E V Ermanyuk

N D Shmakova

J.-B Flór

Internal wave focusing by a horizontally oscillating torus

Keywords: internal waves, stratified flows, topographic effects

This paper presents an experimental study on internal waves emitted by a horizontally oscillating torus in a linearly stratified fluid. Two internal wave cones are generated with the kinetic energy focused at the apices of the cones above and below the torus where the wave amplitude is maximal. Their motion is measured via tracking of distortions of horizontal fluorescein dye planes created prior to the experiments and illuminated by a vertical laser sheet. The distortion of the dye planes gives a direct access to the Lagrangian displacement of local wave amplitudes and slopes, and in particular, allows us to calculate a local Richardson number. In addition particle image velocimetry measurements are used. Maximum wave slopes are found in the focal region and close to the surface of the torus. As the amplitude of oscillations of the torus increases, wave profiles in the regions of maximum wave slopes evolve nonlinearly toward local overturning. A theoretical approximation based on the theory of Hurley & Keady (J. Fluid Mech., vol. 351, 1997, pp. 119-138) is presented and shows, for small amplitudes of oscillation, a very reasonable agreement with the experimental data. For the focal region the internal wave amplitude is found to be overestimated by the theory. The wave breaking in the focal region is investigated as a function of the Keulegan-Carpenter number, Ke = A/a, with A the oscillation amplitude and a the short radius of the torus. A linear wave regime is found for Ke < 0.4, nonlinear effects start at Ke ≈ 0.6 and breaking for Ke > 0.8. For large forcing, the measured wave amplitude normalized with the oscillation amplitude decreases almost everywhere in the wave field, but increases locally in the focal region due to nonlinear effects. Due to geometric focusing the amplitude of the wave increases with √ ǫ, with ǫ = b/a and b is the mean radius of the torus. The relevance of wave focusing due to ocean topography is discussed.

Introduction

In the oceans, the interaction of the tidal motion with the bottom topography is continuously generating internal waves [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF][START_REF] Vlasenko | Baroclinic Tides: Theoretical Modeling and Observational Evidence[END_REF][START_REF] Garrett | Internal tide generation in the deep ocean[END_REF]. There is a reasonable agreement about the global rate of energy of 3.7 TW from lunar and solar tides, of which approximately 1 TW is converted into the baroclinic tide [START_REF] Morozov | Semidiurnal internal wave global field[END_REF][START_REF] Garrett | Internal tide generation in the deep ocean[END_REF]. The transfer of this energy into mixing is relevant to the general circulation in the oceans, and is therefore also of interest to climate modelling [START_REF] Wunsch | Vertical mixing, energy, and the general circulation of the oceans[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: reservoirs, sources, and sinks[END_REF].

Over the past decades, the mechanism of internal wave generation has been studied in some detail for oscillating objects of various idealized geometries. Most well known are the classical flow visualizations of [START_REF] Mowbray | A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid[END_REF] of wave rays emitted by an oscillating cylinder in the shape of a St Andrews cross. Linear theory for two-dimensional internal waves generated by the oscillation of an elliptical cylinder of [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 1. Inviscid solution[END_REF] and [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF] was shown to be in good agreement with quantitative experimental results (see [START_REF] Sutherland | Visualization and measurement of internal waves by 'synthetic schlieren'. Part 1. Vertically oscillating cylinder[END_REF][START_REF] Sutherland | Internal waves revisited[END_REF][START_REF] Sutherland | Internal wave excitation by a vertically oscillating elliptical cylinder[END_REF][START_REF] Ermanyuk | The use of impulse response functions for evaluation of added mass and damping coefficient of a circular cylinder oscillating in linearly stratified fluid[END_REF][START_REF] Ermanyuk | Force on a body in a continuously stratified fluid. Part 1. Circular cylinder[END_REF]. For ridges of Gaussian, exponential or witch-of-Agnesi shape [START_REF] Llewellyn | Conversion of the barotropic tide[END_REF] extended analytic estimates of [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF] to compare internal tidal conversion rate, an approach that later was applied to the waves generated by a vertical barrier [START_REF] Llewellyn | Tidal conversion at a very steep ridge[END_REF]. The ridge-and plateau-type geometries of specific shape are shown to generate no propagating internal waves for certain frequencies and depths [START_REF] Maas | Topographies lacking tidal conversion[END_REF].

Three-dimensional effects of internal waves are particularly considered in [START_REF] King | Tidal flow over three-dimensional topography in a stratified fluid[END_REF]. For a horizontally oscillating hemisphere they revealed the conical structure of wave beams and asymmetric bimodal structure. For moderate forcing, a flow perpendicular to the forcing direction was found, leading to a large-scale horizontal circulation. The internal wave pattern generated by a horizontally oscillating sphere was compared with the three-dimensional linear theory in Voisin, Ermanyuk & Flór (2011). This theory included viscous effects and showed good agreement with experiments at low oscillation amplitude, and also allowed to better investigate the transition from bimodal to unimodal waves. At moderate oscillation amplitude, the propagative first and second harmonics were shown to have radically different horizontal patterns and are respectively of dipole and quadrupole type in the horizontal plane (Ermanyuk, Flór & Voisin 2011). More complex geometries have also been considered. [START_REF] Bühler | Instability and focusing of internal tides in the deep ocean[END_REF] developed the linear theory for the oscillation of a ring with a subcritical Gaussian generatrix, and consider for the first time the effect of geometric focusing of wave energy into localized regions of high wave amplitude. Some examples on this effect of focusing were given. For a circular Gaussian hill and horseshoe topography also the mean flow localized in regions of wave dissipation have been considered [START_REF] Grisouard | Forcing of oceanic mean flows by dissipating internal tides[END_REF]).

As mentioned above, a main interest in internal wave dynamics is the conversion of wave energy into mixing and small-scale dissipation due to a rich variety of mechanisms including wave-wave and wave-current interactions, and overturning motions. Several scenarios have been considered for the energy concentration of internal waves in localized zones, of which we recall the most recurrent ones: (i) nearly critical reflection at continental slope (e.g. [START_REF] Dauxois | Near-critical reflection of internal waves[END_REF][START_REF] Gayen | Turbulence during the generation of internal tide on a critical slope[END_REF], (ii) energy concentration at attractors [START_REF] Maas | Observation of an internal wave attractor in a confined, stably stratified fluid[END_REF][START_REF] Echeverri | Tidally generated internalwave attractors between double ridges[END_REF][START_REF] Scolan | Nonlinear fate of internal wave attractors[END_REF][START_REF] Guo | Internal wave attractors over random, small-amplitude topography[END_REF], (iii) internal wave refraction at horizons of high density gradient (e.g. [START_REF] Mathur | Internal wave beam propagation in non-uniform stratifications[END_REF], (iv) interaction of wave beams in two dimensions [START_REF] Teoh | Laboratory study of the interaction between two internal wave rays[END_REF][START_REF] Xing | On the interaction of internal tides over two adjacent sills in a fjord[END_REF][START_REF] Zhang | Virtual seafloor reduces internal wave generation by tidal flow[END_REF] and, as mentioned above, (v) geometric focusing by three-dimensional topography [START_REF] Bühler | Instability and focusing of internal tides in the deep ocean[END_REF][START_REF] Grisouard | Forcing of oceanic mean flows by dissipating internal tides[END_REF], recently also observed near canyons [START_REF] Dale | Tidal mixing processes amid small-scale, deep-ocean topography[END_REF][START_REF] Vlasenko | Focusing of baroclinic tidal energy in a canyon[END_REF].

These latter studies (v) are novel compared to the various types of two-dimensional focusing because of the convergence of the internal wave rays. Similarly as light waves passing through a convex lens the wave rays converge to a focal point, adding a fundamentally new aspect to the internal wave dynamics. Conservation of energy flux through a surface which reduces with distance, implies a continuous increase of energy density along the rays towards the focal point and a decrease after passing through it.

Geometric focusing of internal wave energy has hardly been investigated experimentally. Wave focusing may occur near an oscillating hemisphere (see figure 3 in [START_REF] King | Tidal flow over three-dimensional topography in a stratified fluid[END_REF] and Gaussian mountain (see figure 3e in [START_REF] King | Tidal flow over three-dimensional topography generates out-of-forcing-plane harmonics[END_REF], but was not discussed. The focusing effect increases with the radius and size of the oscillating object since more energy is transported to the focal zone, but in general its effect is negligible in small-scale laboratory experiments with small oscillating spherical objects and has therefore not been reported before. First experiments with a 60 cm diameter vertically oscillating torus revealed a strong vortical motion with wave overturning and local mixing in the focal zone (unpublished results, Flór 1997). In a similar experiment in a rotating fluid, inertial waves were shown to generate turbulence in the focal zone [START_REF] Duran-Matute | Turbulence and columnar vortex formation through inertial-wave focusing[END_REF]). In the context of the tidal motion in the oceans, a horizontal oscillation is more appropriate. The horizontal oscillation direction also gives a direction to the overturning motions in the focal region, and therefore generates a mean flow that may be relevant to ocean applications (see [START_REF] Bühler | Waves and Mean Flows[END_REF].

In the present paper we investigate the wave pattern generated by a horizontally oscillating torus, the criterion for the onset of wave breaking and observations of the nonlinear aspects of the wave generation in the focal region. We focus on the linear and weakly nonlinear regime, and consider the second harmonic generation and mean flow aspects in a separate contribution. The oscillating torus (or ring) generates 'outward' moving diverging waves and 'inward' converging waves. As is known from previous studies on the oscillation of three-dimensional objects of spherical geometry (see [START_REF] Flynn | Internal wave excitation by a vertically oscillating sphere[END_REF][START_REF] King | Tidal flow over three-dimensional topography in a stratified fluid[END_REF][START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF][START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF], diverging waves gradually decrease in amplitude with distance due to the increasing cross-section of the wave cone and viscous dissipation. For the present flows under consideration, the diverging waves are indeed relatively weak. The converging waves, however, increase in amplitude with distance from the torus toward the focal regions, leading to wave amplification and breaking.

In the next section, § 2, theoretical considerations are presented on the focusing of internal waves. The description of the experimental installation and techniques is given in § 3 of the present paper. The results of experiments with internal waves are described in § 4, and the main results and possible oceanic applications are presented in § 5.

Theoretical considerations

We consider a torus of which the geometrical parameters are defined in figure 1. The torus is submerged into a uniformly stratified fluid with the buoyancy frequency N = [(-g/ρ)∂ρ/∂z] 1/2 , where ρ(z) is the density distribution in z-direction and g is the gravitational acceleration. It has a circular generatrix with a the radius of the tube (i.e. minor radius), and major radius b the distance from the tube centre to the torus centre (see figure 1a). A Cartesian coordinate system (see figure 1b) is introduced, with the z-axis pointing upwards. The origin of the coordinate system O is taken at the mean position of the centre of the torus which undergoes rectilinear harmonic oscillations with frequency ω. The non-dimensional geometry can be characterized by the aspect ratio ǫ = b/a. We assume that the torus is slender, i.e. ǫ is large. The non-dimensional coordinates X, Y and Z are introduced after normalization of x, y and z with the radius of the generatrix, a.

Let us consider first the vertical oscillations of the torus with amplitude A, which is assumed to be small compared to a so that the Keulegan-Carpenter number Ke = A/a ≪ 1. The Stokes number defined as β = ωa 2 /ν, with ν the kinematic viscosity, is assumed to be sufficiently large, i.e. β ≫ 1. In the experiments described below, β = O(100) which guarantees a sufficiently small non-dimensional boundary layer thickness δ/a. To construct an approximate solution we use an approach similar to the strip theory in marine hydrodynamics (Newman 1977a,b).

Suppose that each radial cross-section of the torus oscillates vertically and generates internal wave beams described by (3.7) in [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF]. For brevity this equation is not reproduced here. Indeed, assumptions Ke = A/a ≪ 1 and β ≫ 1 are in agreement with [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF]. This linear solution plays a role of an 'inner' solution, which is approximately valid at each radial cross-section φ = const., where φ is an azimuthal angle in the cylindrical coordinate system (r, z, φ), with r = (x 2 + y 2 ) 1/2 . For vertical oscillations the problem is axisymmetric and all cross-sections φ = const. are equivalent. Without loss of generality we consider oscillations in the plane xOz (see figure 1). The streamfunction for a single cylinder,

ψ 1 HK (x, z, t) = Ψ 1 HK (x, z) exp(iωt), (2.1)
is constructed as a sum of four streamfunctions describing the four beams of the St. Andrew cross wave pattern (see [START_REF] Sutherland | Visualization and measurement of internal waves by 'synthetic schlieren'. Part 1. Vertically oscillating cylinder[END_REF])

Ψ 1 HK (x, z) = Ψ + l + Ψ + r + Ψ - l + Ψ - r , (2.2)
where the superscripts + andrefer, respectively, to the upper and lower half-planes, whereas the subscripts l and r refer, respectively, to the beams propagating to the left and the right.

For the second cylinder the solution Ψ 2 HK (x, z) is analogous. Assuming that b/a is sufficiently large, the solution for the system of two cylinders can be written as

Ψ HK (x, z) = Ψ 1 HK (x, z) + Ψ 2 HK (x, z), (2.3)
where the appropriate choice of signs and phases in (2.2) assure that both cylinders oscillate vertically and in phase. The instantaneous vertical displacement of fluid particles due to the oscillation of two cylinders is then evaluated as

ζ t HK (x, z, t) = (i/ω) exp(iωt) ∂ ∂x Ψ HK (x, z).
(2.4)

The corresponding distribution of wave amplitudes is denoted as ζ HK (x, z). Further, we introduce the geometric correction factor for convergence (divergence) of the wave field. Let us consider a point C with coordinates (x C , z C ) in the twodimensional wave field generated by a system of two cylinders. The density of the energy flux at point C is proportional to the wave amplitude squared

(ζ HK (x C , z C )) 2 .
Since the torus is slender, we can equate the energy fluxes due to internal waves emitted by vertical oscillations of a torus of mean radius b and small radius a, and due to a system of two cylinders (see figure 1) of radius a and length πb. We obtain then

(ζ (r C , z C )) 2 πr C = (ζ HK (x C , z C )) 2 πb, (2.5)
where r C = x C at φ = 0. Finally, we obtain in non-dimensional cylindrical coordinates

ζ (R, Z) = (R/ǫ) -1/2 ζ HK (R, Z), ( 2.6)
where R = r/a. The term (R/ǫ) -1/2 assures a proper decay of the wave amplitude at infinity. Let us note that in the case of vertical oscillations of two cylinders the wave amplitude ζ HK (x, z) is symmetric with the respect to the vertical axis x = 0 and non-zero at x = 0. Therefore the approximate solution predicts a divergence of wave amplitudes of the form of X -1/2 in the focal region at X → 0. This result shows that near-field interactions should be taken into account in this zone in order to develop a more advanced linear theory. Also, in realistic situations, one can expect strong spatially localized nonlinear effects in the form of vertical jets. Such effects have been indeed observed for inertial waves in [START_REF] Duran-Matute | Turbulence and columnar vortex formation through inertial-wave focusing[END_REF] and have been shown to create a localized turbulence zone due to a cascade of instability events. Now, let us adapt the same approach in the spirit of strip theory (Newman 1977a,b) to a slender torus undergoing horizontal oscillations with amplitude A along the x-axis. The local forcing is then taken as a projection of horizontal oscillations on the radial coordinate of the form A cos φ. Obviously, equations (2.2)-(2.4) remain valid in the case of horizontal oscillations of a system of two cylinders, with the important difference that the signs and phases of motions in wave beams should now be chosen such that the wave amplitude ζ HK (x, z) = 0 at x = 0 and the instantaneous wave profiles are antisymmetric with respect to the vertical axis x = 0. The geometric conversion introduced in (2.6) remains valid but now, owing to azimuthal modulation of the wave field, (2.6) transforms into

ζ (R, Z, θ ) = (R/ǫ) -1/2 ζ HK (R, Z)|cos θ|.
(2.7)

For horizontal oscillations, ζ HK (R, Z) is proportional to R at a fixed Z in a small vicinity of R = 0, and therefore ζ (R, Z, θ ) is proportional to R 1/2 as R tends to 0. Thus, the wave amplitude remains limited but the wave slope tends to the vertical. This qualitatively indicates a possibility of overturning in the focal region, which is a qualitatively different behaviour compared to the case of vertical oscillations.

To characterize the forcing of internal waves, the Keulegan-Carpenter number Ke = A/a is often used (see e.g. [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF][START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF]. Although this number characterizes the nonlinearity close to the torus, it does not take into account the geometric effect of focusing at larger distances from the torus. Supposing that in the focal zone the wave motion can be confined to a zone with radius a, the amplitude may increase to ζ corr = ζ HK (b/a) 1/2 = ζ HK ǫ 1/2 i.e. with a factor ǫ 1/2 due to focusing. The same correction factor has been introduced earlier in [START_REF] Bühler | Instability and focusing of internal tides in the deep ocean[END_REF]. Note that the above-described theory neglects the near-field wave interactions in the focal region. Also, the viscous damping in the above solution is adapted from [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF], and therefore neglects additional shear close to the focal region. Therefore it is expected to provide a quantitative estimate with a reasonable accuracy only in the vicinity of the torus. The effects of finite values of Ke, β and ǫ in experiments are discussed below.

When oscillating an obstacle with frequency ω, in addition to the fundamental first harmonic wave oscillating at frequency ω, the dispersion relation admits wave radiation for higher harmonics with frequencies nω, where n = 2, 3, . . . , were ωn < N (see [START_REF] Mowbray | A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid[END_REF]. In contrast, for ω > N/2 all higher harmonics are evanescent. In the experiments described below the value of Ω = ω/N > 0.5 so that the higher harmonics are evanescent and the effect due to focusing of the first harmonic waves emitted by the torus can be studied in isolation.

Experimental set-up and measurement procedure

Experimental set-up

To measure the evolution of wave amplitudes and slopes with distance from the ring we use the laser induced fluorescein (LIF) technique (for details see [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF][START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF].

This technique is based on the accurate tracking of distortions of isodensity surfaces (fluorescent dye planes), and allows us to measure the amplitude of the vertical velocity with a very high precision. Since dye tracers are followed in time the recording is of Lagrangian type, in contrast with Eulerian-type techniques (particle image velocimetry or synthetic schlieren) usually employed in experimental studies on waves. For waves of small amplitude the two types of measurements yield essentially the same results. As the wave amplitude increases, Lagrangian measurements allow us to study the evolution of wave profiles toward overturning via direct measurement of wave slopes. Complementary particle image velocimetry (PIV) measurements provided information about profiles of horizontal velocity and its distribution over a horizontal plane.

Two tori have been used in the present experiments, a 'thin ring' with a = 1.5 cm and b = 13.5 cm, and a 'thick ring' with a = 2 cm and b = 10 cm. The aspect ratio ǫ = b/a is respectively equal to 9 for the 'thin', and 5 for the 'thick' one. At the experimental value of frequency ω = 0.65 rad s -1 the Stokes numbers are β = 150 and 260 respectively for the thin and the thick torus.

The experiments are conducted in a Plexiglas 100 cm cubic tank filled to a working depth of 90 cm with a linearly salt-stratified fluid using the conventional double-bucket technique (see figure 2). The stratification profile is calculated from the density of fluid samples taken at different heights in the fluid and measured with an Anton Paar density meter. The values of the buoyancy frequency N are listed in table 1.

Internal waves are generated by a horizontally oscillating Plexiglas torus that is painted black to avoid laser light reflections. It is attached to the end of a pendulum of length l = 180 cm, and the oscillations of the pendulum at frequency ω are driven by a crank mechanism. The oscillation amplitude, A, of the torus is small compared to the length of the pendulum, and the motion is therefore in good approximation horizontal and sinusoidal. The two side walls perpendicular to the direction of the oscillation of the cylinder are covered with a mesh of 5 cm thickness to avoid wave reflections. In some cases also the bottom and the back wall were covered with wave absorbers, but no difference in results could be noticed. After 10 oscillation periods the wave pattern reaches the steady state in the region of interest. Measurements are taken after 20 oscillations, which is the typical duration used to exclude the effects of internal wave transients (see [START_REF] Voisin | Limit states of internal wave beams[END_REF][START_REF] Ermanyuk | Duration of transient processes in the formation of internal-wave beams[END_REF], 2008;[START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF].

For the method of fluorescein dye planes (first employed in [START_REF] Hopfinger | Internal waves generated by a moving sphere and its wake in a stratified fluid[END_REF][START_REF] Flór | Spin-up from rest in a stratified fluid: boundary flows[END_REF][START_REF] Flór | Spin-up from rest in a stratified fluid: boundary flows[END_REF], a set of equidistant dye planes is generated by slowly displacing a rake of horizontally spanned cotton threads through the fluid. These cotton threads are soaked in a concentrated fluorescein solution and dried before the experiment. The dye planes are illuminated with a vertical laser sheet parallel to the direction of oscillations. The data processing is performed with a version of cross-correlation technique described in [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF] and [START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF], from which the vertical displacements of the dye lines are determined with the accuracy of approximately 0.05 pixel. The laser sheet is perpendicular to the dye planes and shows therefore dye lines on the recordings (see figure 9).

Prior to the onset of the oscillations, the horizontal dye planes are scanned by a laser sheet in the otherwise quiescent fluid to obtain the reference state ζ 0 for each dye plane. The vertical displacement of these dyelines, ζ ′ , is measured at a certain position in time with respect to the reference state ζ 0 , i.e.

ζ ′ (x, t) = ζ ′ -ζ ′
0 (the prime is to note that this value does not necessarily correspond to a particle displacement). The local slopes of wave profiles in plane XZ are measured as s(t) = arctan(dζ (t)/dx). In practice the slope at position (X, Z) is evaluated as

s(t) = arctan( ζ (t)/ x), (3.1)
where x should be sufficiently large compared to the accuracy of measurement of vertical displacements ζ (t) and sufficiently small compared to the radius of the generatrix a to resolve the details of wave profiles. The wave slope represents a sensitive and physically important measure of the baroclinic torque of the wave, caused by the shear. This slope is related to the Richardson number and is therefore an indication for overturning, as is discussed further in § 4.4. As described in [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF] and [START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF], the vertical displacements are evaluated as the average values over vertical stripes of width of several pixels, which defines the horizontal spatial resolution of the experimental data. In the present experiments x is taken equal to the horizontal spatial resolution, with x = 5 (≈0.253 cm) and 8 pixels (≈0.339 cm) for respectively the 'thin' and the 'thick' torus.

To reconstruct a quasi-three-dimensional distribution of the wave amplitudes (note that the velocity in y-direction is not measured), a series of images was acquired for different positions in y-direction. Therefore, the mirror reflecting the laser light (see figure 2) was moved by a computer-controlled step motor with a prescribed increment y = 0.8 cm. At each position the mirror remained fixed during one period of oscillation. This allowed to acquire time series suitable for subsequent Fourier analysis and evaluation of amplitudes of harmonic components.

For the PIV measurement the algorithm of [START_REF] Fincham | Advanced optimization of correlation imaging velocimetry algorithms[END_REF] was used, and the optimum particle size and seeding density of [START_REF] Westerweel | Fundamentals of digital particle image velocimetry[END_REF] were followed. The fluid was seeded with Orgasol 30 µm particles of density ρ = 1.2 kg m -3 , illuminated with a vertical and horizontal laser sheet. The particle displacement was obtained using the UVMAT/CIVx software package developed at LEGI (http://www.legi.cnrs.fr/web/spip.php?article763). Successive images are taken with a time increment of t = t i+1t i = 0.5 s, which is kept constant in all experiments.

Data analysis and parameters

Time series of vertical displacements ζ (t i ) are analysed using three different methods. With the first method, the amplitudes of the nth harmonic components of the signal ζ n and S n are determined via Fourier filtering of time series. Similar analysis has been used in [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF] and [START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF]. This type of data processing is particularly useful for linear or weakly nonlinear processes when the first few harmonics are dominant and can be clearly identified. With the second method, the amplitudes ζ rms and S rms are evaluated as the period-averaged root mean square of the corresponding time series multiplied by 2 1/2 [START_REF] Sutherland | Internal wave excitation by a vertically oscillating elliptical cylinder[END_REF]. In principle, this quantity takes into account the contribution of all harmonics in the signal. However, owing to the period averaging procedure the root-mean-square value cannot precisely capture the extreme values of wave amplitude and wave slope, which may instantly occur at a particular phase of oscillation. Of special interest are the extreme slopes which can trigger incipient overturning in the case wave crests become steep. Therefore, a third method of data processing was introduced. Time series ζ (t i ) and s(t i ) measured at each point were sorted to find the maximum values of ζ max = max |ζ (t i )| and S max = max |s(t i )|.

Thus with these three methods, next to the values ζ n , ζ rms and ζ max , the slopes S n , S rms and S max were measured in degrees allowing to explicitly study the transition of the wave system to overturning. The wave amplitudes are normalized generally with the oscillation amplitude, A, or in case only the focal region is of interest, the generatrix of the torus, a. The governing parameters of experimental runs performed in this study are presented in table 1.

Results and discussion

Wave pattern in XZ-plane

Figures 3(a)-3( f ) show the typical evolution of the first harmonic wave patterns for different oscillation amplitude, represented by the Keulegan-Carpenter number Ke, in terms of contours of wave amplitude ζ 1 /A and contours of wave slope S 1 . For comparison figures 3(g) and 3(h) represent the unfiltered wave fields in terms of ζ max /A and S max . A higher level of noise is observed for these non-filtered max-type quantities. Figures 3(e)-3(h) show a high resemblance indicating that at moderate oscillation amplitude, the dominant contribution to the wave field is represented by the first harmonic wave obtained after Fourier filtering, as can be expected since higher harmonics are evanescent for this forcing.

Wave amplitudes close to the torus are roughly two times smaller than the wave amplitudes in the focal region, but the wave slopes close to the torus and in the focal region are comparable. The onset of overturning occurs therefore not only in the focal region but also near to the torus. It should be noted that this latter overturning is rather due to the steep topography, an effect that would most likely be absent for a Gaussian-shaped torus. At low Ke the amplified wave amplitude in the focal region forms then two symmetric ellipses due to the antisymmetry of the instantaneous wave profiles with respect to YZ plane, where the wave amplitude is zero. As Ke increases, these regions of maximum amplitude slightly shift into the Z-direction away from the torus. The nonlinear evolution of the wave field is well visible in terms of wave slopes. At low Ke the focal region (figure 3b) is nearly elliptic. As Ke increases, the focal region takes a spearhead-like shape (figure 3f,h), indicating the higher shear in the outer region of the focal region and incipient overturning. As a consequence, the waves are hindered by nonlinear effects in the focal region.

Note also that the waves in the focal zone have a unimodal structure, i.e. there is a single wave beam due to the presence of viscosity in contrast to the bimodal structure often observed very near to oscillating objects (for a discussion see e.g. [START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF]. Though the general flow evolution was very similar to that of the thick torus (ǫ = 5), its larger generatrix a implied a larger Stokes number, (260 instead of 150) causing a rather bimodal wave structure near the torus which turned into a single beam near the focal region. In the case of very large tori, however, one may expect a multi focal region because of the bimodal wave structure. In view of the range of accessible Stokes numbers, this was not observed in the present experiments.

Comparison of measurements with linear theory

Figure 4 shows the wave amplitude across the wave beam measured in plane XZ at different distances Z from the centre of the torus. The experimental distributions of the scaled amplitude, ζ 1 (X)/A for the torus with ǫ = 9 are presented for a range of |Z| values up to the middle of the focal zone. Good agreement with the theory described in § 2 is observed at low values of Ke, and close to the torus. Thus, a simple geometric correction is fully sufficient for quantitative description of the main effects due to convergence (divergence) of waves in the vicinity of a curved elongated body. Thereby it provides a consistent framework for quantitative interpretation of the experimental results almost everywhere in the wave field, except the focal region. For the focal region, the theory shows that focusing amplifies the wave amplitude to a factor 3, in qualitative agreement with the correction factor ǫ 1/2 introduced in [START_REF] Bühler | Instability and focusing of internal tides in the deep ocean[END_REF]. This can be observed in figure 4( f ) (|Z| = 12.51) from the comparison of the calculated wave amplitudes for the torus (solid line) and the reference case of the two parallel cylinders of infinite length (dash line). The experimentally measured wave amplitudes in the focal zone are significantly lower (approximately by 30 %) than the theoretical estimate. Indeed the theory presented in § 2 neglects the near-field interactions between the wave components in the focal zone, leading to unrealistic infinite wave slopes at the vertical axis Z.

The experimental data presented in figure 4 provide qualitative information on the nonlinear trends in the wave amplitude as function of the oscillation amplitude Ke. With increasing Ke, the normalized wave amplitude of the first harmonic, ζ 1 /A typically decreases almost everywhere in the wave field except the focal region: the amplitude profiles marked by light grey and grey symbols are consistently lower than the profiles marked by black symbols. This trend is in agreement with the nonlinear trends observed for the two- [START_REF] Zhang | Experimental study of internal gravity waves generated by supercritical topography[END_REF] and three-dimensional [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF][START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF] cases. The opposite trend is seen in figure 4( f ) in the focal region at |Z| = 12.51 where higher normalized wave amplitudes ζ 1 /A correspond to higher Ke. Thus, a linear extrapolation of the low-amplitude experimental data to a high-amplitude case should be done with care: it tends to underestimate the wave amplitudes in localized zones of the focal region and to overestimate the wave amplitudes in all other regions of the wave field. Interestingly, in the middle of the focal region, located at approximately |Z| = 11.2, the linear scenario applies in a remarkably wide range of A as discussed below in § 4.4.

It is worth making a special remark concerning the data presented in figure 4(b). The low-amplitude standing wave pattern seen close to the centre of the torus can hardly be interpreted within a linear approach. The whole inner region delimited by the conical wave beams emitted by the torus seems to undergo a weak resonant motion. Its amplitude is typically an order of magnitude smaller than the amplitude of the main wave beams but at high oscillation amplitude A it is high enough to deform the straight wave beams (compare the form of isolines of wave amplitudes in the first column of figure 3 at low and high amplitudes). The mechanism providing the energy flux to this motion is unclear and will be considered elsewhere.

Wave pattern in XY-plane

To study the azimuthal distribution of the wave amplitude, the wave fields in the XY plane were reconstructed from series of images taken particularly in experiments Exp C and Exp D of table 1. Figures 5 and6 show the contours of wave amplitudes and the normalized radial distributions of the vertical velocity amplitude of the first harmonic wave, ζ 1 (R)/(A cos φ) close to the torus, with φ the angle with respect to the plane of symmetry. For this scaling, the data are found to collapse reasonably well on a more or less common curve, for both tori (see figures 5b, 6b). Also we note that the agreement with the linear theory presented in § 2 is within an error of approximately 10 % for both cases. For the thick torus (see figure 6b) the shape of the wave envelope is slightly different due to its bimodal character. Also, here the theoretical prediction is still quite reasonable (note that the original theory of [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF], used in § 2 takes bimodality and related viscous effects into account). For larger oscillation amplitudes (see figures 5d, 6d), the theoretical prediction is systematically higher than the measured values due to nonlinear effects. With increasing azimuthal angle, φ, the wave amplitude decreases, and therefore also the energy radiated by the torus. The departure of the observed profiles from the theoretical curve at φ tending to 90 • arises partially due to higher noise-to-signal ratio for these waves, and partially due to the fully three-dimensional nature of the flow. For azimuthal angles close to 90 • the cosine variation of wave amplitudes with the azimuthal angle is therefore no longer valid, especially for a thick torus. 1 /a confirms once again the expected dominance of the first harmonic wave for this forcing frequency (ω/N > 0.5). For larger Ke the increasing wave steepness and the increasing difference between S * max and S * 1 (see figure 8b) suggest the presence of nonlinear effects. Incipient overturning is observed at Ke = 0.81 and Ke = 1.1 for tori with ǫ = 5 and ǫ = 9, respectively. From the dye images shown in figure 9, we indeed recognize well-developed overturning regions in the wave pattern. As expected, these regions are located close to the surface of the torus and in the focal region, corresponding to the zones of high local slopes which can be identified in figure 3.

PIV measurements show (see figure 10a) that in the focal region, the horizontal velocity of the first harmonic wave in the direction of oscillation is of the same magnitude as the vertical velocity. With the PIV data in the focal region, the horizontal shear and thus the local Richardson number in the wave can be calculated as

Ri = N 2 (∂ û1 /∂z) 2 , (4.1)
where the stratification N is measured at the start of the experiment; the vertical gradient in velocity is measured from the PIV velocity data over a typical grid The Miles criterion is developed for a flow with vorticity due to the presence of a shear flow, and no baroclinic vorticity in its basic state. For the critical value of Ri = 0.25, vorticity of the shear flow is accumulated by the Kelvin-Helmholtz instability until breaking occurs. In the present experiments there is no shear flow, and the instability is not of the Kelvin-Helmholtz type. However, the amplitude of the waves is here represented by baroclinic vorticity. This amplitude (or baroclinic vorticity) is continuously increased due to focusing of wave energy, until the critical value is reached and breaking occurs. Though the sources of vorticity and mechanism for overturning are essentially different, the present results suggest that the criterion and, as shown below, the critical value are the same.

The LIF method allowed to precisely measure the wave slope of the wave for the different aspect ratio tori. Observations of visualizations showed a nearly linear dependence for Ke < 0.4 with the waves remaining linear and passing almost without modification through the focal region. Small changes with respect to the linear regime were found to occur for Ke around 0.6, whereas for approximately Ke > 0.8 the onset of wave breaking is observed and waves are hindered to pass through the focal region. From this wave steepness one can derive directly the local wave Richardson number which can be defined as the ratio between the buoyancy frequency N 2 , and the y-component of the baroclinic vorticity of the wave. This yields in the Boussinesq approximation,

Ri wave = N 2 ∇p × ∇ρ ρ 2 -1 y ≈ N 2 -gρ ∂ρ ∂x = x z ρ = 1 tan S , (4.2)
where the index y indicates the y component of the baroclinic vorticity vector. In figure 10(b) the wave Richardson number is displayed as a function of Ke. Even though it is derived in a different manner, it shows also that wave breaking is reached when Ri wave ≈ 0.25 at Ke ≈ 0.8 in coherence with the observations and the PIV measurements and shear instability. This local Richardson number indicates the maximum slope which an isopycnal plane may have before it becomes unstable and overturns. This overturning corresponds to a wave slope of θ ≈ 75 • (see figure 8b), the wave slope being measured over a horizontal length scale δx (see § 3.1) near X = 0 (see figure 11).

Conclusions

This paper considers the first experimental results on the geometric focusing of internal waves generated by a horizontally oscillating torus in a linearly stratified fluid. Experiments are conducted in the weakly viscous regime, i.e. for Stokes numbers between 150 and 260. The focusing leads to a strong amplification of the wave amplitude along its ray path with a maximum wave amplitude in the focal point where wave breaking is expected for certain oscillation amplitudes.

A linear wave regime is discerned for Ke < 0.4, whereas nonlinear effects start to occur at values close to Ke ≈ 0.6. Incipient wave breaking occurs around Ke ≈ 0.8, corresponding to a local (shear) Richardson number Ri = 0.25, in accordance with classical theory for shear instability. This value coincides with the Richardson number calculated from the wave steepness. Wave breaking occurs for a wave slope of θ = 75 • . Nonlinear aspects, as well as the generation of higher harmonics, that occur for these amplitudes, and the effect of higher Stokes numbers are presently under consideration and will be presented elsewhere. As a first approximation, the focusing wave field has been approached by making use of the two-dimensional theory of [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF] that was adapted by including a term for the wave convergence. This theory shows qualitative agreement with the observed wave field, and gives reasonable quantitative results up to a vertical distance of 5 times the tore radius from the tore centre. However, there is an increasing discrepancy between the data and the theory when approaching the focal zone (see figure 4), for the entire range of wave amplitudes. This difference is approximately 30 % and indicates that the present theory is not satisfactory for the study of the focal region. A three-dimensional viscous theory is at present in preparation (Bruno Voisin, personal communications).

Focusing of internal wave energy is a rather recent subject, and opens perspectives to new ways of generating turbulence in the ocean. The overturning in the present experiments has been observed at moderate Reynolds numbers when internal waves are strongly damped by viscous effects. At higher Reynolds numbers the overturning is expected to be more likely to occur. Therefore, internal wave breaking over other shapes of curved topography (see [START_REF] Bühler | Instability and focusing of internal tides in the deep ocean[END_REF]) may serve as an effective mechanism for the generation of 'hot spots' responsible for abyssal mixing. The measurements based on wave steepness allow us to make more precise estimations of the relevance of wave breaking due to wave focusing.

For a realistic bottom topography the effects of viscosity are small. Using the geometric correction for focusing, we may consider whether wave breaking is likely to occur or not. For a M2 tidal oscillation frequency of approximately 12 h and a typical stratification in the ocean one obtains Ω ≈ 0.3 corresponding to an angle of wave propagation of 17 • . Considering a large mountain of 2000 m height, and curvature 120 km, i.e. a = 1 km, and b = 60 km, the oscillation amplitude in the focal region is amplified with a factor ǫ 1/2 , implying for a typical tidal excursions of O(100 m) a wave amplitude of approximately 800 m. This amplitude is comparable to the size of the generatrix a, which is in turn comparable to the width of internal wave beams. Since the amplitude and the beam width becomes of the same order of magnitude in the focal zone, an overturning can be expected. Note that topographies with nearly critical local slopes can generate very narrow wave beams, increasing the probability of overturning events even at low Ω typical for the ocean, where Ω ≈ 0.1. Thin large ridges as well as thick small ridges of realistic sizes can thus be expected to cause overturning waves due to focusing. Geometric wave focusing could therefore be a candidate for the explanation of recently observed intense mixing regions near spur-shaped mountains (see e.g. [START_REF] Dale | Tidal mixing processes amid small-scale, deep-ocean topography[END_REF].

FIGURE 1 .

 1 FIGURE 1. Geometry of the torus: in (a) view in vertical plane and (b) three-dimensional view defining the coordinate system.

FIGURE 2 .

 2 FIGURE 2. Sketch of the experimental set-up: (a) front view and (b) side view, with the light grey lines the fluorescein dye planes and the laser plane shown in dark grey.(X, Y, Z) = (x, y, z)/a are the non-dimensional coordinates. The torus oscillates in the X-direction (plane of view), Z is in the vertical direction. The mobility of the laser plane in the Y-direction allows for the measurement of successive planes, and the reconstitution of the horizontal wave field.

  Experimental parameters, with ǫ = b/a the torus aspect ratio, Ke = A/a the Keulegan-Carpenter number and Ω = ω/N, where A and ω are respectively the oscillation amplitude and frequency and Z focal corresponds to the geometrical point of intersection of the wave rays. The buoyancy frequency, N, is represented in dimensional values.

FIGURE 3 .

 3 FIGURE 3. Contour plots of the wave amplitude ζ 1 /A (a,c,e,g) and wave slope S 1 (b,d,f,h) in the XZ-plane, with (a,b) Ke = 0.19; (c,d) 0.41 and (e,f ) 0.65. (g,h) Contour plots for ζ max /A and S max for Ke = 0.65. Exp A, with torus aspect ratio ǫ = 9. Data are obtained from LIF measurements.

FIGURE 4 .

 4 FIGURE 4. Normalized wave amplitude ζ 1 /A for Exp A at different heights Z below the centre of the torus, with (a) Z = -1.84, (b) Z = -1.84 zoomed in including negative values of X (the centre line of the torus is at X = 0); (c) Z = -4.48; (d) -7.17; (e) -9.84 and ( f ) -12.51. The focal zone is located at Z = -11.2. The dashed and solid lines represent, respectively, the theoretical profile (2.6) obtained for two oscillating cylinders, and the profile corrected for convergence; black, grey and light grey dots correspond respectively to Ke = 0.19, 0.41 and 0.65. Data are obtained from LIF experiments.

FIGURE 5 .

 5 FIGURE 5. Contours of the normalized wave amplitude ζ 1 /A (a,c) and the corresponding radial distributions ζ 1 (R)/(A cos ϕ) (b,d) close to the torus for (a,b) Ke = 0.17 and (c,d) Ke = 0.63, where R = r/a is the non-dimensional radial coordinate. The black lines in (b) and (d) corresponds to (2.6). Exp C in table 1 (Z = -3.6, ǫ = 9). Data are obtained from LIF measurements.

FIGURE 6 .FIGURE 7 .

 67 FIGURE 6. As in figure 5 for the thick torus, with (a,b) Ke = 0.13 and (c,d) Ke = 0.29, where R = r/a is the non-dimensional radial coordinate. The black line corresponds to ζ corr . Exp D in table 1 (Z = -3.0, ǫ = 5). Data are obtained from LIF experiments.

FIGURE 8 .FIGURE 9 .

 89 FIGURE 8. Maximum wave amplitude (a) and maximum wave slope (b) against Ke. Filled and open symbols are as in figure 7 with circles for the thin torus with ǫ = 9 (at Z = -12.51), and squares for the thick torus with ǫ = 5 (at Z = -7). The dashed line at θ = 75 • indicates the slope for which overturning is observed.

FIGURE 10 .

 10 FIGURE 10. PIV measurement results with (a) horizontal and vertical velocity amplitude (respectively grey dots and black dots) along the X-axis in the focal zone at Z = -4.7 for ǫ = 5, with Ke = 0.19 (filled dots) and Ke = 0.6 (open dots); (b) Richardson numbers Ri from PIV data (grey symbols) for Exps F and G, and Ri wave from LIF data (black symbols) of experiments A and B in table 1) with squares for ǫ = 5, and circles for ǫ = 9. The vertical dashed line denotes the Ke = 1, and the horizontal line Ri = 0.25 corresponding to the onset of overturning.

  FIGURE 11. (Colour online) Images of dye visualizations of the internal wave field for different oscillation amplitudes A/a: (a) 0.19, (b) 0.41, (c) 0.65, (d) 0.84, (e,f ) 1.1. Images (a-d) were taken after 10 periods of oscillation (stable regime), (e) and ( f ) were taken after 10 and 12 oscillation periods, respectively.

Acknowledgements

The authors acknowledge the referees for a careful reading of the paper, B. Voisin for helpful discussions on the theory and the technical assistance of P.-A. Barraud with the experiments that were conducted at the LEGI. E.V.E. gratefully acknowledges the University of Grenoble Alpes for his appointment as a visiting professor at LEGI in February-March 2011 and partial support from IMC NSU. This work has been supported by LabEx Osug@2020 (Investissements d'avenir ANR10LABX56).


Natalia Shmakova
email: shmakova@hydro.nsc.ru

Evgeny Ermanyuk

Jan-Bert Flór
email: jan-bert.flor@legi.cnrs.fr

Generation of higher harmonic internal waves by oscillating spheroids

Oscillating bodies in stratified fluids may emit higher harmonics in addition to fundamental waves. In the present experimental study, we consider higher harmonics of an internal wave field generated by a horizontally oscillating spheroid in a linearly stratified fluid for moderate to high oscillation amplitudes, i.e., scaled oscillation amplitude A/a 0.5, with a the minor radius of the spheroid. Three different spheroid shapes are tested. The results are discussed in the context of the different theories on the generation of higher harmonics. Higher harmonics are observed at the intersections of fundamental wave beams, and at the critical points of the topography where the topographic slope equals the wave slope. The velocity amplitudes of the fundamental, second, and third harmonic waves grow respectively linearly, quadratically, and with the third power of the scaled oscillation amplitude A/a. Though these amplitudes are generally higher when the object's slope is larger, the increase in amplitude above and below the axisymmetric oscillating objects is found to be due to the effect of focusing. In order to discern the relative importance of the harmonics to the fundamental wave, the horizontal structure of the wave amplitude is measured. The results suggest that the nth harmonic of the internal wave field is associated with a radiation diagram corresponding to a multipole of order 2 n , with 2n directions of propagation.

I. INTRODUCTION

In the Earth's oceans, internal waves generated by the barotropic tide are known to play an important role for its energy balance [START_REF] Garrett | Internal tide generation in the deep ocean[END_REF][START_REF] Morozov | Semidiurnal internal wave global field[END_REF][START_REF] Munk | Abyssal recipes II: Energetics of tidal and wind mixing[END_REF][START_REF] Vlasenko | Baroclinic Tides: Theoretical Modeling and Observational Evidence[END_REF]. Motivated by this application, the oscillation of idealized bodies of different shapes in a linearly stratified fluid has been investigated in a large number of studies. Seen from the reference frame of the fluid, wave generation by an oscillating body in the linear regime is, to a good approximation, comparable to that of the interaction of the barotropic tide with bottom topography [START_REF] Boyer | The interaction of time-dependent rotating and stratified flow with isolated topography[END_REF][START_REF] Boyer | Motion of oscillatory currents past isolated topography[END_REF]. Most well known are the classical experiments of Mowbray and Rarity [START_REF] Mowbray | A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid[END_REF], which show that the dispersion relation for internal waves generated by a cylinder, oscillating with frequency ω in a linearly stratified fluid with buoyancy frequency N, admits for the propagation of an nth harmonic component [START_REF] Mowbray | A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid[END_REF] of the wave field if nω < N . Linear wave theory for elliptic cylinders [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders, part 2: Approximate viscous solution[END_REF][START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders, part 1: Inviscid solution[END_REF] adequately predicts the observed fundamental wave [START_REF] Sutherland | Visualization and measurement of internal waves by synthetic schlieren, part 1: Vertically oscillating cylinder[END_REF], but higher harmonics seem to be due to fully nonlinear dynamics as shown by Sutherland and Linden [START_REF] Sutherland | Internal wave excitation by a vertically oscillating elliptical cylinder[END_REF] and Zhang et al. [START_REF] Zhang | Experimental study of internal gravity waves generated by supercritical topography[END_REF]. Interaction between fundamental waves and higher harmonics lead to a complex fine-scale structure over the entire fluid depth, as has been shown numerically for a two-dimensional tidal motion interacting with a mountain ridge [START_REF] Korobov | Interharmonics in internal gravity waves generated by tide-topography interaction[END_REF].

Higher harmonics also have a three-dimensional distribution of wave amplitudes in the azimuthal direction, when generated by a three-dimensional object. For a spherical geometry, linear theory for FIG. 1. Fundamental wave (large circle) superposed on the horizontal structure of the (a) dipolar, (b) quadrupolar, and (c) octupolar wave patterns. The annular propagation of the waves and the 2 n -polar radiation pattern as cos(nφ), where φ is the azimuthal angle, is represented by light gray and the combination of the fundamental wave and higher harmonic is shown in dark gray.

the fundamental wave has been described and experimentally verified [START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF]. A detailed study of the radiation pattern of the propagative first and second harmonics (see Ermanyuk et al. [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF]) showed that first and second harmonics have dipolar and quadrupolar radiation patterns, respectively. The shapes of these patterns are shown in Fig. 1 together with the octupolar radiation pattern which we might expect for the third harmonic wave. The presence of higher harmonics has also been demonstrated for an isolated Gaussian hill by King et al. [START_REF] King | Tidal flow over three-dimensional topography generates out-of-forcing-plane harmonics[END_REF].

The aim of the present paper is to consider the generation of higher harmonic internal waves near topography of different elongation, here represented by an oscillating sphere, and a prolate spheroid, oriented either horizontally or vertically. Experimental observations are compared with the theoretical predictions of Bell [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF] for waves on curved boundaries and Tabaei et al. [START_REF] Tabaei | Nonlinear effects in reflecting and colliding internal wave beams[END_REF] and Jiang and Marcus [START_REF] Jiang | Selection Rules for the Nonlinear Interaction of Internal Gravity Waves[END_REF] for wave generation at wave beam intersections. The measurements of the horizontal flow field allow us to make a hypothesis about the horizontal structure of the nth harmonic radiation patterns.

Our paper is organized as follows. The experimental setup, measurement techniques, and the filtering methods used to visualize a specific higher harmonic wave are detailed in the next section. In Sec. III, the results obtained for the vertical and horizontal wave structure, wave amplitude, and time-frequency evolution are discussed in light of the theoretical results mentioned above, followed by brief conclusions in Sec. IV.

II. EXPERIMENTAL SETUP AND DATA PROCESSING

Experiments were conducted in an acrylic Plexiglass cubical tank with working depth of 97 cm and horizontal dimensions of 97 × 97 cm 2 , filled to a depth of 90 cm with a linearly stratified fluid [see Figs. to as V spheroid, or horizontal [Fig. 2(e)], referred to as H spheroid. The nondimensional coordinates (X,Y,Z) = (x,y,z)/a are used, with the x and y axes respectively parallel and perpendicular to the direction of oscillation of the spheroid, as shown in Fig. 2(c) and(x,y,z) = (0,0,0) at the center of the object at rest. The surface of the spheroids was painted black to avoid reflections of the laser light. The oscillations of the pendulum were driven by a crank mechanism. The motion was a good approximation horizontal and sinusoidal, with amplitude A and frequency ω.W a v e absorbers, consisting of three layers of plastic grids of 3 mm thickness and square meshes of 3 cm, were placed along the tank walls perpendicular to the plane of oscillation, effectively suppressing wave reflections. The wave pattern generally became steady after 10 oscillation periods. The same experimental apparatus has served for the experiments in Refs. [START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF][START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF].

To measure the velocity field, a two-dimensional particle image velocimetry (PIV) technique was used. The fluid was seeded with Orgasol 30-μm particles of density ρ = 1.035 kg/m 3 , which were illuminated with either a vertical or a horizontal laser sheet. The vertical laser sheet passed through the transparent bottom of the tank and was parallel to the direction of oscillation as shown in Fig. 2(a). Different vertical planes could be illuminated by moving the inclined mirror on the traverse mechanism below the tank. The position of the sheet was computer controlled and allowed for the recording of sequences of planes. For the illumination in the horizontal plane, the horizontal laser sheet was fixed at one height and the laser passed through the side wall of the tank as shown in Fig. 2(b).

In both cases, we measured velocities in the plane of view and assumed that the velocities perpendicular to it are small. To measure period-averaged quantities and to determine the amplitudes of the Fourier components of the signal in a vertical plane, the sheet was kept fixed during one oscillation period and then moved 0.5 cm to the next position, until enough vertical planes were obtained to reconstruct the vertical and horizontal longitudinal velocity components in three dimensions by spline interpolation. In the horizontal plane, long time series were taken in order to process the data with Fourier transform techniques [START_REF] Flandrin | Time-Frequency/Time-Scale Analysis[END_REF][START_REF] Mercier | Reflection and diffraction of internal waves analyzed with the Hilbert transform[END_REF] (and in particular with the Hilbert transform). The size of the visualization window was 36 × 36 cm 2 and 53 × 53 cm 2 for the vertical and horizontal planes, respectively. Images were recorded with a 12-bit DALSA digital camera with charge-coupled device (CCD) matrix of 1024 × 1024 pixels.

In the experiments, successive images of the wave field were taken at time intervals of t = 0.2s. This interval was chosen to be sufficiently small compared to the period of oscillations so that the number of images per period was large, i.e., between 56 and 84. The particle displacement was obtained by cross correlating two successive images using the velocity image correlation algorithm UVMAT/CIVx [START_REF] Fincham | Advanced optimization of correlation imaging velocimetry algorithms[END_REF]. Fourier decompositions of time series of the horizontal, u(t), and vertical, w(t), velocity were used to separate the fundamental wave and its higher harmonics. For the vertical plane, the amplitude of horizontal, u a , and vertical, w a , velocity was defined as root mean square of u(t) and w(t), respectively, multiplied by √ 2 [START_REF] Sutherland | Internal wave excitation by a vertically oscillating elliptical cylinder[END_REF]. The moduli u a and w a of the Hilbert transformation of u(t) and w(t) were taken in the horizontal plane. Nondimensional horizontal (longitudinal) and vertical velocity amplitudes are respectively given by U a = u a /Aω and W a = w a /Aω.

III. RESULTS

In a uniformly stratified fluid, the oscillation of a body generates internal waves with their generatrices inclined at the angle

θ = arccos (1)
to the vertical where = ω/N is the frequency ratio [START_REF] Peacock | The effect of rotation on conical wave beams in a stratified fluid[END_REF]. The dispersion relation (1) admits for the generation of a propagative nth harmonic component of the wave field if n < 1. The horizontal structures of the first and the second harmonics generated by a sphere have been studied by Ermanyuk et al. [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF]. The effective generation of the nth harmonic requires an appropriate choice of the nondimensional oscillation amplitude A/a and oscillation frequency . Obviously, A/a should be large enough for the amplitude of the nth harmonic, proportional to (A/a) n , to be observable. Based on the knowledge of the behavior of the first harmonic, a plausible conjecture for the frequency tuning can be formulated. For circular cylinders and spheres oscillating with a small fixed amplitude, the most efficient transmission of energy into first harmonic waves is found for between 0.7 and 0.8 (see Fig. 26 in Ref. [START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF]). Since the particular geometry of the wave generator has a weak influence on the frequency corresponding to the maximum radiated power, we can make the conjecture that the nth harmonic, irrespective of its actual generation mechanism, is most effectively generated when 0.7 <n <0.8. This condition was used for the frequency tuning in the present experiments.

A. Vertical structure

The wave patterns emitted by a horizontally oscillating prolate spheroid are shown in Fig. 3,for the H and the V spheroids in the left and right panels, respectively. This change in the orientation of the spheroid allows for the study of the interaction of vertically or horizontally elongated topography with the stratification. For both spheroids, schematic illustrations of the first and second harmonic generation is performed in Fig. 4 based on the theoretical predictions described in Tabaei et al. [START_REF] Tabaei | Nonlinear effects in reflecting and colliding internal wave beams[END_REF], Jiang and Marcus [START_REF] Jiang | Selection Rules for the Nonlinear Interaction of Internal Gravity Waves[END_REF], and Bell [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF]. In the left column of Fig. 3, one notices the generation of second harmonic waves at Z = 0, i.e., at the intersections of the fundamental waves. In agreement with the theory of Tabaei et al. [START_REF] Tabaei | Nonlinear effects in reflecting and colliding internal wave beams[END_REF] and Jiang and Marcus [START_REF] Jiang | Selection Rules for the Nonlinear Interaction of Internal Gravity Waves[END_REF], only fundamental and second harmonic waves are generated that propagate away from the obstacle when 1/3 < <1/2 [corresponding with the blue dashed lines in Fig. 4(a)]. The waves below (and above) the spheroid are generated at the boundary and appear to have their origin at the critical points, where the slope of the topography s equals the slope of the harmonic wave, i.e., [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF][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]) by (a) the horizontally oriented and (b) vertically oriented prolate spheroids, noted respectively as H and V spheroids in the text; green and red dots correspond to critical points ε 1,2 = 1, where first and second harmonics are generated. Fundamental (green lines) and second (red dashed lines) harmonic generation from critical points ε i at the boundary agrees with theory [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF]; second harmonic generation from zones of fundamental wave intersection is in agreement with Refs. [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] (blue dashed lines); the harmonic waves that are not observed in the experiment are represented by dashed gray lines. The horizontal dashed gray line determines the vertical position of the fundamental wave beams intersection Z * .

ε i = s ω 2 i / ω 2 i -N 2 = 1, i = 1,
not axially symmetric about the Z axis so that the wave rays located in the plane defined by the gradient to the surface and its horizontal projection do not intersect at a focal point. Instead, the intersections of these rays form a Gaussian-like curve in the XZ plane with the peak located near Z =-3 for the rays corresponding to the second harmonic wave. Focusing of waves emitted by an oscillating elliptically shaped ridge is discussed in more detail by Bühler and Muller [START_REF] Bühler | Instability and focusing of internal tides in the deep ocean[END_REF]. In contrast to the axisymmetric focusing of the fundamental waves on the side of the spheroid at Z = 0, the focal zone of fundamental waves below the spheroid does not generate observable higher harmonics [Fig. 3(e)] due to the three-dimensional spreading of the energy over a curved focal region.

The motion observed just below the H spheroid, close to Z =-1, is generated at the frequency of the second harmonic but does not propagate; i.e., it is not observed at a larger distance from the object [Fig. 3(e)]. The second harmonic waves we observe below are emitted at the critical points and move, due to nonlinear interactions with the sheared boundary layer, along this boundary layer before taking their original direction in agreement with the dispersion relation [Fig. 3(e)]. Video recordings showed that the second harmonic wave first deviated horizontally and therefore reaches its focal point (X,Z) ≈ (0,-3) instead of the predicted focal point in (X,Z) = (0,-4) shown in Fig. 4(a). This observation is consistent with the time-frequency results discussed in the next section.

In the right-hand column of Fig. 3, the results are shown for the vertically oriented V spheroid, for the same stratification, amplitude, and oscillation frequency. Again, fundamental and second harmonics are generated at critical points. For the V spheroid, the energy of the fundamental waves is spread along a curved zone around the spheroid instead of a focal point and therefore has a much lower amplitude. As a consequence, second harmonic waves are not observed. The sketches of the wave patterns in Figs. 4(a) and 4(b) summarize the observations in the vertical plane, showing the second harmonics (1) generated at the critical points (red dashed lines), (2) generated at in the zones of primary beam intersection (blue dashed lines), and (3) that were not observed in experiment (gray dashed lines).

B. Horizontal structure

Complementary experiments were conducted for the H spheroid to determine the position of generation of higher harmonics using a horizontal laser sheet at Z 1 =-1 and in the zone of the primary beam intersection Z * [see the horizontal line in Fig. 4(a)]. The latter value, Z * , is calculated from formula (3.14) in the linear analysis of Hurley [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders, part 1: Inviscid solution[END_REF]. For the sphere, the distance between the object surface and zone of primary wave beam intersection (|Z 1 -Z * |=0.1) appears to be too small for PIV measurements in the horizontal plane, but for the H spheroid oscillating with frequency = 0.4 this distance is large enough (|Z 1 -Z * |=0.33) to visualize the flow with the laser sheet at two different positions and conduct PIV measurements.

Distributions of amplitudes of the first and second harmonics of the horizontal velocity U a in the horizontal plane at Z =-1 are presented in Fig. 5. The second harmonic close to the object (-2 <X<2) has a quadrupolar structure. As mentioned in the previous section, this second harmonic is generated at the critical points ε 2 = 1 and is affected by the shear at the boundary layer of the spheroid. The second harmonics at |X|=4 and |X|=6 correspond respectively to the outgoing wave beams generated at the critical point ε 2 = 1 and the wave beam generated in the zone of fundamental wave intersection at Z = 0. Figure 6 shows that the amplitude of the second harmonic wave has its maximum close to the surface of the spheroid and decreases with the distance from the object, confirming once again the generation of higher harmonic waves according to Bell's [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF] theory at the object boundary. Following the approach of Bourget et al. [START_REF] Bourget | Experimental study of parametric subharmonic instability for internal plane waves[END_REF], we have determined the wave spectrum generated by the H spheroid and its evolution with time using a time-frequency representation described in Flandrin [START_REF] Flandrin | Time-Frequency/Time-Scale Analysis[END_REF]:

S u (t,ω) = +∞ -∞ u(τ )exp -iωτ h(t -τ )dτ 2 xy or xz , (2)
where h(t) is a Hamming window typically of a width of several oscillation periods, while the average is taken over an interrogation area xy or xz. Figure 7(a) shows the time-frequency diagram obtained for the logarithm of S averaged in space inside a small horizontal rectangle (-2 <X<2, -1 <Y <1,Z =-1) for the H spheroid oscillating at frequency = 0.4. Under experimental conditions, the first two harmonics are propagative, with their generation starting at t = 0, and their frequency remaining constant for all 40 oscillation periods; all higher harmonics are evanescent. The transient development of these evanescent waves is visible in Fig. 7(a) 0 <T <10, > 1. Figure 7(b) shows that for the H spheroid oscillating at frequency = 0.45 in addition to the second harmonic frequency (ω 2 = 0.9rad/s) generated below the focal zone, waves are generated at a lower frequency ω 2 ≈ 0.8rad/s above the focal zone in Fig. 3(e).

The normalized kinetic energy E was calculated from the vertical and horizontal longitudinal velocity components for waves generated by the H spheroid in the vertical plane Y = 0, where the transverse velocity component is 0 and averaged over -b<X<b for all values between -6 <Z<-1, i.e.,

E (i) = 1 2 u (i) a 2 + w (i) a 2 x 1 2 (Aω) 2 , (3)
with i = 1,2 for the first and second harmonics, respectively. The distribution of averaged kinetic energy along the vertical axis Z is shown in Fig. 8. The fundamental wave energy has its maximum close to the object boundary and decreases with distance Z. The energy of the second harmonic wave has two maxima: one close to the object (Z =-1) and another one in the zone of secondary beam intersection [Z =-2.3; see Fig. 3(e)]. Because of the focusing, the second harmonic is dominant for -6 <Z<-3. Distributions of the third harmonic component of the horizontal longitudinal velocity U (3) a and the vertical velocity W (3) a emitted by the S sphere are shown in Figs. 9(a) and 9(b), respectively. The third harmonics frequency is estimated as 3 ≈ 0.8. An octupolar structure is clearly visible, 3) of the first (dots) and second (circles) harmonic for (X,Y ) = (0,0) along Z. Experimental parameters: A/a = 0.5and = 0.45. with 6 azimuthal poles implying the amplitude variation as sin(3φ) for the horizontal velocity amplitude and cos(3φ) for the vertical velocity amplitude.

The distribution of the amplitudes of the third harmonic along the X = x/L axis is presented in Fig. 10 for objects of different curvature: S sphere, V spheroid (L = a), and H spheroid (L = b). Dye-line visualizations showed that the fluid moves well over all obstacles, confirming that the fluid motion experienced a steeper slope for the case of the V spheroid. All six azimuthal poles of the third harmonic generated by the sphere have a similar form, size, and amplitude distribution but in the case of the oscillating V spheroid, the amplitude of the azimuthal poles along the X axis is smaller compared to the amplitude of the same poles for the sphere (0.4 <X<1f o rU (3) a and 0.9 <X<1.2forW (3) a ). We notice that the structure of the wave pattern approximately scales with the horizontal radius of the object. The amplitude of the higher harmonic waves generally changes with the slope of the object, but for the V spheroid and S sphere, that possess the rotational symmetry with respect to the vertical axis, an important geometrical effect in terms of ray theory causes the waves to focus at the vertex point. For a horizontally elongated obstacle, such as the H spheroid, the energy focuses at a curve due to the difference of curvatures in the planes parallel to XZ and YZ. Thus, the focal zone (above and below the oscillating object) where the enhanced wave motions take place is not well localized for the H spheroid while it is well localized for the S sphere and V spheroid due to their rotational symmetry. Indeed, the sphere and V spheroid generate waves of the same amplitude, while the slopes are different [Fig. 10;0<X<0.4forU (3) a and 0 <X<0.9for W (3) a ]. Therefore, this focusing effect makes the principal difference between our objects. The effect of focusing internal waves have been discussed in detail by Ermanyuk et al. [START_REF] Ermanyuk | Internal wave focusing by a horizontally oscillating torus[END_REF].

The dependance of extreme values of the nondimensional amplitudes of the first three harmonics on the nondimensional oscillation amplitude is presented in Fig. 11. The logarithmic representation shows that the amplitude of horizontal (U a ) and vertical (W a ) velocity is independent of the oscillation amplitude for the normalized fundamental wave and varies as A/a and (A/a) 2 for the normalized second and third harmonics, respectively. This trend is in agreement with the two-dimensional theory of Bell [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF] and previous experimental studies on fundamental and second harmonic waves generated by an oscillating sphere [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF]. FIG. 11. Variations of the peak magnitudes of the first three harmonics with oscillation amplitude A/a for (a) horizontal (U a ) and (b) vertical (W a ) velocity fields. First, second, and third harmonics are represented with red, green, and blue colors, respectively, and compared with theoretical predictions for the sphere (red line) and linear regressions for U (2) a (W (2) a )andU (3) a (W (3) a ) (green and blue lines). Dots, circles, and squares correspond to results for oscillating S sphere, V spheroid, and H spheroid, respectively. N = 1rad/s, = 0.27.

In view of former observations on the relevance of the fundamental wave and higher harmonics [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF][START_REF] Ghaemsaidi | 3D Stereoscopic PIV visualization of the axisymmetric conical internal wave field generated by an oscillating sphere[END_REF], a note should be made on higher harmonics in three-dimensional flows. In the two-dimensional case, the energy content of the higher harmonics is usually relatively low, as demonstrated for the second harmonic in Zhang et al. [START_REF] Zhang | Experimental study of internal gravity waves generated by supercritical topography[END_REF]. However, in the three-dimensional case, such as for a sphere, the amplitude of the second harmonic can be higher than the amplitude of the first; they can be comparable for a Gaussian topography or a flow over a flattened topography [START_REF] King | Tidal flow over three-dimensional topography generates out-of-forcing-plane harmonics[END_REF]. The reason for this difference is related to the fact that, at low oscillation frequencies and fixed amplitude A/a, the power P of internal waves that are generated by horizontally oscillating bodies varies as 2 for a circular cylinder and 5 for a sphere [START_REF] Voisin | Internal wave generation by oscillation of a sphere, with application to internal tides[END_REF], and similar asymptotic behavior is valid for elliptic cylinders and spheroids [START_REF] Ermanyuk | The rule of affine similitude for the force coefficients of a body oscillating in a uniformly stratified fluid[END_REF]. Accordingly, at low frequency and fixed A/a, the amplitude of the first linear harmonic is much smaller in three dimensions than in two and can more easily be overtaken by the amplitude of the higher, nonlinear harmonics. Therefore, for low oscillation frequencies that are relevant to geophysical applications, we may have a situation where the fundamental wave generated by barotropic flow over three-dimensional topography is weak while a higher nth harmonic is strong. Reciprocally, higher harmonics are of interest for three-dimensional topography because of their high energy content under certain conditions. High oscillation amplitude and frequency can be found in wave-focusing regions at a large distance from the topography. The azimuthal harmonic wave structure may have an influence on the energy and mass fluxes in the vicinity of ocean topography.

IV. CONCLUSIONS

We have studied the generation of higher harmonic waves near oscillating boundaries of varying slope that occur for moderate to high oscillation amplitudes. Fundamental and second harmonic waves are observed at the critical points on the boundaries in agreement with the theory of Bell [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF]. For high amplitude of oscillation A/a > 0.5, overturning motions are generated in the boundary layer of a spheroid. Intersections of the fundamental wave beams lead to the generation of second harmonic waves, confirming the theoretical considerations of Tabaei et al. [START_REF] Tabaei | Nonlinear effects in reflecting and colliding internal wave beams[END_REF] and Jiang and Marcus [START_REF] Jiang | Selection Rules for the Nonlinear Interaction of Internal Gravity Waves[END_REF]. However, second harmonic waves have not been observed when intersections of fundamental wave rays form a curve instead of a point. They form a ring-shaped region in the case of the V spheroid and a curved region below and above the H spheroid. The three-dimensional spreading of the energy over a larger region results in relatively weak secondary beams emitted from these zones. This could explain their absence in the observations. It is also shown that, in agreement with previous observations [START_REF] Ermanyuk | Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere[END_REF] and theoretical predictions [START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF], the amplitudes of the first three harmonics grow respectively linearly, quadratically, and cubically with increasing oscillation amplitudes. This geometric focusing effect is also responsible for the increase in amplitude of the waves generated by axisymmetric objects.

Our observations show that the spatial structures of the radiation patterns of the first three harmonics of the wave field are respectively dipolar, quadrupolar, and octupolar. According to this trend, we expect that the nth harmonic wave radiation pattern of multipolar order 2 n has 2n azimuthal poles in the horizontal plane. Because of the complexity of the structure and in particular the narrowness of the wave, it is more difficult to visualize the radiation patterns of higher wave modes in small laboratory setups. In view of their relatively large amplitudes, we may expect them to be relevant also for the radiation of wave energy as generated by the tidal motion in the oceans.

FIG. 2 .

 2 FIG. 2. Schematic views of the experimental setup: (a) front view with vertical laser plane; (b) side view with horizontal laser plane. Geometry of oscillating objects: (c) S sphere; prolate spheroid with (d) vertical axis of revolution (V spheroid) and (e) horizontal axis of revolution (H spheroid).

FIG. 3 .

 3 FIG. 3. Instantaneous vorticity and velocity vector field of the wave field in the vertical plane of symmetry generated by a H spheroid (left column) and a V spheroid (right column): [(a), (b)] total (fundamental and second harmonic), [(c), (d)] first harmonic, and [(e), (f)] second harmonic. Experimental parameters: N = 0.89 rad/s, = 0.44,A/a= 0.5. Images are presented at the phase ±π/4fortheV and H spheroids, respectively. Dashed black lines in panels (a) and (b) correspond to extreme positions of spheroids.

FIG. 4 .

 4 FIG.[START_REF] Vlasenko | Baroclinic Tides: Theoretical Modeling and Observational Evidence[END_REF]. Schematic representation of the first and second harmonics of waves generated (according to the theory of Refs.[START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF][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]) by (a) the horizontally oriented and (b) vertically oriented prolate spheroids, noted respectively as H and V spheroids in the text; green and red dots correspond to critical points ε 1,2 = 1, where first and second harmonics are generated. Fundamental (green lines) and second (red dashed lines) harmonic generation from critical points ε i at the boundary agrees with theory[START_REF] Bell | Lee waves in stratified flows with simple harmonic time dependance[END_REF]; second harmonic generation from zones of fundamental wave intersection is in agreement with Refs.[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] (blue dashed lines); the harmonic waves that are not observed in the experiment are represented by dashed gray lines. The horizontal dashed gray line determines the vertical position of the fundamental wave beams intersection Z * .

FIG. 5 .

 5 FIG. 5. Radiation patterns for the first (a) and second harmonic wave (b) of the horizontal longitudinal velocity field at the boundary (Z =-1) of the oscillating H spheroid. Experimental parameters A/a = 0.75,N = 1.08 rad/s, and = 0.4.

FIG. 6 .

 6 FIG.[START_REF] Boyer | Motion of oscillatory currents past isolated topography[END_REF]. Profiles of second harmonic of horizontal velocity field at Z =-1 (red dots), Z =-1.47 (green dots), and Z =-1.91 (blue dots) for oscillating H spheroid. Data obtained at Y = 0 with A/a = 0.75,N = 1.08 rad/s, and = 0.4. Here the normalization X = x/L is used, with the length scale L in the direction of oscillation, i.e., L = b in the case of the H spheroid.

5 FIG. 7 .

 57 FIG. 7. Time-frequency diagram for the waves generated by the H spheroid, with (a) log 10 [S u (t,ω)/S u (t,ω 1 )] averaged over -2 <X<2, -1 <Y <1 in the horizontal plane Z =-1f o r the horizontal longitudinal velocity u, experimental parameters A/a = 0.75 and = 0.4; and (b) log 10 [S w (t,ω)/S w (t,ω 1 )] measured in -2.5 <X<2.5,Y = 0a n d-4 <Z<-1 for the vertical velocity w, experimental parameters A/a = 0.5and = 0.45.

5 FIG. 8 .

 58 FIG.8. Distribution of the kinetic energy E defined by Eq. (3) of the first (dots) and second (circles) harmonic for (X,Y ) = (0,0) along Z. Experimental parameters: A/a = 0.5and = 0.45.

FIG. 9 .FIG. 10 .

 910 FIG.9. Octupolar radiation pattern corresponding to the upper half of Fig.1(c) obtained for an oscillating sphere: gray levels visualize (a) the horizontal velocity amplitude of the third harmonic wave U(3) a and (b) the vertical velocity amplitude of the third harmonic wave W(3) a . Experimental parameters: oscillating sphere with A/a = 0.91 at Z =-1.66,N = 1rad/s, = 0.27.

ACKNOWLEDGMENTS

The authors thank B. Voisin for bringing our attention to Ref. [24], explaining the 3D geometry of the focal region and second harmonic generation. N.S. gratefully acknowledges S. Mercier and J.-M. Barnoud for their technical help and assistance with the experimental setup. N.S. thanks the French Ministry of Research for the fellowship that allowed her to conduct this work and her Ph.D. thesis at LEGI, and J.B.F. acknowledges financial support by LabEx OSUG@2020 (Investissements