Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves

A statistical theory of rogue waves is proposed and tested against experimental data collected in a long water tank where random waves with different degrees of nonlinearity are mechanically generated and free to propagate along the flume. Strong evidence is given that the rogue waves observed in the tank are hydrodynamic instantons, that is, saddle point configurations of the action associated with the stochastic model of the wave system. As shown here, these hydrodynamic instantons are complex spatiotemporal wave field configurations which can be defined using the mathematical framework of large deviation theory and calculated via tailored numerical methods. These results indicate that the instantons describe equally well rogue waves created by simple linear superposition (in weakly nonlinear conditions) or by nonlinear focusing (in strongly nonlinear conditions), paving the way for the development of a unified explanation to rogue wave formation.

Popular Summary

Rogue waves are waves of extreme height that appear suddenly on the surface of the ocean. Understanding their origin is a matter of intense research, but researchers have yet to agree upon a definitive explanation. Here, we propose and test a statistical theory that suggests that rogue waves are instantons, or specific realizations of an underlying stochastic process. The remarkable observation is that rogue waves of a certain height always develop in the same predictable way, despite the fact that they are random events.

To develop our theory, we generate random waves in a 270-meter-long wave tank, along which the waves propagate and then break on a smooth synthetic beach. Using various mathematical tools, we analyze the height of the water in the tank as the waves pass and look for extreme events. We find that all rogue waves are remarkably alike: Because they are relatively rare, rogue waves occur by the least unlikely way possible, in the sense that all other ways are so much more unlikely that they are never observed. Everything needs to come together in just the right way for a rogue wave to occur, which allows us to predict their shape.

Our results reconcile two apparently contradictory theories about the origin of rogue waves. One theory suggests that rogue events arise from the linear superposition of smaller waves; the other proposes that local perturbations grow to an extreme size. In our theory, both effects play a role, with their relative contribution determined by the strength of the nonlinearity in the wave equation that describes the water height. These findings pave the way for understanding, once and for all, the origin of rogue waves in the ocean.

Figure 1

Wave flume experiment. The wave maker generates a random wave field with stationary Gaussian statistics with the JONSWAP energy spectrum observed in the oceans. The planar wave fronts propagate along the water tank, where the surface elevation $\eta$ is measured by vertical probes.

Figure 2

JONSWAP spectra from Eq. (3) for the three experimental regimes of Table 1 (lines), compared to experimental measurements at the $x=10\mathrm{m}$ probe (dots). These spectra remain roughly constant through the tank, except for small changes that are the signature of non-Gaussian effects that develop [42].

Figure 3

(a) Extreme wave event selection. At $x=45\mathrm{m}$, we monitor the temporal maximum of the experimental data series of $\eta$, record events reaching above a given threshold, and monitor the evolution of these events at probes located earlier in the channel. This is done within an observation time window centered at the maximum and following the wave packet with group velocity $c_g$; we repeat this for the whole time series to build a collection of extreme events and their evolution. (b) Mean extreme event. The thick line shows the mean extreme event at different points along the channel, the shaded area a 1 standard deviation range around it. The noise-to-signal ratio is small in the focusing region, leading naturally to the question, can we explain the common pathway by which these rogue waves are most likely to arise?

Figure 4

Experimental validation of the instanton. Snapshots of the instanton during its evolution along the channel (black lines) are compared to the mean and standard deviation of the experimental rogue wave (color lines), for different regimes of nonlinearity. The instanton prediction agrees with the experimental mean across all regimes, and captures the whole evolution along the channel. This confirms that typical rogue waves are well represented by instantons, and the typical extreme events collapse onto the most likely event with only small fluctuations around it.

Figure 5

Agreement of the instanton with individual extreme events. The evolution of a single realization of an extreme wave (red lines) is reasonably approximated by the instanton evolution (black and white surface), here for a sample of the highly nonlinear dataset. In order to capture the focusing pattern in an essential way, the envelope $|\psi |$ is plotted instead of the surface elevation $\eta$ to remove carrier-frequency oscillations.

Figure 6

Comparison of the instanton to the predictions of the theory of quasideterminism and the semiclassical theory. (a) The quasilinear instanton converges to the linear prediction, correctly reproducing the rogue waves averaged over the experiments. (b) The highly nonlinear instanton evolution closely follows the averaged rogue wave and converges locally to a Peregrine soliton around its space-time maximum, as predicted by the semiclassical theory, and reproduced by the instanton. The linear prediction instead fails, especially around the maximum. (c) The contour plots show agreement with the two limiting theories and recover the respective dominant length scales. In the linear limit, dominated by dispersion, the rogue waves arise and decay very rapidly. On the contrary, in the semiclassical limit, where nonlinear effects are prevalent, the Peregrine-like structure of the extreme event is persistent, with a very slow decay. The rogue waves in intermediate regimes display both linear and nonlinear features, as shown in the central panel.

Figure 7

Comparison of the PDF of the surface elevation $\rho (z)=-P^{\prime }(z)$ obtained by binning the data from the experiments and the LDT estimates from Eq. (18), showing good agreement in the rogue wave regime (right-hand side of the vertical dashed line, indicating the conventional rogue wave threshold $z=H_s$). The figure refers to the intermediate regime. The blue, green, and red colors indicate data collected at the probes 10, 30, and 45 m away from the wave maker, respectively.