Saturation of the Inverse Cascade in Surface Gravity-Wave Turbulence

We report on the observation of surface gravity-wave turbulence at scales larger than the forcing ones in a large basin. In addition to the downscale transfer usually reported in gravity-wave turbulence, an upscale transfer is observed, interpreted as the inverse cascade of weak turbulence theory. A steady state is achieved when the inverse cascade reaches a scale in between the forcing wavelength and the basin size, but far from the latter. This inverse cascade saturation, which depends on the wave steepness, is probably due to the emergence of nonlinear dissipative structures such as sharp-crested waves.

Figure 1

Experimental setup showing the 48 flaps wave maker, the ending wall, and the locations of the 12 wave probes.

Figure 2

Temporal evolution of the power spectrum density (PSD) of the wave height at $x=20\mathrm{m}$ (averaged on four probes n°5, 10, 11, 12—see Fig. 1). ${\sigma }_{\eta }=1.14\mathrm{cm}$. Gray area: forcing bandwidth. Dash-dotted vertical lines correspond to the maximum of each spectrum. Dash-dotted red line is the prediction in $f^{-11/3}$ of Eq. (1). Dashed black line displays a $f^{-6}$ fit. Inset: same for $x=2\mathrm{m}$ (averaged on probes n°1, 2).

Figure 3

Frequency of the spectrum maximum, $f_{\max }$, over time for different forcing amplitudes. ${\sigma }_{\eta }^{\mathrm{sat}}=(▿)$ 0.59, (square) 1.19, (diamond) 1.54, (triangle) 1.83, and (circle) 1.92 cm. Dash-dotted line is the prediction $t^{-3/11}$ [32]. Dashed lines are best power-law fits, at short times. $x=20\mathrm{m}$.

Figure 4

Left-hand axis: saturation frequency $f_{\max }^{\mathrm{sat}}$ reached by the spectrum versus forcing amplitude at $x=20\mathrm{m}$ (blue open symbols). Right-hand axis: wave steepness $\epsilon$ versus forcing at $x=20\mathrm{m}$ (red full symbols). Inset: $f_{\max }^{\mathrm{sat}}$ for different distances $x$ along the basin and different forcing amplitudes. Same symbols as inset of Fig. 3.

Figure 5

Logarithmic time evolution of ${\sigma }_{\eta }(t)$ for different forcing amplitudes (same symbols as Fig. 3). Dash-dotted lines show logarithmic growth fits. Dashed lines shows the saturation values ${\sigma }_{\eta }^{\mathrm{sat}}$. Inset: saturation times of (circle) ${\sigma }_{\eta }(t)$ and of (diamond) the spectrum evolution (inferred from Fig. 3) versus forcing. Dashed line corresponds to the typical wave packet round-trip (3 min). $x=20\mathrm{m}$.

Figure 6

Temporal evolution of the gravity wave energy. Wave maker is stopped at $t=0$. Dashed line: tangent at $t=0$ of slope $P=5.98\times {10}^{-6}{\mathrm{m}}^3/{\mathrm{s}}^3$. ${\sigma }_{\eta }^{\mathrm{sat}}=1.82\mathrm{cm}$. $x=20\mathrm{m}$. Mean energy flux $P$ (top inset) and mean wave action flux $Q$ (bottom inset) versus wave steepness to the power 6 from decaying experiments (main).