2 Experimental set-up

The experimental set-up used in the present work and sketched in figure 2 is similar to the one described in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) and Brouzet et al. (Reference Brouzet, Sibgatullin, Scolan, Ermanyuk and Dauxois2016b ). Experiments are conducted in the rectangular test tank of size $800\times 170\times 425~\text{mm}^{3}$ filled with uniformly stratified fluid using the conventional double-bucket technique. Salt is used as a stratifying agent. The density profile is measured prior to and after the experiments by a conductivity probe attached to a vertical traverse mechanism. The value of the buoyancy frequency $N$ is evaluated from the measured density profile. The trapezoidal fluid domain of length  $L$ (measured along the bottom) and depth $H$ is delimited by a sliding sloping wall, inclined at the angle $\unicode[STIX]{x1D6FC}$ . The wall is slowly inserted into the fluid after the end of the filling procedure. The global Archimedes number representing the ratio of the Reynolds and Froude numbers based on the fluid depth is kept fixed throughout all the experiments so that $Ar=H^{2}N/\unicode[STIX]{x1D708}=85\,000$ , where $\unicode[STIX]{x1D708}$ is kinematic viscosity. Thus, at the global scale of the experimental set-up, the effect of viscosity is reasonably weak, allowing an energy cascade toward smaller dissipative scales.

Figure 2. Sketch of the experimental set-up showing the wave generator and the sloping wall (inclined at an angle  $\unicode[STIX]{x1D6FC}$ with respect to the vertical) inside the immobile tank of size $800\times 170\times 425~\text{mm}^{3}$ . The working bottom length of the section, the depth and the sloping angle are given within the text.

The input forcing is introduced into the system by an internal wave generator described in Gostiaux et al. (Reference Gostiaux, Didelle, Mercier and Dauxois2007), Mercier et al. (Reference Mercier, Martinand, Mathur, Gostiaux, Peacock and Dauxois2010) and Joubaud et al. (Reference Joubaud, Munroe, Odier and Dauxois2012). The time-dependent vertical profile of the generator is prescribed in the form

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D701}(z,t)=a\sin (\unicode[STIX]{x1D714}_{0}t)\cos (\unicode[STIX]{x03C0}z/H),\end{eqnarray}$$

where $a$ and $\unicode[STIX]{x1D714}_{0}$ are the amplitude and frequency of oscillations, respectively. In a horizontally semi-infinite domain, the motion of the generator would generate the first vertical mode of internal waves. The profile is reproduced in discrete form by the horizontal motion of a stack of $47$ plates.

The whole-field velocity measurements are performed via the standard particle image velocimetry (PIV) technique. The fluid is seeded with light-reflecting hollow glass spheres of size $8~\unicode[STIX]{x03BC}\text{m}$ and density $1100~\text{kg}~\text{m}^{-3}$ . The sedimentation velocity of particles is found to be very low, with negligible effect on results of velocity measurements. The longitudinal $(x,y=0,z)$ mid-plane of the test section is illuminated by a vertical laser sheet coming through the side of the tank. The velocity field is calculated with the help of the cross-correlation technique (Fincham & Delerce 2000) with a typical resolution of 1 velocity vector per area of approximately $3\times 3~\text{mm}^{2}$ . The mesh of measurements is found to be sufficient to resolve the small-scale details of the wave field.

A computer-controlled video AVT (Allied Vision Technologies) Stingray or Pike camera with a charge-coupled device (CCD) matrix of $2452\times 2054$ pixels is used for video recording of the fluid motion. The camera is located at a distance of $2850~\text{mm}$ from the test tank. For the PIV measurements in the short-term experiments, the camera operates at the constant frame rate of 4 Hz. After the PIV treatment, $4$ velocity fields per second are obtained, yielding typically around $40$ fields per period of forcing. In the long-term experiments, the camera operates in burst mode taking two images separated by $0.125$  s and waiting $0.375$  s before the subsequent burst. The PIV treatment in this case yields $2$ velocity fields per second. This sampling rate is found to provide a sufficient resolution of the significant frequency components of the signal.

3 Arnold tongue of a stable (1,1) attractor

To quantify the strength of (1,1) attractor, we consider the total dimensionless kinetic energy of the fluid confined in the trapezoidal domain. Experimentally, this quantity is measured in the vertical longitudinal mid-plane and defined as follows

(3.1) $$\begin{eqnarray}K=\frac{\displaystyle \int _{S}\text{d}x\,\text{d}z\,\frac{1}{2}(v_{x}^{2}+v_{z}^{2})}{\displaystyle \frac{1}{2}(a\unicode[STIX]{x1D714}_{0})^{2}S},\end{eqnarray}$$

where $v_{x}$ and $v_{z}$ are the horizontal and vertical velocity components and $S$ is the area of the trapezoidal domain. Brouzet et al. (Reference Brouzet, Sibgatullin, Scolan, Ermanyuk and Dauxois2016b ) show that the internal wave pattern in the experimental set-up is reasonably two-dimensional. Thus, the measurements performed in the vertical mid-plane are representative for the whole volume except thin near-wall boundary layers.

Figure 3 presents a typical experimental time history of $K$ . It can be seen that after a transient, which has a typical duration of about $25$  periods, the process reaches saturation and kinetic energy oscillates between certain well-defined minimum and maximum values. We denote its corresponding time-averaged value as $\left\langle K\right\rangle$ , performed in the saturated regime. The quantity $\left\langle K\right\rangle$ can be interpreted as a susceptibility, i.e. a dynamic response of the experimental system to the prescribed forcing of unit amplitude $a$ . In addition, one can introduce $R=K_{min}/K_{max}$ , the ratio of minimum to maximum kinetic energy as a measure characterizing a particular wave regime as standing or propagating waves. Similar to $\left\langle K\right\rangle$ , this quantity is defined for the saturated regime. For purely standing waves, $R=0$ , while $R\rightarrow 1$ for purely propagating waves with vanishingly thin wave beams (in the linear regime with a vanishingly small viscosity). In realistic systems with wave beams of finite thickness, we observe $0<R<1$ .

Figure 3. Typical experimental time history of the non-dimensional kinetic energy. The vertical dotted dashed line indicates the instant where the kinetic energy saturates. The average of the signal, computed from this instant and to the end of the experiment, is represented by the dashed white line. The two horizontal dashed black lines represent $K_{min}$ and $K_{max}$ , the minimal and maximal values bounding the oscillations of the kinetic energy during the saturated state. The parameters for this experiment are: $\unicode[STIX]{x1D6FA}_{0}=0.59$ , $H=300~\text{mm}$ , $L=450~\text{mm}$ ,  $\unicode[STIX]{x1D6FC}=27.3^{\circ }$ , $\unicode[STIX]{x1D70F}=1.84$ , $d=0.31$ and $a=1.5~\text{mm}$ . For this experiment, $\left\langle K\right\rangle =5$ and $R=0.65$ .

Table 1. Parameters for the series of experiments represented in figure 4 and studied in this paper. For all these experiments, the working depth is 300 mm, the working bottom length is 455 $\pm$ 10 mm, and the amplitude of the generator is $a=15~\text{mm}$ . $\#$ is the number of the experiment (from 1 to 50), $\unicode[STIX]{x1D6FC}$ the angle (in deg.) of the slope, $\unicode[STIX]{x1D6FA}_{0}$ the non-dimensional forcing frequency, $d$ and $\unicode[STIX]{x1D70F}$ the corresponding non-dimensional parameters of the diagram presented in figure 1.

Using $\langle K\rangle$ and $R$ as variables, we performed a series of 50 short-term experiments (parameters are given in table 1) with stable (1,1) attractors to study the structure of their domain of existence, the so-called Arnold tongue, as a function of the two parameters  $(d,\unicode[STIX]{x1D70F})$ controlling the convergence of wave rays. For the particular case of trapezoidal geometry, Maas et al. (Reference Maas, Benielli, Sommeria and Lam1997) introduce $d=1-(2H/L)\tan \unicode[STIX]{x1D6FC}$ as the control parameter of the slope and $\unicode[STIX]{x1D70F}=(2H/L)\sqrt{1/\unicode[STIX]{x1D6FA}_{0}^{2}-1}$ as the control parameter of the wave-ray pattern. The limiting case of triangle geometry ( $d=-1$ ) is typically characterized by the presence of a point attractor at a vertex of the triangle, while the case of rectangular geometry ( $d=1$ ) corresponds to classic normal modes (McEwan Reference McEwan1971) for a discrete set of $\unicode[STIX]{x1D70F}$ . Originally, $(d,\unicode[STIX]{x1D70F})$ diagrams have been suggested in Maas et al. (Reference Maas, Benielli, Sommeria and Lam1997) as a highly convenient tool for the classification of the inviscid wave regimes in a trapezoidal domain filled with a linearly stratified fluid. Lyapunov exponents have been used as a variable characterizing the convergence of wave rays toward limit cycles. The plot of the Lyapunov exponents as a function of $(d,\unicode[STIX]{x1D70F})$ is shown in figures 1 and 4 in greyscale. The white (respectively grey) domains correspond to strong (respectively weak) convergence onto wave attractors. Wave attractors with high rate of convergence toward a simple well-defined structure are found in white regions (as anticipated, the so-called Arnold tongues). The dark- and light-grey regions have a highly complicated structure where the Lyapunov exponent varies in a wide range, occasionally reaching nearly zero values (quasi-resonances). Ray tracing reveals for these regions a slow convergence toward attractors of high complexity with a high number of boundary reflections. Identically zero Lyapunov exponents are found at a discrete set of lines corresponding to global resonances (seiche modes).

It can be seen that the domain of existence of a (1,1) attractor in figure 4 has a triangular shape. In a linear viscous regime at fixed values of $Ar$ and $H/L$ , the quantities $\left\langle K\right\rangle$ and $R$ are the functions of a coordinate in $(d,\unicode[STIX]{x1D70F})$ plane. Combining the information presented in terms of  $\left\langle K\right\rangle$ and  $R$ , we can conclude that there are two distinct regions with different regimes for $R$ .

1. (i) The central part of the Arnold tongue with high values of  $R$ : it is indicating a propagating wave system (denoted NP in the remainder of the text) corresponding to a classic case of wave attractor with thin well-defined branches. It is clearly seen in figure 4(a) that there exists an optimum range in $(d,\unicode[STIX]{x1D70F})$ -space, where the mean kinetic energy of a well-focused attractor is maximized. As the focusing increases (negative $d$ ), the excitation of high-energy attractors is hindered by increased dissipation in narrow wave beams.

2. (ii) The border regions of the Arnold tongue are typically characterized by low values of  $R$ due to geometric degeneration of the attractor. In that domain referred as NS, waves are nearly standing and the mean kinetic energy of attractor is low except, the right tip of the Arnold tongue, where we observed high values of  $\left\langle K\right\rangle$ and low values of $R$ , typical features for standing waves. Indeed, the right tip of the Arnold tongue corresponds to $d=1$ and $\unicode[STIX]{x1D70F}=2$ , i.e. to an exact standing mode. Thus, the secondary maximum of $\left\langle K\right\rangle$ in the vicinity of the right corner of the Arnold tongue is associated with optimum condition of excitation of a nearly standing wave in a nearly rectangular domain.

Figure 4. Structure of Arnold tongue of a stable (1,1) attractor in terms of the mean normalized kinetic energy  $\left\langle K\right\rangle$ (a) and the ratio of minimum to maximum kinetic energy  $R$ (b). The results are plotted from the 50 experiments described in table 1. Superimposed on this picture are three different symbols corresponding to the location of the three main long-term experiments discussed throughout this paper: $\blacklozenge$ and $\diamondsuit$ for the case of the rectangular domain discussed by McEwan (Reference McEwan1971) (see § 4.1), ▪ for the well-focused (1,1) attractor discussed in Brouzet et al. (Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ) with propagating waves (see § 4.2), $\star$ for the weakly focused (1,1) attractor described here (see § 4.3). The solid black line shows the contour at the value $R=0.5$ , enclosing the region with nearly propagating waves (NP), while nearly standing waves (NS) are outside this region.

4 Three scenarios for the onset of triadic resonance in a trapezoidal domain

In this section, we discuss how the choice of the operating point in the Arnold tongue of (1,1) attractors affects the observed scenario for the onset of triadic resonance instability (TRI). In table 2 we specify the experimental conditions of the three main long-term experiments represented by the symbols $\blacklozenge$ , ▪ and $\star$ in figure 4.

Table 2. Parameters for the three main and long-term experiments discussed in this paper. $\ell _{\ast }$ and $m_{\ast }$ are respectively the horizontal and vertical forcing wavenumbers. The horizontal and vertical wave-vector components are denoted ( $\ell _{i},m_{i}$ ) in which the subscript $i$ refers to the primary ( $i=0$ ) or to the secondary waves ( $i=1$ and 2). $\unicode[STIX]{x1D6FA}_{i}$ is the corresponding non-dimensionalized frequency. In McEwan (Reference McEwan1971), we have chosen the data with the frequency triplet that is close to those considered in Brouzet et al. (Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ) and in the present paper. The values of wave vectors are evaluated from the parameters of standing waves presented in McEwan (Reference McEwan1971).

4.2 TRI with propagating waves: the case of well-focused attractor

This case of TRI has been described in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) and Brouzet et al. (Reference Brouzet, Sibgatullin, Scolan, Ermanyuk and Dauxois2016b ). It is a typical scenario of instability in the central region of the Arnold tongue corresponding to a (1,1) attractor in $(d,\unicode[STIX]{x1D70F})$ diagram. An experiment reported in Brouzet et al. (Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ), also located in the central region of the (1,1) Arnold tongue, shows a similar instability, illustrated on figure 5. This experiment is represented by the symbol ▪ in figure 4 for $a=5~\text{mm}$ . For this regime, $\langle K\rangle$ is large and $R\gtrsim 0.5$ . Accordingly, the (1,1) attractor has thin well-localized branches. Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) and Brouzet et al. (Reference Brouzet, Sibgatullin, Scolan, Ermanyuk and Dauxois2016b ) have shown that the onset of instability in such a wave attractor occurs locally first in the most energetic wave beam which has the largest amplitude but the smallest width. This scenario of instability is consistent with the one discussed in Bourget et al. (Reference Bourget, Dauxois, Joubaud and Odier2013, Reference Bourget, Scolan, Dauxois, Le Bars, Odier and Joubaud2014) and Karimi & Akylas (Reference Karimi and Akylas2014). This is also what is observed in Brouzet et al. (Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ), at the very beginning of the triadic cascade, as shown in figure 5.

Owing to local character of instability (Scolan et al. Reference Scolan, Ermanyuk and Dauxois2013; Brouzet et al. Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ,Reference Brouzet, Sibgatullin, Scolan, Ermanyuk and Dauxois b ), the confinement of fluid domain does not affect the instability directly. The focusing provides the energy transfer from the input perturbation which has the scale of vertical size of the fluid domain to the scale associated with the width of the attractor beams, which serves as a primary wave. Subsequent triadic resonance transfers energy to secondary waves. The overall efficiency of the energy transfer from large- to small-scale motions is remarkably high, even at the laboratory scale. Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) describe the case where the typical wavelength of the primary wave represented by the beam of the attractor is approximately 9 times shorter than the wavelength of the input forcing, and one of the secondary waves has even a wavelength roughly 3 times shorter than the primary wave, leading to a reduction factor of 25. This is also visible in figure 5 where the primary and the two secondary waves have been disentangled using the Hilbert transform (Mercier, Garnier & Dauxois Reference Mercier, Garnier and Dauxois2008).

Figure 5. Horizontal velocity fields for the experiment indicated by the symbol ▪ in figure 4 and table 2 for $a=5~\text{mm}$ , filtered at $\unicode[STIX]{x1D6FA}_{0}$ (a), $\unicode[STIX]{x1D6FA}_{1}$ (b) and $\unicode[STIX]{x1D6FA}_{2}$ (c) in frequency and in space to keep only the relevant directions. The velocity field is displayed only where the wave amplitude is larger than $15\,\%$ of the maximum. The wave vectors are represented by the arrows. Black dashed lines show the billiard geometric prediction of the attractor.

4.3 TRI with nearly standing waves: the case of weakly focused attractor

In this subsection, we describe a new scenario for the onset of instability which is intermediate between those described in §§ 4.1 and 4.2. This new scenario is typical for weak focusing. The observation of triadic resonance has been performed for $(d,\unicode[STIX]{x1D70F})=(0.65,1.80)$ and $a=5~\text{mm}$ . This experiment is represented by the symbol $\star$ in figure 4.

4.3.1 The onset of instability

The snapshot of wave pattern after $200$ periods of oscillations (approximately 30 min of observations) is shown in figure 6(a). The time history of oscillations measured at the point A, defined in figure 6(a), is presented in figure 6(b). After a rather short transient of approximately $25$ periods, the signal is stable: the attractor is set. Then, one can clearly see the apparent instability developing after $400$ forcing periods, i.e. after more than $1$  hour of observations.

Figure 6. Velocity field for the experiment indicated by the symbol $\star$ in figure 4 with $a=5~\text{mm}$ (see table 2 for the parameters). (a) Picture of internal wave field, filtered around $\unicode[STIX]{x1D6FA}_{0}$ , $200~T_{0}$ after the start of the wave maker and black lines show the billiard geometric prediction of the attractor. The picture presents the absolute value of the norm of the velocity vectors. The large value around the upper corner of the attractor is due to the presence of a small homogenous layer above the stratified domain. (b) The time history of oscillations at the point A shown in (a). The vertical dashed line in (b) shows the instant of the picture (a) while the dashed square in (a) shows the region where the time–frequency diagram in figure 7 has been calculated. The initial transient is due to the setting of the attractor, while the instability is noticeable after roughly 400 periods of oscillations.

The development of the spectrum of wave motion over time is presented in figure 7(a) with the use of the time–frequency diagram. It is calculated at each point in space, as in Bourget et al. (Reference Bourget, Dauxois, Joubaud and Odier2013), with the formula

(4.1) $$\begin{eqnarray}S_{r}(\unicode[STIX]{x1D6FA},t)=\left\langle \left|\int _{-\infty }^{+\infty }\!v_{r}(x,z,\unicode[STIX]{x1D70F})\text{e}^{\text{i}\unicode[STIX]{x1D6FA}N\unicode[STIX]{x1D70F}}h(t-\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\right|^{2}\right\rangle _{xz},\end{eqnarray}$$

where $h$ is an Hamming window and $v_{r}$ the component of the velocity field, $r=x$ or $z$ . The calculations are performed with the Matlab toolbox described in Flandrin (Reference Flandrin1999).

Figure 7. Weak TRI. (a) Time–frequency diagram of the internal wave field, obtained from a $50~\text{mm}$ side square region, surrounding the point A shown in figure 6(a); (b) vertical cut of the diagram along the frequency axis at $t=400T_{0}$ , indicated by the white dashed line. $S_{0}$ corresponds to the time average of the frequency component associated with the primary wave $\unicode[STIX]{x1D6FA}_{0}$ . Results are presented for the experiment indicated by the symbol $\star$ in figure 4 (see table 2 for parameters).

The appropriate choice of the length of the Hamming window allows us to tune the resolution in frequency and time. Typically, the data are averaged over a finite analysing area which, as a limit, can extend over the whole fluid domain. Figure 7(a) presents the time–frequency diagram for the case illustrated in figure 6. The diagram is calculated for the data averaged over the rectangle shown in figure 6(a) around point A. It can be seen that, initially, the signal is entirely dominated by the forcing frequency $\unicode[STIX]{x1D6FA}_{0}=0.60$ corresponding to the primary (carrier) wave. Oscillations with frequencies $\unicode[STIX]{x1D6FA}_{1}=0.33$ and $\unicode[STIX]{x1D6FA}_{2}=0.27$ slowly develop with time. These frequencies correspond to two secondary waves generated by TRI. They satisfy the frequency conditions for the triadic resonance:

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{1}+\unicode[STIX]{x1D6FA}_{2}=\unicode[STIX]{x1D6FA}_{0}.\end{eqnarray}$$

In addition, one can also see two peaks satisfying differential conditions:

(4.3a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{3}-\unicode[STIX]{x1D6FA}_{1}=\unicode[STIX]{x1D6FA}_{0}\quad \text{and}\quad \unicode[STIX]{x1D6FA}_{4}-\unicode[STIX]{x1D6FA}_{2}=\unicode[STIX]{x1D6FA}_{0}.\end{eqnarray}$$

4.3.2 The components of the wave pattern

Figure 8 shows the decomposition of the wave field into the components corresponding to the secondary waves while the primary wave has already been exhibited in figure 6(a). The decomposition, performed with the help of Hilbert transform (Mercier et al. Reference Mercier, Garnier and Dauxois2008), reveals node–antinode patterns of amplitudes of secondary waves, similar to Chladni figures. These patterns are present soon in the experiment, but their amplitude increases gradually. Thus, this is very different from TRI presented in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) where the waves are created on a very localized region of the tank. Here, the secondary waves appear directly everywhere in the tank, in a shape presented in figure 8. The sequences of wave profiles measured along the vertical line indicated in the images are also shown in the right panels. It can be seen that the secondary wave motion is represented by standing waves of high vertical modes. They do not correspond to global eigenmodes of the trapezoidal basin (represented by the black lines of the $(d,\unicode[STIX]{x1D70F})$ diagram) but to quasi-global resonances (dark regions of the $(d,\unicode[STIX]{x1D70F})$ diagram, as shown in figure 1(c). Diffusion induces a certain thickness to the branches and thus allows the secondary waves to be quasi-standing and to correspond to quasi-global modes of the trapezoidal basin.

Figure 8. Components of the wave field corresponding to frequencies $\unicode[STIX]{x1D6FA}_{1}$ (a) and $\unicode[STIX]{x1D6FA}_{2}$ (b), obtained with the Hilbert transform around $620~T_{0}$ using a time window $85~T_{0}$ long. Results are presented for the experiment indicated by the symbol $\star$ in figure 4 with $a=5~\text{mm}$ (see table 2 for the parameters). Note that the components oscillating at $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ correspond to nearly standing waves as is clearly seen on the sequences of wave profiles shown on the right of each picture.

The standing-wave patterns can be further decomposed into the sums of propagating waves whose wave vectors can be measured (see data in table 2). All vectors satisfy the dispersion relation individually. The length of the primary wave can be estimated (Mercier et al. Reference Mercier, Garnier and Dauxois2008) as $\unicode[STIX]{x1D706}_{0}=2\unicode[STIX]{x03C0}/|\boldsymbol{k}_{0}|=39.8~\text{cm}$ with horizontal and vertical components $\unicode[STIX]{x1D706}_{x0}=66.1~\text{cm}$ and $\unicode[STIX]{x1D706}_{z0}=49.7~\text{cm}$ . The quantities $\unicode[STIX]{x1D706}_{x0}/2$ and $\unicode[STIX]{x1D706}_{z0}/2$ are reasonably close to the distances from the left top corner of the trapezoidal fluid domain to the reflection points of the attractor. The ray tracing yields the distances to the reflection points 28 and 21.6 cm in horizontal and vertical directions, respectively. Therefore, in the present case, the primary wave has a length scale associated with the global geometry of attractor. The length scale associated with the beam width does not clearly manifest itself. Indeed, weakly focused attractors with elongated shape have two parallel beams which can interact with each other since the width of beams and the inter-beam distance are comparable quantities. As result the notion of the beam width is difficult to define for such attractors, in contrast with spatially localized beams of well-focused attractors. In the latter case described in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) (see § 4.2 and symbol ▪ in figure 4), we have seen that the length of the primary wave is associated with the width of the attractor beam.

4.3.3 The growth rate

Horizontal cuts in figure 7 at $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ are plotted in figure 9. At low amplitude of secondary waves, the linear trend is obscured by the instrumental noise, while toward the end of the experiment there is a trend to saturation of amplitudes of secondary waves. The vertical logarithmic representation emphasizes a remarkably linear growth for about three decades (!) in the magnitude of the spectral peaks corresponding to secondary waves. The growth rate of the instability can be estimated to be around $\unicode[STIX]{x1D70E}=6.5\times 10^{-4}~\text{s}^{-1}$ , which leads to a characteristic growth time of the instability around 1500 s (i.e. half an hour or $150T_{0}$ ). The theoretical value for monochromatic plane wave, taking account of viscosity (Koudella & Staquet Reference Koudella and Staquet2006; Bourget et al. Reference Bourget, Dauxois, Joubaud and Odier2013) is equal to $2.1\times 10^{-2}~\text{s}^{-1}$ , that is two order of magnitudes larger than the experimental value. The difference is due to the spatial confinement of the attractor, which differs clearly from monochromatic plane wave, as underlined by Karimi & Akylas (Reference Karimi and Akylas2014) and Bourget et al. (Reference Bourget, Scolan, Dauxois, Le Bars, Odier and Joubaud2014).

Figure 9. Cuts along the time axis of the time–frequency diagram presented in figure 7 at frequencies $\unicode[STIX]{x1D6FA}_{1}$ (solid red line) and $\unicode[STIX]{x1D6FA}_{2}$ (dashed blue line).

The instability growth time has to be compared with the time scale  $\unicode[STIX]{x1D70F}_{w}$ set by the limited size of the test tank. It can be defined as the time for secondary waves to perform a horizontal return trip in the tank: $\unicode[STIX]{x1D70F}_{w}=2L/c_{gx}$ , where $c_{gx}$ is the group velocity in the horizontal direction. This group velocity can be measured using the wavenumbers of the secondary waves estimated via Hilbert transform and wavenumber filtering. It gives $\unicode[STIX]{x1D70F}_{w}\approx 60~$ s, which is very small in comparison with the instability growth time of $1500~$ s. It is worth to note that the horizontal group velocity is a good measure of the return-trip time of secondary waves because these waves are only slightly inclined. Indeed, their propagation angle, with respect to the horizontal direction, is close to $20^{\circ }$ since $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ are around $\unicode[STIX]{x1D6FA}_{0}/2=0.3$ .

The data presented in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) or in figure 5 clearly show that the triadic resonance develops first in the most energetic branch of the wave attractor, the one connecting the inclined slope to the surface. The instability growth time is of $30$  s. The data on the wave-vector components presented in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) yield $\unicode[STIX]{x1D70F}_{w}$ around $180$ and $320~$ s for the two secondary waves evolved into the triplet which is measured at a time approximately $300~$ s after emergence of detectable secondary waves. Therefore, the secondary waves could make only between 1 and 2 return trips, and owing to the presence of the vertical component of the group velocity they could not return to their generation site. So, the onset of the instability described in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013), and also seen in the experiment represented by the ▪ on the $(d,\unicode[STIX]{x1D70F})$ diagram is to a good approximation a spatially isolated local event, while the onset and subsequent development of instability described in the present section is a global event, which is inherently related to the confinement. This event has many common features with the instability in a rectangular domain described in McEwan (Reference McEwan1971).

5 Well-developed TRI cascade in well-focused attractor: presence of secondary standing waves

In the previous section, we have demonstrated the effect of the ratio between the typical growth time of the primary instability $\unicode[STIX]{x1D70F}_{\ast }$ and the duration of the horizontal return trip $\unicode[STIX]{x1D70F}_{w}$ of secondary waves in the tank. If $\unicode[STIX]{x1D70F}_{\ast }\ll \unicode[STIX]{x1D70F}_{w}$ , the onset of instability is local, while for $\unicode[STIX]{x1D70F}_{\ast }\gg \unicode[STIX]{x1D70F}_{w}$ the secondary waves are more likely to appear in form of standing waves. However, $\unicode[STIX]{x1D70F}_{\ast }$ and $\unicode[STIX]{x1D70F}_{w}$ do not form an exhaustive set of time scales. The evolution of the system to saturated state where the energy injection is in balance with dissipation may have a very long time scale $\unicode[STIX]{x1D70F}_{sat}$ . Over the time span $0<t<\unicode[STIX]{x1D70F}_{sat}$ , the system may experience a cascade of triadic resonance instabilities generating new time scales of the return trips $\unicode[STIX]{x1D70F}_{w}$ . For the full study of dynamics, the system should be observed during the time $\unicode[STIX]{x1D70F}_{obs}>\unicode[STIX]{x1D70F}_{sat}\gg \unicode[STIX]{x1D70F}_{w}$ . Thus, long-term experiments with $\unicode[STIX]{x1D70F}_{obs}\gg \unicode[STIX]{x1D70F}_{w}$ can reveal the effects of confinement for the system with $\unicode[STIX]{x1D70F}_{\ast }\ll \unicode[STIX]{x1D70F}_{w}$ similar to the one described in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013). In this section, we analyse the long-term evolution of the spectrum in the case with strong focusing (▪ on $(d,\unicode[STIX]{x1D70F})$ diagram) and present for the first time the evidence of the effects due to confinement.

5.1 Time–frequency diagram

The time–frequency diagram of the long-term experiment with well-focused attractor (▪) at $a=5~\text{mm}$ is presented in figure 10. One can identify at least eight couples of secondary waves in (a). The four most energetic couples in the spectrum are labelled in panel (b). The secondary frequencies are named using two indices as follows. The first index $i$ indicates the position in the couple, $i=1$ for frequencies higher than $\unicode[STIX]{x1D6FA}_{0}/2$ or $i=2$ for frequencies smaller than $\unicode[STIX]{x1D6FA}_{0}/2$ . The second index $n$ corresponds to the number of the couple. Thus, $n$ varies from $1$ to $8$ , as one can observe at least eight different couples. For each couple  $n$ , one has

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{1,n}+\unicode[STIX]{x1D6FA}_{2,n}=\unicode[STIX]{x1D6FA}_{0}.\end{eqnarray}$$

The index $n$ classifies the different couples from the most intense ( $n=1$ ) to the less intense ( $n=8$ ).

Figure 10. TRI cascade. (a) Time–frequency diagram of internal wave field, obtained from a $5\times 5$  cm $^{2}$ side square region in the most energetic wave beam; (b) vertical cut of the diagram along the frequency axis at $t=400T_{0}$ , indicated by the black dashed line. $S_{0}$ corresponds to the time average of the frequency component associated with the primary wave $\unicode[STIX]{x1D6FA}_{0}$ . The different frequency peaks corresponding to the four first couples are labelled. Results are presented for the experiment indicated by the symbol ▪ in figure 4 with $a=5$  mm (see table 2 for the parameters).

The first couple is generated by the attractor itself, as a standard TRI described in Scolan et al. (Reference Scolan, Ermanyuk and Dauxois2013) or in § 4.2. Then, other couples appear and all the frequencies in the tank are linked by $3$ -wave interactions. The cascade is created as follows. Once the first couple has been created by the attractor, the two frequencies of this couple ( $\unicode[STIX]{x1D6FA}_{1}=\unicode[STIX]{x1D6FA}_{1,1}$ and $\unicode[STIX]{x1D6FA}_{2}=\unicode[STIX]{x1D6FA}_{2,1}$ ) interact together to create a third frequency ( $\unicode[STIX]{x1D6FA}_{2,2}$ ), which belongs to the second couple. The second couple is completed by the interaction between the third frequency and the frequency of the attractor, $\unicode[STIX]{x1D6FA}_{0}$ . These reactions are given by

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}_{1,1}-\unicode[STIX]{x1D6FA}_{2,1}=\unicode[STIX]{x1D6FA}_{2,2}, & \displaystyle\end{eqnarray}$$

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}_{0}-\unicode[STIX]{x1D6FA}_{2,2}=\unicode[STIX]{x1D6FA}_{1,2}. & \displaystyle\end{eqnarray}$$

Thus, the second couple is complete. Now, several interactions are possible between the different frequencies. An interaction between the first and the second couples leads to a third one:

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}_{1,2}-\unicode[STIX]{x1D6FA}_{1,1}=\unicode[STIX]{x1D6FA}_{2,3}, & \displaystyle\end{eqnarray}$$

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}_{0}-\unicode[STIX]{x1D6FA}_{2,3}=\unicode[STIX]{x1D6FA}_{1,3}, & \displaystyle\end{eqnarray}$$

and an interaction between the frequencies of the second couple creates a forth couple:

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}_{1,2}-\unicode[STIX]{x1D6FA}_{2,2}=\unicode[STIX]{x1D6FA}_{1,4}, & \displaystyle\end{eqnarray}$$

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}_{0}-\unicode[STIX]{x1D6FA}_{1,4}=\unicode[STIX]{x1D6FA}_{2,4}. & \displaystyle\end{eqnarray}$$

Here, one has four different couples, so eight frequencies. One can easily continue to create other frequencies by combining the different frequencies of the different couples. This mechanism leads to an infinite set of discrete frequencies and the first eight couples are at least visible in figure 10. Note that each coupling satisfies also the spatial resonant condition and that all the waves fulfil the dispersion relation. At the end, each frequency is linked with a large number of other frequencies by a three wave interaction (Brouzet et al. Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ). Experimentally, we observe the fulfillment of the above equations with the accuracy in non-dimensional frequency of $\pm 0.001$ .

5.2 Observational evidence of standing secondary waves

Although the instability starts and evolves differently in well- and weakly focused attractors, they can exhibit some common features. Figure 8 shows that the secondary waves generated by the weakly focused attractor are standing waves. Some frequencies created by a well-focused attractor in long-term experiments can also be standing waves as shown in figure 11. For example, the fluid motions at frequencies $\unicode[STIX]{x1D6FA}_{2,1}$ and $\unicode[STIX]{x1D6FA}_{2,2}$ are standing waves as it is clear from the wave patterns filtered at these frequencies. Note that not all the frequencies present in figure 10 correspond to standing waves.

Figure 11. Components of the wave field corresponding to frequencies $\unicode[STIX]{x1D6FA}_{2,1}$ (a) and $\unicode[STIX]{x1D6FA}_{2,2}$ (b), obtained with the Hilbert transform around $620~T_{0}$ using a time window $85~T_{0}$ long. Note that the components oscillating at $\unicode[STIX]{x1D6FA}_{2,1}$ and $\unicode[STIX]{x1D6FA}_{2,2}$ correspond to standing waves as is clearly seen on the sequences of wave profiles shown on the right of each picture. Results are presented for the experiment indicated by the symbol ▪ in figure 4 with $a=5~\text{mm}$ (see table 2 for the parameters).

With the frequencies corresponding to the peaks of the spectrum shown in figure 10(b), one can compute the values of parameters $\unicode[STIX]{x1D70F}$ associated with these waves. These frequencies are shown by different symbols in the diagram $(d,\unicode[STIX]{x1D70F})$ presented in figure 1. It can be seen that the $\unicode[STIX]{x1D70F}$ parameters for the standing secondary waves are close to black zones in the $(d,\unicode[STIX]{x1D70F})$ diagram. Thus, at least some frequencies created by the weakly and well-focused attractors could correspond to quasi-resonant modes. In absence of commonly accepted terminology, we can tentatively call the dark regions (see figure 1) as geometric quasi-resonances. Previously quasi-resonances have been discussed assuming non-zero resonance width (see Nazarenko Reference Nazarenko2011). Here we refer to possibility of global quasi-resonances assuming that the Lyapunov exponent can take very small negative non-zero value. In a realistic system with weak viscous dissipation, we can expect that some of the quasi-resonances can exhibit a seiche-like behaviour similar to exact seiche modes discussed in Maas et al. (Reference Maas, Benielli, Sommeria and Lam1997) and Manders & Maas (Reference Manders and Maas2003). Therefore, at the conceptual level, the key driving processes in long-term experiments with unstable wave attractors can be considered as a combination of ‘local’ (Scolan et al. Reference Scolan, Ermanyuk and Dauxois2013) and ‘global-scale’ (present paper) TRI events, with long-term transients between the two.

Interesting parallels can be found in the dynamics of rotating fluids. For instance, Beardsley (Reference Beardsley1970), Thompson (Reference Thompson1970) and Duguet, Scott & Le Penven (Reference Duguet, Scott and Le Penven2006) show rich multi-peak spectra for inertial oscillations in a compressible fluid confined in a rotating cylindrical vessel; they relate the frequencies associated with the observed peaks to global modes of the fluid motion. Similarly, Favier et al. (Reference Favier, Grannan, Le Bars and Aurnou2015) analyse the frequency spectrum of fluid motion generated by libration-driven elliptic instability in a rotating ellipsoid and show the presence of the eigenfrequencies of linear and quadratic inertial modes.

Figure 12. Time–frequency diagrams of two signals recorded in unstable wave attractors corresponding to different operating points. Both attractors are unstable and the wave regime is such that it induces significant mixing. (a) Corresponds to the experiment indicated by the symbol ▪ in figure 4, and (b) by the $\star$ (with $a=10~\text{mm}$ and other parameters listed in table 2).

Figure 13. (a–d) present the probability density functions of the vorticity in the tank, while (e,f) exhibit the ratio of the final (after 700 periods) to initial stratifications. The PDF are calculated from all images between $480$ and $500~T_{0}$ , for the two propagating (a, ▪ in figure 4) or standing (b, $\star$ in figure 4) wave experiments presented in figure 12; (c,d) present the same quantity for the vorticity filtered at $\unicode[STIX]{x1D6FA}_{0}$ . In all panels, dashed black and solid red lines correspond to $a=5~\text{mm}$ and $a=10~\text{mm}$ .

6 The ‘wave mixing box’ regime in NP and NS attractors

In the previous sections, we have shown that the scenario of instability in wave attractors and the parameters of wave triplets involved into resonant interactions strongly depend on the choice of the operating points in $(d,\unicode[STIX]{x1D70F})$ plane. However, one can expect a certain level of universality for fully developed cascades of triadic interactions transferring energy to small-scale internal waves and ultimately to mixing events. A high mixing efficiency of triadic cascades operating in confined domains has been demonstrated so far in McEwan (Reference McEwan1983) for normal modes and in  Brouzet et al. (Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ) for well-focused attractors, in rectangular and trapezoidal domains, respectively.

Below, we compare the case of a well-focused (1,1) attractor described in Brouzet et al. (Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ) (▪) with the case of a weakly focused (1,1) attractor ( $\star$ ). The time–frequency diagrams of two unstable attractors obtained at high amplitude of forcing ( $a=10~\text{mm}$ ) in regimes with measurable mixing are shown in figure 12. Both diagrams have qualitatively similar features: several discrete frequency peaks of large magnitude are embedded into a continuous frequency spectrum, which has significantly smaller magnitude. The magnitudes of discrete peaks strongly fluctuate in time.

Brouzet et al. (Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ) show that the normalized horizontal $y$ -component of the vorticity field,

(6.1) $$\begin{eqnarray}\unicode[STIX]{x1D6EF}_{y}=\frac{\unicode[STIX]{x1D709}_{y}(x,z,t)}{N}=\frac{1}{N}\left(\frac{\unicode[STIX]{x2202}v_{x}}{\unicode[STIX]{x2202}z}-\frac{\unicode[STIX]{x2202}v_{z}}{\unicode[STIX]{x2202}x}\right),\end{eqnarray}$$

represents a useful quantity indicating the occurrence of extreme events in the experimental system. This quantity has a clear physical meaning as a ratio of the destabilizing effect of vorticity to the stabilizing effect of stratification. Obviously, we can expect some mixing in the system when a significant statistics of events with high values of  $|\unicode[STIX]{x1D6EF}_{y}|$ is observed.

Figure 13 represents the PDFs of $\unicode[STIX]{x1D6EF}_{y}$ for (1,1) attractors with weak and strong focusing for two different values of the forcing amplitude $a$ . Comparing the data obtained at fixed $a$ , we see that quantitatively the system with weak focusing shows consistently lower statistics of extreme events than the system with high focusing. Qualitatively, the shape of vorticity PDFs in attractors with high and low focusing is similar. Moreover, the long-term erosion of stratification is also similar as discussed below.

The significant statistics of extreme events with high horizontal vorticity arises from the spontaneous summation of the frequency components of the wave field. It is important to note that the primary (monochromatic) wave alone cannot produce extreme events. This is attested by figure 13(c,d) that represent the PDFs of $\unicode[STIX]{x1D6EF}_{y}$ for the primary wave field filtered at  $\unicode[STIX]{x1D6FA}_{0}$ . It can be seen that these PDFs have no tails extending to the domains with $|\unicode[STIX]{x1D6EF}_{y}|>2$ . The threshold value $|\unicode[STIX]{x1D6EF}_{y}|>2$ has been proposed in Brouzet et al. (Reference Brouzet, Ermanyuk, Joubaud, Sibgatullin and Dauxois2016a ) as extension of the classic Miles–Howard criterion for stability of horizontal shear flows of continuously stratified fluid. In fact, a possibility of extension of the Miles–Howard criterion to a wider context has been discussed in some classic works, e.g. Phillips (Reference Phillips1966). Using $|\unicode[STIX]{x1D6EF}_{y}|=2$ as a threshold value one can make some qualitative conclusions.

In figure 13(e,f), we show the ratios of the final to initial stratifications observed in different experiments described in the present paper. In both cases at large amplitude of forcing we observe a well-measurable mixing when comparing the initial and final density distributions. The probability of having an extreme event with intensity $|\unicode[STIX]{x1D6EF}_{y}|>2$ is given by the integral over the corresponding tails of the PDF. The cases with measurable mixing depicted in figure 13(e,f) correspond to drastically increased probability of extreme events. It increases from 2 ‰, for $a=5~\text{mm}$ to 19 ‰ for $a=10~\text{mm}$ in the ‘propagating waves’ case, and from 0.7 ‰  for $a=5~\text{mm}$ to 5.6 ‰ for $a=10~\text{mm}$ in the ‘standing waves’ case. We observe a comparable increase of probability, by factor 10 and 8, in both cases. Although the statistics of high vorticity events seems to represent a kinematic indicator for the occurrence of mixing, at the present stage it is difficult to relate this statistics directly to the overall mixing efficiency. Additional parameters should be measured, most importantly, the typical scale of overturning structures. As an alternative, one can make an attempt to estimate the turbulent diffusion coefficient in the experimental system similar to Dossmann et al. (Reference Dossmann, Bourget, Brouzet, Dauxois, Joubaud and Odier2016).