Interactions non-linéaires d’ondes et tourbillons en
milieu stratifié ou tournant

Generation of a plane inertial wave

In order to generate a plane inertial wave, we have made use of a wavemaker, introduced by Gostiaux et al. [26], which was originally designed to generate internal gravity waves (see Mercier et al. [48] for a detailed characterization of the wavemaker). This wavemaker consists of a series of oscillating stacked plates, designed to reproduce the fluid motion in the bulk of an internal gravity wave invariant along y. The use of this internal wave generator for the generation of inertial waves is motivated by the similarity of the spatial structure of the two types of waves in the vertical plane (x, z). However, the fluid motion in the internal wave is a simple oscillating translation in the direction of the group velocity, whereas fluid particles describe anticyclonic circular translation in the case of inertial waves. As a consequence, the oscillating plates of the wavemaker only force the longitudinal component of the circular motion of the inertial waves, whereas the lateral component is let to freely adjust according to the spatial structure of the wave solution. The wavemaker is made of a series of 48 parallelepipedic plates stacked around a helical camshaft, with the appropriate shifts between successive cames in order to form a sinusoidal profile at the surface of the generator. We introduce the local coordinate system (ξ, y, η), tilted at an angle θ about y, where ξ is along the wave propagation and η is parallel to the camshaft axis (see Fig. 3.1). The group velocity and the phase velocity of the wave are oriented along ξ and η respectively. As the camshaft rotates at frequency σ0, the plates, which are constrained in the y direction, oscillate back and forth along ξ. The sign of the rotation of the helical camshaft selects the helicity of the excited wave, and hence an upward or downward phase velocity. In the present experiment, the rotation of the camshaft is set to produce a downward phase velocity, resulting in a left-handed inertial wave of negative helicity s0 = −1. The cames are 14 cm wide in the y direction, and their eccentricities are chosen to produce a sinusoidal displacement profile, ξ0(η) = ξo sin(κ0η), of wavelength λ = 2π/κ0 = 7.6 cm and amplitude ξo = 0.5 cm at the center of the beam. The wave beam has a width 30.5 cm with a smooth decrease to 0 at the borders, and contains approximately 4 wavelengths. The generator is only forcing the ξ component of the inertial wave, and the y component is found to adjust according to the inertial wave structure after a distance of order of 2 cm.

INSTABILITÉ PARAMÉTRIQUE SOUS-HARMONIQUE D’ONDES INERTIELLES

The wavemaker is placed in a tank of 120 cm length, 80 cm width and 70 cm depth which is filled with 58 cm of water. The tank is mounted on the precision rotating platform “Gyroflow” of 2 m in diameter. The angular velocity Ω of the platform is set in the range 1.05 to 3.15 rad s−1 , with relative fluctuations ∆Ω/Ω less than 10−3 . A cover is placed at the free surface, preventing from disturbances due to residual surface waves. The rotation of the fluid is set long before each experiment (at least 1 hour) in order to avoid transient spin-up recirculations and to achieve a clean solid body rotation. The propagation angle θ of the inertial wave is varied by changing the rotation rate of the platform, while keeping the wavemaker frequency constant, σ0 = 1.05 rad s−1 . This allows to have a fixed wave amplitude σ0 ξo = 0.52 cm s−1 for all angles. The Coriolis parameter has been varied in the range f = 1.004 σ0 to 3 σ0, corresponding to angles θ from 5o to 70o . For each value of the rotation rate, the axis of the wavemaker camshaft is tilted to the corresponding angle θ = cos−1 (σ0/f), in order to keep the plate oscillation aligned with the fluid motion in the excited wave. As a consequence, the efficiency of the forcing should not depend significantly on the angle θ. For each experiment, the fluid is first reset to a solid body rotation before the wavemaker is started. 2.3 PIV measurements Velocity fields are measured using a 2D particle image velocimetry (PIV) system [23, 51] mounted on the rotating platform. The flow is seeded by 10 µm tracer particles, and illuminated by a vertical laser sheet, generated by a 140 mJ Nd :YAG pulsed laser. A vertical 59×59 cm2 field of view is acquired by a 14 bits 2048×2048 pixels camera synchronized with the laser pulses. For each rotation rate, a set of 3200 images is recorded, at a frequency of 4 Hz, representing 24 images per wavemaker period. This frame rate is set to achieve a typical particle displacement of 5 to 10 pixels between each frame, ensuring an optimal signal-to-noise ratio for the velocity measurement. PIV computations are performed over successive images, on 32×32 pixels interrogation windows with 50% overlap. The spatial resolution is 4.6 mm, which represents 17 points per wavelength of the inertial wave. Figure 3.2 shows typical instantaneous horizontal velocity fields after 2 and 7 periods T = 2π/σ0 from the start of generator, for an experiment performed with σ0/f = 0.84. A well defined truncated plane wave propagates downward, making an angle θ = cos−1 (σ0/f) ≃ 34o to the horizontal. The front of the plane wave is propagating at a velocity 8.3±0.6 mm s−1 , which agrees well with the expected group velocity cg = f sin θ/κ = 8.5 mm s−1 . The phase velocity is downward, normal to the group velocity, and also agrees with the expected value cϕ = σ0/κ = 12.7 mm s−1 . Two sources of noise have been identified, which can be seen in the temporal energy spectrum of the velocity fields (Fig. 3.3, described in the next subsection) : an oscillatory motion at frequency σ = Ω = 0.5f, due to a residual modulation of the rotation rate of the platform, and slowly drifting thermal convection structures at frequency σ → 0, due to slight temperature inhomogeneities in the tank. Both effects contribute to a velocity noise of order of 0.2 mm s−1 , i.e. 25 times lower than the wave amplitude close to the wavemaker. This noise could be safely removed using a temporal Fourier filtering of the velocity fields at the forcing frequency σ0.  Figure 3.2 – Horizontal velocity field after 2 and 7 periods from the start of the wavemaker for σ0/f = 0.84. The wavemaker is on the top-right, forcing a wave propagating along cg with a phase propagating along cϕ. filtering however fails in the particular case where σ0 = Ω, for which the mechanical noise of the platform cannot be filtered out of the inertial wave signal. The wavemaker is found to successfully generate well defined plane waves for frequencies σ0 ≥ 0.65f. For lower frequency, i.e. for steeper angle of propagation [θ = cos−1 (σ0/f) > 50o ], the wave pattern shows significant departure from the expected plane wave profile, which may be attributed to the interference of the incident wave with the reflected wave on the bottom of the tank. 3 Subharmonic instability 

Experimental observations

After a few excitation periods, the front of the inertial wave has travelled outside the region of interest, and the inertial wave can be considered locally in a stationary regime. However, after typically 15 wavemaker periods (the exact value depends on the ratio σ0/f), the inertial wave becomes unstable and show slow disturbances of scale slightly smaller than the excited wavelength. We have characterized this instability using Fourier analysis of the PIV time series. We compute, at each location (x, z) of the PIV field, the temporal Fourier transform of the two velocity components over a temporal window ∆t, uˆσ(x, z) = 1 √ 2π Z t0+∆t t0 u(x, z, t)e iσtdt. (3.13) The temporal energy spectrum is then defined as E(σ) = h|uˆσ| 2 ix,z, (3.14) where h·ix,z is the spatial average over the PIV field. If we compute E(σ) over a temporal window ∆t spanning a few excitation periods, we observe, as t0 is increased, the emergence of two broad peaks at frequencies smaller  Figure 3.3 – Temporal energy spectra for two experiments performed at rotation rate Ω = 0.63 rad s−1 with (continuous line) and without (dashed line) the wavemaker operating at σ0/f = 0.84. The spectrum with the generator working has been computed on the time interval between 24 and 116 periods after the start of the generator. The peak at σ/f = 0.5 present in the two spectra is the trace of the mechanical noise of the platform at the rotation frequency σ = Ω, whereas the low frequencies are due to thermal convection effects (see text). than the excitation frequency σ0, suggesting the growth of a subharmonic instability. These two subharmonic peaks can be seen in Fig. 3.3, for an experiment performed at rotation rate Ω = 0.63 rad s−1 with the wavemaker operating at σ0/f = 0.84. Here, the temporal window ∆t is chosen equal to 92 wavemaker periods, yielding a spectral resolution of ∆σ = 2π/∆t ≃ 9 × 10−3 f. The two secondary peaks are centered on σ1/f = 0.25 ± 0.03 and σ2/f = 0.59 ± 0.03, and their sum matches well with the forcing frequency σ0/f = 0.84, as expected for a subharmonic resonance. The significant width of the secondary peaks, of order 0.07 f, indicates that this resonance is weakly selective. This broad-band selection will be further discussed in Sec. 4.2. The subharmonic instability of the primary wave is found for all forcing frequencies σ0 ranging from 0.65f to f ; the measured frequencies σ1,2 are given in Tab.

The absence of clear subharmonic instability at lower forcing frequency may be due to an intrinsic stability of the primary wave for σ0 < 0.65f, or to the low quality of the plane wave at steep angles because of the interference with the reflected wave beam on the bottom of the tank. Using temporal Hilbert filtering [14, 47], the spatial structure of the wave amplitude uo(x) and phase ϕ(x, t) = k · x − σt can be extracted for each secondary wave. The procedure consists in (i) computing the Fourier transform uˆσ(x, z) of the velocity field according to Eq. (3.13), with a temporal window ∆t of at least 42 excitation periods ; (ii) band-pass filtering uˆσ(x, z) around the frequency of interest σ1 or σ2 with a bandwidth of δσ = 2.0 10−2f, but without including the associated

SUBHARMONIC INSTABILITY

Table 3.1 – Frequencies of the secondary waves σ1/f and σ2/f, determined from the peaks in the temporal energy spectra, as a function of the frequency of the primary wave σ0/f. The uncertainty for σ1/f and σ2/f is ±0.03. negative frequency ; (iii) reconstructing the complex velocity field by computing the inverse Fourier transform (including a factor 2, which accounts for the redundant negative frequency, in order to conserve energy), uH(x, t) = uo(x) e iϕ(x,t) . (3.15) The physical velocity field is finally given by Re(uH). The wave amplitude uo and phase field ϕ are finally obtained from the Hilbert-filtered field uH. In Figs. 3.4(c) and (d), for the experiment at σ0/f = 0.84, we show the maps of the phase of the secondary waves, extracted from Hilbert filtering at frequencies σ1 and σ2 respectively. It is worth to note, as can be verified from Fig. 3.3, that the corresponding typical velocity amplitude is at least ten times smaller than for the primary wave [see Fig. 3.4(a)]. The spatial structures of the phase of these secondary waves are not as clearly defined as for the primary wave [Fig. 3.4(b)]. In particular, dislocations can be distinguished in the phase field. The finite extent of the primary wave and its spatial decay due to viscous attenuation are probably responsible for this departure of the secondary waves from pure plane waves. It is also important to note that the monochromaticity of the first subharmonic wave [Fig. 3.4(c)] is affected by interferences with its reflection on the wavemaker which is due to the fact this secondary wave is propagating toward the wavemaker. However, to a reasonable degree of accuracy, the two secondary waves can be considered locally as plane waves, characterized by local wavevectors k1 and k2.

Helical modes

The approximate plane wave structure of the two secondary waves suggests to analyze the instability in terms of a triadic resonance between the primary wave, of wavevector k0, and the two secondary waves, of wavevectors k1 and k2. This resonance may be conveniently analyzed in the framework of the helical decomposition, introduced by Waleffe [71, 72], which we briefly recall here. Helical modes have been introduced as a general spectral decomposition basis, which is useful to analyze the energy transfers via triadic interactions. Although this decomposition also applies for non-rotating flows, it is particularly relevant for  Figure 3.4 – Hilbert filtered vertical velocity (a) and phase (b) of the primary wave at σ0/f = 0.84, and phase of the Hilbert filtered first [(c), σ1/f = 0.25] and second [(d), σ2/f = 0.59] subharmonic waves. The phase is displayed only where the wave amplitude is larger than 1.3 10−1 σ0ξ0 for (b) and 7.7 10−3 σ0ξ0 for (c) and (d). In (a), the square in dashed lines indicates the region where the primary wave amplitude A0 has been measured. rotating flows, because inertial plane waves have exactly the structure of helical modes [72]. Any velocity field can actually be decomposed as a superposition of helical modes of amplitudes Ask (k, t), u(x, t) = X k X sk=±1 Ask (k, t) hsk (k)e i(k·x−σ k sk t) , (3.16) where σ k sk is the frequency associated to a plane wave of wavevector k and helicity sign sk. The helical mode hsk (k) is normal to k (by incompressibility), and given by hsk (k) = k |k| × k × ez |k × ez| + isk k × ez |k × ez| , (3.17) where sk = ±1 is the sign of the mode helicity 1 . Injecting the decomposition (3.16) into the Navier-Stokes equation (3.1) yields  ∂ ∂t + νκ2  Ak = 1 2 XC skspsq kpq A ∗ pA ∗ q e i(σk+σp+σq)t , (3.18) 1. The definition of the helical mode used here corresponds to the one in Ref. [60]. In Refs. [71] and [72], the helical mode is defined as the complex conjugate of Eq. (3.17), resulting in a sign change of the helicity. 30 3. SUBHARMONIC INSTABILITY with stars denoting complex conjugate, and Ak, σk being short-hands for Ask (k, t), σ k sk . In Eq. (3.18), the sum is to be understood over all wavevectors p and q such that k + p + q = 0 and all corresponding helicity signs sp and sq. In the following, the equation k + p + q = 0 will be referred to as the spatial resonance condition for a triad of helical modes. The interaction coefficient is given by C skspsq kpq = 1 2 [sqκq − spκp]  h ∗ sp (p) × h ∗ sq (q)  · h ∗ sk (k).