From deep to shallow water 2D wave turbulence: Emergence of soliton gas

TL;DR

The paper experimentally investigates the transition between soliton-gas–dominated dynamics and dispersive weak turbulence in 2D gravity waves by tuning dispersion via the forcing peak frequency $f_p$ and nonlinearity via forcing amplitude in a large wave tank. Using time–space resolved stereoscopic measurements and spectral analyses, the study identifies a regime boundary around $Ur=0.25$ where solitons emerge, evidenced by a rising $E_{solitons}/E_{free}$ ratio and energy localization along the shallow-water soliton line, alongside bound-wave signatures and non-Gaussian elevation statistics. The results reveal distinct spectral and PDF changes associated with soliton gas and highlight the significant role of finite-size reflections (Mach-type interactions) in directing soliton energy and shaping the turbulence state. These findings advance understanding of how integrable turbulence and weak turbulence compete in realistic 2D settings and illustrate limitations of classical weak turbulence statistics in solitonic regimes.

Abstract

Experiments on 2D random water wave propagation in a large wave tank are analyzed when the effect of dispersion changes. A stereoscopic profilometry technique is used to measure the water surface displacement resolved in both time and space over a significant fraction of the wave tank. The wave regimes are characterized by analyzing the space-time spectral statistical properties of the wave field. At a given, finite, water depth, the effect of dispersion can be varied by tuning the peak frequency of the wave generation. In shallow water conditions, the waves are only weakly dispersive and this enables the propagation of solitons. {In these conditions random wave forcing produces soliton gases}. In deep water conditions, the waves are dispersive and for wideband spectra, one observes the development of weak turbulence. A transition between these regimes is observed when changing the peak forcing frequency (dispersion) and the wave amplitude (nonlinearity), with a clear threshold between states with solitons and soliton-less states. The development of a soliton gas is associated with a strong change of the wave spectrum as well as a significant evolution of the distribution of the water elevation. We also observed a strong effect of the finite size of the tank due to the peculiar reflection laws of line solitons.

Figures (20)

• Figure 1: (a) View of part of the ARTELIA LHF wave tank with the camera systems. The pairs of cameras can be seen at the top of the aluminum frame at the upper-left part of the image (another pair of cameras were installed but not used in this paper). The water surface appears white due to the floating particles. A few wavemakers can be seen on the right of the image. The cables carry the signal of the capacitive wave probes. A pair of such probes is visible at the bottom right of the image. The field of view of the cameras lies in between the two lines of cables separated by $12\,$meters. A set of spotlights illuminate the water surface at small incidence angles so as to avoid specular reflections into the cameras. (b) Top view sketch of the tank. Blue dots: positions of the capacitive wave probes. The green polygon delineates the measurement region of the stereoscopic system. The green squares are the positions of the cameras. The wavemakers are on the left at $x=0$. (c) View of a region of the water surface with the floating particles.
• Figure 2: Parameters $H_{m0}$ and $T_p$ of the JONSWAP spectra forcings of all experiments (see text for definitions of the parameters). Symbols correspond to different directivity of the forcing: blue disks to $s_{max}=30$ and red stars to $s_{max}=1$.
• Figure 3: Examples of water surface displacement time series for the experiments at $T_p=1.8$, $4$ and $10$ s from bottom to top as described in table \ref{['table_Tp']}. The surface displacement has been measured by a capacitive probe located at $x=19.37$ m and $y=10.02$ m.
• Figure 4: Examples of snapshots of the wave fields for the experiments at $T_p=1.8$, $4$ and $10$ s from left to right as described in table \ref{['table_Tp']}. The water surface displacement is measured by the stereoscopic reconstruction system. The water surface displacement is coded in linear color scale in meters.
• Figure 5: Wave amplification $\eta_{rms}/H_{m0}$ for the experiments with $H_{m0}=3.5$ cm described in table \ref{['table_Tp']}. The red dashed horizontal line is $0.72$.