Scale effects in internal wave attractors

As a necessary preliminary step toward geophysically significant extrapolations, we study the scale effects in internal wave attractors in the linear and nonlinear regimes. We use two geometrically similar experimental setups, scaled to factor 3, and numerical simulations (a spectral element method, based on the Nek5000 open solver) for a range of parameters that is typically accessible in laboratory. In the linear regime, we recover the classical viscous scaling for the beam width, which is not affected by variations of the amplitude of the input perturbation. In the nonlinear regime, we show that the scaling of the width-to-length ratio of the attractor branches is intimately related with the energy cascade from large-scale energy input to dissipation. We present results for the wavelength, amplitude, and width of the beam as a function of time and as a function of the amplitude of the forcing.

Figure 1

Geometric configuration of the experimental setups, with the wave generator prescribing the motion of the vertical wall, and the sloping wall inclined at an angle $\alpha$ with respect to the vertical. The sloping wall delimits the trapezoidal domain with the working length $L$ and the depth $H$.

Figure 2

Ray tracing prediction (dashed line) for an attractor with $(d,\tau )=(0.2,1.6)$. The slope is represented by the thick solid black line. The virtual point source is indicated by the black dot, at the right of the slope. It emits a beam, whose width scales to the power $1/3$ with the distance from the source. Branches grow wider until the focusing reflection on the slope. The distance $c$ and the coordinate $\xi$ are indicated in the figure.

Figure 3

(a) Real part of $\partial {\rho }^{\prime }/\partial {\eta }_1$ of a stable attractor observed with SyS during the steady state, after Hilbert filtering in frequency and space. The black square indicates where the filtering in wave vector space is performed. The cut made through branch 1 is plotted as a dashed black arrow. (b) $\partial {\rho }^{\prime }/\partial {\eta }_1$ along the cut through branch 1. The real part is represented by a dash-dotted blue line and the modulus by the two black lines. (c) Fourier spectrum of the full complex signal $\partial {\rho }^{\prime }/\partial {\eta }_1$. The vertical dotted line shows the peak wave number found using the phase on panel (d). (d) Unwrapped phase (black solid line) and fit of the central part (magenta dash-dotted line). The fit is performed between the two vertical dashed lines, on both panels (b) and (d). The distance between these two lines defines the width $\sigma$ of branch 1.

Figure 4

Time history of the horizontal density gradient fields in a point located at $x=85$ cm and $y=63$ cm, on branch 1. The wave generator is started at $t=0T_0$ and is stopped at $t=73T_0$.

Figure 5

(a): Wavelength of the attractor branch 1 as a function of time. (b) Transverse density gradient fields' amplitude of the attractor branch 1 as a function of time. (c) Transverse density gradient fields' amplitude, normalized with respect to the steady state, as a function of time. The different symbols and colors indicate experiments of different amplitudes: green pentagons for $a=0.7$ mm, magenta hexagons for $a=1.5$ mm, and blue dots for $a=2.2$ mm. The symbols show the instants where the wavelength and the amplitude have been measured. There are two growth and decay experiments (green and blue points) and one growth experiment (magenta points). The vertical dashed line represents the moment when the wave generator was stopped.

Figure 6

(a) Width $\sigma$ of branch 1 as a function of time, for the same three experiments as in Fig. 5. (b) Ratio between the width $\sigma$ and the wavelength $\lambda$ of branch 1 as a function of time. The dashed black lines show the time where the wave maker has been stopped. The symbols are the same as the ones in Fig. 5.

Figure 7

$|\partial {\rho }^{\prime }/\partial x|$ filtered around ${\omega }_0$ and normalized by the amplitude of the branch 1 for attractors made in the small (a) and large (b) tanks. $(d,\tau )=(0.38,1.85)$ for (a) and $(d,\tau )=(0.52,1.83)$ for (b).

Figure 8

(a) Time history of the horizontal density gradient field at one point located on branch 1 for the unstable attractor with $a=4.4$ mm (exp. 6 of Table 1). [(b), (c)] Wavelength and amplitude of branch 1 as a function of time for three stable and three unstable attractors (see Table 1). The three curves for the stable attractors are the ones presented in Fig. 5. In panel (b), for the sake of clarity, the average of the wavelengths of these three stable experiments is plotted as a unique solid black line. The vertical dashed line on the three panels represents the time corresponding to visible onset of TRI in panel (a), which corresponds to red squares in panels (b) and (c).

Figure 9

Final wavelength (a) and amplitude (b) of the branch 1 of attractors as a function of the amplitude of the wave generator $a$. The dashed lines on the two panels delimit the lowest amplitudes, where no TRI is observed, and the highest amplitudes, where TRI is observed.

Figure 10

Final wavelength (a) and amplitude (c) as a function of the amplitude of the wave generator for three different values of the water depth: $H=92$ cm (red circles), $H=46$ cm (blue triangles), and $H=30$ cm (green diamonds). Panel (b) shows on log-log scale the data from panel (a), with the ordinate rescaled as $({\lambda }_f/H){(H/H_{\mathrm{ref}})}^{2/3}$ and the horizontal coordinate rescaled as $A\times (H/H_{\mathrm{ref}})$, where $H_{\mathrm{ref}}=30$ cm is taken as reference value. A black dash-dotted line with slope 1 is drawn to guide the eye. Panel (d) shows on log-log scale the data from panel (c), with the ordinate rescaled as $\left(\right|\partial \widetilde{{\rho }^{\prime }}/\partial \widetilde{{\eta }_1}{|}_f/A){(H_{\mathrm{ref}}/H)}^{4/3}$ and the horizontal coordinate rescaled as $A\times (H/H_{\mathrm{ref}})$. Again, $H_{\mathrm{ref}}=30$ cm is taken as reference value. The black dash-dotted line indicates a $-3/2$ slope. Light gray stripes in the four panels show approximatively the transition between stable and unstable attractors, where the left (respectively right) borders of the stripes are relevant to the larger (resp. smaller) setups.