Large-scale flow driven by turbulently generated internal gravity waves

The generation of large-scale flows by stochastically excited internal gravity waves remains largely unexplored despite numerous applications in geophysical and astrophysical contexts. Here, we investigate this problem experimentally in a cylindrical annulus geometry. Our working fluid is made of two layers. In the top, fresh water layer, turbulence is generated by 12 jets with an oscillating flow rate. Those turbulent fluctuations impinge the interface with the bottom, linearly stratified, salt water layer, where they excite internal gravity waves which propagate, damp viscously, and generate a mean azimuthal flow. The jet structure, wave spectra, and mean-flow properties are addressed using particle image velocimetry. Our measurements validate quantitatively the transfer of momentum from the waves to the mean flow through the wave associated Reynolds stress, as previously validated for a monochromatic forcing. In addition, wave energy decays through time, likely because of the mixing at the interface between the two layers, and the driven mean flow accordingly decreases and eventually vanishes. This has up to now prevented the observation of quasibiennial-oscillation-like reversals in our system.

Figure 1

Left: Schematic of the cylindrical annulus geometry. Turbulence in the top layer is generated by 12 jets emerging from the red and blue holes. Waves are generated at the interface between the turbulent and stratified layers and an azimuthal mean flow is observed. The positions of the green laser sheet and cameras for particle image velocimetry (PIV) are also pictured. The figure is not to scale. Right: Top photography of the setup and injection plate. The flow from the two pumps is divided into 12 pipes directed to the nozzle exits at the top of the tank. The dotted blue (respectively, thick red) pipes are connected to the blue (respectively, red) holes at the top of the tank in the left panel.

Figure 2

Left: Density profile taken before an experiment. In our convention, $z=0$ is the bottom of the stratified layer camera field, so the interface is initially located at $z=27$ cm, with a homogeneous layer of fresh water above and a constant $N=0.21$ Hz stratified layer below. Right: Interface position over time during an experiment. At $t=45$ min, the refilling process is started.

Figure 3

Left: Instantaneous velocity field in the turbulent layer. One jet is present in the camera view field. Right: Time-averaged signal over a 2-min acquisition time.

Figure 4

Left: Flow rate of the two pumps due to the imposed sinusoidal modulation. Right: Observed vertical velocity at a single point located at $x=10$ cm and $z=27$ cm (see Fig. 3). One can notice the imposed period $T=20$ s.

Figure 5

Horizontal profiles of vertical velocities taken from the time-averaged velocity field displayed in Fig. 3. The colors indicate the $z$ position, with dark purple located at the top of the jet and yellow located just below the interface. $x_{\text{jet}}$ is the center of the injection hole. The red line shows the location of maximum velocity for each height.

Figure 6

Left: Power spectral density of the horizontal (blue) and vertical (red) velocities as a function of the horizontal wave number $k_x$. Different slopes characterizing the spectrum for turbulent jets are displayed in a dashed line [27]. Right: Power spectral density of the horizontal (blue) and vertical (red) velocities as a function of the frequency $f$. The dashed line shows the $-5/3$ slope for reference. The vertical dashed line indicates the frequency of the forcing $f=1/T=5\times {10}^{-2}$ Hz.

Figure 7

Left: Instantaneous field in the stratified layer. Top right: 5-min vertical velocity signal at the location $x=10$ cm, $z=16.5$ cm, pictured by the red cross in the left panel. Bottom right: Spatiotemporal diagram showing the vertical velocity $w$ along the $x$ axis at $z=16.5$ cm from a 150-s signal.

Figure 8

Left: PSD of the horizontal velocity $u$ computed at $z=16.5$ cm for each horizontal location and then horizontally averaged. The red dashed line shows the buoyancy frequency $N=0.21$ Hz and the gray dashed curve shows the theoretical spectra computed from the polarization relation and the experimental spectra of $w$: $\mathrm{PSD}\left(u_{\mathrm{th}}\right)=\mathrm{PSD}\left(w\right)[{(N/\omega )}^2-1]$. Right: Same for the vertical velocity $w$. Bottom: The ratio $\mathrm{PSD}\left(u\right)/\mathrm{PSD}\left(w\right)$ is plotted in black. The polarization relation of internal gravity waves ${(u/w)}^2={(N/\omega )}^2-1$ is plotted in red. All PSDs are computed between $t=300$ s and $t=3500$ s.

Figure 9

PSDs of the horizontal velocity $u$ (top) and of the vertical velocity $w$ (bottom) with respect to frequency and height. The buoyancy frequency $N$ is shown by the gray vertical line. The PSDs are computed between $t=300$ s and $t=3500$ s.

Figure 10

PSD integrated between the forcing frequency $1/T$ and the buoyancy frequency $N$ as a function of depth, for the horizontal (blue) and vertical (red) velocities.

Figure 11

Left: Horizontal mean-flow $\overline{u}$ evolution over time. The interface is located at $z\sim 21$ cm. Right: Vertical, time-averaged profiles of the mean-flow viscous dissipation $\nu {\partial }_{zz}\overline{u}$ in gray and of the Reynolds stress forcing term $-{\partial }_z\left(\overline{u^{\prime }{w}^{\prime }}\right)$ in purple.

Figure 12

Top: PSD of the horizontal velocity $u$ (left) and corresponding time evolution of the horizontal mean-flow $\overline{u}$ (right) during 1 h continuing the acquisition shown in Figs. 9 and 11. Bottom: PSD of the horizontal velocity $u$ (left) and corresponding time evolution of the horizontal mean-flow $\overline{u}$ (right) during 1 h continuing the acquisition shown in the upper panels.

Figure 13

Left: Horizontal mean-flow $\overline{u}$ evolution over time in the stratified layer, obtained from 2D direct numerical simulation with an ad hoc stochastic wave maker. The domain height is small compared to the attenuation length of the generated waves. Right: Same for a domain ten times higher. In both cases, the domain is 2D, periodic along the $x$ axis. The buoyancy frequency is constant. Waves with random phases are forced at the top boundary by setting velocity and buoyancy perturbations.