Finite-system size effects in gravity-capillary wave turbulence

TL;DR

This work addresses how finite-system size alters gravity-capillary wave turbulence by using magnetically forced, randomly driven waves in a rectangular tank with tunable confinement. Through spatiotemporal measurements, it reveals discrete sloshing branches in the confined direction and a continuous cascade in the unconfined direction, with the branch structure and spectral exponents tunable via confinement and wave steepness. High-order correlations show suppression of two-dimensional three-wave resonances along the confined axis, indicating a transition from discrete/mesoscopic to continuous turbulence as the system becomes less confined. The findings quantify the interplay between nonlinear broadening and spectral discreteness and highlight geometry as a key control parameter for wave-turbulence dynamics, with relevance to both experiments and natural systems.

Abstract

We experimentally investigate the effects of finite-system size on the dynamics of weakly nonlinear random gravity-capillary surface waves. Experiments are conducted in rectangular tanks with varying aspect ratios, in which the fluid surface is perturbed locally and erratically by small, partially submerged magnets. Driven by an oscillating vertical electromagnetic field, these magnets generate a statistically homogeneous and isotropic random wave field. This setup enables us to probe finite-size effects without the dominant influence of global forcing present in horizontally oscillated tanks. Spatiotemporal measurements of the wave field reveal multiple branches in the wave-energy spectrum along the unconfined direction, corresponding to sloshing modes in the confined direction. We show that the spectral properties of these modes can be tuned by varying either the wave steepness or the confinement. Signatures of discrete wave turbulence in the confined direction and mesoscopic continuous wave turbulence in the unconfined direction are observed. As the confinement is gradually relaxed, we further demonstrate a smooth transition from discrete to continuous wave turbulence, consistent with the nonlinear-to-discreteness timescale ratio. Using high-order correlation analysis, we also show that finite-size effects alter wave dynamics by depleting two-dimensional three-wave resonant interactions along the confined direction.

Figures (8)

• Figure 1: (a) Schematic diagram of the experimental setup used to generate waves in a confined environment. Random waves are generated by two magnets partially submerged in the fluid, and energized by electromagnetic coils located beneath the container. A movable partition changes the container aspect ratio by reducing the length in the $x$ direction. A fringe pattern is projected on the surface of water, and the spatiotemporal deformations of the fringes are recorded using a high-speed camera. (b) Schematic diagram of the case used to suspend a magnet.
• Figure 2: Spatial profile of the wave field gradient $||\boldsymbol{\nabla} \eta(x,y,f^*)||$, near the center of the container, for different confinements $L_x$ (at fixed $L_y=50$ cm) corresponding to different container aspect ratios $\mathrm{AR}\equiv L_y/L_x$: (a) $L_x=100$ cm (unconfined case $\mathrm{AR}=0.5$), (b) $L_x=11$ cm ($\mathrm{AR}=4.54$), (c) $L_x=8$ cm ($\mathrm{AR}=6.25$), and (d) $L_x=5$ cm ($\mathrm{AR}=10$). Only one single Fourier mode ($f^*\equiv \omega^*/2\pi=10$ Hz) is selected to estimate the wave field gradient. Random forcing: $2\pm 0.5$ Hz. $\epsilon=2$%.
• Figure 3: Spatiotemporal spectra of the wave vertical-velocity field, $S_v(k_y,\omega)$, for different confinements $L_x$ at fixed $L_y=50$ cm corresponding to different container aspect ratios $\mathrm{AR}\equiv L_y/L_x$: (a) $L_x=100$ cm (unconfined case $\mathrm{AR}=0.5$), (b) $L_x=11$ cm ($\mathrm{AR}=4.54$), (c) $L_x=8$ cm ($\mathrm{AR}=6.25$), and (d) $L_x=5$ cm ($\mathrm{AR}=10$). Log-scale colorbar. Random forcing: $2\pm 0.5$ Hz. $\epsilon=2$%. The white curve represents the theoretical dispersion relation of linear waves of Eq. \ref{['eq1']}, and the black dashed lines represent different sloshing modes of Eq. \ref{['eq6']}. Dashed white lines in (a) correspond to the nonlinear broadening of the dispersion relation $\omega(k\pm \delta k)$ with $\delta k/(2\pi)=5$ m$^{-1}$.
• Figure 4: Experimental low-frequency cutoffs of sloshing modes [from Fig. \ref{['fig2']}(b-d)], for different confinements (see symbols) and all tested wave steepnesses $\epsilon$, as a function of the theoretical prediction, $f_{\mathrm{theo}}$, of Eq. \ref{['eqfcutoff']} with $n=1$, 2, 3, $\ldots$. The black dashed line has a slope of one.
• Figure 5: Spatiotemporal spectra of the wave vertical-velocity field, $S_v(k_y,\omega)$, for the strongest confinement ($L_x=5$ cm) and different wave steepnesses: (a) $\epsilon=0.36$%, (b) $\epsilon=1.7$%, and (c) $\epsilon=2$% [same as Fig. \ref{['fig2']}(d)]. Log-scale colorbar. Random forcing: $2\pm 0.5$ Hz. Experimental data in Fig. 5(c) are the same as in Fig. 3(d).