Coexistence of solitons and extreme events in deep water surface waves

We study experimentally, in a large-scale basin, the propagation of unidirectional deep water gravity waves stochastically modulated in phase. We observe the emergence of nonlinear localized structures that evolve on a stochastic wave background. Such a coexistence is expected by the integrable turbulence theory for the nonlinear Schrödinger equation (NLSE), and we report the first experimental observation in the context of hydrodynamic waves. We characterize the formation, the properties, and the dynamics of these nonlinear coherent structures (solitons and extreme events) within the incoherent wave background. The extreme events result from the strong steepening of wave train fronts, and their emergence occurs after roughly one nonlinear length scale of propagation (estimated from the NLSE). Solitons arise when nonlinearity and dispersion are weak, and of the same order of magnitude as expected from the NLSE. We characterize the statistical properties of this state. The number of solitons and extreme events is found to increase all along the propagation, the wave-field distribution has a heavy tail, and the surface elevation spectrum is found to scale as a frequency power law with an exponent $-4.5\pm 0.5$. Most of these observations are compatible with the integrable turbulence theory for the NLSE although some deviations (e.g., power-law spectrum, asymmetrical extreme events) result from effects proper to hydrodynamic waves.

Figure 1

Left: Theoretical envelope soliton ${\eta }_{\mathrm{sol}}\left(t\right)$ of Eq. (8) with $f_0=0.9$ Hz, $x=10$ m, and $A_{\mathrm{sol}}=5$ cm. The red solid line shows the envelope of the solution. Right: Dimensionless growth rate of the Benjamin-Feir instablility, $\sigma /{\sigma }_m$, versus dimensionless wave number $K/K_m$. Unstable modes are found below the solid line.

Figure 2

Sketch of a vertical section of the wave basin facility at Ecole Centrale de Nantes and locations of the resistive probes.

Figure 3

Experimental evolution along the basin length $x$ of the wave height $\eta \left(t\right)$ recorded at different probe locations (from bottom to top: $x=0$, 3.4, 7, 12.6, 16.3, 20.1, 24.9, and 29.8 m). $\tau =0.44({\epsilon }_{\eta }=0.08$ and $\mathrm{\Delta }f=0.16$ Hz). Dashed (red) lines have a slope corresponding to the group velocity $c_g$. The dashed (black) line on top of the two upper curves is the theoretical shape of the envelope soliton of Eq. (8).

Figure 4

Phase diagram of stochastic waves (white area) coexisting with envelope solitons (violet area) and/or extreme events (orange area) as a function of the nonlinearity-to-dispersion ratio $\tau$ and the dimensionless distance $x/L_{nl}$. Each performed run is displayed by a symbol corresponding to the observation of $(▿)$ only stochastic waves, $(\circ )$ envelope solitons, and $(\times )$ extreme events coexisting with stochastic waves. $\square$ indicates the coexistence with few spilling breakers (less than 10%) in time series. Right: Typical envelope soliton detected at $x=29.8$ m (zoom in of a part of the top curve in Fig. 3), $\tau =0.44({\epsilon }_{\eta }=0.08,\mathrm{\Delta }f=0.16$ Hz). (—) Detected envelope. (- - -) Theoretical shape of NLS envelope soliton from Eq. (8).

Figure 5

Left: Full width (at half maximum) of detected solitons as a function of their maximum amplitude, $A_{\mathrm{sol}}$. $x=29.8$ m. $\tau =0.55({\epsilon }_{\eta }=0.1,\mathrm{\Delta }f=0.16$ Hz). (—) Theoretical prediction from Eq. (9) with no fitting parameter. Right: Number of detected solitons as a function of the dimensionless distance $x/L_{nl}$ for different ratios $\tau =(\square )0.30({\epsilon }_{\eta }=0.08,\mathrm{\Delta }f=0.24$ Hz), $(▿)0$.$44({\epsilon }_{\eta }=0.08,\mathrm{\Delta }f=0.16$ Hz), $(\circ )0$.$55({\epsilon }_{\eta }=0.1,\mathrm{\Delta }f=0.16$ Hz), and $(\times )0$.$65({\epsilon }_{\eta }=0.12,\mathrm{\Delta }f=0.16$ Hz).

Figure 6

Left: Wave height signal recorded close to the wavemaker at the first probe $(x=3.4$ m). $\tau =0.3({\epsilon }_{\eta }=0.08,\mathrm{\Delta }f=0.24$ Hz). Top: Rescaled phase $\varphi /\pi$ of the signal. Right: Same part of the signal recorded at the second last probe $(x=24.9$ m) showing a structure similar to a Peregrine soliton. Theoretical temporal profile (- - -) and envelope (—) of a Peregrine soliton from Eq. (10) with $f_0=0.9$ Hz, $x=0$, and $k_0{A}_p={\epsilon }_{\eta }$. Top: (—) Rescaled phase of the signal showing a $\pi$ jump at times where the envelope falls to zero as predicted (- - -) by Eq. (10).

Figure 7

Left: Wave height signal recorded close to the wavemaker at the first probe $(x=3.4$ m). $\tau =0.55({\epsilon }_{\eta }=0.1,\mathrm{\Delta }f=0.16$ Hz). Right: Same part of the signal recorded at the last probe $(x=29.8$ m) showing coexistence of two envelope solitons (see full arrows) and three extreme events (see dashed arrows). (- - -) Theoretical shape of the soliton from Eq. (8).

Figure 8

Left: A typical extreme event as function of time at $x=29.8$ m. The wavefront is the left-hand side. The normalized wave height $\eta \left(t\right)/{\sigma }_{\eta }$ is on the left-hand-side axis, and the normalized wave local slope $(d\eta /dt)/{\sigma }_{\dot{\eta }}$ is on the right-hand-side axis. $\tau =0.44({\epsilon }_{\eta }=0.08,\mathrm{\Delta }f=0.16$ Hz). ${\sigma }_{\eta }=2.1$ cm. Right: Number $N_e$ of extreme events detected as a function of the dimensionless distance $x/L_{nl}$ for different ratios $\tau$ (same symbols as in Fig. 5). Inset: Typical temporal signal of the wave local slope for $\tau =0.65({\epsilon }_{\eta }=0.12,\mathrm{\Delta }f=0.16$ Hz) at $x=29.8$ m. Solid lines correspond to $\pm 4{\sigma }_{d\eta /dt}$.

Figure 9

Top: Temporal wave height signal for $\tau =0.44({\epsilon }_{\eta }=0.08,\mathrm{\Delta }f=0.16$ Hz). $x=29.8$ m. Bottom: Time-frequency spectrum $S_{\eta }(f,t)$ of the corresponding wave height $\eta \left(t\right)$. The color bar is a logarithmic scale of the spectrum amplitude. Large amplitude events of the wave signal correspond to maxima of the spectrum at high frequencies (see solid lines).

Figure 10

Left: Power spectrum density of $\eta \left(t\right)$ recorded close to $[x=3.4$ m (blue)] and far from $[x=29.8$ m (red)] the wavemaker. Dashed lines have slopes $\alpha =-6.4$ (blue) and $\alpha =-4.5$ (red). $\tau =0.44$. Vertical dashed lines correspond to theoretical satellites of the Benjamin-Feir instability, $\pm c_g{K}_c/\left(2\pi \right)\simeq \pm 0.1$ Hz. Insets show the corresponding temporal wave height signals close to (bottom) and far from (top) the wavemaker. Right: Dimensionless width $W/\mathrm{\Delta }f$ of the main peak of the spectrum (at one hundredth of its maximum amplitude) as a function of $x/L_{nl}$. $\mathrm{\Delta }f=(\circ )0$.1, $(\times )0$.07, $(▿)0$.04, and $(\square )0$.02 Hz. ${\epsilon }_{\eta }=$ (blue) 0.08, (red) 0.1, (green) 0.12, (black) 0.14. Inset: Unrescaled curve $W$ vs $x$. Same symbols as in the main figure.

Figure 11

Left: Exponent $\alpha$ of the wave spectrum in $f^{\alpha }$ as a function of dimensionless distance $x/L_{nl}$ for different ratios $\tau$ (same symbols as in Fig. 5). Right: Exponent $\alpha$ as a function of the wave steepness for different modulation bandwidths $\mathrm{\Delta }f=0.09(\circ )$, 0.$16(△)$, 0.$24(◃)$ Hz at $x/L_{nl}>1.2$. Dashed lines show the 1D wave turbulent prediction $-9/2$.

Figure 12

Probability density function (PDF) of normalized wave height $\eta /{\sigma }_{\eta }$ recorded at the first (left) and last (right) probes $(x=3.4$ m and $x=29.8$ m, respectively). $\tau =0.55({\epsilon }_{\eta }=0.1,\mathrm{\Delta }f=0.16$ Hz). Solid lines display a Gaussian of zero mean and unit standard deviation. Dashed lines show a Tayfun distribution for a wave steepness of 0.1.

Figure 13

Left: Probability density function (PDF) of the square of the wave envelope, $A^2/\langle A^2\rangle$, at different distances $x/L_{nl}=0.22$, 0.58, 0.82, 1.62, and 1.94 (from bottom to top). $L_{nl}=15.34$ m. $\tau =0.55({\epsilon }_{\eta }=0.1,\mathrm{\Delta }f=0.16$ Hz). Dashed line: Exponential (Rayleigh) distribution. Right: Kurtosis of $\eta \left(t\right)$ as a function of the dimensionless distance $x/L_{nl}$ for constant ${\epsilon }_{\eta }=0.12$ and different $\mathrm{\Delta }f=0.05(\times )$, 0.$09(\circ )$, 0.$16(▿)$, and 0.24 Hz $(\square )$ (i.e., $\tau =2.29$, 1.15, 0.65, and 0.46 respectively). The dashed line displays Gaussian value $\left(K=3\right)$.