2.1 Experimental set-up

The experimental set-up, shown in figure 1(a), is similar to the one described in Bourget et al. (Reference Bourget, Dauxois, Joubaud and Odier2013, Reference Bourget, Scolan, Dauxois, Lebars, Odier and Joubaud2014), with the exception that the whole apparatus is set in rotation on a turntable (PERPET, designed and constructed by GP Concept). The turntable is 1.92 m in diameter and its rotational velocity can be adjusted from $\unicode[STIX]{x1D6FA}=0$ to 60 rpm, which allows variations of the Coriolis parameter $f=2\unicode[STIX]{x1D6FA}$ from 0 to $12.5~\text{rad}~\text{s}^{-1}$ with 0.1 % precision. The axis of rotation is aligned with gravity with $2~\text{mm}~\text{m}^{-1}$ .

Figure 1. (a) Experimental set-up fixed to a rotating table, which permits the production of inertia–gravity waves in the tank, with controlled $N$ and $f$ . Synthetic schlieren and particle image velocimetry (PIV) are used to investigate the wave properties. (b) Density profile along the vertical axis $z$ expressed as the difference between the measured density and the density of pure water. (1) Profile taken before rotation, (2) profile taken after linearly accelerating the table to 9.54 rpm during 15 min and (3) profile taken after 2 h of constant rotation at 9.54 rpm ( $f=2~\text{rad}~\text{s}^{-1}$ ). It should be noted that this rotation rate, used here for the purpose of this test, is much larger than the values used for our experiments in what follows. The dashed line (4) is a linear adjustment leading to a value of $N=1.13~\text{rad}~\text{s}^{-1}$ . For the sake of clarity, curves (2)–(4) are shifted vertically by respectively 5, 10 and 15 cm.

Experiments are conducted in a rectangular tank 80 cm long and 17 cm wide filled with linearly stratified fluid, i.e. a constant buoyancy frequency $N$ , using the standard double-bucket method (Fortuin Reference Fortuin1960; Oster & Yamamoto Reference Oster and Yamamoto1963). Then, the experimental apparatus is set into rotation. In order to avoid mixing and to keep $N$ unchanged, the angular acceleration is less than $10^{-3}~\text{rad}~\text{s}^{-2}$ . Figure 1(b) displays the density profile before rotation, after spin-up of the table and after two hours of constant rotation. On a time scale much larger than the typical experimental time (ca. 20 min), the variation of $N$ is smaller than its accuracy, i.e. $\pm 0.25\,\%$ . The linear fit of the profile gives the value of the buoyancy frequency, which is of the order of $1~\text{rad}~\text{s}^{-1}$ . This is the typical value of $N$ used in all of the experiments presented in this paper. We restricted our study to the case $f<N$ . In this range of rotation rates, isopycnal deformation is very small ( ${<}0.5~\text{cm}$ ), which preserves the alignment of the background density gradient with gravity and with the rotation axis. For all of the experiments presented in this article, solid body rotation is established in the fluid.

The inertia–gravity waves are generated using a configuration identical to the one described in Bourget et al. (Reference Bourget, Dauxois, Joubaud and Odier2013). The wave generator (Gostiaux et al. Reference Gostiaux, Dauxois, Didelle, Sommeria and Viboud2006; Mercier et al. Reference Mercier, Martinand, Mathur, Gostiaux, Peacock and Dauxois2010), consisting of stacked moving plates, is set horizontally at the surface in such a way that the plates move vertically. The time-dependent vertical motion of the generator is prescribed in the form

(2.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}(x,t)=a(x)\cos (\unicode[STIX]{x1D714}_{0}t-\ell _{0}x), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{0}$ is the frequency of the oscillations. This forcing sets the horizontal component of the wavevector, denoted $\ell _{0}$ , and no component in the transverse $(y)$ direction. The amplitude $a(x)$ is constant (denoted $a_{0}$ ) over two wavelengths in the middle of the generator and is smoothly decreased over a half-wavelength on each side to avoid spurious inertia–gravity wave emission. In this paper, $\ell _{0}$ and the amplitude are kept constant ( $\ell _{0}\approx 80~\text{m}^{-1}$ and $a_{0}=5~\text{mm}$ ). The motion of the generator generates a plane wave of finite extent. The length of the tank is chosen to be long enough so that reflected waves are dissipated before coming back into the field of view.

As a preliminary step, particle image velocimetry (PIV) is performed in a horizontal plane 10 cm below the generator to observe the motion of the wave in the $(x,y)$ plane and measure the transverse component of the velocity. The fluid is seeded with $2~\text{mg}~\text{l}^{-1}$ of hollow glass particles (110 p8, sphericell). A horizontal laser sheet is created by a cylindrical lens, and observations are made via a mirror below the tank inclined at $45^{\circ }$ (see figure 1 a). The CIVx algorithm (Fincham & Delerce Reference Fincham and Delerce2000) computes the cross-correlation between two successive images, giving the instantaneous horizontal two-dimensional velocity.

This velocity measurement in a horizontal plane shows that, as expected for linear inertia–gravity waves (Sutherland Reference Sutherland2010), rotation induces a velocity component oriented in the transverse direction, $v_{y}$ , proportional to the Coriolis parameter $f$ . The main result of these measurements is, however, that no dependence in the $y$ direction is observed: the wavevector is contained in the $(x,z)$ plane, i.e. $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{e}_{\boldsymbol{y}}=\mathbf{0}$ . This result remains true in the bulk of the fluid, while in the first centimetre close to the wall a decrease of velocity is observed due to viscous boundary layers, previously observed for pure internal waves ( $f=0$ ) by Brouzet et al. (Reference Brouzet, Sibgatullin, Scolan, Ermanyuk and Dauxois2016).

Figure 2. The dimensionless frequency $\unicode[STIX]{x1D714}_{0}/N$ of an inertia–gravity wave plotted versus the angle of propagation $\unicode[STIX]{x1D703}_{0}$ defined in (2.2) for various values of the dimensionless Coriolis parameter $f/N$ ranging from 0 to 0.85. The lines are theoretical predictions (from (2.2)) over which are plotted experimental measurements. For all of these experiments $N$ was close to $1~\text{rad}~\text{s}^{-1}$ . The inset shows a typical snapshot of a density-gradient field obtained from synthetic schlieren, for the following parameters: $\unicode[STIX]{x1D714}_{0}/N=0.66$ and $f/N=0.35$ .

The absence of transverse gradients allows for the use of a synthetic schlieren technique (Sutherland et al. Reference Sutherland, Dalziel, Hughes and Linden1999; Dalziel, Hughes & Sutherland Reference Dalziel, Hughes and Sutherland2000), which is more accurate for the visualization and study of TRI (Bourget et al. Reference Bourget, Scolan, Dauxois, Lebars, Odier and Joubaud2014). The motion of the fluid is recorded in a vertical plane by a camera at 2 fps. The CIVx algorithm computes the cross-correlation between the real time and the $t=0$ background image, when the fluid is at rest. The outcome is the instantaneous two-dimensional density-gradient field $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D70C}^{\prime }(x,z,t)$ and $\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D70C}^{\prime }(x,z,t)$ where $\unicode[STIX]{x1D70C}^{\prime }(x,z,t)=\unicode[STIX]{x1D70C}(x,z,t)-\unicode[STIX]{x1D70C}_{0}(z)$ , $\unicode[STIX]{x1D70C}(x,z,t)$ and $\unicode[STIX]{x1D70C}_{0}(z)$ being the instantaneous and initial fluid densities respectively.

2.2 Wave characteristics and dispersion relation

A typical snapshot of a wave field produced by the generator and measured using synthetic schlieren is shown in the inset of figure 2. From such an image, the phase of the wave is extracted using the Hilbert transform method developed by Mercier, Garnier & Dauxois (Reference Mercier, Garnier and Dauxois2008) and previously used for pure internal waves (Bourget et al. Reference Bourget, Dauxois, Joubaud and Odier2013). From there, the components $(\ell _{0},m_{0})$ of the wavevector are obtained by linearly fitting the variation of the phase with $x$ and $z$ respectively. The error is estimated by the standard deviation of the difference between the measured values and the values predicted by the linear fit. In addition, from a single-point time series of the density-gradient field, the wave frequency, the peak of the spatial average of the fast Fourier transform of the time signal, can be extracted. Once these characteristics of the waves are obtained, one can check whether they satisfy the dispersion relation

(2.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{0}^{2}=N^{2}\sin ^{2}\unicode[STIX]{x1D703}_{0}+f^{2}\cos ^{2}\unicode[STIX]{x1D703}_{0}=N^{2}\frac{\ell _{0}^{2}}{\unicode[STIX]{x1D705}_{0}^{2}}+f^{2}\frac{m_{0}^{2}}{\unicode[STIX]{x1D705}_{0}^{2}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{0}$ is the angle of propagation such that $\boldsymbol{k}_{0}=(\ell _{0},0,m_{0})=\unicode[STIX]{x1D705}_{0}(\sin \unicode[STIX]{x1D703}_{0},0,\cos \unicode[STIX]{x1D703}_{0})$ , where $\unicode[STIX]{x1D705}_{0}$ is the magnitude of the wavevector $\boldsymbol{k}_{0}$ . Figure 2 displays the wave frequency versus the angle of propagation $\unicode[STIX]{x1D703}_{0}$ , $\arctan (\ell _{0}/m_{0})$ , for several values of the Coriolis parameter. The result is consistent with the dispersion relation (2.2) for inertia–gravity waves, shown in the figure as lines. Even if our wavemaker was designed to generate pure internal waves, these two observations demonstrate that the wavemaker successfully generates well-defined inertia–gravity waves in the vertical plane $(x,z)$ as long as $f$ is smaller than $0.85N$ . For larger values of $f$ , we were unable to obtain a well-defined wave beam: the emitted wave had a much lower amplitude and a poorly defined wavelength. This problem may result from the lack of an imposed transverse velocity from the wavemaker. For $f/N\lesssim 1$ , $v_{y}$ becomes of the same order of magnitude as $v_{x}$ , and there is a loss of efficiency of the wave generator in this range. In the study presented in this article, however, we did not need to consider cases with $f/N>0.45$ .

2.3 An unstable plane wave

Evolving in the $(\unicode[STIX]{x1D714}_{0},f)$ parameter space, the observed wave field is not always as simple as shown in the inset of figure 2. Three examples of density-gradient fields, taken at different locations of the parameter space, are shown in figure 3. In the three cases, the amplitude $a_{0}$ and the horizontal component of the wavevector, $\ell _{0}$ , are the same. The corresponding time signals taken at a given location (indicated by a star in a) are also shown, in the top left corner. The signals are fairly stable after an initial transient of less than five periods, demonstrating that steady states are established for the frequencies and Coriolis parameter values considered.

Figure 3. Snapshots of the vertical density field, $\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D70C}^{\prime }(x,z,t)$ , at $t=50~T_{0}$ displayed in an ( $f/N,\unicode[STIX]{x1D714}_{0}/N$ ) plane. The disposition of the snapshots indicates the relative importance of $f/N$ and $\unicode[STIX]{x1D714}_{0}/N$ : (a) $(\unicode[STIX]{x1D714}_{0}/N,f/N)=(0.66,0.05)$ , (b) $(\unicode[STIX]{x1D714}_{0}/N,f/N)=(0.66,0.2)$ , (c) $(\unicode[STIX]{x1D714}_{0}/N,f/N)=(0.8,0.2)$ . For the three experiments, the wavemaker has the geometrical characteristics ( $\ell _{0}=72~\text{m}^{-1}$ , $a_{0}=5~\text{mm}$ ). The top left panels $(a^{\prime })$ , $(b^{\prime })$ and $(c^{\prime })$ show time series taken in respectively experiment (a), (b) and (c) at a point located 5 cm below the wavemaker, roughly at the centre of the wave beam (indicated by a star in wave field a).

Figure 4. (a) The time–frequency diagram corresponding to the unstable wave presented in figure 3(c) ( $\unicode[STIX]{x1D714}_{0}/N=0.8$ , $f/N=0.2$ , $\ell _{0}=72~\text{m}^{-1}$ , $a_{0}=5~\text{mm}$ ). Here,  $S_{z}$ is the time–frequency spectrum of the vertical density-gradient field, computed on a $30T_{0}$ wide sliding time window. (b) The corresponding experimental measurement of secondary wave dimensionless wavevector components $\ell _{1,2}/\ell _{0}$ and $m_{1,2}/m_{0}$ of the same unstable wave. Ellipses represent the experimental errors as described in § 2.2. The black (grey) lines represent the theoretical resonance locus for $f/N=0.2$ ( $f/N=0$ ). The positions corresponding to the maximum growth rate (3.6) are indicated by two grey circles.

For the lowest frequency and Coriolis parameter (case a), the wave field is described by straight lines for the density-gradient phase, and a single-point time signal oscillates sinusoidally. However, as $f/N$ is increased (case b), there is a modification of the wave field: in the first 5 cm there is a deformation of the phase lines, below which the primary wave amplitude is significantly decreased. This corresponds to the appearance of TRI. This feature is emphasized by the modulation of the time signal (panel b), which is typical of the appearance of new frequencies in the system. When $\unicode[STIX]{x1D714}_{0}/N$ is increased as well (case c), the TRI is fully developed. To obtain more information about TRI, the time–frequency response $S_{z}(\unicode[STIX]{x1D714},t)$ of the corresponding signal (time series c) can be computed (Flandrin Reference Flandrin1999):

(2.3) $$\begin{eqnarray}\displaystyle S_{z}(\unicode[STIX]{x1D714},t)=\left\langle \left|\int _{-\infty }^{+\infty }\,\text{d}u\,\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D70C}^{\prime }(u,x,z)\text{e}^{\text{i}\unicode[STIX]{x1D714}u}h(t-u)\right|^{2}\right\rangle , & & \displaystyle\end{eqnarray}$$

where $h$ is a Hamming time window of chosen length. In figure 4(a), $S_{z}(\unicode[STIX]{x1D714},t)$ indicates the appearance of two secondary waves growing in time, with frequencies $\unicode[STIX]{x1D714}_{1}$ and $\unicode[STIX]{x1D714}_{2}$ such that $\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}=\unicode[STIX]{x1D714}_{0}$ .

The corresponding wavevectors, computed as in the previous subsection, are shown in figure 4(b) in an $(\ell /\ell _{0},m/m_{0})$ plane. They satisfy the spatial resonance condition $\boldsymbol{k}_{0}=\boldsymbol{k}_{1}+\boldsymbol{k}_{2}$ . The corresponding theoretical resonance locus, obtained using the temporal and spatial resonance conditions and the dispersion relation for each of the three waves, is also shown. This curve depends naturally on the Coriolis parameter and is similar to the one presented by McComas & Bretherton (Reference McComas and Bretherton1977). In figure 4(b), we show this locus for two cases, $f/N=0$ (no rotation) and $f/N=0.2$ . We observe that when $f/N$ increases, the allowed range for the secondary vertical wavenumber increases whereas the allowed range for the secondary horizontal wavenumber shrinks, leading to secondary waves propagating more horizontally (recall that the wavevector is perpendicular to the group velocity). The tips of the vectors $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{2}$ are located, within experimental errors, on the curve that shows the theoretical resonance locus for the corresponding Coriolis parameter $f/N=0.2$ .

As a conclusion of this first set of observations, inertia–gravity waves can either be unstable through TRI or remain stable. Rotation seems to play a key role in this picture, by favouring the instability (at least in some cases like in figure 3) but also by changing the characteristics of the secondary waves. To gain insight into the role of rotation in TRI, we first present the theoretical framework for weak TRI of a viscous plane wave and then consider finite size effects on the wave beam.

3.1 Infinite plane wave theoretical growth rate

A general theory for TRI of an inviscid inertia–gravity wave beam was previously obtained by Young et al. (Reference Young, Tsand and Balmforth2008). Viscosity, however, is not negligible in laboratory experiments. The conditions under which TRI can develop have been theoretically presented for internal waves by Koudella & Staquet (Reference Koudella and Staquet2006) and Bourget et al. (Reference Bourget, Dauxois, Joubaud and Odier2013) and for inertial waves by Bordes et al. (Reference Bordes, Moisy, Dauxois and Cortet2012). This framework for weakly nonlinear infinite plane waves including viscous dissipation can be generalized by taking into account both stratification and rotation. We consider an incompressible continuously stratified Boussinesq fluid with constant buoyancy frequency $N$ , rotating around the vertical axis parallel to gravity at a constant angular rotation $\unicode[STIX]{x1D6FA}$ (i.e. the Coriolis parameter is $f=2\unicode[STIX]{x1D6FA}$ ). We assume no variation in the transverse ( $y$ ) direction, which is verified by experimental observation. The velocity is written as ( $v_{x}=\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D713}$ , $v_{y}$ , $v_{z}=-\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713}$ ), $v_{y}$ being the velocity in the transverse direction and $\unicode[STIX]{x1D713}$ the streamfunction of the in-plane flow. After some calculations of weak nonlinearities, fully detailed in appendix A, one can obtain the evolution equations for the amplitudes of waves 0, 1 and 2:

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\unicode[STIX]{x1D6F9}}_{0}=I_{0}\unicode[STIX]{x1D6F9}_{1}\unicode[STIX]{x1D6F9}_{2}-\frac{1}{2}\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}_{0}^{2}\left(1+\frac{f^{2}m_{0}^{2}}{\unicode[STIX]{x1D705}_{0}^{2}\unicode[STIX]{x1D714}_{0}^{2}}\right)\unicode[STIX]{x1D6F9}_{0}, & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\unicode[STIX]{x1D6F9}}_{1}=I_{1}\unicode[STIX]{x1D6F9}_{0}\unicode[STIX]{x1D6F9}_{2}^{\ast }-\frac{1}{2}\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}_{1}^{2}\left(1+\frac{f^{2}m_{1}^{2}}{\unicode[STIX]{x1D705}_{1}^{2}\unicode[STIX]{x1D714}_{1}^{2}}\right)\unicode[STIX]{x1D6F9}_{1}, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\unicode[STIX]{x1D6F9}}_{2}=I_{2}\unicode[STIX]{x1D6F9}_{0}\unicode[STIX]{x1D6F9}_{1}^{\ast }-\frac{1}{2}\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}_{2}^{2}\left(1+\frac{f^{2}m_{2}^{2}}{\unicode[STIX]{x1D705}_{2}^{2}\unicode[STIX]{x1D714}_{2}^{2}}\right)\unicode[STIX]{x1D6F9}_{2}, & \displaystyle\end{eqnarray}$$

where $I_{r}$ is

(3.4) $$\begin{eqnarray}\displaystyle I_{r}=\unicode[STIX]{x1D6FE}_{r}\frac{\ell _{p}m_{q}-m_{p}\ell _{q}}{2\unicode[STIX]{x1D714}_{r}\unicode[STIX]{x1D705}_{r}^{2}}\left[\unicode[STIX]{x1D714}_{r}(\unicode[STIX]{x1D705}_{p}^{2}-\unicode[STIX]{x1D705}_{q}^{2})+\ell _{r}N^{2}\left(\frac{\ell _{p}}{\unicode[STIX]{x1D714}_{p}}-\frac{\ell _{q}}{\unicode[STIX]{x1D714}_{q}}\right)+m_{r}f^{2}\left(\frac{m_{p}}{\unicode[STIX]{x1D714}_{p}}-\frac{m_{q}}{\unicode[STIX]{x1D714}_{q}}\right)\right],\qquad & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D6FE}_{0}=1$ , $\unicode[STIX]{x1D6FE}_{1,2}=-1$ , $(p,q,r)=(0,1,2)$ , or any circular permutation thereof. Here, $I_{r}$ is an interaction coefficient between the primary and the two secondary waves resulting from the nonlinear term in the Navier–Stokes and incompressibility equations. In the dispersion relation (2.2) and in the interaction coefficient (3.4), the products $fm$ and $N\ell$ appear, highlighting the symmetric role played by stratification for the horizontal wavevector component and by rotation for the vertical wavevector component.

Figure 5. (a) The evolution of the theoretical threshold of TRI in the case of an infinite plane inertia–gravity wave of horizontal wavenumber $\ell _{0}=80~\text{m}^{-1}$ . The threshold is represented by the critical Reynolds number $Re_{c}=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6F9}_{0,c}/\unicode[STIX]{x1D708}$ and is plotted versus $f/N$ at constant $\unicode[STIX]{x1D714}_{0}/N$ . (b) The growth rate (3.6) at different values of the Coriolis parameter $f/N$ , renormalized by the maximum growth rate at $f/N=0$ ( $\unicode[STIX]{x1D70E}_{0}=0.11~\text{s}^{-1}$ ) versus the horizontal component $\ell _{1}$ of one of the secondary wavevectors, for a primary wave at ( $\ell _{0}=80~\text{m}^{-1}$ , $\unicode[STIX]{x1D714}_{0}/N=0.8$ , $Re=188$ ).

In § 2, we observed that rotation has an influence on the development of the instability. To understand this effect, the growth rate $\unicode[STIX]{x1D70E}$ of the instability can be extracted using (3.2) and (3.3) assuming a constant amplitude for the primary wave, because we consider only the early times of the instability growth. This amplitude will be defined using the Reynolds number $Re=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6F9}_{0}/\unicode[STIX]{x1D708}$ . The details of the calculation of the generalized growth rate, which takes into account both rotation and stratification, is in appendix A, yielding

(3.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E} & = & \displaystyle -\frac{1}{4}\unicode[STIX]{x1D708}\!\left(\!\unicode[STIX]{x1D705}_{1}^{2}+\unicode[STIX]{x1D705}_{2}^{2}+\frac{f^{2}m_{1}^{2}}{\unicode[STIX]{x1D714}_{1}^{2}}+\frac{f^{2}m_{2}^{2}}{\unicode[STIX]{x1D714}_{2}^{2}}\right)\!+\sqrt{\frac{1}{16}\unicode[STIX]{x1D708}^{2}\!\left(\unicode[STIX]{x1D705}_{1}^{2}-\unicode[STIX]{x1D705}_{2}^{2}+\frac{f^{2}m_{1}^{2}}{\unicode[STIX]{x1D714}_{1}^{2}}-\frac{f^{2}m_{2}^{2}}{\unicode[STIX]{x1D714}_{2}^{2}}\right)^{\!2}+I_{1}I_{2}|\unicode[STIX]{x1D6F9}_{0}|^{2}},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle \displaystyle & = & \displaystyle -\frac{1}{4}\unicode[STIX]{x1D708}\left(\unicode[STIX]{x1D705}_{1}^{2}+\unicode[STIX]{x1D705}_{2}^{2}+\frac{f^{2}m_{1}^{2}}{\unicode[STIX]{x1D714}_{1}^{2}}+\frac{f^{2}m_{2}^{2}}{\unicode[STIX]{x1D714}_{2}^{2}}\right)+\sqrt{I_{1}I_{2}}|\unicode[STIX]{x1D6F9}_{0}|+o(\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}_{1,2}^{2}).\end{eqnarray}$$

The condition for instability is that the growth rate $\unicode[STIX]{x1D70E}$ from (3.6) is positive for at least one combination of $\ell _{1},m_{1},\unicode[STIX]{x1D714}_{1},\ell _{2},m_{2},\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2}$ that respects the dispersion relation and the resonance conditions. The theoretical threshold in amplitude is defined as the value $\unicode[STIX]{x1D6F9}_{0,c}$ of the amplitude when the maximum value of $\unicode[STIX]{x1D70E}$ is zero. The threshold of TRI depends on the amplitude of the primary wave and on its frequency, but not on its wavelength: if $\boldsymbol{k}_{0}$ , $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{2}$ are multiplied by the same coefficient $K$ , the growth rate (3.6) is multiplied by $K^{2}$ , and the threshold is unchanged. In figure 5(a), we show the amplitude threshold ( $Re_{c}=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6F9}_{0,c}/\unicode[STIX]{x1D708}$ ) as a function of $f/N$ for various values of $\unicode[STIX]{x1D714}_{0}/N$ . As underlined by Bourget et al. (Reference Bourget, Scolan, Dauxois, Lebars, Odier and Joubaud2014), if $f=0$ there is no TRI threshold. When rotation is introduced in the system, however, it induces a finite threshold for TRI. This reflects the fact that, as the amplitude of the primary wave vanishes, the frequencies of the secondary waves tend towards zero and $\unicode[STIX]{x1D714}_{0}$ . When there is rotation in the system, a zero-frequency subharmonic wave is no longer allowed, hence the appearance of a threshold in amplitude, which increases with $f/N$ . This threshold tends to infinity as $f$ reaches $\unicode[STIX]{x1D714}_{0}/2$ . A simple explanation for this observation is that for both $\unicode[STIX]{x1D714}_{1,2}>f$ and the temporal resonance condition to be fulfilled, $f$ must be smaller than $\unicode[STIX]{x1D714}_{0}/2$ .

Figure 5(b) shows the associated growth rate (3.6) as a function of $\ell _{1}/\ell _{0}$ for various values of $f/N$ . For each curve, the growth rate reaches a maximum resulting in a selection criterion for TRI. It should be noted that outside the range of $\ell _{1}/\ell _{0}$ presented in the figure, the growth rate tends monotonically towards negative values. In this model, for low values of $f/N$ , rotation has little impact on the maximum growth rate. For larger values of $f/N$ , the maximum growth rate decreases strongly as $f$ increases. This is a general feature, related to the threshold divergence as $f$ approaches $\unicode[STIX]{x1D714}_{0}/2$ .

Within this theoretical framework, one can compute the two wavevectors $\boldsymbol{k}_{\unicode[STIX]{x1D70E}_{max}}$ corresponding to the maximum growth rate of TRI for experiment (c) in figure 3. Their tips are represented by grey circles in figure 4. The measured wavevectors fall on this theoretical prediction, within their experimental errors.

3.2 Finite size effect

In the case of infinite plane waves, we showed in the previous section that the amplitude instability threshold increases continuously with the Coriolis parameter $f/N$ . From an oceanic perspective, this would mean that inertia–gravity waves at the Equator would be more prone to TRI than waves at higher latitude (because of the latitude dependence of the Coriolis parameter $f=2\unicode[STIX]{x1D6FA}_{\mathit{Earth}}\sin \unicode[STIX]{x1D702}$ , where $\unicode[STIX]{x1D702}$ is the latitude). This does not correspond to existing oceanic observations: numerical simulations (MacKinnon Reference Mackinnon2005; MacKinnon et al. Reference Mackinnon, Alford, Sun, Pinkel, Zhao and Klymak2013b ) and field studies (MacKinnon et al. Reference Mackinnon, Alford, Pinkel, Klymak and Zhao2013a ) highlight the existence of a critical latitude at $f=\unicode[STIX]{x1D714}_{0}/2$ where TRI is strongly enhanced. The experiments presented in § 2 show that TRI appears only when rotation is strong enough (compare figures 3 a and b) with other parameters kept constant. The predictions presented in the previous section cannot explain this behaviour and actually seem to show that rotation prevents TRI. The major difference from the experiment lies in the infinite beam width of the plane wave in our theoretical development. The effect of finite size for pure internal waves ( $f=0$ ) was studied theoretically by Karimi & Akylas (Reference Karimi and Akylas2014). Laboratory experiments have also been performed by Bourget et al. (Reference Bourget, Scolan, Dauxois, Lebars, Odier and Joubaud2014), who derived a simple model taking into account the finite size effect of the beam width. The theory by Karimi & Akylas (Reference Karimi and Akylas2014) and the model by Bourget et al. (Reference Bourget, Scolan, Dauxois, Lebars, Odier and Joubaud2014) are based on the same premises: the triadic interaction has to be strong enough in the limited time that subharmonic perturbations overlap with the beam. As schematically shown in figure 6(a), the secondary waves leave the interaction region of width $W$ with a velocity equal to the projection of their group velocity onto the direction $\boldsymbol{n}_{\bot }=\boldsymbol{k}_{0}/\unicode[STIX]{x1D705}_{0}$ , perpendicular to the primary wave propagation direction. The amplitude of this exit velocity is then $|\boldsymbol{v}_{g,i}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|$ .

Figure 6. (a) Schematics of the triadic resonant interaction, highlighting the existence of an interaction zone of width $W$ corresponding to the width of the primary wave. (b) Typical transverse profile of the wave amplitude, for the wave presented in figure 3(a) extracted by Hilbert transform. (c,d) The ratio of the outgoing advection rate of the secondary waves away from the primary wave beam over the maximum growth rate of the instability, as a function of $f/\unicode[STIX]{x1D714}_{0}$ . This ratio is shown individually for each secondary wave and also for the sum of both advection rates, which directly compares the two terms of (3.10): $(1/4)(|\boldsymbol{v}_{g,1}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|+|\boldsymbol{v}_{g,2}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|)/2\unicode[STIX]{x1D706}\unicode[STIX]{x1D70E}_{max}$ . Inset: corresponding maximum growth rate $\unicode[STIX]{x1D70E}_{max}$ (normalized by the maximum growth rate at $f=0$ , $\unicode[STIX]{x1D70E}_{0}$ ). Two different sets of parameters are considered: (c) experimental parameters ( $\unicode[STIX]{x1D714}_{0}/N=0.8$ , $\ell _{0}=80~\text{m}^{-1}$ and $Re=188$ ); (d) oceanic case, where parameter values are taken from Gayen & Sarkar (Reference Gayen and Sarkar2013) and Sun & Pinkel (Reference Sun and Pinkel2013) ( $\unicode[STIX]{x1D714}_{0}/N=0.1$ , $\ell _{0}=6.3\times 10^{-2}~\text{m}^{-1}$ and $Re=3.9\times 10^{5}$ ).

Following the approach of Bourget et al. (Reference Bourget, Scolan, Dauxois, Lebars, Odier and Joubaud2014), the time evolution of the secondary wave amplitude also depends on the rate at which these waves leave the interaction region, which we call the advection rate. This rate can be expressed as the ratio of the exit velocity defined above over the width of the primary wave beam. The amplitude equations (3.2) and (3.3) of the secondary waves thus become

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\unicode[STIX]{x1D6F9}}_{1}=I_{1}\unicode[STIX]{x1D6F9}_{0}\unicode[STIX]{x1D6F9}_{2}^{\ast }-\frac{1}{2}\left[\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}_{1}^{2}\left(1+\frac{f^{2}m_{1}^{2}}{\unicode[STIX]{x1D705}_{1}^{2}\unicode[STIX]{x1D714}_{1}^{2}}\right)+\frac{|\boldsymbol{v}_{g,1}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|}{W}\right]\unicode[STIX]{x1D6F9}_{1}, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\unicode[STIX]{x1D6F9}}_{2}=I_{2}\unicode[STIX]{x1D6F9}_{0}\unicode[STIX]{x1D6F9}_{1}^{\ast }-\frac{1}{2}\left[\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}_{2}^{2}\left(1+\frac{f^{2}m_{2}^{2}}{\unicode[STIX]{x1D705}_{2}^{2}\unicode[STIX]{x1D714}_{2}^{2}}\right)+\frac{|\boldsymbol{v}_{g,2}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|}{W}\right]\unicode[STIX]{x1D6F9}_{2}. & \displaystyle\end{eqnarray}$$

Increasing the rotation $f$ affects the time spent by the secondary wave in the primary wave beam in two different ways. First, as $f$ increases, the group velocity is reduced, vanishing at $f=\unicode[STIX]{x1D714}$ . Second, changing the rotation affects the direction of propagation. If the primary and the secondary waves were collinear, the secondary waves would never leave the primary beam. In contrast, when the group velocity of the secondary waves $\boldsymbol{v}_{\boldsymbol{g},(\mathbf{1},\mathbf{2})}$ is not perpendicular to the primary wavevector $\boldsymbol{k}_{0}$ , the secondary waves will leave the primary beam. From (3.7) and (3.8), one can then extract a modified instability growth rate, in the same way as the growth rate (3.6) was extracted in the infinite beam case:

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4} & = & \displaystyle -\frac{1}{4}\left(\unicode[STIX]{x1D708}\left(\unicode[STIX]{x1D705}_{1}^{2}+\unicode[STIX]{x1D705}_{2}^{2}+\frac{f^{2}m_{1}^{2}}{\unicode[STIX]{x1D714}_{1}^{2}}+\frac{f^{2}m_{2}^{2}}{\unicode[STIX]{x1D714}_{2}^{2}}\right)+\frac{|\boldsymbol{v}_{g,1}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|}{W}+\frac{|\boldsymbol{v}_{g,2}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|}{W}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\sqrt{\frac{1}{16}\left(\unicode[STIX]{x1D708}\left(\unicode[STIX]{x1D705}_{1}^{2}-\unicode[STIX]{x1D705}_{2}^{2}+\frac{f^{2}m_{1}^{2}}{\unicode[STIX]{x1D714}_{1}^{2}}-\frac{f^{2}m_{2}^{2}}{\unicode[STIX]{x1D714}_{2}^{2}}\right)+\frac{|\boldsymbol{v}_{g,1}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|}{W}-\frac{|\boldsymbol{v}_{g,2}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|}{W}\right)^{2}+I_{1}I_{2}|\unicode[STIX]{x1D6F9}_{0}|^{2}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Taking into account the experimental values, $I_{1}I_{2}|\unicode[STIX]{x1D6F9}_{0}|^{2}$ is the leading term inside the square root in this equation, so that

(3.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}=\unicode[STIX]{x1D70E}-\frac{1}{4W}(|\boldsymbol{v}_{g,1}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|+|\boldsymbol{v}_{g,2}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|)+o(\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}_{1,2}^{2})+o\left(\frac{|\boldsymbol{v}_{g,1,2}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|}{W}\right). & & \displaystyle\end{eqnarray}$$

A beam will be considered wide when its width $W$ is large compared with the wavelength. In such a case, one recovers expression (3.6) from (3.10). To get an idea of the actual width of our experimental beam, generated as described in § 2.1, figure 6(b) shows a typical transverse profile of the beam (measured perpendicular to the direction of propagation of the wave). The effective width of the beam lies between $1.5\unicode[STIX]{x1D706}$ and $3\unicode[STIX]{x1D706}$ .

The relative importance of the advection rate of each secondary wave, $1/4W|\boldsymbol{v}_{g,i}\boldsymbol{\cdot }\boldsymbol{n}_{\bot }|$ , versus the growth rate is shown in figure 6(c,d), where the ratio between the two is plotted as a function of $f/\unicode[STIX]{x1D714}_{0}$ , for two given sets of parameters. Based on figure 6(b), the typical beam width was chosen to be $W=2\unicode[STIX]{x1D706}$ . Small values of this ratio correspond to cases where finite size is not a limiting process for TRI, since the secondary waves stay long enough in the primary beam. Figure 6(c) corresponds to values used in our experiments. First, we notice that although the advection term is not dominant, which justifies the first-order approximation leading to (3.10), it is not negligible. In addition, we observe the existence of a particular range of $f/\unicode[STIX]{x1D714}_{0}$ where TRI is more likely to occur. In the range $0<f/\unicode[STIX]{x1D714}_{0}<0.45$ , $\unicode[STIX]{x1D70E}_{max}$ is fairly constant (see the inset of figure 6 c), but, in the range $0.35<f/\unicode[STIX]{x1D714}_{0}<0.45$ , the interaction time is larger (low advection rate), which enhances the instability. Figure 6(d) shows the same ratio for typical oceanic inertia–gravity waves (characteristics extracted from Gayen & Sarkar (Reference Gayen and Sarkar2013) and Sun & Pinkel (Reference Sun and Pinkel2013)). In this case, the advection rate is the dominant term and the range of $f/\unicode[STIX]{x1D714}_{0}$ favouring TRI is roughly reduced to a single value, close to 0.5. In the ocean, wave beams can be extremely narrow (Lien & Gregg Reference Lien and Gregg2001; Gostiaux & Dauxois Reference Gostiaux and Dauxois2007). In such cases, we can make the assumption that the finite size effect is paramount and that TRI will only appear in the most favourable case, i.e. $f\simeq \unicode[STIX]{x1D714}_{0}/2$ from figure 6(d). This result is consistent with MacKinnon et al. (Reference Mackinnon, Alford, Pinkel, Klymak and Zhao2013a ,Reference Mackinnon, Alford, Sun, Pinkel, Zhao and Klymak b ), and gives a sensible explanation for the predominance of TRI at a critical latitude rather than at the Equator.

The approach presented in this section shows that TRI is favoured for a given range of Coriolis parameters. Let us now return to laboratory experiments and check whether this feature can be observed.

4.1 Experimental threshold

We now focus on estimating the threshold of the triadic instability experimentally. To do so, we generate inertia–gravity waves with a given horizontal wavelength ( $\ell _{0}=80~\text{m}^{-1}$ ) for a given generator displacement amplitude of 5 mm but with varying frequency $\unicode[STIX]{x1D714}_{0}$ and Coriolis parameter $f$ . The parameter space is not reduced by keeping a constant amplitude. Indeed, in our experiments $Re$ decreases as $\unicode[STIX]{x1D714}_{0}/N$ decreases. A good approximation is to assume that the Reynolds number is simply proportional to $(\unicode[STIX]{x1D714}_{0}/N)^{2}$ , in agreement with the experimental measurements. Because $\ell _{0}$ is fixed by the generator, in this configuration, the wave can be described with only two parameters ( $\unicode[STIX]{x1D714}_{0}/N$ , $f/N$ ). For this reason, the threshold that will be discussed here is a threshold in frequency rather than a threshold in amplitude. In figure 7, we show the numerous experimental measurements in this parameter space. Stable or unstable cases, with respect to TRI, are indicated by a different symbol. For example, for $f/N=0$ the primary wave is stable at $\unicode[STIX]{x1D714}_{0}/N=0.66$ , but is unstable at $\unicode[STIX]{x1D714}_{0}/N=0.71$ , in agreement with previous studies (Bourget et al. Reference Bourget, Scolan, Dauxois, Lebars, Odier and Joubaud2014). These measurements allow one to locate the instability frequency threshold, which lies between the highest frequency for which no TRI is detected and the lowest frequency for which TRI is observed.

Figure 7. Experimental observations showing the triadic resonance stability or instability of a primary wave of dimensionless frequency $\unicode[STIX]{x1D714}_{0}/N$ at Coriolis parameter $f/N$ . The fixed parameters are $\ell _{0}=80~\text{m}^{-1}$ , $e_{0}=5~\text{mm}$ , leading to $Re>100$ for each experiment. The predicted threshold is shown by a dotted line in the infinite beam case, by a dashed line in the finite beam case with a 3-wavelength-wide beam and by a dash-dotted line in the finite beam case with a 1.5-wavelength-wide beam. Our prediction uncertainty due to the uncertainty in the width of the beam is displayed as a light green zone. The hatched zone corresponds to a forbidden zone for TRI: if $\unicode[STIX]{x1D714}_{1}$ and $\unicode[STIX]{x1D714}_{2}$ are larger than $f$ , then according to the space resonance condition, $\unicode[STIX]{x1D714}_{0}$ is larger than $2f$ . The larger symbols help to locate on this graph the experiments described in other figures of this article, with the figure number indicated. The horizontal dashed line indicates $\unicode[STIX]{x1D714}_{0}/N=0.8$ , which corresponds to the experiments presented in figure 8.

For low values of the Coriolis parameter ( $f/N<0.12$ ), the threshold remains fairly constant. It then decreases to reach a minimal value around $f/N=0.2$ . At larger values of the Coriolis parameter, the threshold follows a limit defined by the line $\unicode[STIX]{x1D714}_{0}=2f$ . Below this line (hatched zone in the figure), TRI is forbidden. Indeed, the two secondary frequencies have to be larger than $f$ , so that $\unicode[STIX]{x1D714}_{0}=\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}$ has to be larger than $2f$ for the temporal resonance condition to be satisfied.

In the previous section, we wrote a simplified expression $\unicode[STIX]{x1D6F4}$ for the growth rate of the instability, which takes into account the finite size effect. This growth rate can be used to estimate the threshold of the instability from the model and compare it with experimental observations.

An experimental criterion is needed to determine whether or not the frequency is above the threshold. To find this criterion, we consider the case $f/N=0$ . From figure 7, we estimate that the threshold in this case is located around $\unicode[STIX]{x1D714}_{0}/N=0.68$ . For these values of $f$ and $\unicode[STIX]{x1D714}_{0}$ , we then calculate the maximum value of $\unicode[STIX]{x1D6F4}(\ell _{1})$ (with respect to $\ell _{1}$ ), later referred to as $\unicode[STIX]{x1D6F4}^{0}$ . In what follows, we assume for a given primary wave in the parameter space ( $\unicode[STIX]{x1D714}_{0}/N$ , $f/N$ ) that any corresponding value of the growth rate $\unicode[STIX]{x1D6F4}$ that is larger than $\unicode[STIX]{x1D6F4}^{0}$ leads to an unstable case, whereas any value below gives a stable primary wave. To estimate the evolution of the threshold with increasing $f/N$ , we operate as follows.

1. (i) We choose a given $f/N$ value.

2. (ii) At $\unicode[STIX]{x1D714}_{0}/N=0.8$ (a region always unstable experimentally, whatever the value of $f$ , except in the hatched zone), the maximum value of $\unicode[STIX]{x1D6F4}(\ell _{1})$ , later referred to as $\unicode[STIX]{x1D6F4}^{max}$ , is calculated.

3. (iii) If $\unicode[STIX]{x1D6F4}^{max}>\unicode[STIX]{x1D6F4}^{0}$ , $\unicode[STIX]{x1D714}_{0}/N$ is decreased by a chosen step (in our study this step is $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D714}_{0}/N)=0.02$ ).

4. (iv) For the new value of frequency and corresponding amplitude, the new $\unicode[STIX]{x1D6F4}^{max}$ is calculated.

5. (v) The procedure is repeated until $\unicode[STIX]{x1D6F4}^{max}<\unicode[STIX]{x1D6F4}^{0}$ . The threshold is therefore determined within a given error (in our study a 2 % error on $\unicode[STIX]{x1D714}_{0}/N$ ).

As discussed earlier, because the exact width $W$ of the beam is not known, an additional error is introduced. For this reason, we show in figure 7 the threshold curve in the two cases $W=1.5\unicode[STIX]{x1D706}$ and $W=3\unicode[STIX]{x1D706}$ , which are the limiting values for $W$ according to figure 6(b). The band between these two threshold curves represents the margin of error in our estimation of the threshold. We note that this band lies roughly in the region that separates experimental points where TRI takes place from experimental points where the primary wave is stable. The corresponding curve for infinitely wide beams is also shown to emphasize the importance of the finite size effect.

4.2 Frequencies of the secondary waves

In figure 7, the good agreement between the model and the experimental threshold prevails until $f/N\approx 0.3$ . Above this value, unstable events are observed below the threshold predicted by the model. To get more insight into this feature, we focus on a particular value of the frequency of the primary wave, $\unicode[STIX]{x1D714}_{0}/N=0.8$ , for different values of $f/N$ . For each run, a temporal fast Fourier transform of the vertical density gradient $\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D70C}^{\prime }$ is performed on the signal recorded during the steady state ( $t=90T_{0}$ ) using a $30T_{0}$ time window. In figure 8, showing the corresponding spectrum, there is a strong peak at $\unicode[STIX]{x1D714}_{0}/N=0.8$ , corresponding to the forcing frequency, and a pair of twin peaks, as expected, for $0<f/N<0.4$ , corresponding to secondary waves of frequencies $\unicode[STIX]{x1D714}_{1}$ and $\unicode[STIX]{x1D714}_{2}$ , smaller than $\unicode[STIX]{x1D714}_{0}$ . Above 0.4, the primary wave is stable, as expected. For every value of $f/N$ , there is a symmetry with respect to the position $\unicode[STIX]{x1D714}_{0}/2N$ , and the temporal resonance condition is fulfilled: $\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}=\unicode[STIX]{x1D714}_{0}$ . As expected from the dispersion relation, as $f/N$ increases the main secondary peaks move closer to $\unicode[STIX]{x1D714}_{0}/2N$ , bringing TRI closer to what is commonly known as PSI. Interestingly, rotation not only has an influence on secondary wave characteristics but may also foster the instability. In this context, the experiment at $f/N=0.25$ is particularly interesting. One can see that the instability is stronger, with several secondary frequencies, which are much weaker in the cases $f/N=0.2$ and $f/N=0.3$ and do not appear in the other cases. This enhanced instability can be explained by considering the secondary wave 1. Its frequency ( $\unicode[STIX]{x1D714}_{1}/N\simeq 0.51$ ) is close to $2f$ . We demonstrated in the theoretical analysis of the finite size effect that a wave at this frequency, now considered as the primary wave, is most likely to undergo a secondary instability, thereby creating the observed multiple TRI.

Figure 8. Renormalized amplitude of the fast Fourier transform of the vertical gradient of density over a $30T_{0}$ time window at $t\approx 90T_{0}$ , for a primary wave propagating at $\unicode[STIX]{x1D714}_{0}/N\simeq 0.8$ and for various values of $f/N$ , which correspond to $63.2^{\circ }<\unicode[STIX]{x1D703}<68.9^{\circ }$ . For the sake of clarity, the spectra are shifted vertically by a multiple of 3, the vertical axis values corresponding to the case $f/N=0$ . On each spectrum, a red dot indicates the value of $f/N$ and separates it into two parts, one in green on the left where $\unicode[STIX]{x1D714}<f$ and one on the right where $\unicode[STIX]{x1D714}>f$ . A vertical dashed line indicates $\unicode[STIX]{x1D714}/N=0.4$ , which corresponds roughly to $\unicode[STIX]{x1D714}_{0}/N/2$ . On each spectrum, the peak on the right represents the primary wave, and daughter waves correspond to the other peaks, identified with a cross. For the sake of clarity, the value $f/N$ is displayed with full precision only when it is necessary, for the experiments just above and just below the threshold.

Let us now turn to the particular case $0.35<f/N<0.4$ . We observe in figure 8 that one of the secondary waves (later referred to as 1) has a frequency above $f$ and the other one (later referred to as 2) has a frequency below $f$ . In this case, the secondary wave 2 is a sub-inertial wave, and its existence is not taken into account in our theoretical approach, where we assumed, for $i=(0,1,2)$ , that $f<\unicode[STIX]{x1D714}_{i}<N$ . This observation may be related to the discrepancy seen in figure 7 for $f>0.3$ .

Snapshots of an experimental vertical density-gradient field at different times are presented in figure 9 for the particular case $f/N=0.39$ . At early time, a plane wave propagates from the top left corner to the bottom right corner of the field of view (figure 9 a). A few oscillating periods later, perturbations of the wave field appear below the wave maker (figure 9 b). These perturbations continue to evolve until a stationary state is reached (figure 9 c). This behaviour is similar (even at long times) to what was observed when $\unicode[STIX]{x1D714}_{1}$ and $\unicode[STIX]{x1D714}_{2}$ were larger than $f$ (for example, figure 3 c). The experimental time series at a point 5 cm below the wavemaker filtered at $\unicode[STIX]{x1D714}_{0}$ , $\unicode[STIX]{x1D714}_{1}$ and $\unicode[STIX]{x1D714}_{2}$ respectively are shown in figure 9(d–f), confirming that a stationary regime is reached for the configuration under consideration.

Figure 9. (a,b,c) Snapshots of the vertical density-gradient field for (a) $t=4T_{0}$ , (b) $t=10T_{0}$ and (c) $t=100T_{0}$ , with $T_{0}$ the primary wave period. The wave is propagating from the top left to the bottom right. The Coriolis parameter is $f/N=0.39$ , the wave frequency is $\unicode[STIX]{x1D714}_{0}/N=0.8$ and the motion amplitude of the generator is 0.5 cm. (d,e,f) Vertical density gradient as a function of time for a point located 5 cm below the wavemaker (indicated by a star in a), filtered at $\unicode[STIX]{x1D714}_{0}$ (d), $\unicode[STIX]{x1D714}_{1}$ (e) and $\unicode[STIX]{x1D714}_{2}$ (f).

From a theoretical point of view, the existence of sub-inertial waves in the case of TRI was predicted for $\unicode[STIX]{x1D714}_{0}$ slightly smaller than $2f$ by Young et al. (Reference Young, Tsand and Balmforth2008). A comparison between this theory and the experiments is not possible because the viscosity and finite size effect of the primary wave are of paramount importance in the laboratory case. A possible explanation for this type of sub-inertial waves could be a misalignment between the background density gradient and the axis of rotation. As mentioned in § 2.1, however, this alignment was carefully controlled in the building of the rotating platform, leading to a maximum angle of $0.1^{\circ }$ , too small to explain the observed phenomenon. A formal theory that incorporated our experimental configuration would be interesting for the understanding of this phenomenon for which the interpretation currently remains open.