M A N I K A N D A N M A T H U R

L O U I S G O S T I A U X

T H E O G E R K

E M M A G A L H Ã

C B D A S I L V A 5 A N D T H I E R R Y D A U X O

Soliton generation by internal tidal beams impinging on a pycnocline : laboratory experiments

 (Received 7 April 2015)In this paper, we present the first laboratory experiments that show the generation of internal solitary waves by the impingement of a quasi two-dimensional internal wave beam on a pycnocline. These experiments were inspired by observations of internal solitary waves in the deep ocean from Synthetic Aperture Radar (SAR) imagery, where this socalled mechanism of 'local generation' was argued to be at work: here in the form of internal tidal beams hitting the thermocline. Nonlinear processes involved here are found to be of two kinds. First, we observe the generation of a mean flow and higher harmonics at the location where the principal beam reflects from the surface and pycnocline; their characteristics are examined using Particle Image Velocimetry measurements. Second, the appearance of internal solitary waves in the pycnocline, detected with ultrasonic probes; they are further characterized by a bulge in the frequency spectrum, distinct from the higher harmonics. Finally, the relevance of our results for understanding ocean observations is discussed.

Introduction

Internal solitary waves (ISWs) are among the most noticeable kinds of internal waves in the ocean. Their surface manifestation may be visible even from spacecraft (see, for example, [START_REF] Jackson | Internal wave detection using the moderate resolution imaging spectroradiometer (modis)[END_REF]. They often appear in groups, and the groups themselves usually appear regularly, every tidal period. This points to a tidal origin: barotropic tidal flow over topography creates internal tides, which, while propagating away from their source, may steepen and split up into ISWs. This is the common picture, in which the internal tide is regarded as a horizontally propagating, interfacial wave. In the early nineties, [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 first evidence of a very different generation mechanism of ISWs. The origin is still tidal, but the mechanism involves an internal tidal beam, generated over the continental slope, which first propagates downward, then reflects from the ocean floor, and finally, as its energy goes upward, impinges on the seasonal thermocline. Here it creates a depression, which, while propagating away, steepens and evolves into ISWs. This interpretation has been corroborated by Synthetic Aperture Radar (SAR) imagery, showing a great increase of ISWs in the central Bay of Biscay, just beyond the area where the beam is expected to approach the surface [START_REF] New | Remote-sensing evidence for the local generation of internal soliton packets in the central Bay of Biscay[END_REF][START_REF] Da Silva | On the role of SAR for observing "local generation" of internal solitary waves off the Iberian Peninsula[END_REF]. Recently, the same mechanism has been proposed to explain ISWs off Portugal [START_REF] Da Silva | On the role of SAR for observing "local generation" of internal solitary waves off the Iberian Peninsula[END_REF]) and in the Mozambique Channel [START_REF] Da Silva | Internal solitary waves in the Mozambique Channel: observations and interpretation[END_REF].

In this paper we present results on laboratory experiments -carried out at the LEGI Coriolis Platform in Grenoble in 2008 -that were set up to create ISWs by this very mechanism, i.e. by an internal wave beam impinging on a pycnocline. At the beginning of the generation process lies the internal 'tidal' beam, generated here with a recently designed wavemaker (Gostiaux et al. 2007;[START_REF] Mercier | New wave generation[END_REF], which creates a well-defined monochromatic unidirectional beam. We varied the stratification, i.e. the strength, depth and thickness of the pycnocline which was set up above a constantly stratified lower layer. For each stratification we carried out experiments with different forcing frequencies, thus changing the angle the beam makes with the vertical. We note here that the wave beam generated by the wavemaker directly models the upward-propagating tidal beam that impinges on the ocean pycnocline.

In the set-up of the experiments, we were guided by theoretical studies on the subject. The process starts at the wavemaker, from which the internal-wave beam originates. So long as it propagates through the layer of uniform stratification, no significant changes are expected to occur, because a unidirectional wave beam propagating in a uniformly stratified Boussinesq fluid satisfies not only the linear but also the nonlinear equations of motion [START_REF] Tabaei | Nonlinear internal gravity wave beams[END_REF]. However, when wave beams propagating in multiple directions intersect, nonlinear effects such as generation of higher harmonics occur [START_REF] Tabaei | Nonlinear effects in reflecting and colliding internal wave beams[END_REF][START_REF] Jiang | Selection rules for the nonlinear interaction of internal gravity waves[END_REF]. This situation occurs when beams cross or reflect from boundaries.

In our set-up, and in the ocean too, the first deformation of the beam is expected to happen when it leaves the layer of near-constant stratification, i.e. when it encounters the pycnocline. This strong inhomogeneity of the medium causes the beam to reflect partially from and within the pycnocline, a linear process that 'scatters' the beam. As a result, some of the energy stays behind in the pycnocline. It has been shown in theoretical studies that this forms the basis from which ISWs may later evolve [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF][START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF]. At the same time, the partial reflections (that occur when a unidirectional wave beam propagates through a region of non-constant stratification) within the pycnocline, as well as the full reflection from the upper surface, create junctions of crossing beams, and hence form a source of nonlinear generation of higher harmonics. In our laboratory experiments, we thus focused on these two features: first, the evolution of the pycnocline after the impact of the beam, and second, higher harmonics. In a way, they are contrasting phenomena: ISWs in the pycnocline are the nonlinear result of a depression that steepens, the depression itself originating from an essentially linear process (internal reflections in the pycnocline). On the other hand, higher harmonics waves behave as linear waves but find their origin in a nonlinear process at the junction of the main beam and its reflected counterpart. This distinction in behaviour has consequences for the tools by which they may be analysed; in particular, the application of harmonic analysis is well-suited to extract the essentially linear higher harmonic beams from the measurements, but cannot expect to yield anything clear on the genuinely nonlinear ISWs. The same is true for wave spectra. We discuss this further in sections 3 & 4.

The linear regime of the problem we address, an internal wave beam impinging a continuously stratified pycnocline in a finite-depth tank, was studied in [START_REF] Mathur | Internal wave beam propagation in nonuniform stratifications[END_REF]. Their experimental results on wave beam ducting, a scenario where the incident energy tends to remain trapped in the pycnocline even after multiple reflections within the pycnocline, were in excellent agreement with the viscous theory. This linear process, as discussed in [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF] and [START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF], serves as a precursor to the formation of solitons.

Previous experimental and theoretical studies have identified the key parameters governing the response of the pycnocline to the incident beam. [START_REF] Delisi | On the role of density jumps in the reflexion and breaking of internal gravity beams[END_REF] derived a parameter β which they interpreted as the square of the ratio of the phase speed for interfacial waves and the horizontal phase speed of the incident beam. This interpretation is open for some debate (see [START_REF] Thorpe | Nonlinear reflection of internal waves at a density discontinuity at the base of the mixed layer[END_REF]). The experimental results of [START_REF] Delisi | On the role of density jumps in the reflexion and breaking of internal gravity beams[END_REF] show that the response of the pycnocline is largest when |β| ≈ 1. In a different setting, in which the interfacial waves were assumed to be long rather than short, a similar criterion was identified in terms of a parameter γ, defined as the ratio of the phase speed of interfacial waves and that of the first mode of the uniformly stratified lower layer [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF]. We amplify on this below. Finally, [START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF], in a setting closer to that of [START_REF] Delisi | On the role of density jumps in the reflexion and breaking of internal gravity beams[END_REF] -their lower uniformly stratified layer being infinitely deep -recovered a parameter similar to β, which they call α. In either setting, the parameter g plays a key role, and we study its influence by examining different shapes and strengths of the pycnocline.

Recently, in a configuration similar to the present setup, [START_REF] Grisouard | Generation of internal solitary waves in a pycnocline by an internal wave beam: a numerical study[END_REF] modelled the local generation of ISWs with a nonlinear, non-hydrostatic numerical code (MIT-gcm) and showed that higher mode ISWs (namely mode-2 and 3) may also be generated by this mechanism. Initially, the idea was to carry out numerical runs concurrently with the laboratory experiments, and the basic setting for the numerical model was chosen to be the same as in the laboratory. But, as shown in the rest of this paper, we observed some important qualitative differences (between the experiments and the simulations) that rendered a direct comparison impossible.

The present paper is organised as follows. In section 2, a description of the experimental set-up and the measurement techniques is given. An analysis of the results from Particle Image Velocimetry is given in Section 3, where we focus on the mean flow and harmonics. Results from the ultrasonic probes are presented in Section 4; they provide a quantitative view of the pycnocline displacements and the ISWs. Finally, we end with a discussion of our results and conclusions in section 5.

Description of the experimental set-up

The experiments were carried out at the Coriolis Platform of LEGI, Grenoble (see Figure 1 for a schematic of the experimental setup). This facility is a 13 m diameter rotating basin that can be filled with salty water; during the process of filling, the salinity can be changed, so that a vertically stratified layer is formed. The size of the experimental tank allowed visualisation of the wave propagation over a large horizontal distance. Furthermore, the platform can serve as a turntable, but the experiments discussed in this paper were done without rotation.

Experimental parameters

A vertically stratified layer of height 80 cm was created by filling the tank from below with water whose salinity was gradually increased; this resulted in a layer whose density ρ decreased from 1040 kg/m 3 at the bottom to 1010 kg/m 3 at the top. Thus, a layer of linearly stratified fluid was formed with a constant value N 0 = 0.58 rad/s of the Brunt-Väisälä frequency,

N = - g ρ 0 dρ dz , (2.1)
also referred to as the buoyancy frequency later on. The first experiments were carried out with this layer alone. It must be noted that the buoyancy frequency in this layer remained constant in subsequent days, except for the lowest part of the tank where it became slightly weaker with time, as can be seen in Figure 10 (a), due to molecular diffusion.

Then, a thin layer (viz. 4 cm) of fresh water (ρ ≈ 1000 kg/m 3 ) was added from above on top of the uniformly stratified layer. This served as the upper mixed layer, while the sharp transition of density between the two layers ensured the presence of a pycnocline. After a day, the pycnocline tended to lose its sharpness, and so new fresh layers were added after experiments, or sometimes the upper part of the water column was removed before adding fresh water. In this way, a whole range of different pycnoclines (in terms of peak value N max , thickness, and depth) was created, serving as varying background conditions during which the experiments were carried out. Examples of stratification profiles are provided in Figure 10 (a), and characteristic values are presented in Table 1. Density measurements done after each experiment showed no significant changes compared to the profile taken before the experiment.

The ratios between ω and the stratification frequencies, N 0 and N max , are both of the order of 10 -1 in our experiments, which is larger than the ones in the ocean. Typical values for the ocean are N 0 = 10 -3 rad s -1 and N max = 10 -2 rad s -1 , which give for the semi-diurnal tide (ω = 1.4 10 -4 rad s -1 ) ratios of the order of 10 -1 and 10 -2 respectively. This implies differences in the steepness of the incident internal wave beam, and the experimental results might not be directly applicable to the ocean.

Incident internal wave beam

In the basin, an internal wave beam directed upwards was generated by a wavemaker similar to the one developed by Gostiaux et al. (2007), to which we refer for a detailed description. As described in [START_REF] Mercier | New wave generation[END_REF], the internal-wave beam profile thus obtained is very similar to the solution derived by [START_REF] Thomas | A similarity solution for viscous internal waves[END_REF]. Their profile has previously been shown to describe wave beams in the ocean after they propagate far from the continental shelf (Gostiaux & Dauxois 2007). In our specific setup, we can directly reproduce such a 'tidal' beam without topography and without barotropic flow.

Briefly, the device consists of 24 stacked PVC sheets of 2 cm thickness, shifted with respect to each other so as to form a wave pattern similar to the Thomas-Stevenson wave beam described in § 4 in [START_REF] Mercier | New wave generation[END_REF]. The PVC sheets in the wavemaker are connected via two eccentric camshafts, which, when subject to rotation, results in downward (phase) propagation of the wave pattern. At both ends of the stacking, there are sheets with zero amplitude of oscillation so that when measured along the wavemaker, the width of the generated beam is of the order of 33 cm. The amplitude of oscillation of the plate in the center of the beam is 3.5 cm. Placed in a stratified fluid at an angle of 30 • with respect to the vertical, as depicted in Figure 1 (a), this wave motion is imparted to the fluid, thus producing a unidirectional internal wave beam at a single frequency. The device had a finite width of 1.5 m along the y-direction, resulting in three-dimensional effects close to the edges.

Quantitative measurements of the velocity field and the pycnocline displacements were made using Particle Image Velocimetry and ultrasonic probes, respectively. From the PIV velocity measurements described below, several properties of the beam can be estimated. We measured the maximum amplitude of the velocity U max in the direction of propagation of the beam, and the dominant wavenumber k η associated with the profile. These properties, along with other relevant parameters in the experiments, are listed in Table 1. It is interesting to notice that although our experiments are at quite low values for the Reynolds number, nonlinearity is expected to play an important part since the excursion parameter, A k = U max k η /ω, which compares the fluid particle displacement with the characteristic wavelength of the problem, is close to 1. 1. Name of the experiment; the maximum buoyancy frequency in the pycnocline Nmax; the forcing period T ; the maximum velocity in the direction of propagation of the initial beam Umax; the dominant wavenumber of the initial beam kη; its horizontal wavelength, λx = 2π/kη sin θ, where θ is the angle of the beam with the horizontal; and finally, the corresponding horizontal phase speed Cx = λx/T . Relevant non-dimensional parameters are also provided: γ, a measure of the strength of the pycnocline (as explained in section 4.2); the ratio of forcing and stratification frequencies; the Reynolds number Re = Umax/νkη and the excursion parameter A k = Umaxkη/ω.

EXP Nmax T Umax kη λx Cx γ ω/N0 ω/Nmax Re A k # (rad s -1 ) (s) (cm s -1 ) (rad m -1 ) (cm) (cm

Particle Imagery Velocimetry (PIV)

Velocity fields have been obtained through PIV measurements done in the (xOz) plane as shown in Figure 1 (c) (grey dashed line). Two CCD 12-bit cameras of resolution 1024 x 1024 record at 3 Hz images of particles illuminated by a 6 W continuous laser.

Although the spatial resolution obtained is good, smaller than 1 mm/pix, it is only possible to visualise the linearly stratified region due to the strong optical distortions caused by the presence of the pycnocline. Furthermore, the seeding of particles in the homogeneous surface layer is insufficient since the particles used here rather match the density range of the stratified layer. Some information can be extracted from floating particles at the free surface, as we will see in section 3.1, which allow us to extrapolate the velocity field from the the top of the linearly stratified region up to the free surface.

Experimental data are processed using open source CIVx algorithms by [START_REF] Fincham | Advanced optimization of correlation imaging velocimetry algorithms[END_REF] and the free Matlab toolbox UVMAT developed at LEGI. Some overlap between the fields of view of the two cameras allows for a horizontal merging of the velocity fields, leading to visualisation windows of 1.6 x 0.8 m.

Ultrasonic probes

The ultrasonic probes designed at LEGI [START_REF] Michallet | Ultrasonic probes and data processing to study interfacial solitary waves[END_REF]) consisted of a sound emitter, which also acts as a receiver, and a reflector that were 0.05m apart. Ten such probes were positioned at x = 0.18, 0.39, 0.62, 0.89, 1.20, 1.254, 1.59, 2.09, 2.80 and 4.0 m from the wave generator, respectively (see Figure 1 (a)). Each probe was mounted vertically on a plane that was slightly off-centered so as to not interfere with the PIV measurements, at a height that ensured that the emitter was in the mixed layer and the reflector in the lower constant stratification layer. The voltage output from the probe is proportional to the time taken for the sound to traverse its path from the emitter to the reflector and back to the emitter/receiver. This traverse time is directly proportional to the relative distance between the emitter and the pycnocline, if the pycnocline is infinitesimally sharp. In realistic scenarios, like the stratification in our experiments, where the pycnocline is of finite thickness, the output from the ultrasonic probe is a measure of the average (over z) displacement of the pycnocline at the specific x location of the probe.

To calibrate each ultrasonic probe, it is moved using a linear traverse mechanism up and down by known distances in a quiescent setting where there is no flow, and hence no pycnocline displacements. The output voltage is continuously recorded as a function of the position of the probe. This allows us to get a voltage V p vs. relative position of the pycnocline z p relation for the probe (see Figure 1 (b)), and the procedure is done for every probe. Converting the output voltage from experiments to equivalent pycnocline displacements involves interpolation on the respective calibration curves.

It is important to note that the values thus found are a measure of the mean vertical displacements between the emitter and receiver. A mode trapped in the pycnocline will have maximum vertical excursions at a certain depth, while the excursions decline rapidly above and below. The acoustic probes, then, will not yield this maximum excursion, but a lower value, because it also registers over parts in which the amplitude is declining. This reduction will affect the signal associated with even modes (i.e., modenumber 2, 4 etc.), in particular.

Mean flow and harmonics

Using PIV, we visualise what happens when the incident wave beam encounters the pycnocline and subsequently reflects from it. To demonstrate the significance of the presence of a pycnocline, we first present a qualitative comparison between settings without and with a pycnocline, see Figure 2. The effect of the pycnocline, as seen in the unfiltered PIV data presented in the lower panel of Figure 2, is twofold. First, it shifts the beam slightly to the right because the beam propagates at a smaller angle with respect to the horizontal in the layer of strong stratification (as indicated by the dashed line which represents the theoretical path of energy propagation). Second, the reflected beam is broadened, and therefore less intense, when it emerges from the pycnocline. This is caused by (multiple) internal reflections: the beam reflects not only from the upper free surface but also from and within the pycnocline itself [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF][START_REF] Mathur | Internal wave beam propagation in nonuniform stratifications[END_REF].

As mentioned in the introduction, two kinds of nonlinear phenomena are expected to take place during reflection at the pycnocline: the nonlinear evolution of the pycnocline displacements excited by the incoming beam, and the generation of higher harmonics at the junction between the incident and reflected beams. The latter occurs even if no pycnocline is present, namely upon reflection from the surface. As noted earlier, the upper most part of the water column produces no visible signal with PIV in the experiments with a pycnocline. Therefore, we resort to the acoustic probes for visualising and analysing the response at the pycnocline, the subject of section 4. In this section, we focus on the features extracted from the PIV data, notably the principal beam and its harmonics. The harmonics cannot be properly interpreted unless we take into account another phenomenon that we found in our experiments: the occurrence of a mean flow in the upper layer. We therefore proceed to discuss this phenomenon first.

Mean flow

Apart from extracting the higher harmonics from the PIV data, harmonic analysis also reveals the presence of a mean Eulerian flow. The mean flow for EXP08, in which the cameras were placed close to the wavemaker, is shown in Figure 3. It is found to originate from the area of reflection of the forced beam at the pycnocline, and to spread further in the direction of wave propagation, but it remains restricted to the upper layer. The horizontal mean velocity displayed here is obtained by averaging over 12 forcing periods, starting at t = 360 s after the start-up of the forcing.

A complete description of the mean-flow field is not directly available as the PIV data lacks information in the upper few centimetres, typically 5 cm, with the exact value depending on the experiment. This is partially remedied by determining, from the raw PIV images, the strength of the mean flow at the free surface (z = 0) by following the particles trapped there by capillary effects. This allows us to establish the profile of the mean flow over the entire vertical, without having to extrapolate; we only interpolate (cubically) over the interval where data is lacking. Furthermore, we also impose the constraint of no shear at the free surface during interpolation. Figure 3 (a) and (b) illustrate the resulting profiles. The second vertical derivative of the mean horizontal velocity, plotted in Figure 3 (c), is an important quantity as it features in equation (3.1), determining the vertical modal structure. A noticeable characteristic of the mean flow is that its amplitude varies with x, being maximum where it originates (at the area of reflection of the forced beam at the pycnocline) and then decaying slightly as you go away from the wavemaker. This evolution must be a signature of a three-dimensional structure of the flow; the amplitudes of the wave field at the forcing and higher harmonic frequencies, however, did not exhibit a similar decay in x. Hence, for the present purposes we assume the flow to be quasi-two-dimensional.

Interestingly, the mean flow is present in all our experiments, including the ones without a pycnocline. In Figure 4, we plot the vertical profile of the mean flow extracted at 0.4 m away from the point of impingement of the incident beam on the free surface for all experiments presented in Table 1, and two extra for different frequencies in the cases with γ = 0. The mean flow maximum velocity is of the same order than the maximum velocity of the incoming wave beam, but no clear effect of the varying pycnocline strength can be observed. So, it is the reflection from the surface, rather than the passage through the pycnocline, which lies at the origin of the mean flow. Furthermore, we verified that the mean flow is not present at the beginning of the experiment and is noticeable only after the beam impinges on the pycnocline and surface.

In none of the experiments is an appreciable mean flow found near the bottom of the tank, even though there, too, beam reflection takes place. However, since PIV measurements are not possible in the bottom 5 mm of the fluid, we cannot determine the flow in the vicinity of the wave-induced bottom boundary-layer of thickness ν/ω, of the order of 2 mm in our case. One could argue that nonlinear effects are stronger at the pycnocline due to the strengthening of the wave beam in regions of stronger stratification, and this may result in a net deposit of momentum. In Figures 2 (a) & (c), the first reflected beam coming from the pycnocline/free surface, when arriving at the tank bottom, is not much weaker than the incident forced beam. Hence, the contrast between the presence of a mean flow near the surface and its absence near the bottom is presumably due to the different nature of the boundaries, a free surface versus a solid boundary (i.e., free-slip versus no-slip). These explanations, however, must remain hypotheses since our experiments do not allow us to draw any firm conclusions. As a matter of fact, mean flow generation near a solid boundary has already been observed in experiments and numerical simulations of barotropic flow over a three-dimensional topography [START_REF] King | Tidal flow over three-dimensional topography in a stratified fluid[END_REF]). Nevertheless, the mechanism at play in our case is different since we do not have any topography, there is no barotropic forcing, and the mean flow generated is in the same plane as the incoming wave beam.

Other experiments dedicated to this mean flow are needed to better understand its cause and its possibly three dimensional structure, as well as its dependence on various parameters (impinging beam angle/profile, strength of the pycnocline etc.). As discussed in the following sections, incorporating the effects of the mean flow plays a crucial role in the interpretation of the higher harmonics and other nonlinear waves in our experiments.

Higher harmonics

Here, we present a harmonic analysis of the PIV data from three experiments that were carried out on the same day, hence with nearly identical density profiles (all corresponding to γ = 0.17), but at different forcing frequencies: ω f = 0.15 rad/s (EXP21), ω f = 0.29 rad/s (EXP22), ω f = 0.37 rad/s (EXP25). These three experiments had the cameras placed further away from the wavemaker than in EXP08, and hence provide a more complete view of the region after reflection from the pycnocline. In EXP25 (ω f /N 0 = 0.64), presented in Figure 5, the principal beam at the forcing frequency reflects from the pycnocline and surface, and broadens due to internal reflections. The dominant wavenumber in the reflected beam, k (r) x 10.7 rad/m is much smaller than the corresponding value in the incident wave beam, k

(i)
x = 20.7 rad/m. The reflected beam at the forcing frequency, which radiates away from the pycnocline into the lower layer, is in stark contrast with the higher harmonics 2ω f (and 3ω f , not shown in the figure), which are trapped in the upper part of the water column and propagate purely horizontally with a well-defined horizontal periodicity. For the 2ω f -signal, we find a horizontal wavelength of 22 cm. The fact that the harmonics are trapped is readily understood from linear internal-wave theory, since their frequencies 2ω f , 3ω f etc. exceed the buoyancy frequency of the lower layer (N 0 ); hence the harmonics cannot propagate into it and are constrained to the pycnocline.

The experimentally measured horizontal wavelength of the trapped higher harmonics may be compared with the theoretical estimates from the eigenvalue problem for the vertical modal structure

W + N 2 (U -c) 2 - U U -c -k 2 W = 0 , (3.1)
with the vertical velocity w(x, z, t) = [W (z) exp[i(kxωt)]], and the boundary condition W = 0 at the surface and bottom (LeBlond & Mysak 1978, Eq. 41.8). Here, c = ω/k is the horizontal phase speed, U (z) is the background mean flow profile, U (z) its second derivative and N (z) the buoyancy frequency profile. We note that this equation is different from the one in terms of the vertical isopycnal displacement, η. In the literature, the two forms are sometimes confused. The relation between the two follows from w = η t + U η x , so that W = ik(Uc)φ, where η = {φ(z) exp[i(kxωt)]}. For ω = 2ω f , and observed N (z) and U (z), we solve (3.1) to obtain the modes W n and corresponding wavenumbers k n or wavelengths λ n = 2π/k n . For EXP25, we find λ 1 = 48.9, λ 2 = 20.2 and λ 3 = 14.8 cm, using the background mean flow profile measured at x = 1.0m. The observed horizontal wavelength (i.e., 22cm) of the signal at ω = 2ω f is the closest to that of the theoretical 2 nd mode. This is confirmed by the excellent agreement between the experimental and theoretical profiles of W 2 and U 2 , as shown in Figure 6, at least so far as the comparison can be made (as mentioned earlier, PIV data is not available in the upper few cm). Exactly why the first harmonic (i.e., 2ω f ) manifests itself as a second mode, and not as some other mode, is not clear. For instance, we note that the horizontal phase speed of the incident internal wave beam, 1.8 cm s -1 , is not particularly close to either the measured (2.6 cm s -1 ) or theoretically calculated (2.4 cm s -1 ) mode-2 phase speed for ω = 2ω f . In fact, it is closest to the theoretically calculated horizontal mode-3 phase speed for ω = 2ω f (c 3 = 2ω f /k 3 = 1.8 cm s -1 ). This suggests that a phase speed matching mechanism, such as the one discussed by [START_REF] Grisouard | Generation of internal solitary waves in a pycnocline by an internal wave beam: a numerical study[END_REF], who examined the local generation of internal solitary waves with a numerical model but without observing any mean flow, is not in play here. Two possible explanations may be advanced. In a weakly nonlinear case where there is just advection of the harmonics by the mean flow, we could search for a criterion with Doppler effect on the phase speed of the harmonics. Another more complex scenario corresponds to fully nonlinear coupling of the mean flow and the harmonics. However our experiments cannot provide an answer to this point of high interest.

Finally, the importance of the shear flow to the modal propagation of the higher harmonics can be illustrated by solving the modes from (3.1) with U = 0; then, for ω = 2ω f , we find much smaller horizontal wavelengths: λ 1 = 41, λ 2 = 13 and λ 3 = 8 cm, none of which matches the observed wavelength.

For EXP22 (ω f /N 0 = 0.50), we again find that the measured horizontal wavelength (35 cm) of the harmonic 2ω f is closest to the corresponding theoretical wavelength of mode-2 (λ 1 = 74, λ 2 = 29 and λ 3 = 20 cm). We note, however, that the wavelengths are in this case (2ω f ≈ N 0 ) sensitively dependent on the frequency ω = 2ω f and N 0 , because they are very close. For example, the mode-2 wavelength for ω = 0.5 and 0.6 rad/s for the stratification in EXP22 are λ 2 = 73 and 24.5 cm, respectively. Thus the quantitative agreement between experiment and theory could be better or worse for even small changes in ω or N 0 . As it was the case for EXP25, the vertical structure of the first harmonic signal (2ω f ) agrees well with the theoretically computed mode-2 shape, except in the deep parts of the lower layer. In the lower layer, where the measured background shear flow is negligible, equation (3.1) predicts a linear variation of W with z for ω = 2ω f ≈ N 0 (since the equations boils down to W ∼ 0); however, we observe a more rapid decay of the signal in the lower layer, the cause of which remains unclear.

In the case of EXP21 (ω f /N 0 = 0.26), for which 3ω f < N 0 , the first two harmonics are not trapped but are allowed to propagate into the lower layer; hence they radiate from the pycnocline as beams and cannot be identified with any one mode, as shown in Figure 7. However, a different simple regularity appears from Figure 7. For each harmonic (n), the beam pattern is clearly periodic (see panels on the left); the horizontal wavenumber associated with this periodicity (k (n) , say) obeys the simple rule k (n) = nk (1) (see dashed lines in panels on the right of Figure 7). This implies that the horizontal phase speeds are the same for all harmonic beams.

Finally, it is evident from Figure 5 and Figure 7 that the amplitude of the harmonics at nω f is weaker than the one at (n -1)ω f (for n > 1), and that these higher harmonics originate from the region of interaction between the incident wave beam and the pycnocline. These observations confirm that the higher harmonics are a result of weakly nonlinear interactions between the incoming and reflected beams. Since the Reynolds number varies along with other parameters through the experiments (cf. Table 1), we are unable to discriminate its role on the generation of harmonics, more specifically its effect on the saturation of the harmonics amplitude as already observed in [START_REF] King | Tidal flow over three-dimensional topography in a stratified fluid[END_REF] for instance. Varying the forcing frequency across various experiments results in the value of ω f /N 0 go from values smaller to larger than 0.5, and thus influencing the propagation in the lower constant stratification layer of higher harmonics; freely propagating for ω f /N 0 < 0.5 and trapped in the pycnocline for ω f /N 0 > 0.5. The change in the ratio ω f /N max however, remains weak and does not seem to play any role in our experiments. A detailed study of the influence of the amplitude of the incoming beam on the amplitude of the generated harmonics is recommended for future research.

Evolution of the pycnocline

In this section, we discuss the evolution of the pycnocline, as measured by the acoustic probes. As mentioned in Section 2.4, ten acoustic probes were positioned at various positions in the range 0 to 4 m from the wave generator. The proximity of two probes located at x = 1.20 & 1.254 m, respectively, allows us to determine the horizontal phase speed of the signal, for its structure remains similar between the two, so that specific peaks can be identified in both. In every experiment, the acoustic probe measurements were initiated well before the wave generator was started, allowing us to visualise the transients before a steady state was reached.

In figures 8 (a) & (b), we plot the pycnocline displacements ζ (i.e., vertical averages over the thickness of the pycnocline, as described in section 2.4) at x = 0.62 m and x = 1.2 m, respectively. In a qualitative way, the results from the probes indicate the formation of ISWs in the pycnocline: the wave pattern is nearly sinusoidal at the region of impact (x = 0.62 m), steepens as it propagates away, and at some point its steep front splits up into higher-frequency peaks, which are, at least initially (x = 1.2 m), ordered in amplitude, the larger ones being ahead. Furthermore, as shown in figure 8 (c), the power spectrum of the time series at x = 1.2 m has a distinct bulge around the frequencies in the neighbourhood of ω = N max , a feature we discuss in detail later in the current section. In a quantitative sense, however, it is not obvious that nonlinear effects are at work; after all, the amplitude of the peaks is small (a few mm) compared to the depth of the pycnocline (a few cm).

We note, however, that the actual peak amplitudes of interfacial displacements must be larger than the measured outputs from the ultrasonic probes as the probes measure the mean amplitudes over the layer between the transmitter and the receiver. This effect is particularly pronounced for the high-frequency waves that are trapped in the pycno-cline, for their amplitude diminishes rapidly outside of the pycnocline. We also note that the shape and amplitude of the ISWs we find are similar to those observed in earlier experiments on internal solitons in a two-layer system (Horn et al. 2001).

Interpretation of spectra

The generation of higher harmonics due to nonlinear interaction at the junction of two beams was discussed in Section 3.2. Being relatively weak, their behaviour must be close to linear as they propagate away from their source. This means that they can be properly identified in spectra, as seen in Figure 8 (c), which after all amounts to treating the signal as if it were linear. For ISWs it is less clear how they might manifest themselves in spectra. Being genuinely nonlinear waves, they are not associated with any particular harmonic; neither can they be described as a superposition of independently propagating harmonics. Still, technically, one can calculate the spectrum of a signal containing ISWs. As a test case, we use results from a set of KdV-type equations for interfacial waves, which include forcing terms due to barotropic tidal flow over topography [START_REF] Gerkema | A unified model for the generation and fission of internal tides in a rotating ocean[END_REF].The model is weakly non-hydrostatic and weakly nonlinear; besides, it contains terms describing the advection of baroclinic velocity fields by the barotropic flow. In one model calculation, we switched off the nonlinear terms while retaining the barotropic advection terms. In this case, the internal-wave field is linear but contains higher harmonics due to the barotropic advection terms. In another calculation, the nonlinear terms were included, giving rise to internal solitons. In this case, there are higher harmonics as well as ISWs, and so this is qualitatively the situation we supposedly have in our laboratory experiments.

If we now calculate the spectra for the results of both model runs, we find important and qualitative differences (see Figure 9). For the quasi-linear case, peaks at higher harmonics are found, and, as expected, they get lower and lower for higher frequencies. In the nonlinear case, the same qualitative behaviour is found at the lowest frequencies, but then a bulge appears, which is not associated with any one harmonic, but more wide-spread, involving a group of harmonics. This is easily understood from the fact that solitons are not associated with harmonics per se, but have nonetheless certain time scales within the range of some of the harmonics. Harmonics in the neighbourhood of these time scales show marked peaks in the spectrum. This does not mean that the solitons are formed by a superposition of independently propagating higher harmonics (after all, the signal is nonlinear), but are the result of applying a linear technique (harmonic analysis) to a nonlinear signal. In any case, the presence of a bulge helps to distinguish the physically distinct cases of pure higher harmonics on the one hand, and solitons on the other.

Such a bulge appears in our experiments too, as is illustrated in figure 8 (c). This confirms the direct visual impression from the probe results shown in figure 8 (b), in which groups of peaked waves are clearly visible. We proceed, in the next section, to investigate the effect of the pycnocline properties (depth, vertical extent, strength etc.) on the bulge and hence propose a way to capture all the properties into one parameter, the magnitude of which determines the appearance of solitons.

Effect of the pycnocline strength

By removing the upper layer, and refilling it with fresh water, we varied the depth, the amplitude, and the width of the peak of the pycnocline (see Figure 10 (a) for the various stratification profiles that were set up during the course of all our experiments). In this section, we present results from three experiments with the same forcing frequency (such that the incident internal wave beam propagates at 30 • with respect to the horizontal) but with different pycnocline properties.

A key parameter for the response of the pycnocline, due to an impinging internal-wave beam, was identified by Gerkema (2001) using linear theory. In that study, a "2c-layer" stratification was adopted, consisting of an upper mixed layer of thickness d, an interfacial pycnocline of strength g , beneath which lies a layer of constant buoyancy frequency, N c . The total water depth being H, the parameter in question is defined by

γ = (g d) 1/2 N c H . (4.1)
In the numerator of equation 4.1, we recognise the phase speed of long interfacial waves in a 2-layer system, while the expression in the denominator is a measure of the phase speed of vertical modes in a constantly stratified layer of depth H. These modes add up to form beams, so the parameter γ reflects the ratio of the phase speeds associated with the interface (i.e. pycnocline) and the uniformly stratified lower layer. It was found by [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF] that for γ either very small or very large, there is no significant transfer of energy from beams to interfacial waves. Such a transfer occurs only for intermediate values, characterised by γ ≈ 0.1. This would therefore be expected to serve as a necessary condition for having locally generated ISWs which emerge from the initial interfacial perturbation as it propagates and steepens. It is, however, not a sufficient condition, for the forcing frequency, which does not feature in (4.1), must also affect the evolution into ISWs.

In the present situation we have a more complicated profile of stratification, but the parameters in (4.1) are still meaningful if interpreted in the following way. First, we look at the density profile to identify the base and top of the pycnocline; an example is shown in figures 10 (b) & (c). As explained in [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF], the parameter g equals the area enclosed by the pycnocline in the N 2 versus z graph; so, g = N 2 dz, where the integral is taken between the previously identified base and top of the pycnocline. The parameter d corresponds to the depth of the pycnocline, but this depth is of course not uniquely defined here. To take the depth of the peak would not always make sense because it may not be representative of where the bulk of the pycnocline lies. Therefore, we calculated a 'weighted depth', as d = zN 2 dz/( N 2 dz), where the boundaries of the integrals are again the base and top of the pycnocline. The depth d thus found is illustrated in Figure 10 (c). Finally, the parameter H is simply the total water depth, and N c is taken equal to the mean of N from the bottom to the base of the pycnocline. We have thus obtained all the parameters needed to calculate γ, which is listed for different profiles in Figure 10 (a) and also in Table 1.

In figures 11 (a)-(c), the pycnocline displacements at x = 1.59 m from three different experiments (all corresponding to ω f = 0.29 rad/s) with γ = 0.11, 0.17, 0.19, respectively, are presented. As is evident from the figures, groups of peaked waves are most clearly visible for γ = 0.11 (figure 11 (a), EXP08). The corresponding power spectra, presented in figures 12 (a)-(c), show that the bulge at frequencies larger than the forcing frequency is most pronounced for EXP08, confirming that the bulges in the power spectra are indeed directly correlated with the appearance of solitons. The bulge is relatively weaker in EXP22, and is the weakest in EXP16, for which the higher harmonics dominate over solitary-wave-like features. Finally, the solitons being most evident for γ = 0.11 is consistent with the theoretical criterion of γ ≈ 0.1 given by [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF] for optimal excitation of solitons in the pycnocline, and also with oceanic observations. Indeed, re- The two closely spaced probes at x = 1.2 m and x = 1.254 m were introduced only after EXP08 was performed. Analysing the time series data from these two closely-spaced acoustic probes in EXP22 (2ω f ≈ N 0 and γ = 0.17), we find that the higher-frequency peaks move with the same speed as the main depression (or the envelope) on which they sit. They move at 2.3 cm/s; the speed being larger than the mode-1 wave speed (c 1 = 1.6 cm/s, calculated by numerically solving (3.1) with the measured N (z) and U (z) at x = 1.15 m)for ω = 2.5 rad/s (frequency representative of the bulge observed in figure 12(b)). This suggests that the observed solitary waves were mode-1, with the solitons moving faster than the corresponding linear mode-1 wave because of nonlinear effects. Finally, the soliton speed is neither very close to the horizontal phase speed of the incident wave beam (1.83 cm/s, derived from the PIV data, again ruling out the phase speed matching mechanism in our experiments) nor close to that of the 2nd-mode structure of the the 2nd harmonic discussed in section 3 (with observed λ 2 = 35 cm, we obtain 3.24 cm/s, suggesting the solitons are rather independent from the higher harmonics).

Discussion

Our experiments were inspired by oceanic observations [START_REF] New | Local generation of internal soliton packets in the central Bay of Biscay[END_REF]; [START_REF] New | Remote-sensing evidence for the local generation of internal soliton packets in the central Bay of Biscay[END_REF]; [START_REF] Azevedo | On the generation and propagation of internal solitary waves in the Southern Bay of Biscay[END_REF][START_REF] Da Silva | On the role of SAR for observing "local generation" of internal solitary waves off the Iberian Peninsula[END_REF], 2009)), theoretical works [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF]; [START_REF] Akylas | Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline[END_REF]) and numerical studies [START_REF] Maugé | Generation of weakly nonlinear nonhydrostatic internal tides over large topography: a multi-modal approach[END_REF]; [START_REF] Grisouard | Generation of internal solitary waves in a pycnocline by an internal wave beam: a numerical study[END_REF]). Apart from the expected result of the generation of groups of solitary waves, the full picture that emerges turns out to be intricate, with the additional features of a mean current and higher harmonics.

We used two measurement techniques, which were complementary. Particle Image Velocimetry measurements were particularly suitable for obtaining data on the harmonics and mean flow in the layer beneath the pycnocline. Ultrasonic probes were used to measure the vertical movement of the pycnocline itself. At the base of the pycnocline there is an overlap: both the techniques provide reliable measurements in this region. A discussion of the consistency between the two techniques is presented in appendix A.

We have found higher harmonics being generated at the junction of the incident and reflected beams, as expected from theory by [START_REF] Tabaei | Nonlinear internal gravity wave beams[END_REF]. Moreover we found that these harmonics are trapped in the pycnocline when their frequency exceeds, or is close to that of the constantly stratified lower layer. These trapped harmonics are each uni-modal, with a well-defined wavelength. The wavelengths and vertical profiles of these modes are significantly affected by a residual current in the upper layer, which, too, finds its origin in the region where the main beam reflects. Theoretical studies on internal-wave reflection do predict residual currents [START_REF] Tabaei | Nonlinear effects in reflecting and colliding internal wave beams[END_REF], for example), but only in the very region of reflection. Our experiments, however, demonstrate that they can be much more extensive horizontally, in the direction of wave propagation. We are not aware of any theoretical work that may explain this phenomenon. The importance of the mean flow on the generation of harmonics, and also the apparently three-dimensional structure of the flow, both deserve separate studies.

We observe the generation of internal solitary waves only when the stratification is such that the parameter γ is close to 0.1, which is consistent with the theoretical criterion put forward by [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF]. Although weak in amplitude, we demonstrated that these waves are intrinsically nonlinear as they manifest themselves in spectra not as a superposition of harmonics but as a bulge. This interpretation of the spectra is suggested by looking at spectra from a KdV-type model. It is also to be noted that we did not perform experiments at non-zero values smaller than γ = 0.11. We therefore suggest investigations in this regime in future experiments to conclusively establish the optimality of γ = 0.1 for the generation of solitary waves.

Although our experiments clearly show the generation of trapped 2nd-mode higher harmonics in the pycnocline, the ISWs seem to be of 1st-mode, in line with most oceanic observations. Yet, exceptions may occur, as is clear from the numerical work by [START_REF] Grisouard | Generation of internal solitary waves in a pycnocline by an internal wave beam: a numerical study[END_REF] and also from oceanic observations such as the one shown in Figure 13. Here we see a SAR image from the Mozambique Channel with surface signatures of ISWs. While these signatures look like many other typical SAR signatures of ISWs, with larger amplitude waves leading the wave trains in their direction of propagation, close inspection of their bright and dark patterns reveals that they are mode-2 solitary like waves, as explained in appendix B. Indeed, a dark band pattern preceding a bright band (in the propagation direction) reveals a surface current divergence preceding a convergence, which is consistent with a mode-2 solitary-like wave [START_REF] Da Silva | On the structure and propagation of internal solitary waves generated at the Mascarene Plateau in the Indian Ocean[END_REF]). In addition, the ISWs in figure 13 are located close to (but ahead of) the surfacing of an internal (tidal) ray emanating from critical slopes off the Mozambique shelf break. The image in Figure 13 provides evidence of mode-2 solitary like waves that are consistent with Local Generation after the impact of an internal (tide) wave beam on the pycnocline. As mentioned in section 1, it was not possible for us to make a direct comparison between our experiments and the numerical simulations of [START_REF] Grisouard | Generation of internal solitary waves in a pycnocline by an internal wave beam: a numerical study[END_REF]. The mean flow observed in our experiments, discussed in section 3, was not observed in the numerical studies. Since the mean flow affects the phase speeds of the modes, it cannot be established from the numerical study how this might affect the criterion for mode-selection proposed by [START_REF] Grisouard | Generation of internal solitary waves in a pycnocline by an internal wave beam: a numerical study[END_REF]. The same is true for the higher harmonics. So, our laboratory experiments are an inspiration for a next step in the numerical modelling on local generation: especially to study the origin and effect of a mean flow.

Finally, we are not aware of oceanic observations in which trapped higher harmonics were found in a pycnocline. In the context of internal tidal beams impinging on a seasonal thermocline, this phenomenon is in any case not be expected, for the semidiurnal tidal frequency (1.4 × 10 -4 rad/s) is much lower than the typical value of N in the seasonal thermocline (1 × 10 -2 rad/s); in other words, the harmonic would have to be of an extremely high multiple for it to be trapped. The mean flow, on the other hand, has not been observed either, as far as we are aware, but there seems to be no a priori reason why it should not occur. In any case, in oceanic observations, too, one would typically find a mixture of higher harmonics (not trapped) and internal solitary waves. The idea we have put forward here to distinguish them in internal-wave spectra, may provide a useful tool in the oceanographic context as well.

Appendix A. Comparing PIV and ultrasonic measurements

Two different techniques were used to study the internal wave beam impinging on a pycnocline. PIV data from the lower layer of constant stratification provides velocity fields in a vertical plane, with a sampling frequency of 3 Hz. An array of ultrasonic probes, on the other hand, gives times series of local mean vertical displacements of the pycnocline at a much higher sampling frequency of 240 Hz. The two techniques are complementary since the PIV measurements offer a large field of view and the ultrasonic probes focus on the pycnocline where PIV cannot be used because of strong optical distortion and sparse particle seeding. They can also be compared to a certain extent and hence be used to reaffirm the conclusions drawn from the data and to demonstrate their consistency.

The first time-derivative of the mean pycnocline displacements measured by the ultrasonic probes should qualitatively to the vertical velocity obtained from PIV data from just below the pycnocline. The comparison can only be qualitative because the acoustic probes provide a vertical average of displacements within the pycnocline, not a value at a specific depth. To illustrate this point, we present in Figure 14 time series of the vertical velocity of the pycnocline obtained with the two techniques at x = 0.62 m and x = 1.2 m for EXP22. The continuous lines are obtained by taking the time derivative of the probe signals; the dashed lines represent the mean vertical velocities (over 5 mm 2 ) derived from the PIV data centered at these x-locations and at a depth z = -5.25 cm. It is clear that the low-frequency oscillations, of the order of the forcing frequency ω f , are similar in both datasets, whereas the higher frequencies (i.e. larger than ω f ) are more pronounced in the ultrasonic probes. This is mainly due to the fact that these high frequencies are localised in the pycnocline itself (ω f N 0 /2 for EXP22) and have weak signatures below it. Finally, an important feature is the presence of out-of-phase oscillations of the vertical velocity of the pycnocline in between its base and its mean position, around t = 170 s or 220 s in figure 14 (b) for instance. These are evidence of mode-2 (or higher) internal waves in the pycnocline, in agreement with our findings in section 3.2.

Figure 1 .

 1 Figure 1. Schematic of the experiment, (a) side view, (b) zoom-in on the ultrasonic probe and its relative positioning with respect to the stratification, and (c) top view. A typical calibration curve obtained by traversing the probe in quiescent medium is shown in the inset in (b).

Figure 2 .

 2 Figure 2. Comparison between experiments without (upper panels, EXP03) and with a pycnocline (lower panels, EXP08). The stratification profiles are shown in (b) and (d), and the corresponding measured velocity vectors and magnitude at t = 240 s after the start-up of the forcing in (a) and (c), respectively. The dashed lines represent the theoretical direction of internal-wave energy propagation, starting from the center of the wavemaker, which is indicated by the solid line on the left. Insufficient overlap between the two cameras, around x = 0.9 m, leads to a gap in the velocity field in (a). The two inverted triangles in (c) indicate the positions of the ultrasonic probes at x = 0.62 m and x = 1.20 m.

Figure 3 .

 3 Figure 3. EXP08, (a) Mean horizontal velocity U and vertical profiles of (b) U and (c) ∂ 2 U/∂z 2 extracted at x = 1.2 m (dashed-dotted line), 1.4 m (continuous line) and 1.6 m (dotted line) from the wavemaker. The thick dashed line represents the theoretical direction of internal-wave energy propagation, starting from the center of the wavemaker. Insufficient overlap between the two cameras, around x = 0.9 m, leads to a gap in the velocity field.

Figure 4 .

 4 Figure 4. Vertical profiles of the mean horizontal velocity U (z) extracted at 0.4 m away from the point of the impingement of the incident beam on the free surface, for various experiments. The two profiles (thin lines) not associated with any experiments are with the same pycnocline strength as EXP03.

Figure 5 .

 5 Figure 5. EXP25, harmonic analysis for the forcing frequency ω f (upper panels), and the first harmonic 2ω f (lower panels). Panels (a) and (c) display the velocity vector fields and its amplitude. The arrow indicating the scale of the velocity vectors in (c) is also valid for (a). Panels (b) and (d) show the horizontal Fourier spectra of the vertical velocity w as a function of the horizontal wavenumber kx and its vertical variation (for each, amplitudes are normalised by the maximum value).

Figure 6 .

 6 Figure 6. EXP25, Comparison between the mode-2 shape from experimental velocity profiles at x = 1.0 m, filtered at ω = 2ω f , and the ones calculated from (3.1).

Figure 7 .

 7 Figure 7. EXP21, harmonics analysis as in Figure 5 for the frequency nω f , with n = 1, 2, 3. The dashed lines in (b), (d) and (f) represent, for harmonic ω = nω f , the value k (n) = nk (1) with k (1) = 7.1 rad m -1 .

Figure 8 .

 8 Figure 8. EXP08 (γ = 0.11), steady state pycnocline displacements at (a) x = 0.62 m and (b) x = 1.2m. (c) Corresponding Fourier spectra.

Figure 9 .

 9 Figure 9. Examples from a nonlinear non-hydrostatic KdV type model for interfacial tidally generated waves. Panel (a) shows the wave profile over two tidal periods, created from the quasi-linear version of the model, where higher harmonics are generated by barotropic advection of the baroclinic field; these harmonics manifest themselves as peaks in the corresponding power spectrum, shown in (c). In (b), genuinely nonlinear effects of the baroclinic field itself are included, giving rise to solitons; the corresponding spectrum in (d) is now markedly different, showing a bulge enveloping higher frequencies.

Figure 10 .

 10 Figure 10. (a) Different profiles of stratification with which experiments were carried out. (b) & (c) Determination of γ from the profile for experiments EXP21 to 25. The black dots indicate the (approximate) positions of the top and base of the pycnocline, delineating the vertical interval where density changes strongly in (b). The depth of the weighted center of the pycnocline, i.e. d, is indicated by the horizontal dotted line in (c).

Figure 11 .

 11 Figure 11. Steady-state pycnocline displacements at x = 1.59 m in (a) EXP08 (γ = 0.11), (b)EXP22 (γ = 0.17) and (c) EXP16 (γ = 0.19).

Figure 12 .

 12 Figure 12. Power spectra of the steady-state pycnocline displacements at x = 0.62 m and x = 1.20 m in (a) EXP08 (γ = 0.11), (b) EXP22 (γ = 0.17) and (c) EXP16 (γ = 0.19), with all the three plots normalized by the same constant.

Figure 13 .

 13 Figure 13. (a) SAR image of the Mozambique Channel dated 24 September 2001 at 7:39 UTC, showing a packet of ISWs consistent with mode-2 vertical structure. The star symbol on the image indicates the predicted position where an internal tide beam impinges the thermocline. (b) On the right hand side a vertical section (perpendicular to the ISW crests) is shown with a simulated beam trajectory based on local stratification (represented by the dashed line). Dark bold circle shows critical slopes from where the beam admittedly originates. The dark star symbol shows the location where the beam impacts the thermocline (as well as in part a), and the arrows indicate the direction of energy propagation. Inset shows geographic location of the observation; M stands for Madagascar and A for the African continent (the arrow in the inset denotes the location of the image).

Figure 14 .

 14 Figure14. EXP 22, comparison of time series of the vertical velocity dζ/dt extracted from the ultrasonic probes and from the PIV images at the closest location to the ultrasonic probes, for probes located at (a) x = 0.62 m and (b) x = 1.20 m. The data do oscillate around 0 but are arbitrary shifted of ±0.1 cm s -1 to make the comparison easier.

† Email address for correspondence: mmercier@mit.edu

We thank Samuel Viboud and Henri Didelle for their help during the experiments, Marc Moulin for the cams design and acknowledge helpful discussions with Joel Sommeria. We also thank the anonymous referees for useful suggestions concerning the mean flow. The experiments were supported by funds from the Hydralab III Transnational Access Program (6th FP) and ANR PIWO (contract number ANR-08-BLAN-0113). We also thank MIT-France for partially funding the travel expenses of Manikandan Mathur.

Appendix B. Modal identification from SAR images

We present in Figure 15 two examples of SAR images with signatures of mode-1 and mode-2 ISWs. The internal waves are moving from right to left.

From top to bottom, the horizontal schematic profiles on the right-hand-side of the SAR images represent the following features: the SAR image intensity profile along ISW propagation direction, with bright enhanced backscatter (b) preceding of following dark reduced backscatter (d) in the direction of propagation; the surface roughness representation indicating how rough (r) and smooth (s) varies along an internal wave train in relation to isopycnal displacements, below; the surface current variability induced by internal waves (note the surface field convergence over the leading slopes of ISWs of depression, and the divergence over the rear slopes); and finally the isopycnal displacements produced by mode-1 or mode-2 internal wave propagation (note the indications of convergence and divergence fields near the surface).

The horizontal profiles represent the same features in both cases. However, one can note how the SAR signatures, roughness field, current field and convergence and divergence patterns are reversed for mode-2 ISWs in relation to mode-1 ISWs, leading to a clear discrimination of the modal structure of the propagating ISWs.


Jesus Gonzalez-Feliu

Logistics Pooling for City Supply

Although often seen as essential for keeping city centers alive and thriving, urban logistics is usually associated with various nuisances, such as noise, pollution, and/or safety problems. Numerous solutions and actions of different natures have been experimented with around the world during recent years. Even if improvements have been measured, many of the actions highlight the difficulties facing stakeholders who want to collaborate. Collaborative logistics still appears to be a sweet dream, and goods are not yet consolidated in a way to rationalize transport flows and reduce the nuisance attributed to urban goods transport. However, Since 2010, collaboration and resource pooling are seen as potential subjects of research to advance a better rationalization of good transport flows in cities [START_REF] Gonzalez-Feliu | Design and scenario assessment for collaborative logistics and freight transport systems[END_REF].

It is important to recall that the supply chain links raw materials to a final product located at the final consumption place (or to its sales place) in an interconnected set of processes, stages, goods, and information flows and with their related stakeholders. In this context, an action or modification made during the last miles of this supply chain can have impacts in all stages [START_REF] Allen | Sustainability strategies for city logistics[END_REF]. For that reason, it is important to approach the question of resource pooling and collaborative logistics using a systemic viewpoint, focusing on the different stages of the supply chain and not only on the final distribution to customers.

These collaboration and resource-sharing strategies can be of two types [START_REF] Gonzalez-Feliu | Design and scenario assessment for collaborative logistics and freight transport systems[END_REF]: horizontal collaboration takes place among stakeholders who act at a same stage (or echelon) of the supply chain and belong (or can belong) to different supply chains and are (or can be) in competition; vertical collaboration takes place among stakeholders who act in different (but correlated) echelons of the same supply chain and are complementaryin that coordination and collaboration seem more natural.

Collaborative logistics is a popular subject in supply chain management research [START_REF] Barratt | Understanding the meaning of collaboration in the supply chain[END_REF][START_REF] Jain | Supply chain collaboration: A governance perspective[END_REF][START_REF] Simatupang | The collaborative supply chain[END_REF][START_REF] Stefansson | Collaborative logistics management and the role of third-party service providers[END_REF]. However, collaboration in city logistics historically has been addressed in public-private collaboration, in a public planner's viewpoint, without entering in detail into private-private relations and supply chain interactions [START_REF] Gonzalez-Feliu | Sustainable urban logistics: Concepts, methods and information systems[END_REF]Taniguchi & Thompson, 2014). In the last five years, communities of practice have highlighted the need to address collaborative urban logistics and resource pooling also using a supply chain viewpoint for better efficiency of actions and practices, at both research and practice levels [START_REF] Gonzalez-Feliu | Collaborative transportation sharing: from theory to practice via a case study from France[END_REF]. This increasing interest by research and practice stakeholders led us to develop the present special issue of Supply Chain Forum: An International Journal entitled "Logistics Pooling for City Supply." The issue focuses on logistics pooling actions and methods created to improve the city's supply efficiency, not only reflecting the viewpoint of last-mile logistics but also taking a systemic viewpoint of collaborative logistics and, more precisely, of resource pooling. This choice takes into account that urban logistics need to be planned and managed in relation to supply chains, and that supply chain choices also can affect urban logistics. This issue focuses then on proposals that facilitate collaborative and pooling-based logistics in order to implement freight flow pooling actions to deliver to cities. Such approaches can take part at different echelons of the supply chain but have as an ultimate target delivering to final customers in urban zones. The refereeing process has resulted in eight articles representing different natures and disciplinary provenances.

One of the most interesting topics that is nowadays less explored in city logistics is the question of quantifying the flows that enter the city. In this context, Guerrero, Nierat, and Proulhac aim to identify the role of urban areas in the freight system and examine the nature and intensity of interurban freight flows in France from the French shippers' survey in a vision of urban goods needs. The authors sketch a first picture of the geography of freight flows with regard to the light they shed on economic interactions between urban centers. Their analysis shows the hierarchical nature of inter-urban freight flows to supply cities in France: it appears from their work that large urban areas are supplied by smaller ones. The authors also propose practical implications of their work. Also focusing on goods that have to be transported to cities, that is, on inbound flows, Lendjel and Fischman analyze the transaction costs formation of river transport, mainly through canalizing flows to cities. Based on the analysis of existing container barge transport (CBT) chains, the authors point out the following obstacles impeding their use for urban river logistics: the complexity of these chains on the one hand and the level of specificity of assets involved in the loading-unloading phases on the other hand. On the basis of transaction costs economics theories, the authors show how several innovations involving barge transport to supply French cities share a common aim to diminish transaction costs. Finally, the article points out the need to coordinate and pool resources to help integrated governance structures organize regular inland shipping lines necessary to supply dense French urban areas.

In the proximity of cities, and mainly in small and medium urban areas, transport is organized by regional or inter-regional strategies, and in some supply chains transportation routes can contain items for urban customers from different cities, which have to be delivered the same day with the same vehicle and with time and demand variability constraints that make it difficult to tactically plan this type of distribution. Mancini addresses this difficulty to optimize urban delivery routes by taking into account that they are inserted into regional logics by putting in synergy all those constraints resulting in a unified framework. First, the author defines and characterizes the main types of constraints related to this type of distribution and, via a literature analysis, presents the main combinatorial optimization problems developed to deal separately with these constrains. After that, the article introduces a real-life vehicle-routing problem, as well as a real case related to regional freight distribution in Piedmont (northwest Italy), which is assessed by a fast heuristic method. The article concludes by discussing the portability of the proposed framework.

On the last echelon of supply chains we find interface with end consumers. In this context, Blanquart and Carbone analyze collaboration in supply chains in terms of proximity, focusing on the needs of urban consumers. The authors start by reviewing the literature, which is completed by the deployment and analysis of 23 semi-directive interviews to define a typology of proximity after examining the collaboration opportunities and limitations for each category. The authors state that the nature of the relationships among the different actors in the collaboration shows different degrees of importance related to environmental concerns and potential sustainable performance of the SC. The article concludes by presenting and discussing several would-be preconditions for developing effectively collaborative supply chains.

Another level of collaboration can be that of warehousing. Indeed, consumers in urban areas need goods but the management of their supply is often steered by production or inventorying strategies. Pellicanò, Gatusso, and Cassone propose a framework to simulate and assess collaborative warehouse management policies. This work is based in the concept of intelligent warehouses, ITS, and specific automation. After setting the background for the functional management of logistics platforms using ITS and specific automation, the authors propose the specification, calibration, and test of a stochastic, dynamic, discrete-event micro-simulation model to analyze inbound trucks and goods management. The final goal is to reduce the time costs related to receiving goods. After calibrating the model, the authors propose a case to illustrate possible applications of the proposed framework.

The three last articles of this issue are related to one of the key subjects in urban logistics, which deals directly with the question of resource pooling and collaboration: urban consolidation. First, a simulation framework that addresses the questions of collaboration among carriers using urban consolidation centers (UCCs) is presented. Then, two complementary articles address the question of collaboration and economic viability of UCCs via a set of case studies.

Battaia, Faure, Marquès, Guillaume, and Montoya-Torres contribute to the assessment of UCC's impacts by examining the collaborative aspects of carriers in competition that (or potentially can) use the UCC. Authors mobilize several fields of knowledge, such as operational research, game theory, and transportation science among others, into an integrated assessment framework that aims to anticipate the activity level of a UCC and determine the condition under which it generates benefits for carriers. The aim is to provide a suitable support to public decision makers in their territorial management choices. The authors apply the proposed framework in a test case for the city of Saint-Etienne, France, as an illustration, and conclude by addressing the implications of the obtained results for public decision makers.

Gonzalez-Feliu, Malhéné, Morganti, and Morana also address the question of urban consolidation viability by a systematic qualitative analysis to study the levers that affect the success factor of UCC deployment. The authors focus on two macro categories of UCCs: those deployed to deliver to the city (or city center) and located in peripheral areas and those located close to a specific area to serve it. Starting from an initial typology of experiments,

obtained from a systematic analysis of 34 experiences from two countries, France and Italy, the authors orient the study toward the analysis of six models of cooperation strategies set up to reach an economic viability of these platforms. Those models are then explored in depth on the basis of a four-element grid: steering, organization, development, and financing. From the proposed typology and the analysis of the six case experiences, the authors reflect on the main elements that influence the search for economic viability in urban consolidation and distribution platforms. The article concludes describing the need for collaboration and practical implications of the analysis for UCC deployment.

Finally, Allen, Browne, Woodburn, and Leonardi also examine UCC viability through the viewpoint of interfacing with the supply chain. If the two preceding works regard horizontal collaboration in UCCs, this article focuses on vertical collaboration, that is, how UCCs can be used in the supply chain to reduce last-mile traffic and its associated environmental impacts while also helping to make supply chains more responsive and efficient and thereby generate commercial benefits. After presenting the role of UCCs and discussing their various types, the authors present six case studies of UCC schemes and trials, comparing them on the basis of the following three-element grid: objectives, operational characteristics, and impacts. The authors examine critical success factors associated with UCCs and conclude with practical implications for the development of those types of platforms.