, they may transfer energy to internal waves propagating in the lower layer through a downward leaking process (Gerkema, [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF] Akylas et al. [START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF] ).

Nonlinear interactions between the pycnocline and the complex internal wave field in the lower layer can explain the enhancement of transverse effects in the interfacial wave structure as time advances.

Such interactions are limited in the case of a stronger pycnocline corresponding to a primary generation configuration. The oscillation of the ridge generate interfacial waves in the pycnocline, that organize into ISW1s trains due to the nonlinearity/dispersion balance. As the lower layer stratification is too small to support the propagation of internal waves, ISW1s are not likely to undergo a downward leaking of energy and propagate in a stable manner (Gerkema 14 ). As the topographic forcing does not vary in the transverse direction, the weak transverse structures observed are possibly induced by turbulent motions and the effects of viscosity close to the tank walls.

VI. CONCLUSION

Laboratory experiments have been used to study two configurations of internal solitary waves generation inspired by oceanic observations. First, the primary generation configuration where the fluid consists in two layers of constant densities separated by a sharp pycnocline has been studied. Measurements issued from one probe were used to observe the generation of large amplitude, stable trains of propagating troughs in various regimes of nonlinearity and dispersion. Comparisons with previous analytical and numerical studies permitted to associate them with ISW1s.

In the secondary generation configuration where the pycnocline of moderate strength lies on the top of a stratified layer, the interfacial dynamics has been studied using probes disposed longitudinally in one experiment. The generation of a train of troughs by the impact of the internal wave beam on the pycnocline has been observed. The propagating troughs exhibit the dynamics features of ISW2s, including an amplitude decay due to a downward leaking of energy (Gerkema, [START_REF] Gerkema | Internal and interfacial tides: Beam scattering and local generation of solitary waves[END_REF] Akylas et al., [START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF] Dossmann, Auclair, and Paci, [START_REF] Dossmann | Topographically induced internal solitary waves in a pycnocline: Secondary generation and spatial selection criteria[END_REF] Grisouard and Staquet [START_REF] Grisouard | Numerical simulations of the local generation of internal solitary waves in the Bay of Biscay[END_REF] ).

Recent developments of the stereo correlation technique have been used to investigate the three-dimensional dynamics of interfacial waves in the two configurations.

ISW1s trains are observed to propagate in a stable manner and are associated with weak transverse structures. On the contrary, the dynamics of the interfacial waves in the secondary generation regime is more complex. In fact, propagating troughs are observed, yet they are not clearly organized in trains of depth-ordered ISW2s. Transverse structures of magnitude on the order of centimetres are observed while they propagate. They are twice as large as in the primary generation case. Interactions of the interfacial troughs with internal waves propagating in the lower layer can be a cause of the generation of transverse structures. Future studies could combine stereo-correlation measurements of the pycnocline displacement with vertical PIV measurements in various (x,z) planes to assess the role played by internal waves propagating in the lower layer.

It appears that two-dimensional numerical simulations in a vertical plane are relevant to capture the dynamics of ISW1s and of ISW2s to a lesser extent. Three-dimensional simulations are necessary to study in details the interactions between ISW2s and internal waves in the lower layer. In fact, the latter process appears intrinsically linked to the generation of transverse structures in the pycnocline.

FIG. 1 .

 1 FIG. 1. (a) Configuration for primary generation experiments and the associated characteristic density profile measured by a densitymeter ( ρ = 63 kg/m 3 , PG027). ISW1s are generated above the ridge crest, located 5 cm away from the pycnocline rest position. (b) Configuration for secondary generation experiments and the associated characteristic density profile ( ρ = 33 kg/m 3 in the pycnocline and N 0 = 0.94 rad/s in the lower layer, SG023 to SG026). ISW2s are generated by the impingement of topographic internal wave beams on the pycnocline.

FIG. 2 .

 2 FIG. 2. (a) Scheme of an ultrasonic probe, composed of a transducer (emitter/receiver) located in the upper, lighter layer and a reflector located in the lower, heavier layer. Acoustic pulses are emitted vertically from the transducer to the reflector, where they propagate back to the transducer. c 1 and c 2 > c 1 are the speeds of sound in the upper and lower layers. The timelag θ varies with the mean vertical displacement between the transducer and the reflector. (b) Raw Voltage output for an experiment with forcing amplitude A = 8 cm and period T = 25 s. Arrows indicate the displacement/Voltage correspondence issued from the calibration.

FIG. 3 .

 3 FIG. 3. Experimental arrangement viewed from above. Ultrasonic probes (triangles) are placed on a bench located downstream from the seeded stereo field (square of 3 m side) during most of the experiments. For seven experiments in the secondary generation configuration, the probes are placed in the tank axis (dashed line).

FIG. 4 .

 4 FIG. 4. Simultaneous interfacial displacement measurements with an ultrasonic probe (a) and with the stereo-correlation technique (b). The pycnocline dynamics is similar in the two measurements sets.

FIG. 5 .

 5 FIG. 5. Pycnocline displacement and corresponding Fourier spectrum in a series of experiments PG031 (a) and (e), PG032 (b) and (f), PG033 (c) and (g), and PG034 (d) and (h) with λ nd = 1.01 and increasing amplitude from top to bottom. The time origin corresponds to the beginning of the calibration.

FIG. 6 .

 6 FIG. 6. Pycnocline displacement in a series of experiments PG045 (a), PG044 (b), PG042 (c), PG043 (d), PG041 (e), and PG040 (f) with = 0.57 and decreasing δ (increasing λ nd ) from (a) to (f).

  FIG.7. Evolution of the first ISW1 depth in a train with respect to the dispersion parameter δ. Triangles (resp. crosses) show the depth measurements with respect to the front (resp. to the back) of the wave. + show the depth average values. For each experiment, the first number is the number of ISW1s observed in a train. The second number is the expected number of ISW1s per train using Eq. (6).

  194.167.203.152 On: Fri, 09 Jan 2015 15:08:41

FIG. 8 .

 8 FIG. 8. Pycnocline displacement and corresponding Fourier spectra obtained from eight ultrasonic probes positioned at x 1 = -0.23 m (a) and (i); x 2 = -0.88 m (b) and (j); x 4 = -1.68 m (c) and (k); x 5 = -2.08 m (d) and (l); x 6 = -2.53 m (e) and (m); x 7 = -2.98 m (f) and (n); x 8 = -3.43 m (g) and (o); x 9 = -5.38 m (h) and (p), where x = 0 is the ridge rest position. The internal wave beam impacts the pycnocline at x = x i, 1 = -0.75 ± 0.05 m.

FIG. 9 .

 9 FIG. 9. (a) Evolution of the fundamental (+ signs), first (× signs), second (diamonds), third (triangles), fourth (dots), and fifth (squares) harmonics amplitude with respect to the ridge/probe distance. (b) Schematic paths of the initially upgoing (solid lines) and the hypothetical initially downgoing (dashed lines) internal wave beams emitted over the ridge. Arrows indicate the probes positions.

  FIG. 10. (a) Transverse mean isopycnal displacement in the pycnocline in PG052 for t = 8.75; 8.80; . . . ; 9.75 T. Solid lines show the propagation of the first and second ISW1s of a train. (b) Transverse mean isopycnal displacement in the pycnocline in SG019 for t = 8; 8.125; . . . ; 9 T. Solid lines show the propagation of ISWs and nonlinear interfacial waves indicated by solid arrows. Dashed arrows indicate a loss of amplitude in a propagating trough.

FIG. 11 .

 11 FIG. 11. Stereo correlation measurements of the interfacial displacement in exp. PG052, at t = 8.75 T, 8.85 T, . . . , 9.75 T. Time evolves in the counterclockwise direction. The ridge rest position is at 90 cm to the right of the field. The first ISW1 (blue circle in the first picture) propagates from right to left with a speed of 0.25 ± 0.03 m/s. The solid black arrow shows a second ISW1.

FIG. 12 .

 12 FIG. 12. Stereo correlation measurements of the interfacial displacement in exp. SG019 at t = 8 T, 8.125 T,. . . , 9 T. Time evolves in the counterclockwise direction. Troughs propagate to the left of the field at a speed of 0.11 m/s. The blue circle highlights propagating troughs at t = 8.75 T.In the case of a moderate pycnocline (γ ≈ 0.1), interfacial troughs can be generated by the impact of internal wave beams on the pycnocline. In a linear regime( 1), the oscillation of the ridge would generate internal wave beams propagating in the direction of the forcing, and non-varying in the transverse direction. In the present configuration, turbulent effects generated by the ridge oscillation

FIG. 13 .

 13 FIG. 13. Space-time diagram of the interfacial displacement at x = -150 cm in SG019. Dark blue contours correspond to propagating troughs. Transverse variations in the interfacial wave structure increase with time.

TABLE I .

 I Experimental parameters.

	Series Experiment Measurement type	ρ (kg/m 3 ) N 0 (rad/s) A (cm) T (s)	δ	λ nd	γ
	9	PG031	Probe	60	1	4	6	0.28 3.97 1.01	1
	9	PG032	Probe	60	1	6	6	0.42 3.97 1.01	1
	9	PG033	Probe	60	1	8	6	0.57 3.97 1.01	1
	9	PG034	Probe	60	1	10	6	0.71 3.97 1.01	1
	12	PG040	Probe	90	1	8	25	0.57 0.15 5.15	1
	12	PG041	Probe	90	1	8	18.5	0.57 0.28 3.81	1
	12	PG042	Probe	90	1	8	13	0.57 0.56 2.69	1
	12	PG043	Probe	90	1	8	15	0.57 0.42 3.09	1
	12	PG044	Probe	90	1	8	12	0.57 0.66 2.47	1
	12	PG045	Probe	90	1	8	8	0.57 1.49 1.65	1
	18	PG052	Stereo	90	1	8	8	0.57 1.49 1.65	1
	19	SG019	Probe/Stereo	30	1.23	4	12	0.28 N/A N/A 0.21
	21	SG033	Probes	33	0.93	8	13	0.28 N/A N/A 0.15
	nondimensional wavelength λ nd

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ACKNOWLEDGMENTS

This work has been supported by LEFE-IDAO Programme "ondes et marées internes dans l'océan" (LEFE-IDAO-07/2) and ANR "PIWO" contract (ANR-08-BLAN-0113). Y. Dossmann's Ph.D. thesis was funded by a MNERT scholarship. We thank B. Beaudoin, A. Belleudy, J.-C. Canonici, F. Murguet, H. Schaffner, and V. Valette of the CNRM-GAME (UMR 3589, METEO-FRANCE, and CNRS) fluid mechanics laboratory for their kind support. S. Cazin and L. Lacaze from IMFT and S. Viboud from LEGI are also thanked for their help during the experimental campaign, regarding, respectively, the stereo-correlation technique and the ultrasonic probes.