Three-dimensionality of the triadic resonance instability of a plane inertial wave

We analyze theoretically and experimentally the triadic resonance instability (TRI) of a plane inertial wave in a rotating fluid. Building on the classical triadic interaction equations between helical modes, we show by numerical integration that the maximum growth rate of the TRI is found for secondary waves that do not propagate in the same vertical plane as the primary wave (the rotation axis is parallel to the vertical). In the inviscid limit, we prove this result analytically, in which case the change in the horizontal propagation direction induced by the TRI evolves from ${60}^{\circ }$ to ${90}^{\circ }$ depending on the frequency of the primary wave. Due to a wave generator with a large spatial extension in the horizontal direction of invariance of the forced wave, we are able to report experimental evidence that the TRI of a plane inertial wave is three dimensional. The wave vectors of the secondary waves produced by the TRI are shown to match the theoretical predictions based on the maximum growth rate criterion. These results reveal that the triadic resonant interactions between inertial waves are very efficient at redistributing energy in the horizontal plane, normal to the rotation axis.

Figure 1

Sketch of a plane inertial wave of wave vector $\mathbf{k}$ with polarity (a) $s=-1$ and (b) $s=+1$. The wave is invariant in the horizontal $y$ direction ($k_y=0$). The fluid motions consist in an anticyclonic circular translation in the planes of constant phase, normal to $\mathbf{k}$, which are tilted by an angle $\theta ={cos}^{-1}\left({\sigma }^{*}\right)={cos}^{-1}(k_z/k)$ with respect to the horizontal. The phase of the wave propagates normally to these constant phase planes, but the energy of the wave propagates parallel to these planes along the group velocity. The vectors ${\mathbf{c}}_g$ and ${\mathbf{c}}_{\phi }$ indicate the direction of the group and phase velocities, respectively. The amplitude of the fluid motions is damped along the energy propagation direction ${\mathbf{c}}_g$ at a rate $\nu k^2$.

Figure 2

Resonance surfaces of (a) ${\mathbf{k}}_1$ and (b) ${\mathbf{k}}_2$ for the combination of wave polarities $(-,+,-)$ and a primary wave defined by $k_0=0.83$ $\text{rad}{\text{cm}}^{-1}$, $s_0=-1$, ${\sigma }_0^{*}=0.84$, and $b_0=0.39$ $\text{cm}{\text{s}}^{-1}$. In (a) the vertical axis shows $-k_{z,1}$ for the sake of clarity. The two 3D plots also show the cone of apex at ${\mathbf{k}}_i=\mathbf{0}$ and of semiangle ${\theta }_0={cos}^{-1}\left({\sigma }_0^{*}\right)$. This cone represents the waves at the forcing frequency according to the dispersion relation. Below each resonance surface, we also report the map of the growth rate $\gamma$ of the instability in the plane $(k_{x,i},k_{y,i})$ in which the locations of the maximum growth rate are shown by black dots. The value $\nu =1.20\times {10}^{-6}{\mathrm{m}}^2{\text{s}}^{-1}$ is used for the kinematic viscosity in order to match the experimental value in the next section.

Figure 3

(a) Resonance surface of ${\mathbf{k}}_1$ (in pink) and (b) 3D view of the instability growth rate $\gamma$ for the combination of polarities $(-,-,-)$. In (a) we also show the cone of apex at ${\mathbf{k}}_1=\mathbf{0}$ and of semiangle ${\theta }_0={cos}^{-1}\left({\sigma }_0^{*}\right)$. The surfaces are computed for a primary wave defined by $k_0=0.83$ $\text{rad}{\text{cm}}^{-1}$, $s_0=-1$, ${\sigma }_0^{*}=0.84$, and $b_0=0.39$ $\text{cm}{\text{s}}^{-1}$ in a rotating fluid of kinematic viscosity $\nu =1.20\times {10}^{-6}{\mathrm{m}}^2{\text{s}}^{-1}$.

Figure 4

(a) Maximum normalized instability growth rate ${\gamma }^{\left(\max \right)}/k_0{b}_0$ for polarity combinations $(-,+,-)$ and $(-,-,-)$ as a function of the primary wave Reynolds number ${\text{Re}}_0=b_{0}2\pi /\nu k_0$ for ${\sigma }_0^{*}=0.84$. The horizontal blue straight line reports the value ${\gamma }^{\left(\max \right)}/k_0{b}_0$ predicted analytically in the inviscid limit in Sec. 2d [see Eq. (15)]. (b) Wave-vector components $k_{x,1}^{\left(\max \right)}$ and $k_{y,1}^{\left(\max \right)}$ corresponding to the maximum growth rate for each mode as a function of ${\text{Re}}_0$ (again for ${\sigma }_0^{*}=0.84$). The red curve shows the wave number $\sqrt{3/(1-{{\sigma }_0^{*}}^2)}\left|k_{x,1}^{\left(\max \right)}\right|$, which is predicted in the inviscid limit (see Sec. 2d) to match the wave-vector component $k_{y,1}^{\left(\max \right)}$ for the instability mode $(-,+,-)$. (c) Corresponding wave numbers. The blue straight line shows a power law ${\text{Re}}_0^{1/4}$.

Figure 5

Sketch of the experimental setup seen from (a) the side and (b) the top.

Figure 6

Temporal power spectral density $E(\sigma ,t=150T,\mathrm{\Delta }T=300T)$ as a function of the normalized frequency ${\sigma }^{*}=\sigma /2\mathrm{\Omega }$ for experiments at ${\sigma }_0^{*}=0.84$ and $\mathrm{\Omega }=18\mathrm{rpm}$ for a forcing wavelength ${\lambda }_f=7.6$ cm and four forcing Reynolds numbers ${\text{Re}}_f=0$ (pink curve), ${\text{Re}}_f=230$ (red curve), ${\text{Re}}_f=300$ (blue curve), and ${\text{Re}}_f=420$ (black curve). The spectrum at ${\text{Re}}_f=0$ corresponds to an experiment with the wave generator off. A vertical shift by a factor of 20 has been introduced between successive spectra.

Figure 7

Snapshots of the vertical velocity component computed with the Hilbert filtering procedure for the experiment at ${\text{Re}}_f=300$ for frequencies (a) ${\sigma }_0^{*}$, (b) ${\sigma }_1^{*}$, and (c) ${\sigma }_2^{*}$: (a) temporal filter at ${\sigma }_0^{*}=0.84$ and spatial filter keeping the wave-vector quadrant $(k_x<0,k_z<0)$, (b) ${\sigma }_1^{*}=0.285$ and $(k_x>0,k_z>0)$, and (c) ${\sigma }_2^{*}=0.555$ and $(k_x<0,k_z<0)$. In each panel, the dotted line indicates the theoretical tilt angle ${\theta }_i={cos}^{-1}\left({\sigma }_i^{*}\right)$ with the horizontal predicted for the constant phase planes for a wave at the considered frequency ${\sigma }_i^{*}$ and propagating in the vertical measurement plane $(x,z)$, i.e., with $k_y=0$. The black rectangle (more precisely, its underside) indicates the mean position of the wave generator surface. In (b) and (c) the dashed line indicates the direction normal to the projection in the measurement plane of the wave vectors computed in Sec. 4b for the modes at ${\sigma }_1^{*}$ and ${\sigma }_2^{*}$ in the instability region (dashed rectangle).

Figure 8

Natural logarithm of the temporal energy spectra $E(\sigma ,t,\mathrm{\Delta }T)$ normalized by its maximum as a function of time $t$ and of the normalized frequency ${\sigma }^{*}=\sigma /2\mathrm{\Omega }$ for the experiments at (a) ${\text{Re}}_f=300$ and (b) ${\text{Re}}_f=420$, computed using a sliding time window of $\mathrm{\Delta }T=15T$. Also shown is the corresponding time evolution of the temporal energy spectra $E(\sigma ,t,\mathrm{\Delta }T)$ normalized by its maximum for four specific frequencies ${\sigma }_0^{*}=0.84$, ${\sigma }_1^{*}$, ${\sigma }_2^{*}$ (reported in Table 1), and ${\sigma }^{*}={\sigma }_0^{*}/2$, for the experiments at (c) ${\text{Re}}_f=300$ and (d) ${\text{Re}}_f=420$.

Figure 9

Cuts of the theoretical resonance surfaces for ${\mathbf{k}}_1$ computed for a primary wave of frequency ${\sigma }_0^{*}=0.84$, wavelength ${\lambda }_f=7.6\mathrm{cm}$, and amplitude $b_0=3.9$ $\text{mm}{\text{s}}^{-1}$ matching the features of the experimental primary wave at ${\text{Re}}_f=300$ in its unstable region (dashed rectangle in Fig. 7). Superimposed on the resonance curves, we show the projection on the considered plane of the wave vectors ${\mathbf{k}}_0^{\mathrm{expt}}$, ${\mathbf{k}}_1^{\mathrm{expt}}$, and ${\mathbf{k}}_2^{\mathrm{expt}}$ experimentally estimated for the experiment at ${\text{Re}}_f=300$. We also report the measurement error corresponding to each wave-vector component via a rectangle around each wave-vector tip. (a) Cut in the vertical plane $k_{y,1}=0$. (b) Cut in the vertical plane $k_{y,1}=k_{y,1}^{\mathrm{expt}}$. (c) Cut in the vertical plane $k_{y,1}/k_{y,1}^{\mathrm{expt}}=k_{x,1}/k_{x,1}^{\mathrm{expt}}$ $[k_{{\varphi }_1,1}=\sqrt{k_{x,1}^2+k_{y,1}^2}=k_{x,1}\sqrt{1+{(k_{y,1}^{\mathrm{expt}}/k_{x,1}^{\mathrm{expt}})}^2}]$.

Figure 10

Map of the growth rate $\gamma$ normalized by its maximum ${\gamma }^{\left(\max \right)}$ as a function of $(k_{x,1},k_{y,1})$ for the $(-,+,-)$ instability computed theoretically for a primary wave with features ${\sigma }_0^{*}=0.84$, ${\lambda }_f=7.6\mathrm{cm}$, and $b_0=3.9$ $\text{mm}{\text{s}}^{-1}$ matching the experimental primary wave at ${\text{Re}}_f=300$ (same map as in Fig. 2). Superimposed on the $\gamma /{\gamma }^{\left(\max \right)}$ map, we show the projection in the $(k_x,k_y)$ plane of the experimental wave vectors ${\mathbf{k}}_0^{\mathrm{expt}}$, ${\mathbf{k}}_1^{\mathrm{expt}}$, and ${\mathbf{k}}_2^{\mathrm{expt}}$. As in Fig. 9, the measurement errors on the wave vectors are reported via a rectangle around each wave-vector tip.

Figure 11

(a) and (b) Cuts of the theoretical resonance surfaces for ${\mathbf{k}}_1$ computed for a primary wave of frequency ${\sigma }_0^{*}=0.84$, wavelength ${\lambda }_f=7.6\mathrm{cm}$, and amplitude $b_0=5.1$ $\text{mm}{\text{s}}^{-1}$ matching the features of the experimental primary wave at ${\text{Re}}_f=420$. Superimposed on the resonance curves, we show the projection on the considered plane of the wave vectors ${\mathbf{k}}_0^{\mathrm{expt}}$, ${\mathbf{k}}_1^{\mathrm{expt}}$, and ${\mathbf{k}}_2^{\mathrm{expt}}$ measured for the experiment at ${\text{Re}}_f=420$. (a) Cut in the vertical plane $k_{y,1}=0$. (b) Cut in the vertical plane $k_{y,1}/k_{y,1}^{\mathrm{expt}}=k_{x,1}/k_{x,1}^{\mathrm{expt}}$. (c) Map of the growth rate $\gamma$ normalized by its maximum ${\gamma }^{\left(\max \right)}$ as a function of $(k_{x,1},k_{y,1})$ for the $(-,+,-)$ instability computed theoretically for a primary wave matching the features of the experimental primary wave at ${\text{Re}}_f=420$. Superimposed on the $\gamma /{\gamma }^{\left(\max \right)}$ map, we show the projection in the $(k_x,k_y)$ plane of the experimental wave vectors ${\mathbf{k}}_0^{\mathrm{expt}}$, ${\mathbf{k}}_1^{\mathrm{expt}}$, and ${\mathbf{k}}_2^{\mathrm{expt}}$.

Figure 12

Map of the growth rate $\gamma$ normalized by its maximum ${\gamma }^{\left(\max \right)}$ as a function of $(k_{x,1},k_{y,1})$ for the $(-,+,-)$ instability computed theoretically for a primary wave with features ${\sigma }_0^{*}=0.84$, ${\lambda }_f=7.6\mathrm{cm}$, and $b_0=3.9$ $\text{mm}{\text{s}}^{-1}$, matching the experimental primary wave at ${\text{Re}}_f=300$. Superimposed on the $\gamma /{\gamma }^{\left(\max \right)}$ map, we show the projection in the $(k_x,k_y)$ plane of the wave vectors ${\mathbf{k}}_0^{\mathrm{expt}}$, ${\mathbf{k}}_1^{\mathrm{expt}}$, and ${\mathbf{k}}_2^{\mathrm{expt}}$ estimated experimentally. (a) Growth rate map for an (infinite) plane wave (same map as in Fig. 10). (b) Growth rate map for a wave beam with four wavelengths in its width $W=4{\lambda }_f$ following Eq. (B1).