Linear and nonlinear regimes of an inertial wave attractor

We present an experimental analysis of the linear and nonlinear regimes of an attractor of inertial waves in a trapezoidal cavity under rotation. Varying the rotation rate and the forcing amplitude and wavelength, we identify the scaling laws followed by the attractor amplitude and wavelength in both regimes. In particular, we show that the nonlinear scaling laws can be well described by replacing the fluid viscosity in the linear model by a turbulent viscosity, a result that could help in extrapolating attractor theory to geophysically and astrophysically relevant situations. We further study the triadic resonance instability of the attractor which is at the origin of the turbulent viscosity. We show that the typical frequencies of the subharmonic waves produced by the instability behave very differently from previously reported numerical results and from the prediction of the theory of triadic resonance. This behavior might be related to the deviation from the horizontal invariance of the attractor in our experiment in relation to the presence of vertical walls of the cavity, an effect that should be at play in all practical situations.

Figure 1

Scheme of the water tank (in blue) containing the trapezoidal cavity (in red). The upper wall of the cavity is made of a series of $n=23$ horizontal bars, each of them connected to a linear motor able to drive it in a vertical motion. The truncated dashed rectangle shows the region over which the velocity amplitude reported in Fig. 3 is averaged.

Figure 2

Time series of the spatially averaged kinetic energy $K=\langle u_x^2+u_z^2\rangle /2$ for $\mathrm{\Omega }=18$ rpm, ${\lambda }_f=52.4$ cm, and three forcing amplitudes $A=0.09$, 0.19, and 0.38 mm. Time $t$ is normalized by the period $T=2\pi /{\sigma }_0$ of the wave maker. For each time series the black thin line shows the original data and the thick line its moving average using a time window of $5T$.

Figure 3

Velocity amplitude $U_{\theta }$ of the wave excited by the generator in the cavity in the absence of the tilted plane for $\mathrm{\Omega }=3$ and 18 rpm, ${\lambda }_f=52.4$ cm, and three forcing amplitudes $A=0.09$, 0.19, and 0.36 mm. The data are reported as a function of $U_f=A{\sigma }_0/sin\theta$. The solid line indicates the identity function.

Figure 4

Reflection of an inertial wave of angular frequency $\sigma =2\mathrm{\Omega }cos\theta$ on a plane tilted by an angle $\alpha$. The vectors ${\mathbf{c}}_{\phi }$ and ${\mathbf{c}}_{\mathbf{g}}$ show the directions of the phase and group velocities for the incident and reflected beams.

Figure 5

Scheme of the theoretical inviscid (in green) and viscous (in blue) attractors in the trapezoidal cavity, for $\theta =32.0^{\circ }$. The blue lines more precisely show the width at midheight of the velocity envelope of the Moore-Saffman self-similar wave beam [Eq. (6)]. The vectors ${\mathbf{c}}_{\phi }$ and ${\mathbf{c}}_{\mathbf{g}}$ show for each attractor branch the direction of the phase and group velocities.

Figure 6

Temporal power spectral density $E\left(\sigma \right)$ [Eq. (12)] as a function of the normalized frequency ${\sigma }^{*}=\sigma /2\mathrm{\Omega }$ for $A=0.09$ mm, ${\lambda }_f=52.4$ cm, and $\mathrm{\Omega }=18$ rpm. We have highlighted five peaks at frequencies ${\sigma }_0^{*}={\sigma }_0/2\mathrm{\Omega }$, ${\sigma }^{*}=0.5$ (i.e., $\sigma =\mathrm{\Omega }$), ${\sigma }^{*}=1$ (i.e., $\sigma =2\mathrm{\Omega }$), ${\sigma }^{*}={\sigma }_0^{*}-0.5$, and ${\sigma }^{*}=1-{\sigma }_0^{*}$.

Figure 7

Snapshots of the velocity field after a temporal moving average over a time window of eight wave-maker periods (i.e., a cutoff frequency of ${\sigma }^{*}\simeq 0.11$) in the vertical plane $y=y_0=L_y/3$ for ${\lambda }_f=52.4$ cm and the lowest forcing amplitude $A=0.09$ mm at $\mathrm{\Omega }=18$ rpm.

Figure 8

Attractor in the linear regime: snapshot of the velocity field in the vertical plane $y=y_0=L_y/3$ Fourier filtered at the forcing frequency ${\sigma }_0$ for ${\lambda }_f=52.4$ cm and the lowest forcing amplitude $A=0.09$ mm at $\mathrm{\Omega }=18$ rpm. A sketch of the theoretical attractor is superimposed on the experimental field: The dashed line shows the inviscid skeleton and the two solid lines delineate the width at midheight of the viscous beam longitudinal velocity amplitude [Eq. (6)].

Figure 9

Attractor in the linear regime: snapshot of the out-of-plane vorticity of the Hilbert filtered field at the forcing frequency ${\sigma }_0$ for $A=0.09$ mm, ${\lambda }_f=52.4$ cm, and $\mathrm{\Omega }=18$ rpm. The reported field is actually a combination of four regions in which different wave-vector quadrants have been selected by the Hilbert filtering in agreement with the direction expected for the wave vector in each attractor branch (cf. Fig. 5): In region 1, we keep the wave-vector quadrant $(k_x>0,k_z<0)$; in 2, $(k_x>0,k_z>0)$; in 3, $(k_x<0,k_z>0)$; and in 4, $(k_x<0,k_z<0)$. A sketch of the theoretical attractor is superimposed (same layout as in Fig. 8). The inset shows the experimental transverse profile (blue line with data markers) of the $y$ component of the vorticity ${\omega }_y(\xi =s+L_0,\eta ,\phi )$ of the ${\sigma }_0$ and $(k_x>0,k_z<0)$ Hilbert-filtered field (region 1). It is taken at coordinate $s=49.7$ cm along the attractor axis (corresponding to the straight line in the snapshot) and at a given arbitrary phase. We also report in the inset the experimental wave-beam envelope (red dashed line) computed from the experimental transverse profiles as ${\omega }_{y,0}=\sqrt{2{\langle {\omega }_y{(\xi ,\eta ,\phi )}^2\rangle }_{\phi }}$ where ${\langle \rangle }_{\phi }$ stands for the average on the phase $\phi \in [0,2\pi ]$.

Figure 10

Vorticity amplitude $W\left(\xi \right)$ of the attractor as a function of the coordinate $\xi =s+L_0$ for $A=0.09$ mm, ${\lambda }_f=52.4$ cm, and $\mathrm{\Omega }=18$ rpm. Here $W\left(\xi \right)$ is normalized by the forcing vorticity $2\pi U_f/{\lambda }_f$ and the position $\xi$ by the viscous length scale $\ell$ given by Eq. (8). Vertical red lines show the reflections on the cavity walls and gray regions delineate zones in which PIV measurement is not possible. Following Eq. (7), a power law with an exponent $-2/3$ is also shown as a guide for the eyes.

Figure 11

Normalized wavelength $\lambda /\ell$ in the attractor beam as a function of the normalized distance from the virtual source $\xi /\ell$ for $A=0.09$ mm, ${\lambda }_f=52.4$ cm, and $\mathrm{\Omega }=18$ rpm. Vertical red lines show the reflections on the cavity walls and gray regions delineate zones in which boundary effect prevents measurements. The solid line shows the theoretical predictions, in ${(\xi /\ell )}^{1/3}$ and with no adjustable parameter, for the wavelength $\lambda$.

Figure 12

(a) Normalized wavelength $\lambda /\ell$ and (b) vorticity amplitude $W\left(\xi \right){\lambda }_f/2\pi U_f$ as a function of the normalized distance from the virtual source $\xi /\ell$ for each forcing amplitude $A$ at $\mathrm{\Omega }=18$ rpm and ${\lambda }_f=52.4$ cm. For both figures, vertical red lines show the reflections of the theoretical attractor on the cavity walls and gray regions zones in which the boundary effect prevents measurements. In (a), the solid line shows the theoretical power law, in ${(\xi /\ell )}^{1/3}$ and with no adjustable parameter, for the linear attractor. In (b), a power law with an exponent $-2/3$ is shown, corresponding to the theoretical linear attractor.

Figure 13

Attractor in the nonlinear regime: snapshot of the velocity field in the vertical plane $y=y_0=L_y/3$ Fourier filtered at the forcing frequency ${\sigma }_0$ for $A=3.00$ mm, ${\lambda }_f=52.4$ cm, and $\mathrm{\Omega }=18$ rpm. As in Fig. 8, a sketch of the theoretical linear attractor is superimposed on the experimental field.

Figure 14

(a) Normalized wavelength $\lambda /{\ell }^{2/3}{L}_0^{1/3}$ and (b) normalized vorticity amplitude $W{\ell }^{4/3}{L}_0^{2/3}/{\lambda }_f{U}_f$ averaged over the first branch of the attractor (the one following the focusing reflection) as a function of the forcing Reynolds number ${\text{Re}}_f=U_f{\lambda }_f/\nu$. Squares correspond to experiments at $\mathrm{\Omega }=3$ rpm and ${\lambda }_f=52.4$ cm, triangles to $\mathrm{\Omega }=18$ rpm and ${\lambda }_f=26.2$ cm, and circles to $\mathrm{\Omega }=18$ rpm and ${\lambda }_f=52.4$ cm. The dashed lines in (a) and in (b) show, respectively, the scaling laws $\lambda /{\ell }^{2/3}{L}_0^{1/3}={\text{Re}}_f^{1/3}$ and $W{\ell }^{4/3}{L}_0^{2/3}/{\lambda }_f{U}_f={\text{Re}}_f^{-2/3}$ predicted when replacing the fluid viscosity by the turbulent viscosity ${\nu }_t=U_f{\lambda }_f$ in the viscous length $\ell$ in the linear attractor model. In (a) the horizontal dash-dotted line shows the theoretical value $\mathrm{\Lambda }$ for $\lambda /{\ell }^{2/3}{L}_0^{1/3}$ predicted by the linear attractor model described in Sec. 3 (average of the theoretical value over the first branch). In (b) the horizontal dash-dotted line shows the corresponding numerical value $1/{\mathrm{\Lambda }}^2$, which stands as an estimate for $W{\ell }^{4/3}{L}_0^{2/3}/{\lambda }_f{U}_f$ (see the main text). The solid horizontal lines, black, red, and blue, show the theoretical wavelength ${\lambda }_f/\gamma$ of the beam excited by the wave maker after one reflection on the sloping wall for the three experimental configurations, $\mathrm{\Omega }=18$ rpm and ${\lambda }_f=52.4$ cm, $\mathrm{\Omega }=3$ rpm and ${\lambda }_f=52.4$ cm, and $\mathrm{\Omega }=18$ rpm and ${\lambda }_f=26.2$ cm, respectively.

Figure 15

Experiments without attractor: snapshots of two velocity fields in the vertical plane $y=y_0=L_y/3$ at ${\text{Re}}_f=9500$, Fourier filtered at ${\sigma }_0$ for (a) $A=6$ mm, ${\lambda }_f=26.2$ cm, and $\mathrm{\Omega }=18$ rpm and (b) $A=18$ mm, ${\lambda }_f=52.4$ cm, and $\mathrm{\Omega }=3$ rpm. No attractor is observed in these fields because the wavelength of the theoretical nonlinear attractor is larger than the forcing wavelength. The absence of an attractor is therefore a combined effect of the nonlinearities and the forcing. As in Fig. 8, a sketch of the corresponding theoretical linear attractor is superimposed on each experimental field.

Figure 16

Attractor Reynolds number ${\text{Re}}_W=W{\lambda }^2/\nu$ averaged over the first branch of the attractor (the one following the focusing reflection) as a function of the forcing Reynolds number ${\text{Re}}_f=U_f{\lambda }_f/\nu$. Squares correspond to experiments at $\mathrm{\Omega }=3$ rpm and ${\lambda }_f=52.4$ cm, triangles to $\mathrm{\Omega }=18$ rpm and ${\lambda }_f=26.4$ cm, and circles to $\mathrm{\Omega }=18$ rpm and ${\lambda }_f=52.4$ cm.

Figure 17

Temporal power spectral density $E\left(\sigma \right)$ [Eq. (12)] as a function of the normalized frequency ${\sigma }^{*}=\sigma /2\mathrm{\Omega }$ for all forcing amplitudes $A$ at $\mathrm{\Omega }=18$ rpm and ${\lambda }_f=52.4$ cm. For each spectrum, the horizontal error bars indicate the frequency intervals around which the subharmonic bumps are centered. These intervals correspond to the frequencies reported in Table 1. We have highlighted with vertical lines three other energy peaks at frequencies ${\sigma }_0^{*}={\sigma }_0/2\mathrm{\Omega }$, ${\sigma }^{*}=0.5$ (i.e., $\sigma =\mathrm{\Omega }$), and ${\sigma }^{*}={\sigma }_0^{*}-0.5$.

Figure 18

Spatiotemporal power spectral density $E^{\prime }(k_x,k_z,\sigma )$ [Eq. (16)] for the experiments at $\mathrm{\Omega }=18$ rpm, ${\lambda }_f=52.4$ cm, and (a) and (b) $A=0.75$ mm (${\text{Re}}_f\simeq 2400$) and (c) and (d) $A=3.00$ mm (${\text{Re}}_f\simeq 9500$). The selected frequencies for each experiment correspond to the center frequencies of the subharmonic bumps in their temporal spectrum, i.e., (a) ${\sigma }^{*}=0.32$ and (b) ${\sigma }^{*}=0.53$ for $A=0.75$ mm and (c) ${\sigma }^{*}=0.29$ and (d) ${\sigma }^{*}=0.62$ for $A=3.00$ mm (see Table 1). In each panel, black lines correspond to the dispersion relation $|{\sigma }^{*}|=|k_z|/{(k_x^2+k_z^2)}^{1/2}$ of inertial waves with $k_y=0$.

Figure 19

Spatiotemporal power spectral density $E^{\prime }(k_x,k_z,\sigma )$ [Eq. (16)] at the forcing frequency $\sigma ={\sigma }_0$ for the experiments at $\mathrm{\Omega }=18$ rpm, ${\lambda }_f=52.4$ cm, and (a) $A=0.75$ mm (${\text{Re}}_f\simeq 2400$) and (b) $A=3.00$ mm (${\text{Re}}_f\simeq 9500$). In each panel, black lines correspond to the dispersion relation $|{\sigma }^{*}|=|k_z|/{(k_x^2+k_z^2)}^{1/2}$ of inertial waves with $k_y=0$.